# Chapter 22. Using the Solver

```Chapter 22.
Using the Solver
This chapter describes the FLUENT solver and how to use it. Details about the solver algorithms used by FLUENT are provided in Sections 22.1–22.5. Section 22.6 provides an overview of how to use the
solver, and the remaining sections provide detailed instructions.
• Section 22.1: Overview of Numerical Schemes
• Section 22.2: Discretization
• Section 22.3: Segregated Solver
• Section 22.4: Coupled Solver
• Section 22.5: Multigrid Method
• Section 22.6: Overview of How to Use the Solver
• Section 22.7: Choosing the Discretization Scheme
• Section 22.8: Choosing the Pressure-Velocity Coupling Method
• Section 22.9: Setting Under-Relaxation Factors
• Section 22.10: Changing the Courant Number
• Section 22.11: Turning on FAS Multigrid
• Section 22.12: Setting Solution Limits
• Section 22.13: Initializing the Solution
• Section 22.14: Performing Steady-State Calculations
• Section 22.15: Performing Time-Dependent Calculations
• Section 22.16: Monitoring Solution Convergence
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Using the Solver
• Section 22.17: Animating the Solution
• Section 22.18: Executing Commands During the Calculation
• Section 22.19: Convergence and Stability
22.1
Overview of Numerical Schemes
FLUENT allows you to choose either of two numerical methods:
• segregated solver
• coupled solver
Using either method, FLUENT will solve the governing integral equations
for the conservation of mass and momentum, and (when appropriate) for
energy and other scalars such as turbulence and chemical species. In both
cases a control-volume-based technique is used that consists of:
• Division of the domain into discrete control volumes using a computational grid.
• Integration of the governing equations on the individual control
volumes to construct algebraic equations for the discrete dependent
variables (“unknowns”) such as velocities, pressure, temperature,
and conserved scalars.
• Linearization of the discretized equations and solution of the resultant linear equation system to yield updated values of the dependent variables.
The two numerical methods employ a similar discretization process (finitevolume), but the approach used to linearize and solve the discretized
equations is different.
The general solution methods are described first in Sections 22.1.1 and
22.1.2, followed by a discussion of the linearization methods in Section 22.1.3.
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22.1 Overview of Numerical Schemes
22.1.1
Segregated Solution Method
The segregated solver is the solution algorithm previously used by FLUENT 4. Using this approach, the governing equations are solved sequentially (i.e., segregated from one another). Because the governing
equations are non-linear (and coupled), several iterations of the solution
loop must be performed before a converged solution is obtained. Each
iteration consists of the steps illustrated in Figure 22.1.1 and outlined
below:
1. Fluid properties are updated, based on the current solution. (If
the calculation has just begun, the fluid properties will be updated
based on the initialized solution.)
2. The u, v, and w momentum equations are each solved in turn using
current values for pressure and face mass fluxes, in order to update
the velocity field.
3. Since the velocities obtained in Step 2 may not satisfy the continuity equation locally, a “Poisson-type” equation for the pressure correction is derived from the continuity equation and the linearized momentum equations. This pressure correction equation
is then solved to obtain the necessary corrections to the pressure
and velocity fields and the face mass fluxes such that continuity is
satisfied.
4. Where appropriate, equations for scalars such as turbulence, energy, species, and radiation are solved using the previously updated
values of the other variables.
5. When interphase coupling is to be included, the source terms in
the appropriate continuous phase equations may be updated with
a discrete phase trajectory calculation.
6. A check for convergence of the equation set is made.
These steps are continued until the convergence criteria are met.
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Using the Solver
Update properties.
Solve momentum equations.
Solve pressure-correction (continuity) equation.
Update pressure, face mass flow rate.
Solve energy, species, turbulence, and other
scalar equations.
Converged?
Stop
Figure 22.1.1: Overview of the Segregated Solution Method
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22.1 Overview of Numerical Schemes
22.1.2
Coupled Solution Method
The coupled solver solves the governing equations of continuity, momentum, and (where appropriate) energy and species transport simultaneously (i.e., coupled together). Governing equations for additional scalars
will be solved sequentially (i.e., segregated from one another and from
the coupled set) using the procedure described for the segregated solver
in Section 22.1.1. Because the governing equations are non-linear (and
coupled), several iterations of the solution loop must be performed before a converged solution is obtained. Each iteration consists of the steps
illustrated in Figure 22.1.2 and outlined below:
1. Fluid properties are updated, based on the current solution. (If
the calculation has just begun, the fluid properties will be updated
based on the initialized solution.)
2. The continuity, momentum, and (where appropriate) energy and
species equations are solved simultaneously.
3. Where appropriate, equations for scalars such as turbulence and
radiation are solved using the previously updated values of the
other variables.
4. When interphase coupling is to be included, the source terms in
the appropriate continuous phase equations may be updated with
a discrete phase trajectory calculation.
5. A check for convergence of the equation set is made.
These steps are continued until the convergence criteria are met.
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Using the Solver
Update properties.
Solve continuity, momentum, energy, and
species equations simultaneously.
Solve turbulence and other scalar equations.
Converged?
Stop
Figure 22.1.2: Overview of the Coupled Solution Method
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22.1 Overview of Numerical Schemes
22.1.3
Linearization: Implicit vs. Explicit
In both the segregated and coupled solution methods the discrete, nonlinear governing equations are linearized to produce a system of equations
for the dependent variables in every computational cell. The resultant
linear system is then solved to yield an updated flow-field solution.
The manner in which the governing equations are linearized may take
an “implicit” or “explicit” form with respect to the dependent variable
(or set of variables) of interest. By implicit or explicit we mean the
following:
• implicit: For a given variable, the unknown value in each cell is
computed using a relation that includes both existing and unknown
values from neighboring cells. Therefore each unknown will appear
in more than one equation in the system, and these equations must
be solved simultaneously to give the unknown quantities.
• explicit: For a given variable, the unknown value in each cell is computed using a relation that includes only existing values. Therefore
each unknown will appear in only one equation in the system and
the equations for the unknown value in each cell can be solved one
at a time to give the unknown quantities.
In the segregated solution method each discrete governing equation is
linearized implicitly with respect to that equation’s dependent variable.
This will result in a system of linear equations with one equation for
each cell in the domain. Because there is only one equation per cell,
this is sometimes called a “scalar” system of equations. A point implicit
(Gauss-Seidel) linear equation solver is used in conjunction with an algebraic multigrid (AMG) method to solve the resultant scalar system
of equations for the dependent variable in each cell. For example, the
x-momentum equation is linearized to produce a system of equations in
which u velocity is the unknown. Simultaneous solution of this equation
system (using the scalar AMG solver) yields an updated u-velocity field.
In summary, the segregated approach solves for a single variable field
(e.g., p) by considering all cells at the same time. It then solves for the
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Using the Solver
next variable field by again considering all cells at the same time, and
so on. There is no explicit option for the segregated solver.
In the coupled solution method you have a choice of using either an implicit or explicit linearization of the governing equations. This choice
applies only to the coupled set of governing equations. Governing equations for additional scalars that are solved segregated from the coupled
set, such as for turbulence, radiation, etc., are linearized and solved implicitly using the same procedures as in the segregated solution method.
Regardless of whether you choose the implicit or explicit scheme, the
solution procedure shown in Figure 22.1.2 is followed.
If you choose the implicit option of the coupled solver, each equation
in the coupled set of governing equations is linearized implicitly with
respect to all dependent variables in the set. This will result in a system
of linear equations with N equations for each cell in the domain, where
N is the number of coupled equations in the set. Because there are N
equations per cell, this is sometimes called a “block” system of equations.
A point implicit (block Gauss-Seidel) linear equation solver is used in
conjunction with an algebraic multigrid (AMG) method to solve the
resultant block system of equations for all N dependent variables in
each cell. For example, linearization of the coupled continuity, x-, y-, zmomentum, and energy equation set will produce a system of equations
in which p, u, v, w, and T are the unknowns. Simultaneous solution
of this equation system (using the block AMG solver) yields at once
updated pressure, u-, v-, w-velocity, and temperature fields.
In summary, the coupled implicit approach solves for all variables (p, u,
v, w, T ) in all cells at the same time.
If you choose the explicit option of the coupled solver, each equation
in the coupled set of governing equations is linearized explicitly. As in
the implicit option, this too will result in a system of equations with
N equations for each cell in the domain. And likewise, all dependent
variables in the set will be updated at once. However, this system of
equations is explicit in the unknown dependent variables. For example,
the x-momentum equation is written such that the updated x velocity
is a function of existing values of the field variables. Because of this,
a linear equation solver is not needed. Instead, the solution is updated
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22.2 Discretization
using a multi-stage (Runge-Kutta) solver. Here you have the additional
option of employing a full approximation storage (FAS) multigrid scheme
to accelerate the multi-stage solver.
In summary, the coupled explicit approach solves for all variables (p, u,
v, w, T ) one cell at a time.
Note that the FAS multigrid is an optional component of the explicit
approach, while the AMG is a required element in both the segregated
and coupled implicit approaches.
22.2
Discretization
FLUENT uses a control-volume-based technique to convert the governing
equations to algebraic equations that can be solved numerically. This
control volume technique consists of integrating the governing equations
about each control volume, yielding discrete equations that conserve each
quantity on a control-volume basis.
Discretization of the governing equations can be illustrated most easily
by considering the steady-state conservation equation for transport of
a scalar quantity φ. This is demonstrated by the following equation
written in integral form for an arbitrary control volume V as follows:
I
~=
ρφ ~v · dA
I
~+
Γφ ∇φ · dA
Z
V
Sφ dV
(22.2-1)
where
ρ
~v
~
A
Γφ
∇φ
Sφ
=
=
=
=
=
=
density
velocity vector (= u ı̂ + v ̂ in 2D)
surface area vector
diffusion coefficient for φ
gradient of φ (= ∂φ/∂x) ı̂ + (∂φ/∂y) ̂ in 2D)
source of φ per unit volume
Equation 22.2-1 is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure 22.2.1 is an example of such a control volume. Discretization of
Equation 22.2-1 on a given cell yields
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Using the Solver
NX
faces
~f =
ρf ~vf φf · A
f
NX
faces
~ f + Sφ V
Γφ (∇φ)n · A
(22.2-2)
f
where
Nfaces
φf
~f
ρf ~vf · A
~f
A
(∇φ)n
V
=
=
=
=
=
=
number of faces enclosing cell
value of φ convected through face f
mass flux through the face
area of face f , |A| (= |Ax ı̂ + Ay ̂| in 2D)
magnitude of ∇φ normal to face f
cell volume
The equations solved by FLUENT take the same general form as the
one given above and apply readily to multi-dimensional, unstructured
meshes composed of arbitrary polyhedra.
c1
A
c0
Figure 22.2.1: Control Volume Used to Illustrate Discretization of a
Scalar Transport Equation
FLUENT stores discrete values of the scalar φ at the cell centers (c0
and c1 in Figure 22.2.1). However, face values φf are required for the
convection terms in Equation 22.2-2 and must be interpolated from the
cell center values. This is accomplished using an upwind scheme.
Upwinding means that the face value φf is derived from quantities in the
cell upstream, or “upwind,” relative to the direction of the normal velocity vn in Equation 22.2-2. FLUENT allows you to choose from several
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22.2 Discretization
upwind schemes: first-order upwind, second-order upwind, power law,
and QUICK. These schemes are described in Sections 22.2.1–22.2.4.
The diffusion terms in Equation 22.2-2 are central-differenced and are
always second-order accurate.
22.2.1
First-Order Upwind Scheme
When first-order accuracy is desired, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell; the face
quantities are identical to the cell quantities. Thus when first-order upwinding is selected, the face value φf is set equal to the cell-center value
of φ in the upstream cell
22.2.2
Power-Law Scheme
The power-law discretization scheme interpolates the face value of a
variable, φ, using the exact solution to a one-dimensional convectiondiffusion equation
∂
∂ ∂φ
(ρuφ) =
Γ
∂x
∂x ∂x
(22.2-3)
where Γ and ρu are constant across the interval ∂x. Equation 22.2-3 can
be integrated to yield the following solution describing how φ varies with
x:
exp(Pe Lx ) − 1
φ(x) − φ0
=
φL − φ0
exp(Pe) − 1
(22.2-4)
where
φ0
φL
=
=
φ|x=0
φ|x=L
and Pe is the Peclet number:
Pe =
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ρuL
Γ
(22.2-5)
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Using the Solver
The variation of φ(x) between x = 0 and x = L is depicted in Figure 22.2.2 for a range of values of the Peclet number. Figure 22.2.2 shows
that for large Pe, the value of φ at x = L/2 is approximately equal to
the upstream value. This implies that when the flow is dominated by
convection, interpolation can be accomplished by simply letting the face
value of a variable be set equal to its “upwind” or upstream value. This
is the standard first-order scheme for FLUENT.
φL
Pe < -1
Pe = -1
φ
Pe= 0
Pe = 1
Pe > 1
φ0
0
L
X
Figure 22.2.2: Variation of a Variable φ Between x = 0 and x = L
(Equation 22.2-3)
If the power-law scheme is selected, FLUENT uses Equation 22.2-4 in an
equivalent “power law” format [172], as its interpolation scheme.
As discussed in Section 22.2.1, Figure 22.2.2 shows that for large Pe, the
value of φ at x = L/2 is approximately equal to the upstream value.
When Pe=0 (no flow, or pure diffusion), Figure 22.2.2 shows that φ
may be interpolated using a simple linear average between the values at
x = 0 and x = L. When the Peclet number has an intermediate value,
the interpolated value for φ at x = L/2 must be derived by applying the
“power law” equivalent of Equation 22.2-4.
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22.2 Discretization
22.2.3
Second-Order Upwind Scheme
When second-order accuracy is desired, quantities at cell faces are computed using a multidimensional linear reconstruction approach [7]. In
this approach, higher-order accuracy is achieved at cell faces through a
Taylor series expansion of the cell-centered solution about the cell centroid. Thus when second-order upwinding is selected, the face value φf
is computed using the following expression:
φf = φ + ∇φ · ∆~s
(22.2-6)
where φ and ∇φ are the cell-centered value and its gradient in the upstream cell, and ∆~s is the displacement vector from the upstream cell
centroid to the face centroid. This formulation requires the determination of the gradient ∇φ in each cell. This gradient is computed using
the divergence theorem, which in discrete form is written as
∇φ =
Nfaces
1 X
~
φ̃f A
V f
(22.2-7)
Here the face values φ̃f are computed by averaging φ from the two cells
adjacent to the face. Finally, the gradient ∇φ is limited so that no new
maxima or minima are introduced.
22.2.4
QUICK Scheme
For quadrilateral and hexahedral meshes, where unique upstream and
downstream faces and cells can be identified, FLUENT also provides the
QUICK scheme for computing a higher-order value of the convected variable φ at a face. QUICK-type schemes [133] are based on a weighted
average of second-order-upwind and central interpolations of the variable. For the face e in Figure 22.2.3, if the flow is from left to right, such
a value can be written as
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Using the Solver
φe = θ
Sd
Su + 2Sc
Sc
Sc
φP +
φE + (1 − θ)
φP −
φW
Sc + Sd
Sc + Sd
Su + Sc
Su + Sc
(22.2-8)
Su
Sc
Sd
W
∆x w P
∆x e E
w
e
Figure 22.2.3: One-Dimensional Control Volume
θ = 1 in the above equation results in a central second-order interpolation
while θ = 0 yields a second-order upwind value. The traditional QUICK
scheme is obtained by setting θ = 1/8. The implementation in FLUENT
uses a variable, solution-dependent value of θ, chosen so as to avoid
introducing new solution extrema.
The QUICK scheme will typically be more accurate on structured grids
aligned with the flow direction. Note that FLUENT allows the use of
the QUICK scheme for unstructured or hybrid grids as well; in such
cases the usual second-order upwind discretization scheme (described
in Section 22.2.3) will be used at the faces of non-hexahedral (or nonquadrilateral, in 2D) cells. The second-order upwind scheme will also be
used at partition boundaries when the parallel solver is used.
22.2.5
Central-Differencing Scheme
A second-order-accurate central-differencing discretization scheme is available for the momentum equations when you are using the LES turbulence
model. This scheme provides improved accuracy for LES calculations.
The central-differencing scheme calculates the face value for a variable
(φf ) as follows:
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22.2 Discretization
φf,CD =
1
1
(φ0 + φ1 ) + (∇φr,0 · ~r0 + ∇φr,1 · ~r1 )
2
2
(22.2-9)
where the indices 0 and 1 refer to the cells that share face f , ∇φr,0 and
∇φr,1 are the reconstructed gradients at cells 0 and 1, respectively, and
~r is the vector directed from the cell centroid toward the face centroid.
It is well known that central-differencing schemes can produce unbounded
solutions and non-physical wiggles, which can lead to stability problems for the numerical procedure. These stability problems can often be
avoided if a deferred approach is used for the central-differencing scheme.
In this approach, the face value is calculated as follows:
φf =
φf,UP
| {z }
+
implicit part
(φf,CD − φf,UP )
|
{z
}
(22.2-10)
explicit part
where UP stands for upwind. As indicated, the upwind part is treated
implicitly while the difference between the central-difference and upwind
values is treated explicitly. Provided that the numerical solution converges, this approach leads to pure second-order differencing.
22.2.6
Linearized Form of the Discrete Equation
The discretized scalar transport equation (Equation 22.2-2) contains the
unknown scalar variable φ at the cell center as well as the unknown
values in surrounding neighbor cells. This equation will, in general, be
non-linear with respect to these variables. A linearized form of Equation 22.2-2 can be written as
aP φ =
X
anb φnb + b
(22.2-11)
nb
where the subscript nb refers to neighbor cells, and aP and anb are the
linearized coefficients for φ and φnb .
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Using the Solver
The number of neighbors for each cell depends on the grid topology, but
will typically equal the number of faces enclosing the cell (boundary cells
being the exception).
Similar equations can be written for each cell in the grid. This results
in a set of algebraic equations with a sparse coefficient matrix. For
scalar equations, FLUENT solves this linear system using a point implicit
(Gauss-Seidel) linear equation solver in conjunction with an algebraic
multigrid (AMG) method which is described in Section 22.5.3.
22.2.7
Under-Relaxation
Because of the nonlinearity of the equation set being solved by FLUENT,
it is necessary to control the change of φ. This is typically achieved by
under-relaxation, which reduces the change of φ produced during each
iteration. In a simple form, the new value of the variable φ within a cell
depends upon the old value, φold , the computed change in φ, ∆φ, and
the under-relaxation factor, α, as follows:
φ = φold + α∆φ
22.2.8
(22.2-12)
Temporal Discretization
For transient simulations, the governing equations must be discretized in
both space and time. The spatial discretization for the time-dependent
equations is identical to the steady-state case. Temporal discretization
involves the integration of every term in the differential equations over a
time step ∆t. The integration of the transient terms is straightforward,
as shown below.
A generic expression for the time evolution of a variable φ is given by
∂φ
= F (φ)
∂t
(22.2-13)
where the function F incorporates any spatial discretization. If the time
derivative is discretized using backward differences, the first-order accurate temporal discretization is given by
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22.2 Discretization
φn+1 − φn
= F (φ)
∆t
and the second-order discretization is given by
3φn+1 − 4φn + φn−1
= F (φ)
2∆t
(22.2-14)
(22.2-15)
where
φ
n+1
n
n−1
=
=
=
=
a scalar quantity
value at the next time level, t + ∆t
value at the current time level, t
value at the previous time level, t − ∆t
Once the time derivative has been discretized, a choice remains for evaluating F (φ): in particular, which time level values of φ should be used
in evaluating F ?
Implicit Time Integration
One method is to evaluate F (φ) at the future time level:
φn+1 − φn
= F (φn+1 )
∆t
(22.2-16)
This is referred to as “implicit” integration since φn+1 in a given cell is
related to φn+1 in neighboring cells through F (φn+1 ):
φn+1 = φn + ∆tF (φn+1 )
(22.2-17)
This implicit equation can be solved iteratively by initializing φi to φn
and iterating the equation
φi = φn + ∆tF (φi )
(22.2-18)
for the first-order implicit formulation, or
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Using the Solver
φi = 4/3φn − 1/3φn−1 + 2/3∆tF (φi )
(22.2-19)
for the second-order implicit formulation, until φi stops changing (i.e.,
converges). At that point, φn+1 is set to φi .
The advantage of the fully implicit scheme is that it is unconditionally
stable with respect to time step size.
Explicit Time Integration
A second method is available when the coupled explicit solver is used.
This method evaluates F (φ) at the current time level:
φn+1 − φn
= F (φn )
∆t
(22.2-20)
and is referred to as “explicit” integration since φn+1 can be expressed
explicitly in terms of the existing solution values, φn :
φn+1 = φn + ∆tF (φn )
(22.2-21)
(This method is also referred to as “global time stepping”.)
Here, the time step ∆t is restricted to the stability limit of the underlying
solver (i.e., a time step corresponding to a Courant number of approximately 1). In order to be time-accurate, all cells in the domain must use
the same time step. For stability, this time step must be the minimum
of all the local time steps in the domain.
The use of explicit time stepping is fairly restrictive. It is used primarily to capture the transient behavior of moving waves, such as shocks,
because it is more accurate and less expensive than the implicit time
stepping methods in such cases. You cannot use explicit time stepping
in the following cases:
• Calculations with the segregated or coupled implicit solver. The
explicit time stepping formulation is available only with the coupled
explicit solver.
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22.2 Discretization
• Incompressible flow. Explicit time stepping cannot be used to compute time-accurate incompressible flows (i.e., gas laws other than
ideal gas). Incompressible solutions must be iterated to convergence within each time step.
• Convergence acceleration. FAS multigrid and residual smoothing
cannot be used with explicit time stepping because they destroy
the time accuracy of the underlying solver.
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Using the Solver
22.3
The Segregated Solver
In this section, special practices related to the discretization of the momentum and continuity equations and their solution by means of the segregated solver are addressed. These practices are most easily described
by considering the steady-state continuity and momentum equations in
integral form:
I
I
~=−
ρ~v ~v · dA
I
~=0
ρ ~v · dA
~+
pI · dA
I
~+
τ · dA
(22.3-1)
Z
F~ dV
(22.3-2)
V
where I is the identity matrix, τ is the stress tensor, and F~ is the force
vector.
22.3.1
Discretization of the Momentum Equation
The discretization scheme described in Section 22.2 for a scalar transport equation is also used to discretize the momentum equations. For
example, the x-momentum equation can be obtained by setting φ = u:
aP u =
X
anb unb +
X
pf A · ı̂ + S
(22.3-3)
nb
If the pressure field and face mass fluxes were known, Equation 22.3-3
could be solved in the manner outlined in Section 22.2, and a velocity
field obtained. However, the pressure field and face mass fluxes are not
known a priori and must be obtained as a part of the solution. There
are important issues with respect to the storage of pressure and the
FLUENT uses a co-located scheme, whereby pressure and velocity are
both stored at cell centers. However, Equation 22.3-3 requires the value
of the pressure at the face between cells c0 and c1, shown in Figure 22.2.1.
Therefore, an interpolation scheme is required to compute the face values
of pressure from the cell values.
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22.3 The Segregated Solver
Pressure Interpolation Schemes
The default scheme in FLUENT interpolates the pressure values at the
faces using momentum equation coefficients [192]. This procedure works
well as long as the pressure variation between cell centers is smooth.
When there are jumps or large gradients in the momentum source terms
between control volumes, the pressure profile has a high gradient at the
cell face, and cannot be interpolated using this scheme. If this scheme is
used, the discrepancy shows up in overshoots/undershoots of cell velocity.
Flows for which the standard pressure interpolation scheme will have
trouble include flows with large body forces, such as in strongly swirling
flows, in high-Rayleigh-number natural convection and the like. In such
cases, it is necessary to pack the mesh in regions of high gradient to
Another source of error is that FLUENT assumes that the normal pressure
gradient at the wall is zero. This is valid for boundary layers, but not in
the presence of body forces or curvature. Again, the failure to correctly
account for the wall pressure gradient is manifested in velocity vectors
pointing in/out of walls.
Several alternate methods are available for cases in which the standard
pressure interpolation scheme is not valid:
• The linear scheme computes the face pressure as the average of the
pressure values in the adjacent cells.
• The second-order scheme reconstructs the face pressure in the manner used for second-order accurate convection terms (see Section
22.2.3). This scheme may provide some improvement over the standard and linear schemes, but it may have some trouble if it is used
at the start of a calculation and/or with a bad mesh. The secondorder scheme is not applicable for flows with discontinuous pressure
gradients imposed by the presence of a porous medium in the domain or the use of the VOF or mixture model for multiphase flow.
• The body-force-weighted scheme computes the face pressure by assuming that the normal gradient of the difference between pressure
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Using the Solver
and body forces is constant. This works well if the body forces are
known a priori in the momentum equations (e.g., buoyancy and
axisymmetric swirl calculations).
• The PRESTO! (PREssure STaggering Option) scheme uses the
discrete continuity balance for a “staggered” control volume about
the face to compute the “staggered” (i.e., face) pressure. This procedure is similar in spirit to the staggered-grid schemes used with
structured meshes [172]. Note that for triangular and tetrahedral
meshes, comparable accuracy is obtained using a similar algorithm.
See Section 22.7.3 for recommendations on when to use these alternate
schemes.
22.3.2
Discretization of the Continuity Equation
Equation 22.3-1 may be integrated over the control volume in Figure 22.2.1
to yield the following discrete equation
NX
faces
Jf Af = 0
(22.3-4)
f
where Jf is the mass flux through face f , ρvn .
As described in Section 22.1, the momentum and continuity equations are
solved sequentially. In this sequential procedure, the continuity equation
is used as an equation for pressure. However, pressure does not appear
explicitly in Equation 22.3-4 for incompressible flows, since density is
not directly related to pressure. The SIMPLE (Semi-Implicit Method
for Pressure-Linked Equations) family of algorithms [172] is used for
introducing pressure into the continuity equation. This procedure is
outlined in Section 22.3.3.
In order to proceed further, it is necessary to relate the face values of
velocity, ~vn , to the stored values of velocity at the cell centers. Linear
interpolation of cell-centered velocities to the face results in unphysical
checker-boarding of pressure. FLUENT uses a procedure similar to that
outlined by Rhie and Chow [192] to prevent checkerboarding. The face
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22.3 The Segregated Solver
value of velocity is not averaged linearly; instead, momentum-weighted
averaging, using weighting factors based on the aP coefficient from equation 22.3-3, is performed. Using this procedure, the face flux, Jf , may
be written as
Jf = Jˆf + df (pc0 − pc1 )
(22.3-5)
where pc0 and pc1 are the pressures within the two cells on either side
of the face, and Jˆf contains the influence of velocities in these cells (see
Figure 22.2.1). The term df is a function of āP , the average of the
momentum equation aP coefficients for the cells on either side of face f .
Density Interpolation Schemes
For compressible flow calculations (i.e., calculations that use the ideal
gas law for density), FLUENT applies upwind interpolation of density at
cell faces. (For incompressible flows, FLUENT uses arithmetic averaging.) Three interpolation schemes are available for the density upwinding at cell faces: first-order upwind (default), second-order-upwind, and
QUICK.
The first-order upwind scheme (based on [109]) sets the density at the
cell face to be the upstream cell-center value. This scheme provides
stability for the discretization of the pressure-correction equation, and
gives good results for most classes of flows. The first-order scheme is the
default scheme for compressible flows.
The second-order upwind scheme provides stability for supersonic flows
and captures shocks better than the first-order upwind scheme. The
QUICK scheme for density is similar to the QUICK scheme used for
other variables. See Section 22.2.4 for details.
! The second-order upwind and QUICK schemes for density are not available for compressible multiphase calculations; the first-order upwind
scheme is used for the compressible phase, and arithmetic averaging is
used for the incompressible phases.
See Section 22.7.4 for recommendations on choosing an appropriate density interpolation scheme for your compressible flow.
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Using the Solver
22.3.3
Pressure-Velocity Coupling
Pressure-velocity coupling is achieved by using Equation 22.3-5 to derive
an equation for pressure from the discrete continuity equation (Equation 22.3-4) FLUENT provides the option to choose among three pressurevelocity coupling algorithms: SIMPLE, SIMPLEC, and PISO. See Section 22.8 for instructions on how to select these algorithms.
SIMPLE
The SIMPLE algorithm uses a relationship between velocity and pressure
corrections to enforce mass conservation and to obtain the pressure field.
If the momentum equation is solved with a guessed pressure field p∗ , the
resulting face flux, Jf∗ , computed from Equation 22.3-5
Jf∗ = Jˆf∗ + df (p∗c0 − p∗c1 )
(22.3-6)
does not satisfy the continuity equation. Consequently, a correction Jf0
is added to the face flux Jf∗ so that the corrected face flux, Jf
Jf = Jf∗ + Jf0
(22.3-7)
satisfies the continuity equation. The SIMPLE algorithm postulates that
Jf0 be written as
Jf0 = df (p0c0 − p0c1 )
(22.3-8)
where p0 is the cell pressure correction.
The SIMPLE algorithm substitutes the flux correction equations (Equations 22.3-7 and 22.3-8) into the discrete continuity equation (Equation 22.3-4) to obtain a discrete equation for the pressure correction p0
in the cell:
aP p0 =
X
anb p0nb + b
(22.3-9)
nb
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22.3 The Segregated Solver
where the source term b is the net flow rate into the cell:
b=
NX
faces
Jf∗ Af
(22.3-10)
f
The pressure-correction equation (Equation 22.3-9) may be solved using
the algebraic multigrid (AMG) method described in Section 22.5.3. Once
a solution is obtained, the cell pressure and the face flux are corrected
using
p = p∗ + αp p0
(22.3-11)
Jf = Jf∗ + df (p0c0 − p0c1 )
(22.3-12)
Here αp is the under-relaxation factor for pressure (see Section 22.2.7
for information about under-relaxation). The corrected face flux, Jf ,
satisfies the discrete continuity equation identically during each iteration.
SIMPLEC
A number of variants of the basic SIMPLE algorithm are available in
the literature. In addition to SIMPLE, FLUENT offers the SIMPLEC
(SIMPLE-Consistent) algorithm [243]. SIMPLE is the default, but many
problems will benefit from the use of SIMPLEC, as described in Section 22.8.1.
The SIMPLEC procedure is similar to the SIMPLE procedure outlined
above. The only difference lies in the expression used for the face flux
correction, Jf0 . As in SIMPLE, the correction equation may be written
as
Jf = Jf∗ + df (p0c0 − p0c1 )
(22.3-13)
P
However, the coefficient df is redefined as a function of (aP − nb anb )
The use of this modified correction equation has been shown to accelerate
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22-25
Using the Solver
convergence in problems where pressure-velocity coupling is the main
deterrent to obtaining a solution.
PISO
The Pressure-Implicit with Splitting of Operators (PISO) pressure-velocity coupling scheme, part of the SIMPLE family of algorithms, is based
on the higher degree of the approximate relation between the corrections
for pressure and velocity. One of the limitations of the SIMPLE and SIMPLEC algorithms is that new velocities and corresponding fluxes do not
satisfy the momentum balance after the pressure-correction equation is
solved. As a result, the calculation must be repeated until the balance
is satisfied. To improve the efficiency of this calculation, the PISO algorithm performs two additional corrections: neighbor correction and
skewness correction.
Neighbor Correction
The main idea of the PISO algorithm is to move the repeated calculations required by SIMPLE and SIMPLEC inside the solution stage
of the pressure-correction equation [97]. After one or more additional
PISO loops, the corrected velocities satisfy the continuity and momentum equations more closely. This iterative process is called a momentum
correction or “neighbor correction”. The PISO algorithm takes a little
more CPU time per solver iteration, but it can dramatically decrease
the number of iterations required for convergence, especially for transient problems.
Skewness Correction
For meshes with some degree of skewness, the approximate relationship
between the correction of mass flux at the cell face and the difference
of the pressure corrections at the adjacent cells is very rough. Since the
components of the pressure-correction gradient along the cell faces are
not known in advance, an iterative process similar to the PISO neighbor
correction described above is desirable [64]. After the initial solution
of the pressure-correction equation, the pressure-correction gradient is
recalculated and used to update the mass flux corrections. This process,
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22.3 The Segregated Solver
which is referred to as “skewness correction”, significantly reduces convergence difficulties associated with highly distorted meshes. The PISO
skewness correction allows FLUENT to obtain a solution on a highly
skewed mesh in approximately the same number of iterations as required
for a more orthogonal mesh.
Special Treatment for Strong Body Forces in Multiphase Flows
When large body forces (e.g., gravity or surface tension forces) exist
in multiphase flows, the body force and pressure gradient terms in the
momentum equation are almost in equilibrium, with the contributions of
convective and viscous terms small in comparison. Segregated algorithms
converge poorly unless partial equilibrium of pressure gradient and body
forces is taken into account. FLUENT provides an optional “implicit body
force” treatment that can account for this effect, making the solution
more robust.
The basic procedure involves augmenting the correction equation for
the face flow rate, Equation 22.3-12, with an additional term involving
corrections to the body force. This results in extra body force correction
terms in Equation 22.3-10, and allows the flow to achieve a realistic
pressure field very early in the iterative process.
This option is available only for multiphase calculations, but it is turned
off by default. Section 20.6.11 includes instructions for turning on the
implicit body force treatment.
In addition, FLUENT allows you to control the change in the body forces
through the use of an under-relaxation factor for body forces.
22.3.4
The governing equations for the segregated solver do not contain any
time-dependent terms if you are performing a steady-state calculation.
For time-accurate calculations, an implicit time stepping scheme is used.
See Section 22.2.8 for details.
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22-27
Using the Solver
22.4
The Coupled Solver
The coupled solver in FLUENT solves the governing equations of continuity, momentum, and (where appropriate) energy and species transport
simultaneously as a set, or vector, of equations. Governing equations for
additional scalars will be solved sequentially (i.e., segregated from one
another and from the coupled set).
22.4.1
Governing Equations in Vector Form
The system of governing equations for a single-component fluid, written
to describe the mean flow properties, is cast in integral, Cartesian form
for an arbitrary control volume V with differential surface area dA as
follows:
∂
∂t
Z
V
W dV +
I
[F − G] · dA =
Z
V
H dV
(22.4-1)
where the vectors W , F , and G are defined as
W =
 
ρ 








ρu 

ρv







ρw 



ρE
, F =


ρv










ρvu
+
pı̂


ρvv + p̂







ρvw + pk̂



ρvE + pv
, G=

0





τ xi







yi



τ

 zi







τ
τ ij vj + q
and the vector H contains source terms such as body forces and energy
sources.
Here ρ, v, E, and p are the density, velocity, total energy per unit mass,
and pressure of the fluid, respectively. τ is the viscous stress tensor, and
q is the heat flux.
Total energy E is related to the total enthalpy H by
E = H − p/ρ
(22.4-2)
where
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22.4 The Coupled Solver
H = h + |v|2 /2
(22.4-3)
The Navier-Stokes equations as expressed in Equation 22.4-1 become
(numerically) very stiff at low Mach number due to the disparity between
the fluid velocity v and the acoustic speed c (speed of sound). This is
also true for incompressible flows, regardless of the fluid velocity, because
acoustic waves travel infinitely fast in an incompressible fluid (speed
of sound is infinite). The numerical stiffness of the equations under
these conditions results in poor convergence rates. This difficulty is
overcome in FLUENT’s coupled solver by employing a technique called
(time-derivative) preconditioning [261].
22.4.2
Preconditioning
Time-derivative preconditioning modifies the time-derivative term in Equation 22.4-1 by pre-multiplying it with a preconditioning matrix. This has
the effect of re-scaling the acoustic speed (eigenvalue) of the system of
equations being solved in order to alleviate the numerical stiffness encountered in low Mach number and incompressible flow.
Derivation of the preconditioning matrix begins by transforming the dependent variable in Equation 22.4-1 from conserved quantities W to
primitive variables Q using the chain-rule as follows:
∂W ∂
∂Q ∂t
Z
V
Q dV +
I
[F − G] · dA =
Z
V
H dV
(22.4-4)
where Q is the vector {p, u, v, w, T }T and the Jacobian ∂W /∂Q is given
by



∂W

=

∂Q

ρp
ρp u
ρp v
ρp w
ρp H − δ
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0
ρ
0
0
ρu
0
0
ρ
0
ρv
0
0
0
ρ
ρw
ρT
ρT u
ρT v
ρT w
ρT H + ρCp







(22.4-5)
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Using the Solver
where
∂ρ ∂ρ ρp =
, ρT =
∂p T
∂T p
and δ = 1 for an ideal gas and δ = 0 for an incompressible fluid.
The choice of primitive variables Q as dependent variables is desirable
for several reasons. First, it is a natural choice when solving incompressible flows. Second, when we use second-order accuracy we need to
reconstruct Q rather than W in order to obtain more accurate velocity
inviscid fluxes. And finally, the choice of pressure as a dependent variable allows the propagation of acoustic waves in the system to be singled
out [245].
We precondition the system by replacing the Jacobian matrix ∂W /∂Q
(Equation 22.4-5) with the preconditioning matrix Γ so that the preconditioned system in conservation form becomes
∂
Γ
∂t
Z
V
Q dV +
I
[F − G] · dA =
Z
V
H dV
(22.4-6)
where




Γ=


Θ
Θu
Θv
Θw
ΘH − δ
0
ρ
0
0
ρu
0
0
ρ
0
ρv
0
0
0
ρ
ρw
ρT
ρT u
ρT u
ρT u
ρT H + ρCp







(22.4-7)
The parameter Θ is given by
Θ=
1
ρT
−
2
Ur
ρCp
!
(22.4-8)
The reference velocity Ur appearing in Equation 22.4-8 is chosen locally
such that the eigenvalues of the system remain well conditioned with
respect to the convective and diffusive time scales [261].
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22.4 The Coupled Solver
The resultant eigenvalues of the preconditioned system (Equation 22.4-6)
are given by
u, u, u, u0 + c0 , u0 − c0
(22.4-9)
where
u = v · n̂
u0 = u (1 − α)
c0 =
q
α2 u2 + Ur2
α =
1 − βUr2 /2
β =
ρT
ρp +
ρCp
!
For an ideal gas, β = (γRT )−1 = 1/c2 . Thus, when Ur = c (at sonic
speeds and above), α = 0 and the eigenvalues of the preconditioned
system take their traditional form, u ± c. At low speed, however, as
Ur → 0, α → 1/2 and all eigenvalues become of the same order as u.
For constant-density flows, β = 0 and α = 1/2 regardless of the values
of Ur . As long as the reference velocity is of the same order as the local
velocity, all eigenvalues remain of the order u. Thus, the eigenvalues of
the preconditioned system remain well conditioned at all speeds.
Note that the non-preconditioned Navier-Stokes equations are recovered
exactly from Equation 22.4-6 by setting 1/Ur2 to ρp , the derivative of
density with respect to pressure. In this case Γ reduces exactly to the
Jacobian ∂W /∂Q.
Although Equation 22.4-6 is conservative in the steady state, it is not,
in a strict sense, conservative for time-dependent flows. This is not a
problem, however, since the preconditioning has already destroyed the
time accuracy of the equations and we will not employ them in this form
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Using the Solver
Flux-Difference Splitting
The inviscid flux vector F appearing in Equation 22.4-6 is evaluated
by a standard upwind, flux-difference splitting [194]. This approach acknowledges that the flux vector F contains characteristic information
propagating through the domain with speed and direction according to
the eigenvalues of the system. By splitting F into parts, where each part
contains information traveling in a particular direction (i.e., characteristic information), and upwind differencing the split fluxes in a manner
consistent with their corresponding eigenvalues, we obtain the following
expression for the discrete flux at each face:
F =
1
1
(F R + F L ) − Γ |Â| δQ
2
2
(22.4-10)
Here δQ is the spatial difference QR −QL . The fluxes F R = F (QR ) and
F L = F (QL ) are computed using the (reconstructed) solution vectors
QR and QL on the “right” and “left” side of the face. The matrix |Â| is
defined by
|Â| = M |Λ| M −1
(22.4-11)
where Λ is the diagonal matrix of eigenvalues and M is the modal matrix
that diagonalizes Γ−1 A, where A is the inviscid flux Jacobian ∂F /∂Q.
For the non-preconditioned system (and an ideal gas) Equation 22.4-10
reduces to Roe’s flux-difference splitting [194] when Roe-averaged values
are used to evaluate Γ |Â|. At present, arithmetic averaging of states QR
and QL is used.
In its current form, Equation 22.4-10 can be viewed as a second-order
dissipation term is not only responsible for producing an upwinding of the
convected variables, and of pressure and flux velocity in supersonic flow,
but it also provides the pressure-velocity coupling required for stability
and efficient convergence of low-speed and incompressible flows.
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22.4 The Coupled Solver
22.4.3
The coupled set of governing equations (Equation 22.4-6) in FLUENT
is discretized in time for both steady and unsteady calculations. In the
steady case, it is assumed that time marching proceeds until a steadystate solution is reached. Temporal discretization of the coupled equations is accomplished by either an implicit or an explicit time-marching
scheme. These two algorithms are described below.
Explicit Scheme
In the explicit scheme a multi-stage, time-stepping algorithm [101] is
used to discretize the time derivative in Equation 22.4-6. The solution
is advanced from iteration n to iteration n + 1 with an m-stage RungeKutta scheme, given by
Q 0 = Qn
∆Qi = −αi ∆tΓ−1 Ri−1
Qn+1 = Qm
where ∆Qi ≡ Qi − Qn and i = 1, 2, . . . , m is the stage counter for
the m-stage scheme. αi is the multi-stage coefficient for the ith stage.
The residual Ri is computed from the intermediate solution Qi and, for
Equation 22.4-6, is given by
i
R =
NX
faces
F (Qi ) − G(Qi ) · A − V H
(22.4-12)
The time step ∆t is computed from the CFL (Courant-Friedrichs-Lewy)
condition
∆t =
CFL∆x
λmax
(22.4-13)
where λmax is the maximum of the local eigenvalues defined by Equation 22.4-9.
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22-33
Using the Solver
The convergence rate of the explicit scheme can be accelerated through
use of the full-approximation storage (FAS) multigrid method described
in Section 22.5.4.
Implicit Residual Smoothing
The maximum time step can be further increased by increasing the support of the scheme through implicit averaging of the residuals with their
neighbors. The residuals are filtered through a Laplacian smoothing
operator:
R̄i = Ri + X
(R̄j − R̄i )
(22.4-14)
This equation can be solved with the following Jacobi iteration:
P
R̄im
Ri + R̄jm−1
P
=
1+ 1
(22.4-15)
Two Jacobi iterations are usually sufficient to allow doubling the time
step with a value of = 0.5.
Implicit Scheme
In the implicit scheme, an Euler implicit discretization in time of the
governing equations (Equation 22.4-6) is combined with a Newton-type
linearization of the fluxes to produce the following linearized system in
delta form [259]:

D +
NX
faces

Sj,k  ∆Qn+1 = −Rn
(22.4-16)
j
The center and off-diagonal coefficient matrices, D and Sj,k are given by
D =
22-34
NX
faces
V
Sj,i
Γ+
∆t
j
(22.4-17)
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22.4 The Coupled Solver
Sj,k =
∂F j
∂Gj
−
∂Qk ∂Qk
(22.4-18)
and the residual vector Rn and time step ∆t are defined as in Equation 22.4-12 and Equation 22.4-13, respectively.
Equation 22.4-16 is solved using a point Gauss-Seidel scheme in conjunction with an algebraic multigrid (AMG) method (see Section 22.5.3)
adapted for coupled sets of equations.
22.4.4
For time-accurate calculations, explicit and implicit time-stepping
schemes are available. (The implicit approach is also referred to as “dual
time stepping”.)
Explicit Time Stepping
In the explicit time stepping approach, the explicit scheme described
above is employed, using the same time step in each cell of the domain,
and with preconditioning disabled.
Dual Time Stepping
To provide for efficient, time-accurate solution of the preconditioned
equations, we employ a dual time-stepping, multi-stage scheme. Here
we introduce a preconditioned pseudo-time-derivative term into Equation 22.4-1 as follows:
∂
∂t
Z
V
W dV + Γ
∂
∂τ
Z
V
Q dV +
I
[F − G] · dA =
Z
V
H dV
(22.4-19)
where t denotes physical-time and τ is a pseudo-time used in the timemarching procedure. Note that as τ → ∞, the second term on the LHS
of Equation 22.4-19 vanishes and Equation 22.4-1 is recovered.
The time-dependent term in Equation 22.4-19 is discretized in an implicit
fashion by means of either a first- or second-order accurate, backward
difference in time. This is written in semi-discrete form as follows:
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22-35
Using the Solver
Γ
0 ∂W
∆Qk+1 +
+
∆τ
∆t ∂Q
=H +
I
1
[F − G] · dA
V
1 0 W k − 1 W n + 2 W n−1
∆t
where {0 = 1 = 1/2, 2 = 0} gives first-order time accuracy, and {0 =
3/2, 1 = 2, 2 = 1/2} gives second-order. k is the inner iteration counter
and n represents any given physical-time level.
The pseudo-time-derivative is driven to zero at each physical time level
by a series of inner iterations using either the implicit or explicit timemarching algorithm. Throughout the (inner) iterations in pseudo-time,
W n and W n−1 are held constant and W k is computed from Qk . As
τ → ∞, the solution at the next physical time level W n+1 is given by
W (Qk ).
Note that the physical time step ∆t is limited only by the level of desired
temporal accuracy. The pseudo-time-step ∆τ is determined by the CFL
condition of the (implicit or explicit) time-marching scheme.
22.5
Multigrid Method
The FLUENT solver contains two forms of multigrid: algebraic (AMG)
and full-approximation storage (FAS). As discussed in Section 22.1.3,
AMG is an essential component of both the segregated and coupled implicit solvers, while FAS is an important, but optional, component of the
coupled explicit solver. (Note that when the coupled explicit solver is
used, since the scalar equations (e.g., turbulence) are solved using the
segregated approach, AMG will also be used.)
This section describes the mathematical basis of the multigrid approach.
Common aspects of AMG and FAS are presented first, followed by separate sections that provide details unique to each method. Information
about user inputs and controls for the multigrid solver is provided in
Sections 22.19.3 and 22.19.4.
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22.5 Multigrid Method
22.5.1
Approach
FLUENT uses a multigrid scheme to accelerate the convergence of the
solver by computing corrections on a series of coarse grid levels. The
use of this multigrid scheme can greatly reduce the number of iterations
and the CPU time required to obtain a converged solution, particularly
when your model contains a large number of control volumes.
The Need for Multigrid
Implicit solution of the linearized equations on unstructured meshes is
complicated by the fact that there is no equivalent of the line-iterative
methods that are commonly used on structured grids. Since direct matrix inversion is out of the question for realistic problems and “wholefield” solvers that rely on conjugate-gradient (CG) methods have robustness problems associated with them, the methods of choice are point
implicit solvers like Gauss-Seidel. Although the Gauss-Seidel scheme
rapidly removes local (high-frequency) errors in the solution, global (lowfrequency) errors are reduced at a rate inversely related to the grid size.
Thus, for a large number of nodes, the solver “stalls” and the residual
reduction rate becomes prohibitively low.
The multi-stage scheme used in the coupled explicit solver can efficiently
remove local (high-frequency) errors as well. That is, the effect of the
solution in one cell is communicated to adjacent cells relatively quickly.
However, the scheme is less effective at reducing global (low-frequency)
errors—errors which exist over a large number of control volumes. Thus,
global corrections to the solution across a large number of control volumes occur slowly, over many iterations. This implies that performance
of the multi-stage scheme will deteriorate as the number of control volumes increases.
Multigrid techniques allow global error to be addressed by using a sequence of successively coarser meshes. This method is based upon the
principle that global (low-frequency) error existing on a fine mesh can be
represented on a coarse mesh where it again becomes accessible as local
(high-frequency) error: because there are fewer coarse cells overall, the
global corrections can be communicated more quickly between adjacent
cells. Since computations can be performed at exponentially decaying
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Using the Solver
expense in both CPU time and memory storage on coarser meshes, there
is the potential for very efficient elimination of global error. The fine-grid
relaxation scheme or “smoother”, in this case either the point-implicit
Gauss-Seidel or the explicit multi-stage scheme, is not required to be particularly effective at reducing global error and can be tuned for efficient
reduction of local error.
The Basic Concept in Multigrid
Consider the set of discretized linear (or linearized) equations given by
A φe + b = 0
(22.5-1)
where φe is the exact solution. Before the solution has converged there
will be a defect d associated with the approximate solution φ:
Aφ + b = d
(22.5-2)
We seek a correction ψ to φ such that the exact solution is given by
φe = φ + ψ
(22.5-3)
Substituting Equation 22.5-3 into Equation 22.5-1 gives
A (φ + ψ) + b = 0
(22.5-4)
A ψ + (A φ + b) = 0
(22.5-5)
Now using Equations 22.5-2 and 22.5-5 we obtain
Aψ + d = 0
(22.5-6)
which is an equation for the correction in terms of the original fine level
operator A and the defect d. Assuming the local (high-frequency) errors
have been sufficiently damped by the relaxation scheme on the fine level,
the correction ψ will be smooth and therefore more effectively solved on
the next coarser level.
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22.5 Multigrid Method
Restriction and Prolongation
Solving for corrections on the coarse level requires transferring the defect
down from the fine level (restriction), computing corrections, and then
transferring the corrections back up from the coarse level (prolongation).
We can write the equations for coarse level corrections ψ H as
AH ψ H + R d = 0
(22.5-7)
where AH is the coarse level operator and R the restriction operator
responsible for transferring the fine level defect down to the coarse level.
Solution of Equation 22.5-7 is followed by an update of the fine level
solution given by
φnew = φ + P ψ H
(22.5-8)
where P is the prolongation operator used to transfer the coarse level
corrections up to the fine level.
Unstructured Multigrid
The primary difficulty with using multigrid on unstructured grids is the
creation and use of the coarse grid hierarchy. On a structured grid, the
coarse grids can be formed simply by removing every other grid line from
the fine grids and the prolongation and restriction operators are simple
to formulate (e.g., injection and bilinear interpolation).
The difficulties of applying multigrid on unstructured grids are overcome
in separate fashion by each of the two multigrid methods used in FLUENT. While the basic principles discussed so far and the cycling strategy
described in Section 22.5.2 are the same, the techniques for construction
of restriction, prolongation, and coarse grid operators are different, as
discussed in Section 22.5.3 and Section 22.5.4 for the AMG and FAS
methods, respectively.
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Using the Solver
22.5.2
Multigrid Cycles
A multigrid cycle can be defined as a recursive procedure that is applied
at each grid level as it moves through the grid hierarchy. Four types
of multigrid cycles are available in FLUENT: the V, W, F, and flexible
(“flex”) cycles. The V and W cycles are available in both AMG and FAS,
while the F and flexible cycles are restricted to the AMG method only.
(The W and flex AMG cycles are not available for solving the coupled
equation set due to the amount of computation required.)
The V and W Cycles
Figures 22.5.1 and 22.5.2 show the V and W multigrid cycles (defined
below). In each figure, the multigrid cycle is represented by a square,
and then expanded recursively to show the individual steps that are
performed within the cycle. The individual steps are represented by a
circle, one or more squares, and a triangle, connected by lines: circlesquare-triangle for a V cycle, or circle-square-square-triangle for a W
cycle. The squares in this group expand again, into circle-square-triangle
or circle-square-square-triangle, and so on. You may want to follow along
in the figures as you read the steps below.
For the V and W cycles, the traversal of the hierarchy is governed by
three parameters, β1 , β2 , and β3 , as follows:
1. First, iterations are performed on the current grid level to reduce the high-frequency components of the error (local error). For
AMG, one iteration consists of one forward and one backward
Gauss-Seidel sweep. For FAS, one iteration consists of one pass
of the multi-stage scheme (described in Section 22.4.3).
These iterations are referred to as pre-relaxation sweeps because
they are performed before moving to the next coarser grid level.
The number of pre-relaxation sweeps is specified by β1 .
In Figures 22.5.1 and 22.5.2 this step is represented by a circle
and marks the start of a multigrid cycle. The high-wave-number
components of error should be reduced until the remaining error is
expressible on the next coarser mesh without significant aliasing.
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22.5 Multigrid Method
grid
level
multigrid cycle
0
pre-relaxation sweeps
0
post-relaxation sweeps and/or
Laplacian smoothings
1
0
1
2
0
1
2
3
0
1
2
3
Figure 22.5.1: V-Cycle Multigrid
grid
level
multigrid cycle
0
pre-relaxation sweeps
0
post-relaxation sweeps and/or
Laplacian smoothings
1
0
1
2
0
1
2
3
0
1
2
3
Figure 22.5.2: W-Cycle Multigrid
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Using the Solver
If this is the coarsest grid level, then the multigrid cycle on this
level is complete. (In Figures 22.5.1 and 22.5.2 there are 3 coarse
grid levels, so the square representing the multigrid cycle on level
3 is equivalent to a circle, as shown in the final diagram in each
figure.)
!
In the AMG method, the default value of β1 is zero (i.e., no prerelaxation sweeps are performed).
2. Next, the problem is “restricted” to the next coarser grid level
using Equation 22.5-7.
In Figures 22.5.1 and 22.5.2, the restriction from a finer grid level
to a coarser grid level is designated by a downward-sloping line.
3. The error on the coarse grid is reduced by performing a specified
number (β2 ) of multigrid cycles (represented in Figures 22.5.1 and
22.5.2 as squares). Commonly, for fixed multigrid strategies β2
is either 1 or 2, corresponding to V-cycle and W-cycle multigrid,
respectively.
4. Next, the cumulative correction computed on the coarse grid is “interpolated” back to the fine grid using Equation 22.5-8 and added
to the fine grid solution. In the FAS method, the corrections are additionally smoothed during this step using the Laplacian smoothing
operator discussed in Section 22.4.3.
In Figures 22.5.1 and 22.5.2 the prolongation is represented by an
upward-sloping line.
The high-frequency error now present at the fine grid level is due
to the prolongation procedure used to transfer the correction.
5. In the final step, iterations are performed on the fine grid to remove the high-frequency error introduced on the coarse grid by the
multigrid cycles. These iterations are referred to as post-relaxation
sweeps because they are performed after returning from the next
coarser grid level. The number of post-relaxation sweeps is specified by β3 .
In Figures 22.5.1 and 22.5.2, this relaxation procedure is represented by a single triangle.
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22.5 Multigrid Method
For AMG, the default value of β3 is 1. Since the default value for β1
is 0 (i.e., pre-relaxation sweeps are not performed), this procedure
is roughly equivalent to using the solution from the coarse level
as the initial guess for the solution at the fine level. For FAS,
the default value of β3 is zero (i.e., post-relaxation sweeps are not
performed); post-relaxation sweeps are never performed at the end
of the cycle for the finest grid level, regardless of the value of β3 .
This is because for FAS, post-relaxation sweeps at the fine level
are equivalent to pre-relaxation sweeps during the next cycle.
22.5.3
Algebraic Multigrid (AMG)
This algorithm is referred to as an “algebraic” multigrid scheme because,
as we shall see, the coarse level equations are generated without the use
of any geometry or re-discretization on the coarse levels; a feature that
makes AMG particularly attractive for use on unstructured meshes. The
advantage being that no coarse grids have to be constructed or stored,
and no fluxes or source terms need be evaluated on the coarse levels.
This approach is in contrast with FAS (sometimes called “geometric”)
multigrid in which a hierarchy of meshes is required and the discretized
equations are evaluated on every level. In theory, the advantage of FAS
over AMG is that the former should perform better for non-linear problems since non-linearities in the system are carried down to the coarse
levels through the re-discretization; when using AMG, once the system is
linearized, non-linearities are not “felt” by the solver until the fine level
operator is next updated.
AMG Restriction and Prolongation Operators
The restriction and prolongation operators used here are based on the additive correction (AC) strategy described for structured grids by Hutchinson and Raithby [96]. Inter-level transfer is accomplished by piecewise
constant interpolation and prolongation. The defect in any coarse level
cell is given by the sum of those from the fine level cells it contains, while
fine level corrections are obtained by injection of coarse level values. In
this manner the prolongation operator is given by the transpose of the
restriction operator
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Using the Solver
P = RT
(22.5-9)
The restriction operator is defined by a coarsening or “grouping” of fine
level cells into coarse level ones. In this process each fine level cell is
grouped with one or more of its “strongest” neighbors, with a preference given to currently ungrouped neighbors. The algorithm attempts
to collect cells into groups of fixed size, typically two or four, but any
number can be specified. In the context of grouping, strongest refers to
the neighbor j of the current cell i for which the coefficient Aij is largest.
For sets of coupled equations Aij is a block matrix and the measure of
its magnitude is simply taken to be the magnitude of its first element. In
addition, the set of coupled equations for a given cell are treated together
and not divided amongst different coarse cells. This results in the same
coarsening for each equation in the system.
AMG Coarse Level Operator
The coarse level operator AH is constructed using a Galerkin approach.
Here we require that the defect associated with the corrected fine level
solution must vanish when transferred back to the coarse level. Therefore
we may write
R dnew = 0
(22.5-10)
Upon substituting Equations 22.5-2 and 22.5-8 for dnew and φnew we have
h
R [A φnew + b] = 0
R A φ + P ψH + b
i
= 0
(22.5-11)
Now rearranging and using Equation 22.5-2 once again gives
R A P ψ H + R (A φ + b) = 0
R A P ψH + R d = 0
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(22.5-12)
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22.5 Multigrid Method
Comparison of Equation 22.5-12 with Equation 22.5-7 leads to the following expression for the coarse level operator:
AH = R A P
(22.5-13)
The construction of coarse level operators thus reduces to a summation
of diagonal and corresponding off-diagonal blocks for all fine level cells
within a group to form the diagonal block of that group’s coarse cell.
The F Cycle
The multigrid F cycle is essentially a combination of the V and W cycles
described in Section 22.5.2.
Recall that the multigrid cycle is a recursive procedure. The procedure
is expanded to the next coarsest grid level by performing a single multigrid cycle on the current level. Referring to Figures 22.5.1 and 22.5.2,
this means replacing the square on the current level (representing a single cycle) with the procedure shown for the 0-1 level cycle (the second
diagram in each figure). We see that a V cycle consists of:
pre sweep → restrict → V cycle → prolongate → post sweep
and a W cycle:
pre sweep → restrict → W cycle → W cycle → prolongate → post sweep
An F cycle is formed by a W cycle followed by a V cycle:
pre sweep → restrict → W cycle → V cycle → prolongate → post sweep
As expected, the F cycle requires more computation than the V cycle,
but less than the W cycle. However, its convergence properties turn out
to be better than the V cycle and roughly equivalent to the W cycle.
The F cycle is the default AMG cycle type for the coupled equation set.
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Using the Solver
The Flexible Cycle
For the flexible cycle, the calculation and use of coarse grid corrections
is controlled in the multigrid procedure by the logic illustrated in Figure 22.5.3. This logic ensures that coarser grid calculations are invoked
when the rate of residual reduction on the current grid level is too slow.
In addition, the multigrid controls dictate when the iterative solution of
the correction on the current coarse grid level is sufficiently converged
and should thus be applied to the solution on the next finer grid. These
two decisions are controlled by the parameters α and β shown in Figure 22.5.3, as described in detail below. Note that the logic of the multigrid procedure is such that grid levels may be visited repeatedly during
a single global iteration on an equation. For a set of 4 multigrid levels, referred to as 0, 1, 2, and 3, the flex-cycle multigrid procedure for
solving a given transport equation might consist of visiting grid levels as
0-1-2-3-2-3-2-1-0-1-2-1-0, for example.
level
return R 0i < α R 00 or
i > i max,fine
Solve for φ on level 0 (fine) grid
0
R0
relaxation
0
0
R i > β R i-1
1
1
R i < α R 0 or
i > i max,coarse
Solve for φ′ on level 1 grid
1
1
R i > β R i-1
2
2
R i < α R 0 or
i > i max,coarse
Solve for φ′ on level 2 grid
2
2
R i > β R i-1
3
3
R i < α R 0 or
i > i max,coarse
etc.
Figure 22.5.3: Logic Controlling the Flex Multigrid Cycle
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22.5 Multigrid Method
The main difference between the flexible cycle and the V and W cycles is
that the satisfaction of the residual reduction tolerance and termination
criterion determine when and how often each level is visited in the flexible
cycle, whereas in the V and W cycles the traversal pattern is explicitly
defined.
The Residual Reduction Rate Criteria
The multigrid procedure invokes calculations on the next coarser grid
level when the error reduction rate on the current level is insufficient, as
defined by
Ri > βRi−1
(22.5-14)
Here Ri is the absolute sum of residuals (defect) computed on the current
grid level after the ith relaxation on this level. The above equation states
that if the residual present in the iterative solution after i relaxations is
greater than some fraction, β (between 0 and 1), of the residual present
after the (i−1)th relaxation, the next coarser grid level should be visited.
Thus β is referred to as the residual reduction tolerance, and determines
when to “give up” on the iterative solution at the current grid level
and move to solving the correction equations on the next coarser grid.
The value of β controls the frequency with which coarser grid levels are
visited. The default value is 0.7. A larger value will result in less frequent
visits, and a smaller value will result in more frequent visits.
The Termination Criteria
Provided that the residual reduction rate is sufficiently rapid, the correction equations will be converged on the current grid level and the result
applied to the solution field on the next finer grid level.
The correction equations on the current grid level are considered sufficiently converged when the error in the correction solution is reduced
to some fraction, α (between 0 and 1), of the original error on this grid
level:
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Using the Solver
Ri < αR0
(22.5-15)
Here, Ri is the residual on the current grid level after the ith iteration
on this level, and R0 is the residual that was initially obtained on this
grid level at the current global iteration. The parameter α, referred to as
the termination criterion, has a default value of 0.1. Note that the above
equation is also used to terminate calculations on the lowest (finest) grid
level during the multigrid procedure. Thus, relaxations are continued on
each grid level (including the finest grid level) until the criterion of this
equation is obeyed (or until a maximum number of relaxations has been
completed, in the case that the specified criterion is never achieved).
22.5.4
Full-Approximation Storage (FAS) Multigrid
FLUENT’s approach to forming the multigrid grid hierarchy for FAS is
simply to coalesce groups of cells on the finer grid to form coarse grid
cells. Coarse grid cells are created by agglomerating the cells surrounding
a node, as shown in Figure 22.5.4. Depending on the grid topology, this
can result in cells with irregular shapes and variable numbers of faces.
The grid levels are, however, simple to construct and are embedded,
resulting in simple prolongation and relaxation operators.
Figure 22.5.4: Node Agglomeration to Form Coarse Grid Cells
It is interesting to note that although the coarse grid cells look very irregular, the discretization cannot “see” the jaggedness in the cell faces.
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22.5 Multigrid Method
The discretization uses only the area projections of the cell faces and
therefore each group of “jagged” cell faces separating two irregularlyshaped cells is equivalent to a single straight line (in 2D) connecting the
endpoints of the jagged segment. (In 3D, the area projections form an irregular, but continuous, geometrical shape.) This optimization decreases
the memory requirement and the computation time.
FAS Restriction and Prolongation Operators
FAS requires restriction of both the fine grid solution φ and its residual
(defect) d. The restriction operator R used to transfer the solution to the
next coarser grid level is formed using a full-approximation scheme [26].
That is, the solution for a coarse cell is obtained by taking the volume
average of the solution values in the embedded fine grid cells. Residuals
for the coarse grid cell are obtained by summing the residuals in the
embedded fine grid cells.
The prolongation operator P used to transfer corrections up to the fine
level is constructed to simply set the fine grid correction to the associated
coarse grid value.
The coarse grid corrections ψ H , which are brought up from the coarse
level and applied to the fine level solution, are computed from the difference between the solution calculated on the coarse level φH and the
initial solution restricted down to the coarse level Rφ. Thus correction
of the fine level solution becomes
φnew = φ + P
φH − Rφ
(22.5-16)
FAS Coarse Level Operator
The FAS coarse grid operator AH is simply that which results from a
re-discretization of the governing equations on the coarse level mesh.
Since the discretized equations presented in Sections 22.2 and 22.4 place
no restrictions on the number of faces that make up a cell, there is no
problem in performing this re-discretization on the coarse grids composed
of irregularly shaped cells.
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Using the Solver
There is some loss of accuracy when the finite-volume scheme is used on
the irregular coarse grid cells, but the accuracy of the multigrid solution
is determined solely by the finest grid and is therefore not affected by
the coarse grid discretization.
In order to preserve accuracy of the fine grid solution, the coarse level
equations are modified to include source terms [100] which insure that
corrections computed on the coarse grid φH will be zero if the residuals
on the fine grid dh are zero as well. Thus, the coarse grid equations are
formulated as
AH φH + dH = dH (Rφ) − Rdh
(22.5-17)
Here dH is the coarse grid residual computed from the current coarse
grid solution φH , and dH (Rφ) is the coarse grid residual computed from
the restricted fine level solution Rφ. Initially, these two terms will be the
same (because initially we have φH = Rφ) and cancel from the equation,
leaving
AH φH = −Rdh
(22.5-18)
So there will be no coarse level correction when the fine grid residual dh
is zero.
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22.6 Overview of How to Use the Solver
22.6
Overview of How to Use the Solver
Once you have defined your model and specified which solver you want
to use (see Section 1.6), you are ready to run the solver. The following
steps outline a general procedure you can follow:
1. Choose the discretization scheme and, for the segregated solver,
the pressure interpolation scheme (see Section 22.7).
2. (segregated solver only) Select the pressure-velocity coupling method
(see Section 22.8).
3. Set the under-relaxation factors (see Section 22.9).
4. (coupled explicit solver only) Turn on FAS multigrid (see Section 22.11).
5. Make any additional modifications to the solver settings that are
suggested in the chapters or sections that describe the models you
are using.
6. Initialize the solution (see Section 22.13).
7. Enable the appropriate solution monitors (see Section 22.16).
8. Start calculating (see Section 22.14 for steady-state calculations,
or Section 22.15 for time-dependent calculations).
9. If you have convergence trouble, try one of the methods discussed
in Section 22.19.
The default settings for the first three items listed above are suitable for
most problems and need not be changed. The following sections outline
how these and other solution parameters can be changed, and when you
may wish to change them.
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Using the Solver
22.7
Choosing the Discretization Scheme
FLUENT allows you to choose the discretization scheme for the convection terms of each governing equation. (Second-order accuracy is
automatically used for the viscous terms.) When the segregated solver
is used, all equations are, by default, solved using the first-order upwind discretization for convection. When one of the coupled solvers is
used, the flow equations are solved using the second-order scheme by
default, and the other equations use the first-order scheme by default.
See Section 22.2 for a complete description of the discretization schemes
available in FLUENT.
In addition, when you use the segregated solver, you can specify the
pressure interpolation scheme. See Section 22.3.1 for a description of the
pressure interpolation schemes available in FLUENT.
22.7.1
First Order vs. Second Order
When the flow is aligned with the grid (e.g., laminar flow in a rectangular
duct modeled with a quadrilateral or hexahedral grid) the first-order upwind discretization may be acceptable. When the flow is not aligned with
the grid (i.e., when it crosses the grid lines obliquely), however, first-order
convective discretization increases the numerical discretization error (numerical diffusion). For triangular and tetrahedral grids, since the flow
is never aligned with the grid, you will generally obtain more accurate
results by using the second-order discretization. For quad/hex grids,
you will also obtain better results using the second-order discretization,
especially for complex flows.
In summary, while the first-order discretization generally yields better
convergence than the second-order scheme, it generally will yield less
accurate results, especially on tri/tet grids. See Section 22.19 for information about controlling convergence.
For most cases, you will be able to use the second-order scheme from
the start of the calculation. In some cases, however, you may need to
scheme after a few iterations. For example, if you are running a highMach-number flow calculation that has an initial solution much different
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22.7 Choosing the Discretization Scheme
than the expected final solution, you will usually need to perform a few
iterations with the first-order scheme and then turn on the second-order
scheme and continue the calculation to convergence.
For a simple flow that is aligned with the grid (e.g., laminar flow in
a rectangular duct modeled with a quadrilateral or hexahedral grid),
the numerical diffusion will be naturally low, so you can generally use
the first-order scheme instead of the second-order scheme without any
significant loss of accuracy.
Finally, if you run into convergence difficulties with the second-order
scheme, you should try the first-order scheme instead.
22.7.2
Other Discretization Schemes
The QUICK discretization scheme may provide better accuracy than the
second-order scheme for rotating or swirling flows solved on quadrilateral
or hexahedral meshes. In general, however, the second-order scheme is
sufficient and the QUICK scheme will not provide significant improvements in accuracy.
A power law scheme is also available, but it will generally yield the same
accuracy as the first-order scheme.
The central-differencing scheme is available only when you are using
the LES turbulence model, and it should be used only when the mesh
spacing is fine enough so that the magnitude of the local Peclet number
(Equation 22.2-5) is less than 1.
22.7.3
Choosing the Pressure Interpolation Scheme
As discussed in Section 22.3.1, a number of pressure interpolation schemes
are available when the segregated solver is used in FLUENT. For most
cases the “standard” scheme is acceptable, but some types of models
may benefit from one of the other schemes:
• For problems involving large body forces, the body-force-weighted
scheme is recommended.
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Using the Solver
• For flows with high swirl numbers, high-Rayleigh-number natural convection, high-speed rotating flows, flows involving porous
media, and flows in strongly curved domains, use the PRESTO!
scheme.
• For compressible flows, the second-order scheme is recommended.
• Use the second-order scheme for improved accuracy when one of
the other schemes is not applicable.
!
The second-order scheme cannot be used with porous media or
with the VOF or mixture model for multiphase flow.
Note that you will not specify the pressure interpolation scheme if you
are using the Eulerian multiphase model. FLUENT will use the solution
method described in Section 20.4.8 for Eulerian multiphase calculations.
22.7.4
Choosing the Density Interpolation Scheme
As discussed in Section 22.3.2, three density interpolation schemes are
available when the segregated solver is used to solve a single-phase compressible flow.
The first-order upwind scheme (the default) provides stability for the
discretization of the pressure-correction equation, and gives good results
for most classes of flows. If you are calculating a compressible flow with
shocks, the first-order upwind scheme may tend to smooth the shocks;
you should use the second-order-upwind or QUICK scheme for such flows.
For compressible flows with shocks, using the QUICK scheme for all
variables, including density, is highly recommended for quadrilateral,
hexahedral, or hybrid meshes.
! Note that you will not be able to specify the density interpolation scheme
for a compressible multiphase calculation; the first-order upwind scheme
will be used for the compressible phase.
22.7.5
User Inputs
You can specify the discretization scheme and, for the segregated solver,
the pressure interpolation scheme in the Solution Controls panel (Fig22-54
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22.7 Choosing the Discretization Scheme
ure 22.7.1).
Solve −→ Controls −→Solution...
Figure 22.7.1: The Solution Controls Panel for the Segregated Solver
For each scalar equation listed under Discretization (Momentum, Energy,
Turbulence Kinetic Energy, etc. for the segregated solver or Turbulence
Kinetic Energy, Turbulence Dissipation Rate, etc. for the coupled solvers)
you can choose First Order Upwind, Second Order Upwind, Power Law,
QUICK, or (if you are using the LES turbulence model) Central Differencing in the adjacent drop-down list. For the coupled solvers, you can
choose either First Order Upwind or Second Order Upwind for the Flow
equations (which include momentum and energy). Note that the panel
shown in Figure 22.7.1 is for the segregated solver.
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Using the Solver
If you are using the segregated solver, select the pressure interpolation scheme under Discretization in the drop-down list next to Pressure.
You can choose Standard, Linear, Second Order, Body Force Weighted,
or PRESTO!. If you are using the segregated solver and your flow is
compressible (i.e., you are using the ideal gas law for density), select
the density interpolation scheme under Discretization in the drop-down
list next to Density. You can choose First Order Upwind, Second Order
Upwind, or QUICK. (Note that Density will not appear for incompressible
or multiphase flows.)
If you change the settings for Discretization, but you then want to return
to FLUENT’s default settings, you can click on the Default button. FLUENT will change the settings to the defaults and the Default button will
become the Reset button. To get your settings back again, you can click
on the Reset button.
22.8
Choosing the Pressure-Velocity Coupling Method
FLUENT provides three methods for pressure-velocity coupling in the
segregated solver: SIMPLE, SIMPLEC, and PISO. Steady-state calculations will generally use SIMPLE or SIMPLEC, while PISO is recommended for transient calculations. PISO may also be useful for steadystate and transient calculations on highly skewed meshes.
! Pressure-velocity coupling is relevant only for the segregated solver; you
will not specify it for the coupled solvers.
22.8.1
SIMPLE vs. SIMPLEC
In FLUENT, both the standard SIMPLE algorithm and the SIMPLEC
(SIMPLE-Consistent) algorithm are available. SIMPLE is the default,
but many problems will benefit from the use of SIMPLEC, particularly
because of the increased under-relaxation that can be applied, as described below.
For relatively uncomplicated problems (laminar flows with no additional
models activated) in which convergence is limited by the pressure-velocity
coupling, you can often obtain a converged solution more quickly using
SIMPLEC. With SIMPLEC, the pressure-correction under-relaxation
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22.8 Choosing the Pressure-Velocity Coupling Method
factor is generally set to 1.0, which aids in convergence speed-up. In some
problems, however, increasing the pressure-correction under-relaxation
to 1.0 can lead to instability. For such cases, you will need to use a
more conservative under-relaxation value or use the SIMPLE algorithm.
For complicated flows involving turbulence and/or additional physical
models, SIMPLEC will improve convergence only if it is being limited
by the pressure-velocity coupling. Often it will be one of the additional
modeling parameters that limits convergence; in this case, SIMPLE and
SIMPLEC will give similar convergence rates.
22.8.2
PISO
The PISO algorithm (see Section 22.3.3) with neighbor correction is
highly recommended for all transient flow calculations, especially when
you want to use a large time step. (For problems that use the LES
turbulence model, which usually requires small time steps, using PISO
may result in increased computational expense, so SIMPLE or SIMPLEC
should be considered instead.) PISO can maintain a stable calculation
with a larger time step and an under-relaxation factor of 1.0 for both
momentum and pressure. For steady-state problems, PISO with neighbor correction does not provide any noticeable advantage over SIMPLE
or SIMPLEC with optimal under-relaxation factors.
PISO with skewness correction is recommended for both steady-state
and transient calculations on meshes with a high degree of distortion.
When you use PISO neighbor correction, under-relaxation factors of 1.0
or near 1.0 are recommended for all equations. If you use just the PISO
skewness correction for highly-distorted meshes, set the under-relaxation
factors for momentum and pressure so that they sum to 1 (e.g., 0.3 for
pressure and 0.7 for momentum). If you use both PISO methods, follow the under-relaxation recommendations for PISO neighbor correction,
above.
22.8.3
User Inputs
You can specify the pressure-velocity coupling method in the Solution
Controls panel (Figure 22.7.1).
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Solve −→ Controls −→Solution...
Choose SIMPLE, SIMPLEC, or PISO in the Pressure-Velocity Coupling
drop-down list under Discretization.
If you choose PISO, the panel will expand to show the PISO Parameters.
By default, both Skewness Correction and Neighbor Correction are on. If
you want to use just neighbor correction or just skewness correction, you
can turn off the appropriate option. The default number of Iterations is
set to 1; you should not need to change this value.
22.9
Setting Under-Relaxation Factors
As discussed in Section 22.2.7, the segregated solver uses under-relaxation
to control the update of computed variables at each iteration. This
means that all equations solved using the segregated solver, including
the non-coupled equations solved by the coupled solvers (turbulence and
other scalars, as discussed in Section 22.1.2), will have under-relaxation
factors associated with them.
In FLUENT, the default under-relaxation parameters for all variables are
set to values that are near optimal for the largest possible number of
cases. These values are suitable for many problems, but for some particularly nonlinear problems (e.g., some turbulent flows or high-Rayleighnumber natural-convection problems) it is prudent to reduce the underrelaxation factors initially.
It is good practice to begin a calculation using the default under-relaxation
factors. If the residuals continue to increase after the first 4 or 5 iterations, you should reduce the under-relaxation factors.
Occasionally, you may make changes in the under-relaxation factors and
resume your calculation, only to find that the residuals begin to increase.
This often results from increasing the under-relaxation factors too much.
A cautious approach is to save a data file before making any changes to
the under-relaxation factors, and to give the solution algorithm a few
iterations to adjust to the new parameters. Typically, an increase in the
under-relaxation factors brings about a slight increase in the residuals,
but these increases usually disappear as the solution progresses. If the
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22.9 Setting Under-Relaxation Factors
residuals jump by a few orders of magnitude, you should consider halting
the calculation and returning to the last good data file saved.
Note that viscosity and density are under-relaxed from iteration to iteration. Also, if the enthalpy equation is solved directly instead of the
temperature equation (i.e., for non-premixed combustion calculations),
the update of temperature based on enthalpy will be under-relaxed. To
see the default under-relaxation factors, you can click on the Default
button in the Solution Controls panel.
For most flows, the default under-relaxation factors do not usually require modification. If unstable or divergent behavior is observed, however, you need to reduce the under-relaxation factors for pressure, momentum, k, and from their default values to about 0.2, 0.5, 0.5, and
0.5. (It is usually not necessary to reduce the pressure under-relaxation
for SIMPLEC.) In problems where density is strongly coupled with temperature, as in very-high-Rayleigh-number natural- or mixed-convection
flows, it is wise to also under-relax the temperature equation and/or density (i.e., use an under-relaxation factor less than 1.0). Conversely, when
temperature is not coupled with the momentum equations (or when it is
weakly coupled), as in flows with constant density, the under-relaxation
factor for temperature can be set to 1.0.
For other scalar equations (e.g., swirl, species, mixture fraction and variance) the default under-relaxation may be too aggressive for some problems, especially at the start of the calculation. You may wish to reduce
the factors to 0.8 to facilitate convergence.
User Inputs
You can modify the under-relaxation factors in the Solution Controls
panel (Figure 22.7.1).
Solve −→ Controls −→Solution...
You can set the under-relaxation factor for each equation in the field
next to its name under Under-Relaxation Factors.
! If you are using the segregated solver, all equations will have an associated under-relaxation factor. If you are using one of the coupled solvers,
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Using the Solver
only those equations that are solved sequentially (see Section 22.1.2) will
have under-relaxation factors.
If you change under-relaxation factors, but you then want to return to
FLUENT’s default settings, you can click on the Default button. FLUENT
will change the factors to the default values and the Default button will
become the Reset button. To get your settings back again, you can click
on the Reset button.
22.10
Changing the Courant Number
For FLUENT’s coupled solvers, the main control over the time-stepping
scheme is the Courant number (CFL). The time step is proportional to
the CFL, as defined in Equation 22.4-13.
Linear stability theory determines a range of permissible values for the
CFL (i.e., the range of values for which a given numerical scheme will
remain stable). When you specify a permissible CFL value, FLUENT will
compute an appropriate time step using Equation 22.4-13. In general,
taking larger time steps leads to faster convergence, so it is advantageous
to set the CFL as large as possible (within the permissible range).
The stability limits of the coupled implicit and explicit solvers are significantly different. The explicit solver has a more limited range and requires
lower CFL settings than does the coupled implicit solver. Appropriate
choices of CFL for the two solvers are discussed below.
22.10.1
Courant Numbers for the Coupled Explicit Solver
Linear stability analysis shows that the maximum allowable CFL for the
multi-stage scheme used in the coupled explicit solver will depend on the
number of stages used and how often the dissipation and viscous terms
are updated (see Section 22.19.5). But in general, you can assume that
the multi-stage scheme is stable for Courant numbers up to 2.5. This
stability limit is often lower in practice because of nonlinearities in the
governing equations.
The default CFL for the coupled explicit solver is 1.0, but you may be
able to increase it for some 2D problems. You should generally not use
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22.10 Changing the Courant Number
a value higher than 2.0.
If your solution is diverging, i.e., if residuals are rising very rapidly, and
your problem is properly set up and initialized, this is usually a good
sign that the Courant number needs to be lowered. Depending on the
severity of the startup conditions, you may need to decrease the CFL to
a value as low as 0.1 to 0.5 to get started. Once the startup transients
are reduced you can start increasing the Courant number again.
22.10.2
Courant Numbers for the Coupled Implicit Solver
Linear stability theory shows that the point Gauss-Seidel scheme used in
the coupled implicit solver is unconditionally stable. However, as with
the explicit solver, nonlinearities in the governing equations will often
limit stability.
The default CFL for the coupled implicit solver is 5.0. It is often possible
to increase the CFL to 10, 20, 100, or even higher, depending on the
complexity of your problem. You may find that a lower CFL is required
during startup (when changes in the solution are highly nonlinear), but
it can be increased as the solution progresses.
The coupled AMG solver has the capability to detect divergence of the
multigrid cycles within a given iteration. If this happens, it will automatically reduce the CFL and perform the iteration again, and a message
will be printed to the screen. Five attempts are made to complete the
iteration successfully. Upon successful completion of the current iteration the CFL is returned to its original value and the iteration procedure
proceeds as required.
22.10.3
User Inputs
The Courant number is set in the Solution Controls panel (Figure 22.10.1).
Solve −→ Controls −→Solution...
Enter the value for Courant Number under Solver Parameters. (Note that
the panel shown in Figure 22.10.1 is for the coupled explicit solver. For
the coupled implicit solver, the Courant Number is the only item that
will appear under Solver Parameters.)
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Figure 22.10.1: The Solution Controls Panel for the Coupled Explicit
Solver
When you select the coupled explicit solver in the Solver panel, FLUENT will automatically set the Courant Number to 1; when you select
the coupled implicit solver, the Courant Number will be changed to 5
automatically.
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22.11 Turning On FAS Multigrid
22.11
Turning On FAS Multigrid
As discussed in Section 22.5, FAS multigrid is an optional component of
the coupled explicit solver. (AMG multigrid is always on, by default.)
Since nearly all coupled explicit calculations will benefit from the use of
the FAS multigrid convergence accelerator, you should generally set a
non-zero number of coarse grid levels before beginning the calculation.
For most problems, this will be the only FAS multigrid parameter you
will need to set. Should you encounter convergence difficulties, consider
applying one of the methods discussed in Section 22.19.4.
! Note that you cannot use FAS multigrid with explicit time stepping
(described in Section 22.2.8) because the coarse grid corrections will
destroy the time accuracy of the fine grid solution.
22.11.1
Setting Coarse Grid Levels
As discussed in Section 22.5.4, FAS multigrid solves on successively
coarser grids and then transfers corrections to the solution back up to
the original fine grid, thus increasing the propagation speed of the solution and speeding convergence. The most basic way you can control the
multigrid solver is by specifying the number of coarse grid levels to be
used.
As explained in Section 22.5.4, the coarse grid levels are formed by agglomerating a group of adjacent “fine” cells into a single “coarse” cell.
The optimal number of grid levels is therefore problem-dependent. For
most problems, you can start out with 4 or 5 levels. For large 3D problems, you may want to add more levels (although memory restrictions
may prevent you from using more levels, since each coarse grid level
requires additional memory). If you believe that multigrid is causing
convergence trouble, you can decrease the number of levels.
If FLUENT reaches a coarse grid with one cell before creating as many
levels as you requested, it will simply stop there. That is, if you request
5 levels, and level 4 has only 1 cell, FLUENT will create only 4 levels,
since levels 4 and 5 would be the same.
To specify the number of grid levels you want, set the number of Multigrid Levels in the Solution Controls panel (Figure 22.10.1) under Solver
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Parameters.
Solve −→ Controls −→Solution...
You can also set the Max Coarse Levels under FAS Multigrid Controls in
the Multigrid Controls panel.
Solve −→ Controls −→Multigrid...
Changing the number of coarse grid levels in one panel will automatically
update the number shown in the other.
Coarse grid levels are created when you first begin iterating. If you
want to check how many cells are in each level, request one iteration
and then use the Grid/Info/Size menu item (described in Section 5.6.1)
to list the size of each grid level. If you are satisfied, you can continue
the calculation; if not, you can change the number of coarse grid levels
and check again.
For most problems, you will not need to modify any additional multigrid
parameters once you have settled on an appropriate number of coarse
grid levels. You can simply continue your calculation until convergence.
22.12
Setting Solution Limits
In order to keep the solution stable under extreme conditions, FLUENT
provides limits that keep the solution within an acceptable range. You
can control these limits with the Solution Limits panel (Figure 22.12.1).
Solve −→ Controls −→Limits...
FLUENT applies limiting values for pressure, temperature, and turbulence quantities. The purpose of these limits is to keep the absolute
pressure or the temperature from becoming 0, negative, or excessively
large during the calculation, and to keep the turbulence quantities from
becoming excessive. FLUENT also puts a limit on the rate of reduction
of temperature to prevent it from becoming 0 or negative.
! Typically, you will not need to change the default solution limits. If pressure, temperature, or turbulence quantities are being reset to the limiting
value repeatedly (as indicated by the appropriate warning messages in
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22.12 Setting Solution Limits
Figure 22.12.1: The Solution Limits Panel
the console window), you should check the dimensions, boundary conditions, and properties to be sure that the problem is set up correctly and
try to determine why the variable in question is getting so close to zero
or so large. You can use the “marking” feature (used to mark cells for
adaption) to identify which cells have a value equal to the limit. (Use
the Iso-Value Adaption panel, as described in Section 23.5.) In very rare
cases, you may need to change the solution limits, but only do so if you
are sure that you understand the reason for the solver’s unusual behavior. (For example, you may know that the temperature in your domain
will exceed 5000 K. Be sure that any temperature-dependent properties are appropriately defined for high temperatures if you increase the
maximum temperature limit.)
Limiting the Values of Solution Variables
The limiting minimum and maximum values for absolute pressure are
shown in the Minimum and Maximum Absolute Pressure fields. If the FLUENT calculation predicts a value less than the Minimum Absolute Pressure
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or greater than the Maximum Absolute Pressure, the corresponding limiting value will be used instead. Similarly, the Minimum and Maximum
Temperature are limiting values for energy calculations.
The Maximum Turb. Viscosity Ratio and the Minimum Turb. Kinetic
Energy are limiting values for turbulent calculations. If the calculation
predicts a k value less than the Minimum Turb. Kinetic Energy, the limiting value will be used instead. For the viscosity ratio limit, FLUENT
uses the limiting maximum value of turbulent viscosity (Cµ k2 /) in the
flow field relative to the laminar viscosity. If the ratio calculated by
FLUENT exceeds the limiting value, the ratio is set to the limiting value
by limiting to the necessary value.
Limiting the Reduction Rate for Temperature
In FLUENT’s coupled solvers, the rate of reduction of temperature is controlled by the Positivity Rate Limit. The default value of 0.2, for example,
means that temperature is not allowed to decrease by more than 20%
of its previous value from one iteration to the next. If the temperature
change exceeds this limit, the time step in that cell is reduced to bring
the change back into range and a “time step reduced” warning is printed.
(This reduced time step will be used for the solution of all variables in
the cell, not just for temperature.) Rapid reduction of temperature is an
indication that the temperature may become negative. Repeated “time
step reduced” warnings should alert you that something is wrong in your
problem setup. (If the warning messages stop appearing, the calculation
may have “recovered” from the time-step reduction.)
Resetting Solution Limits
If you change and save the value of one of the solution limits, but you
then want to return to the default limits set by FLUENT, you can reopen
the Solution Limits panel and click on the Default button. FLUENT will
change the values to the defaults and the Default button will become the
Reset button. To get your values back again, you can click on the Reset
button.
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22.13 Initializing the Solution
22.13
Initializing the Solution
Before starting your CFD simulation, you must provide FLUENT with an
initial “guess” for the solution flow field. In many cases, you must take
extra care to provide an initial solution that will allow the desired final
solution to be attained. A real-life supersonic wind tunnel, for example,
will not “start” if the back pressure is simply lowered to its operating
value; the flow will choke at the tunnel throat and will not transition
to supersonic. The same holds true for a numerical simulation: the flow
must be initialized to a supersonic flow or it will simply choke and remain
subsonic.
There are two methods for initializing the solution:
• Initialize the entire flow field (in all cells).
• Patch values or functions for selected flow variables in selected cell
zones or “registers” of cells. (Registers are created with the same
functions that are used to mark cells for adaption.)
! Before patching initial values in selected cells, you must first initialize the
entire flow field. You can then patch the new values over the initialized
values for selected variables.
22.13.1
Initializing the Entire Flow Field
Before you start your calculations or patch initial values for selected variables in selected cells (Section 22.13.2) you must initialize the flow field
in the entire domain. The Solution Initialization panel (Figure 22.13.1)
allows you to set initial values for the flow variables and initialize the
solution using these values.
Solve −→ Initialize −→Initialize...
You can compute the values from information in a specified zone, enter
them manually, or have the solver compute average values based on all
zones. You can also indicate whether the specified values for velocities
are absolute or relative to the velocity in each cell zone. The steps for
initialization are as follows:
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Figure 22.13.1: The Solution Initialization Panel
1. Set the initial values:
• To initialize the flow field using the values set for a particular
zone, select the zone name in the Compute From drop-down
list. All values under the Initial Values heading will automatically be computed and updated based on the conditions
defined at the selected zone.
• To initialize the flow field using computed average values, select all-zones in the Compute From drop-down list. FLUENT
will compute and update the Initial Values based on the conditions defined at all boundary zones.
• If you wish to change one or more of the values, you can
enter new values manually in the fields next to the appropriate
variables. If you prefer to enter all values manually, you can
do so without selecting a zone in the Compute From list.
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22.13 Initializing the Solution
2. If your problem involves moving reference frames or sliding meshes,
indicate whether the initial velocities are absolute velocities or velocities relative to the motion of each cell zone by selecting Absolute
or Relative to Cell Zone under Reference Frame. (If no zone motion
occurs in the problem, the two options are equivalent.) The default
reference frame for velocity initialization in FLUENT is relative. If
the solution in most of your domain is rotating, using the relative
option may be better than using the absolute option.
3. Once you are satisfied with the Initial Values displayed in the panel,
you can click on the Init button to initialize the flow field. If solution data already exist (i.e., if you have already performed some
calculations or initialized the solution), you must confirm that it
is OK to overwrite those data.
Saving and Resetting Initial Values
When you initialize the solution by clicking on Init, the initial values
will also be saved; should you need to reinitialize the solution later, you
will find the correct values in the panel when you reopen it. If you wish
to define initial values now, but you are not yet ready to initialize the
solution, you can follow the instructions above for setting the values and
then click Apply instead of Init. This will save the currently displayed
values without initializing the solution. You can return to the panel later
on and perform the initialization.
If you accidentally select the wrong zone from the Compute From list or
manually set a value incorrectly, you can use the Reset button to reset
all fields to their “saved” values. Values are saved each time the panel
is opened, before Compute From is executed, and after Init or Apply is
executed.
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22.13.2
Patching Values in Selected Cells
Once you have initialized (or calculated) the entire flow field, you may
patch different values for particular variables into different cells. If you
have multiple fluid zones, for example, you may want to patch a different
temperature in each one. You can also choose to patch a custom field
function (defined using the Custom Field Function Calculator panel) instead of a constant value. If you are patching velocities, you can indicate
whether the specified values are absolute velocities or velocities relative
to the cell zone’s velocity. All patching operations are performed with
the Patch panel (Figure 22.13.2).
Solve −→ Initialize −→Patch...
Figure 22.13.2: The Patch Panel
1. Select the variable to be patched in the Variable list.
2. In the Zones To Patch and/or Registers To Patch lists, choose the
zone(s) and/or register(s) for which you want to patch a value for
the selected variable.
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22.13 Initializing the Solution
3. If you wish to patch a constant value, simply enter that value in the
Value field. If you want to patch a previously-defined field function,
turn on the Use Field Function option and select the appropriate
function in the Field Function list.
4. If you selected a velocity in the Variable list, and your problem involves moving reference frames or sliding meshes, indicate whether
the patched velocities are absolute velocities or velocities relative
to the motion of each cell zone by selecting Absolute or Relative to
Cell Zone under Reference Frame. (If no zone motion occurs in the
problem, the two options are equivalent.) The default reference
frame for velocity patching in FLUENT is relative. If the solution
in most of your domain is rotating, using the relative option may
be better than using the absolute option.
5. Click on the Patch button to update the flow-field data. (Note that
patching will have no effect on the iteration or time-step count.)
Using Registers
The ability to patch values in cell registers gives you the flexibility to
patch different values within a single cell zone. For example, you may
want to patch a certain value for temperature only in fluid cells with
a particular range of concentrations for one species. You can create a
cell register (basically a list of cells) using the functions that are used to
mark cells for adaption. These functions allow you to mark cells based
on physical location, cell volume, gradient or isovalue of a particular
variable, and other parameters. See Chapter 23 for information about
marking cells for adaption. Section 23.9 provides information about manipulating different registers to create new ones. Once you have created
a register, you can patch values in it as described above.
Using Field Functions
By defining your own field function using the Custom Field Function Calculator panel, you can patch a non-constant value in selected cells. For
example, you may want to patch varying species mass fractions throughout a fluid region. To use this feature, simply create the function as
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described in Section 27.5, and then perform the function-patching operation in the Patch panel, as described above.
Using Patching Later in the Solution Process
Since patching affects only the variables for which you choose to change
the value, leaving the rest of the flow field intact, you can use it later in
the solution process without losing calculated data. (Initialization, on
the other hand, resets all data to the initial values.) For example, you
might want to start a combustion calculation from a cold-flow solution.
You can simply read in (or calculate) the cold-flow data, patch a high
temperature in the appropriate cells, and continue the calculation.
Patching can also be useful when you are solving a problem using a
step-by-step technique, as described in Section 22.19.2.
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22.14
For steady-state calculations, you will request the start of the solution
process using the Iterate panel (Figure 22.14.1).
Solve −→Iterate...
Figure 22.14.1: The Iterate Panel
In this panel, you will supply the number of additional iterations to be
performed in the Number of Iterations field. (For unsteady calculation
inputs, see Section 22.15.1.) If no calculations have been performed yet,
FLUENT will begin calculations starting at iteration 1, using the initial
solution. If you are starting from current solution data, FLUENT will
begin at the last iteration performed, using the current solution data as
its starting point.
By default, FLUENT will update the convergence monitors (described in
Section 22.16) after each iteration. If you increase the Reporting Interval
from the default of 1 you can get reports less frequently. For example,
if you set the Reporting Interval to 2, the monitors will print or plot
reports at every other iteration. Note that the Reporting Interval also
specifies how often FLUENT should check if the solution is converged.
For example, if your solution converges after 40 iterations, but your
Reporting Interval is set to 50, FLUENT will continue the calculation for
an extra 10 iterations before checking for (and finding) convergence.
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When you click on the Iterate button, FLUENT will begin to calculate.
During iteration, a Working dialog is displayed. Clicking on the Cancel
button or typing <Control-C> in the FLUENT console window will interrupt the iteration, as soon as it is safe to stop. (See below for more
details.)
Updating UDF Profiles
If you have used a user-defined function (UDF) to define any boundary
conditions, properties, etc., you can control the frequency with which the
function is updated by modifying the value of the UDF Profile Update
Interval. If UDF Profile Update Interval is set to n, the function will be
updated after every n iterations.
By default, the UDF Profile Update Interval is set to 1. You might want
to increase this value if your profile computation is expensive. See the
separate UDF Manual for details about creating and using UDFs.
Interrupting Iterations
As mentioned above, you can interrupt the calculation by clicking on the
Cancel button in the Working dialog box that appears while the solver
is calculating. In addition, on most, but not all, computer systems you
will be able to interrupt calculations using a control sequence, usually
<Control-C>. This allows you to stop the calculation process before
proceeding with the remainder of the requested iterations.
Resetting Data
After you have performed some iterations, if you decide to start over
again from the first iteration (e.g., after making some changes to the
problem setup), you can reinitialize the solution using the Solution Initialization panel, as described in Section 22.13.1.
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22.15 Performing Time-Dependent Calculations
22.15
Performing Time-Dependent Calculations
FLUENT can solve the conservation equations in time-dependent form,
to simulate a wide variety of time-dependent phenomena, such as
• vortex shedding and other time-periodic phenomena
• compressible filling and emptying problems
• transient heat conduction
• transient chemical mixing and reactions
Figures 22.15.1 and 22.15.2 illustrate the time-dependent vortex shedding flow pattern in the wake of a cylinder.
Activating time dependence is sometimes useful when attempting to
solve steady-state problems which tend toward instability (e.g., natural convection problems in which the Rayleigh number is close to the
transition region). It is possible in many cases to reach a steady-state
solution by integrating the time-dependent equations.
See Section 22.2.8 for details about temporal discretization.
22.15.1
User Inputs for Time-Dependent Problems
To solve a time-dependent problem, you will follow the procedure outlined below:
1. Enable the Unsteady option in the Solver panel (Figure 22.15.3),
and specify the desired Unsteady Formulation.
Define −→ Models −→Solver...
The 1st-Order Implicit formulation is sufficient for most problems.
If you need improved accuracy, you can use the 2nd-Order Implicit
formulation instead. The Explicit formulation (available only if the
coupled explicit solver is selected under Solver and Formulation at
the top of the panel) is used primarily to capture the transient
behavior of moving waves, such as shocks. See Section 22.2.8 for
details.
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1.92e+01
1.76e+01
1.60e+01
1.44e+01
1.28e+01
1.12e+01
9.60e+00
8.00e+00
6.40e+00
4.80e+00
3.20e+00
1.60e+00
0.00e+00
Contours of Stream Function (kg/s) (Time=3.6600e+01)
Figure 22.15.1: Time-Dependent Calculation of Vortex Shedding (t=3.66
sec)
1.92e+01
1.76e+01
1.60e+01
1.44e+01
1.28e+01
1.12e+01
9.60e+00
8.00e+00
6.40e+00
4.80e+00
3.20e+00
1.60e+00
0.00e+00
Contours of Stream Function (kg/s) (Time=4.1600e+01)
Figure 22.15.2: Time-Dependent Calculation of Vortex Shedding (t=41.6
sec)
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22.15 Performing Time-Dependent Calculations
Figure 22.15.3: The Solver Panel for an Unsteady Calculation
2. Define all relevant models and boundary conditions. Note that any
boundary conditions specified using user-defined functions can be
made to vary in time. See the separate UDF Manual for details.
3. If you are using the segregated solver, choose PISO as the PressureVelocity Coupling scheme under Discretization in the Solution Controls panel.
Solve −→ Controls −→Solution...
In general, you will not need to modify the PISO Parameters from
using the PISO algorithm.
!
If you are using the LES turbulence model with small time steps,
the PISO scheme may be too computationally expensive. It is
therefore recommended that you use SIMPLE or SIMPLEC instead
of PISO.
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4. (optional) If you are using the explicit unsteady formulation or if
you are using the adaptive time stepping method (described below and in Section 22.15.2) it is recommended that you enable the
printing of the current time (for the explicit unsteady formulation) or the current time step size (for the adaptive time stepping
method) at each iteration, using the Statistic Monitors panel.
Solve −→ Monitors −→Statistic...
Select time (for the current time)
time step size) in the Statistics list
When FLUENT prints the residuals
iteration, it will include a column
current time step size.
or delta time (for the current
and turn on the Print option.
to the console window at each
with the current time or the
5. (optional) Use the Force Monitors panel or the Surface Monitors
panel to monitor (and/or save to a file) time-varying force coefficient values or the average, mass average, integral, or flux of a
field variable or function on a surface as it changes with time. See
Section 22.16 for details.
6. Set the initial conditions (at time t = 0) using the Solution Initialization panel.
Solve −→ Initialize −→Initialize...
You can also read in a steady-state data file to set the initial conditions.
7. Use the automatic saving feature to specify the file name and frequency with which case and data files should be saved during the
solution process.
File −→ Write −→Autosave...
See Section 3.3.4 for details about the use of this feature.
You may also want to request automatic execution of other commands using the Execute Commands panel. See Section 22.18 for
details.
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22.15 Performing Time-Dependent Calculations
8. (optional) If you want to create a graphical animation of the solution over time, you can use the Solution Animation panel to set up
the graphical displays that you want to use in the animation. See
Section 22.17 for details.
9. (optional) If you want FLUENT to gather data for time statistics
(i.e., time-averaged and root-mean-square values for solution variables) during the calculation, follow these steps:
(a) Turn on the Data Sampling for Time Statistics option in the
Iterate panel.
Solve −→Iterate...
Enabling this option will allow you to display and report
both the mean and the root-mean-square (RMS) values, as
described in Section 22.15.3.
!
Note that gathering data for time statistics is not meaningful
inside a moving cell zone (i.e., a sliding zone in a sliding mesh
problem).
(b) Initialize the flow statistics.
solve −→ initialize −→init-flow-statistics
Note that you can also use the init-flow-statistics text
command to reset the flow statistics after you have gathered
some data for time statistics. If you perform, say, 10 time
steps with the Data Sampling for Time Statistics option enabled, check the results, and then continue the calculation for
10 more time steps, the time statistics will include the data
gathered in the first 10 time steps unless you reinitialize the
flow statistics.
10. Specify time-dependent solution parameters and start the calculation, as described below for the implicit and explicit unsteady
formulations:
• If you have chosen the 1st-Order or 2nd-Order Implicit formulation, the procedure is as follows:
(a) Set the time-dependent solution parameters in the Iterate
panel (Figure 22.15.4).
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Solve −→Iterate...
Figure 22.15.4: The Iterate Panel for Implicit Unsteady Calculations
Solution parameters for the implicit unsteady formulations are as follows:
– Max Iterations per Time Step: When FLUENT solves
the time-dependent equations using the implicit formulation, iteration is necessary at each time step.
This parameter sets a maximum for the number of
iterations per time step. If the convergence criteria
are met before this number of iterations is performed,
the solution will advance to the next time step.
– Time Step Size: The time step size is the magnitude
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of ∆t. Since the FLUENT formulation is fully implicit, there is no stability criterion that needs to be
met in determining ∆t. However, to model transient
phenomena properly, it is necessary to set ∆t at least
one order of magnitude smaller than the smallest time
constant in the system being modeled. A good way
to judge the choice of ∆t is to observe the number
of iterations FLUENT needs to converge at each time
step. The ideal number of iterations per time step is
10–20. If FLUENT needs substantially more, the time
step is too large. If FLUENT needs only a few iterations per time step, ∆t may be increased. Frequently
a time-dependent problem has a very fast “startup”
transient that decays rapidly. It is thus often wise
to choose a conservatively small ∆t for the first 5–10
time steps. ∆t may then be gradually increased as
the calculation proceeds.
For time-periodic calculations, you should choose the
time step based on the time scale of the periodicity. For a rotor/stator model, for example, you might
want 20 time steps between each blade passing. For
vortex shedding, you might want 20 steps per period.
By default, the size of the time step is fixed (as indicated
by the selection of Fixed under Time Stepping Method).
To have FLUENT modify the size of the time step as the
calculation proceeds, select Adaptive and specify the parameters under Adaptive Time Stepping in the expanded
Iterate panel. See Section 22.15.2 for details.
With the Adaptive time stepping method, the value you
specify for the Time Step Size will be the initial size of the
time step. As the calculation proceeds, the Time Step Size
shown in the Iterate panel will be the size of the current
time step.
(b) Specify the desired Number of Time Steps in the Iterate
panel and click Iterate.
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As it calculates a solution, FLUENT will print the current
time at the end of each time step.
• If you have chosen the Explicit unsteady formulation, you will
(a) Use the default settings for the Solver Parameters in the
Solution Controls panel.
Solve −→ Controls −→Solution...
If you have modified the Solver Parameters, you can click
on the Default button to retrieve the default settings.
(b) Specify the desired Number of Iterations and click Iterate.
Solve −→Iterate...
Remember that when the explicit unsteady formulation
is used, each iteration is a time step. When FLUENT
prints the residuals to the console window, it will include
a column with the current time (if you requested this in
step 4, above).
11. Save the final data file (and case file, if you have modified it) so
that you can continue the unsteady calculation later, if desired.
File −→ Write −→Data...
The procedures for setting the reporting interval, updating UDF profiles, interrupting iterations, and resetting data are the same as those for
steady-state calculations. See Section 22.14 for details.
! If you are using a user-defined function in your time-dependent calculation, note that, in addition to being updated after every n iterations
(where n is the value of the UDF Profile Update Interval), the function
will also be updated at the first iteration of each time step.
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22.15.2
As mentioned in Section 22.15.1, it is possible to have the size of the time
step change as the calculation proceeds, rather than specifying a fixed
size for the entire calculation. This section provides a brief description of
the algorithm that FLUENT uses to compute the time step size, as well
as an explanation of each of the parameters that you can set to control
! Adaptive time stepping is available only with the segregated and coupled
implicit solvers; it cannot be used with the coupled explicit solver. In
addition, it cannot be used with the VOF or discrete phase model.
The automatic determination of the time step size is based on the estimation of the truncation error associated with the time integration scheme
(i.e., first-order implicit or second-order implicit). If the truncation error
is smaller than a specified tolerance, the size of the time step is increased;
if the truncation error is greater, the time step size is decreased.
An estimation of the truncation error can be obtained by using a predictorcorrector type of algorithm [82] in association with the time integration
scheme. At each time step, a predicted solution can be obtained using a computationally inexpensive explicit method (forward Euler for
the first-order unsteady formulation, Adams-Bashford for the secondorder unsteady formulation). This predicted solution is used as an initial condition for the time step, and the correction is computed using
the non-linear iterations associated with the implicit (segregated or coupled) algorithm. The norm of the difference between the predicted and
corrected solutions is used as a measure of the truncation error. By comparing the truncation error with the desired level of accuracy (i.e., the
truncation error tolerance), FLUENT is able to adjust the time step size
by increasing it or decreasing it.
Specifying Parameters for Adaptive Time Stepping
The parameters that control the adaptive time stepping appear in the
Iterate panel, as described in Section 22.15.1. These parameters are as
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follows:
Truncation Error Tolerance specifies the threshold value to which the computed truncation error is compared. Increasing this value will lead
to an increase in the size of the time step and a reduction in the
accuracy of the solution. Decreasing it will lead to a reduction in
the size of the time step and an increase in the solution accuracy,
although the calculation will require more computational time. For
most cases, the default value of 0.01 is acceptable.
Ending Time specifies an ending time for the calculation. Since the ending time cannot be determined by multiplying the number of time
steps by a fixed time step size, you need to specify it explicitly.
Minimum/Maximum Time Step Size specify the upper and lower limits
for the size of the time step. If the time step becomes very small,
the computational expense may be too high; if the time step becomes very large, the solution accuracy may not be acceptable to
you. You can set the limits that are appropriate for your simulation.
Minimum/Maximum Step Change Factor limit the degree to which the
time step size can change at each time step. Limiting the change
results in a smoother calculation of the time step size, especially
when high-frequency noise is present in the solution. If the time
step change factor, f , is computed as the ratio between the specified truncation error tolerance and the computed truncation error,
the size of time step ∆tn is computed as follows:
• If 1 < f < fmax , ∆tn is increased to meet the desired tolerance.
• If 1 < fmax < f , ∆tn is increased, but its maximum possible
value is fmax ∆tn−1 .
• If fmin < f < 1, ∆tn is unchanged.
• If f < fmin < 1, ∆tn is decreased.
Number of Fixed Time Steps specifies the number of fixed-size time steps
that should be performed before the size of the time step starts to
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change. The size of the fixed time step is the value specified for
Time Step Size in the Iterate panel.
It is a good idea to perform a few fixed-size time steps before
switching to the adaptive time stepping. Sometimes spurious discretization errors can be associated with an impulsive start in time.
These errors are dissipated during the first few time steps, but they
can adversely affect the adaptive time stepping and result in extremely small time steps at the beginning of the calculation.
Specifying a User-Defined Time Stepping Method
the method described above, you can create a user-defined function for
your method and select it in the User-Defined Time Step drop-down list.
The other inputs under Adaptive Time Stepping will not be used when
you select a user-defined function.
See the separate UDF Manual for details about creating and using userdefined functions.
22.15.3
Postprocessing for Time-Dependent Problems
The postprocessing of time-dependent data is similar to that for steadystate data, with all graphical and alphanumeric commands available.
You can read a data file that was saved at any point in the calculation
(by you or with the autosave option) to restore the data at any of the
time levels that were saved.
FLUENT will label any subsequent graphical or alphanumeric output
with the time value of the current data set.
If you save data from the force or surface monitors to files (see step 5 in
Section 22.15.1), you can read these files back in and plot them to see a
time history of the monitored quantity. Figure 22.15.5 shows a sample
plot generated in this way.
If you enabled the Data Sampling for Time Statistics option in the Iterate
panel, FLUENT will compute the time average (mean) and root-mean-
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-5.00e+00
-5.10e+00
-5.20e+00
-5.30e+00
-5.40e+00
Cl
-5.50e+00
-5.60e+00
-5.70e+00
-5.80e+00
-5.90e+00
-6.00e+00
0
Y
X
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Time
Z
Cl
Figure 22.15.5: Lift Coefficient Plot for a Time-Periodic Solution
squares of the instantaneous values sampled during the calculation. The
mean and root-mean-square (RMS) values for all solution variables will
be available in the Unsteady Statistics... category of the variable selection
drop-down list that appears in postprocessing panels.
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22.16
Monitoring Solution Convergence
During the solution process you can monitor the convergence dynamically by checking residuals, statistics, force values, surface integrals, and
volume integrals. You can print reports of or display plots of lift, drag,
and moment coefficients, surface integrations, and residuals for the solution variables. For unsteady flows, you can also monitor elapsed time.
Each of these monitoring features is described below.
22.16.1
Monitoring Residuals
At the end of each solver iteration, the residual sum for each of the conserved variables is computed and stored, thus recording the convergence
history. This history is also saved in the data file. The residual sum is
defined below.
On a computer with infinite precision, these residuals will go to zero
as the solution converges. On an actual computer, the residuals decay
to some small value (“round-off”) and then stop changing (“level out”).
For “single precision” computations (the default for workstations and
most computers), residuals can drop as many as six orders of magnitude
before hitting round-off. Double precision residuals can drop up to twelve
orders of magnitude. Guidelines for judging convergence can be found
in Section 22.19.1.
Definition of Residuals for the Segregated Solver
After discretization, the conservation equation for a general variable φ
at a cell P can be written as
aP φP =
X
anb φnb + b
(22.16-1)
nb
Here aP is the center coefficient, anb are the influence coefficients for the
neighboring cells, and b is the contribution of the constant part of the
source term Sc in S = Sc + SP φ and of the boundary conditions. In
Equation 22.16-1,
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Using the Solver
aP =
X
anb − SP
(22.16-2)
nb
The residual Rφ computed by FLUENT’s segregated solver is the imbalance in Equation 22.16-1 summed over all the computational cells P .
This is referred to as the “unscaled” residual. It may be written as
X
Rφ =
|
cells P
X
anb φnb + b − aP φP |
(22.16-3)
nb
In general, it is difficult to judge convergence by examining the residuals defined by Equation 22.16-3 since no scaling is employed. This is
especially true in enclosed flows such as natural convection in a room
where there is no inlet flow rate of φ with which to compare the residual.
FLUENT scales the residual using a scaling factor representative of the
flow rate of φ through the domain. This “scaled” residual is defined as
P
Rφ =
cells P
|
P
anb φnb + b − aP φP |
Pnb
cells P
|aP φP |
(22.16-4)
For the momentum equations the denominator term aP φP is replaced
by aP vP , where vP is the magnitude of the velocity at cell P .
The scaled residual is a more appropriate indicator of convergence for
most problems, as discussed in Section 22.19.1. This residual is the
default displayed by FLUENT. Note that this definition of residual is
also used by Fluent Inc.’s structured grid solver FLUENT 4.
For the continuity equation, the unscaled residual for the segregated
solver is defined as
Rc =
X
|rate of mass creation in cell P|
(22.16-5)
cells P
The segregated solver’s scaled residual for the continuity equation is
defined as
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c
Riteration
N
c
Riteration
5
(22.16-6)
The denominator is the largest absolute value of the continuity residual
in the first five iterations.
The scaled residuals described above are useful indicators of solution
convergence. Guidelines for their use are given in Section 22.19.1. It is
sometimes useful to determine how much a residual has decreased during
calculations as an additional measure of convergence. For this purpose,
FLUENT allows you to normalize the residual (either scaled or unscaled)
by dividing by the maximum residual value after M iterations, where M
is set by you in the Residual Monitors panel in the Iterations field under
Normalization.
R̄φ =
φ
Riteration
N
φ
Riteration
M
(22.16-7)
Normalization in this manner ensures that the initial residuals for all
equations are of O(1) and is sometimes useful in judging overall convergence.
By default, M = 5. You can also specify the normalization factor (the denominator in Equation 22.16-7) manually in the Residual Monitors panel.
Definition of Residuals for the Coupled Solvers
A residual for the coupled solvers is simply the time rate of change of
the conserved variable (W ). The RMS residual is the square root of the
average of the squares of the residuals in each cell of the domain:
R(W ) =
s
X ∂W 2
∂t
(22.16-8)
Equation 22.16-8 is the unscaled residual sum reported for all the coupled
equations solved by FLUENT’s coupled solvers.
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Using the Solver
pled solvers (turbulence and other scalars, as discussed in Section 22.1.2)
are the same as those described above for the segregated solver.
In general, it is difficult to judge convergence by examining the residuals defined by Equation 22.16-8 since no scaling is employed. This is
especially true in enclosed flows such as natural convection in a room
where there is no inlet flow rate of φ with which to compare the residual.
FLUENT scales the residual using a scaling factor representative of the
flow rate of φ through the domain. This “scaled” residual is defined as
R(W )iteration N
R(W )iteration 5
(22.16-9)
The denominator is the largest absolute value of the residual in the first
five iterations.
The scaled residuals described above are useful indicators of solution
convergence. Guidelines for their use are given in Section 22.19.1. It is
sometimes useful to determine how much a residual has decreased during
calculations as an additional measure of convergence. For this purpose,
FLUENT allows you to normalize the residual (either scaled or unscaled)
by dividing by the maximum residual value after M iterations, where M
is set by you in the Residual Monitors panel in the Iterations field under
Normalization.
Normalization of the residual sum is accomplished by dividing by the
maximum residual value after M iterations, where M is set by you in
the Residual Monitors panel in the Iterations field under Normalization:
R̄(W ) =
R(W )iteration N
R(W )iteration M
(22.16-10)
Normalization in this manner ensures that the initial residuals for all
equations are of O(1) and is sometimes useful in judging overall convergence.
By default, M = 5, making the normalized residual equivalent to the
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scaled residual. You can also specify the normalization factor (the denominator in Equation 22.16-10) manually in the Residual Monitors panel.
Overview of Using the Residual Monitors Panel
All inputs controlling the monitoring of residuals are entered using the
Residual Monitors panel (Figure 22.16.1).
Solve −→ Monitors −→Residual...
or
Plot −→Residuals...
Figure 22.16.1: The Residual Monitors Panel
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Using the Solver
In general, you will only need to enable residual plotting and modify the
convergence criteria using this panel. Additional controls are available
for disabling monitoring of particular residuals, and modifying normalization and plot parameters.
Printing and Plotting Residuals
By default, residual values for all relevant variables are printed in the
text (console) window after each iteration. If you wish to disable this
printout, turn off Print under Options. To enable the plotting of residuals
after each iteration, turn on Plot under Options. Residuals will be plotted
in the graphics window (with the window ID set in the Window field)
during the calculation.
If you wish to display a plot of the current residual history, simply click
on the Plot push button.
Modifying Convergence Criteria
In addition to plotting and printing residual values during the calculation, FLUENT will also check for convergence. If convergence is being
monitored, the solution will stop automatically when each variable meets
its specified convergence criterion. Convergence checks can be performed
only for variables for which you are monitoring residuals (i.e., variables
for which the Monitor option is enabled).
You can choose whether or not you want to check the convergence for
each variable by turning on or off the Check Convergence option for it in
the Residual Monitors panel. To modify the convergence criterion for a
particular variable, enter a new value in the corresponding Convergence
Criterion field.
Plot Parameters
If you choose to plot the residual values (either interactively during the
solution or using the Plot button after calculations are complete), there
are several display parameters you can modify.
In the Window field under Plotting, you can specify the ID of the graphics
window in which the plot will be drawn. When FLUENT is iterating, the
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active graphics window is temporarily set to this window to update the
residual plot, and then returned to its previous value. Thus, the residual
plot can be maintained in a separate window that does not interfere with
other graphical postprocessing.
By changing the Iterations entry under Plotting, you can modify the number of residual history points to be displayed in the plot. If you specify
n points, FLUENT will display the last n history points. Since the y
axis is scaled by the minimum and maximum values of all points in the
plot, you can “zoom in” on the end of the residual history by setting
Iterations to a value smaller than the number of iterations performed.
If, for example, the residuals jumped early in the calculation when you
turned on turbulence, that peak broadens the overall range in residual
values, making the smaller fluctuations later on almost indistinguishable.
By setting the value of Iterations so that the plot does not include that
early peak, your y-axis range is better suited to the values that you are
interested in seeing.
You can also modify the attributes of the plot axes and the residual
curves. Click on the Axes... or Curves... button to open the Axes panel
or Curves panel. See Sections 25.8.8 and 25.8.9 for details.
Disabling Monitoring
If your problem requires the solution of many equations (e.g., turbulence
quantities and multiple species), a plot that includes all residuals may
be difficult to read. In such cases, you may choose to monitor only a
subset of the residuals, perhaps those that affect convergence the most.
You can indicate whether or not you want to monitor residuals for each
variable by enabling or disabling the relevant check box in the Monitor
list of the Residual Monitors panel.
Controlling Normalization
By default, scaling of residuals (see Equations 22.16-4 and 22.16-9) is
enabled and the default convergence criterion is 10−6 for energy and P-1
equations and 10−3 for all other equations. Residual normalization (i.e.,
dividing the residuals by the largest value during the first few iterations)
is also available but disabled by default.
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Normalization can be used with both scaled and unscaled residuals. Note
that if normalization is enabled, the convergence criterion may need to be
adjusted appropriately. See Section 22.19.1 for information about judging convergence based on the different types of residual reports. (Both
the raw residuals and scaling factors are stored in the data file, so you
can switch between scaled and unscaled residuals.) To report unscaled
residuals, simply turn off the Scale option under Normalization.
! If you switch from scaled to unscaled residuals (or vice versa) and you
are normalizing the residuals (as described below), you must click on the
Renorm button to recompute the normalization factors.
If you wish to normalize the residuals (see Equation 22.16-7 or 22.16-10),
turn on the Normalize option under Normalize. The Normalization Factor
column will be added to the panel at this time. FLUENT will normalize
the printed or plotted residual for each variable by the value indicated
as the Normalization Factor for that variable. The default Normalization
Factor is the maximum residual value after the first 5 iterations. To use
the maximum residual value after a different number of iterations (i.e.,
specify a different value for M in Equation 22.16-7 or 22.16-10), you can
modify the Iterations entry under Normalization.
In some cases, the maximum residual may occur sometime after the
iteration specified in the Iterations field. If this should occur, you can click
on the Renorm button to set the normalization factors for all variables
to the maximum values in the residual histories. Subsequent plots and
printed reports will use the new normalization factor.
You can also specify the normalization factor (the denominator in Equation 22.16-7 or 22.16-10) explicitly. To modify the normalization factor
for a particular variable, enter a new value in the corresponding Normalization Factor field in the Residual Monitors panel.
If you wish to report unnormalized, unscaled residuals (Equation 22.16-3
or 22.16-8), turn off the Normalize and Scale options under Normalization in the Residual Monitors panel. Note that unnormalized, unscaled
residuals are stored in the data file regardless of whether the reported
residuals are normalized or scaled.
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Storing Residual History Points
Residual histories for each variable are automatically saved in the data
file, regardless of whether they are being monitored. You can control
the number of history points to be stored by changing the Iterations
entry under Storage. By default, up to 1000 points will be stored. If
more than 1000 iterations are performed (i.e., the limit is reached), every
other point will be discarded—leaving 500 history points—and the next
500 points will be stored. When the total hits 1000 again, every other
point will again be discarded, etc. If you are performing a large number
of iterations, you will lose a great deal of residual history information
about the beginning of the calculation. In such cases, you should increase
the Iterations value to a more appropriate value. Of course, the larger
this number is, the more memory you will need, the longer the plotting
will take, and the more disk space you will need to store the data file.
Postprocessing Residual Values
If you are having solution convergence difficulties, it is often useful to
plot the residual value fields (e.g., using contour plots) to determine
where the high residual values are located. When you use one of the
coupled solvers, the residual values for all solution variables are available
in the Residuals... category in the postprocessing panels. (If you read
case and data files into FLUENT, you will need to perform at least one
iteration before the residual values are available for postprocessing.) For
the segregated solver, however, only the mass imbalance in each cell is
available by default.
If you want to plot residual value fields for a segregated solver calculation,
you will need to do the following:
1. Read in the case and data files of interest (if they are not already
in the current session).
2. Use the expert command in the solve/set/ text menu to enable
the saving of residual values.
solve −→ set −→expert
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Among other questions, FLUENT will ask if you want to save cell
residuals for postprocessing. Enter yes or y, and keep the default
settings for all of the other questions (by pressing the <RETURN>
key).
3. Perform at least one iteration.
The solution variables for which residual values are available will appear
in the Residuals... category in the postprocessing panels. Note that
residual values are not available for the radiative transport equations
solved by the discrete ordinates radiation model.
22.16.2
Monitoring Statistics
If you are solving a fully-developed periodic flow, you may want to monitor the pressure gradient or the bulk temperature ratio, as discussed in
Section 8.3.
If you are solving an unsteady flow (especially if you are using the explicit
time stepping option), you may want to monitor the “time” that has
elapsed during the calculation. The physical time of the flow field starts
at zero when you initialize the flow. (See Section 22.15 for details about
If you are using the adaptive time stepping method described in Section 22.15.2, you may want to monitor the size of the time step, ∆t.
You can use the Statistic Monitors panel (Figure 22.16.2) to print or plot
these quantities during the calculation.
Solve −→ Monitors −→Statistic...
The procedure for setting up this monitor is listed below:
1. Indicate the type of report you want by turning on the Print option
for a printout or the Plot option for a plot. You can enable these
options simultaneously.
2. Select the appropriate quantity in the Statistics list.
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Figure 22.16.2: The Statistic Monitors Panel
3. If you are plotting the quantities, you can set any of the plotting
options discussed below.
Plot Parameters
If you choose to plot the statistics, there are several display parameters
you can modify.
In the First Window field, you can specify the ID of the graphics window
in which the plot will be drawn (or in which the first plot will be drawn,
if you are plotting more than one quantity.) When FLUENT is iterating,
the active graphics window is temporarily set to this window to update
the plot, and then returned to its previous value. Thus, the statistics
plot can be maintained in a separate window that does not interfere
with other graphical postprocessing. Note that additional quantities
that you have selected in the Statistics list will be plotted in windows
with incrementally higher IDs.
You can also modify the attributes of the plot axes and curves. Click on
the Axes... or Curves... button to open the Axes panel or Curves panel.
See Sections 25.8.8 and 25.8.9 for details.
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22.16.3
Monitoring Forces and Moments
At the end of each solver iteration, the lift, drag, and/or moment coefficient can be computed and stored to create a convergence history. You
can print and plot this convergence data, and also save it to an external file. The external file is written in the FLUENT XY plot file format
described in Section 25.8.5. Monitoring forces can be useful when you
are calculating external aerodynamics, for example, and you are especially interested in the forces. Sometimes the forces converge before the
residuals have dropped three orders of magnitude, so you can save time
by stopping the calculation earlier than you would if you were monitoring only residuals. (You should be sure to check the mass flow rate
and heat transfer rate as well, using the Flux Reports panel described
in Section 26.2, to ensure that the mass and energy are being suitably
conserved.)
! The force and moment coefficients use the reference values described in
Section 26.8. Specifically, the force coefficients use the reference area,
density, and velocity, and the moment coefficients use the reference area,
density, velocity and length.
! Only the processed force coefficient data is saved. If you decide to change
any of the parameters controlling the force monitoring, such as the reference values, force vector, moment center, moment axis, or wall zones,
you may see a discontinuity in the data: the previous data is not updated. Usually, if you have made changes you will want to delete the
previous force coefficient data before continuing to iterate.
Monitoring Forces in Unsteady Flow Calculations
If you are calculating unsteady flow, the specified force reports will be
updated after each time step, rather than after each iteration. All other
features of the force monitor and the related setup procedures are unchanged.
Overview of Using the Force Monitors Panel
You can use the Force Monitors panel (Figure 22.16.3) to print, plot, and
save the convergence history of the drag, lift, and moment coefficients
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on specified wall zones.
Solve −→ Monitors −→Force...
Figure 22.16.3: The Force Monitors Panel
In this panel you will indicate the types of reports that you want (printouts, plots, or files), and specify which coefficients and wall zones are of
interest. Additional information will be entered for each coefficient that
is monitored. You can also modify the plot parameters.
! Remember to click Apply after making the desired modifications to the
setup for each coefficient report.
Specifying the Force Coefficient Report
For each coefficient that you choose to monitor, you will set all appropriate parameters in the Force Monitors panel and click Apply. You can
monitor one, two, or all three of the coefficients (drag, lift, and moment vector component) while iterating. When you select the desired
coefficient, the current (or default) panel settings are shown for that coefficient. Clicking the Apply button will save the current panel settings
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for the selected coefficient.
The procedure for specifying a force coefficient report is listed below:
1. Indicate the type of report you want (printout, plot, or file), as
described below.
2. If you want to monitor the force or moment on individual walls in
a single printout, plot, or file, turn on the Per Zone option. See
below for details.
3. Choose the coefficient of interest by selecting Drag, Lift, or Moment
in the Coefficient drop-down list.
4. In the Wall Zones list, select the wall zone(s) on which the selected
coefficient is to be computed. If you are monitoring more than
one coefficient, the selected wall zones are often the same for each
coefficient. If you want, however, you can have each coefficient
computed on a different set of zones.
5. Depending on the coefficient selected, do one of the following:
• If you are monitoring the drag or lift coefficient, enter the
X, Y, and Z components of the Force Vector along which the
forces will be computed. The Force Vector heading will appear
only if you have selected Drag or Lift in the Coefficient dropdown list. By default, drag is computed in the x direction
and lift in the y direction.
• If you are monitoring the moment coefficient, enter the Cartesian coordinates (X, Y, and Z) of the Moment Center about
which moments will be computed. The Moment Center heading will appear only if you have selected Moment in the Coefficient drop-down list. The default moment center is (0,0,0).
You will also need to specify which component of the moment
vector you wish to monitor. Presently, you can monitor only
one component of the moment vector at a time. Select XAxis, Y-Axis, or Z-Axis in the About drop-down list. (This list
is active only if you have selected Moment in the Coefficient
drop-down list.) For two dimensional flows, only the moment
vector about the z-coordinate axis exists.
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6. Click Apply and repeat the process for additional coefficients, if
desired.
Printing, Plotting, and Saving Force Coefficient Histories
There are three methods available for reporting the selected force coefficients. To print the coefficient value(s) in the text (console) window
after each iteration, turn on the Print option under Options in the Force
Monitors panel. To plot the coefficient in the graphics window indicated
in Plot Window, turn on the Plot option. If you want to save the values
to a file, turn on the Write option and specify the File Name. You can
enable any combination of these options simultaneously.
! If you choose not to save the force coefficient data in a file, this information will be lost when you exit the current FLUENT session.
If you wish to display a plot of the current force-coefficient history, simply
click on the Plot button.
Plot Parameters
If you choose to plot the force coefficients (either interactively during the
solution or using the Plot button after calculations are complete), there
are several display parameters you can modify.
In the Plot Window field, you can specify the ID of the graphics window
in which the plot for each force coefficient will be drawn. When FLUENT
is iterating, the active graphics window is temporarily set to this window
to update the plot, and then returned to its previous value. Thus, the
force-coefficient plots can be maintained in separate windows that do not
interfere with other graphical postprocessing.
You can also modify the attributes of the plot axes and the coefficient
curves. The same attributes apply to all force-monitor plots. Click on
the Axes... or Curves... button to open the Axes panel or Curves panel.
See Sections 25.8.8 and 25.8.9 for details.
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Monitoring Forces and Moments on Individual Walls
By default, FLUENT will compute and monitor the total force or moment
for all of the selected walls combined together. If you have selected
multiple walls and you want to monitor the force or moment on each wall
separately, you can turn on the Per Zone option in the Force Monitors
panel. The specified force vector or moment axis will apply to all selected
walls.
If the monitor results are printed to the console (using the Print option),
the force or moment on each wall will be printed in a separate column.
If the results are plotted (using the Plot option), a separate curve for
each wall will be drawn in the specified plot window. If the results
are written to a file (using the Write option), the file will be in a tabseparated column format based on the XY plot file format described in
Section 25.8.5.
Should you decide that the data gathered by the force monitor are not
useful (e.g., if you are restarting the calculation or you changed one of
the reference values), you can discard the accumulated data by clicking
on the Clear button. The Clear button will delete all monitoring data
for the coefficient selected in the Coefficient drop-down list, including
the associated history file (with the name in the File Name field). When
you use the Clear button, you will need to confirm the data discard in
a Question dialog box. Only the force-monitoring data is removed using
this operation; the solution data is not affected.
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22.16.4
Monitoring Surface Integrals
At the end of each solver iteration or time step, the average, mass average, integral, flow rate, or other integral report of a field variable or
function can be monitored on a surface. You can print and plot these
convergence data, and also save them in an external file. The external file
is written in the FLUENT XY plot file format described in Section 25.8.5.
The report types available are the same as those in the Surface Integrals
panel, as described in Section 26.5.
Monitoring surface integrals can be used to check for both iteration convergence and grid independence. For example, you can monitor the
average value of a certain variable on a surface. When this value stops
changing, you can stop iterating. You can then adapt the grid and reconverge the solution. The solution can be considered grid-independent
when the average value on the surface stops changing between adaptions.
Overview of Defining Surface Monitors
You can use the Surface Monitors panel (Figure 22.16.4) to create surface monitors and indicate whether and when each one’s history is to
be printed, plotted, or saved. The Define Surface Monitor panel (Figure 22.16.5), opened from the Surface Monitors panel, allows you to define what each monitor tracks (i.e., the average, integral, flow rate, mass
average, or other integral report of a field variable or function on one or
more surfaces).
Defining Surface Monitors
You will begin the surface monitor definition procedure in the Surface
Monitors panel (Figure 22.16.4).
Solve −→ Monitors −→Surface...
The procedure is as follows:
1. Increase the Surface Monitors value to the number of surface monitors you wish to specify. As this value is increased, additional
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Figure 22.16.4: The Surface Monitors Panel
monitor entries in the panel will become editable. For each monitor, you will perform the following steps.
2. Enter a name for the monitor under the Name heading, and use
the Plot, Print, and Write check buttons to indicate the report(s)
you want (plot, printout, or file), as described below.
3. Indicate whether you want to update the monitor every Iteration
or every Time Step by selecting the appropriate item in the dropdown list below Every. Time Step is a valid choice only if you
are calculating unsteady flow. If you specify every Iteration, and
the Reporting Interval in the Iterate panel is greater than 1, the
monitor will be updated at every reporting interval instead of at
each iteration (e.g., for a reporting interval of 2, the monitor will
be updated after every other iteration). If you specify every Time
Step, the reporting interval will have no effect; the monitor will
always be updated after each time step.
4. Click on the Define... button to open the Define Surface Monitor
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panel (Figure 22.16.5). Since this is a modal panel, the solver will
not allow you to do anything else until you perform steps 5–10,
below.
Figure 22.16.5: The Define Surface Monitor Panel
5. In the Define Surface Monitor panel, choose the integration method
for the surface monitor by selecting Integral, Area-Weighted Average, Flow Rate, Mass Flow Rate, Mass-Weighted Average, Sum, Facet
Average, Facet Minimum, Facet Maximum, Vertex Average, Vertex
Minimum, or Vertex Maximum in the Report Type drop-down list.
These methods are described in Section 26.5.
6. In the Surfaces list, choose the surface or surfaces on which you
wish to integrate.
7. Specify the variable or function to be integrated in the Report Of
drop-down list. First select the desired category in the upper dropdown list. You can then select one of the related quantities in the
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lower list. (See Chapter 27 for an explanation of the variables in
the list.)
8. If you are plotting the data or writing them to a file, specify the
parameter to be used as the x-axis value (the y-axis value corresponds to the monitored data). In the X Axis drop-down list, select
Iteration, Time Step, or Flow Time as the x-axis function against
which monitored data will be plotted or written. Time Step and
Flow Time are valid choices only if you are calculating unsteady
flow. If you choose Time Step, the x axis of the plot will indicate
the time step, and if you choose Flow Time, it will indicate the
elapsed time.
9. If you are plotting the monitored data, specify the ID of the graphics window in which the plot will be drawn in the Plot Window
field. When FLUENT is iterating, the active graphics window is
temporarily set to this window to update the plot, and then returned to its previous value. Thus, each surface-monitor plot can
be maintained in a separate window that does not interfere with
other graphical postprocessing.
10. If you are writing the monitored data to a file, specify the File
Name.
11. Remember to click OK in the Surface Monitors panel after you finish
defining all surface monitors.
Printing, Plotting, and Saving Surface Integration Histories
There are three methods available for reporting the selected surface integration. To print the surface integration in the text (console) window
after each iteration, turn on the Print option in the Surface Monitors
panel. To plot the integrated values in the graphics window indicated
in Plot Window (in the Define Surface Monitor panel), turn on the Plot
option in the Surface Monitors panel. If you want to save the values to a
file, turn on the Write option in the Surface Monitors panel and specify
the File Name in the Define Surface Monitor panel. You can enable any
combination of these options simultaneously.
! If you choose not to save the surface integration data in a file, this infor22-106
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mation will be lost when you exit the current FLUENT session.
Plot Parameters
You can modify the attributes of the plot axes and curves used for each
surface-monitor plot. Click on the Axes... or Curves... button in the
Define Surface Monitor panel for the appropriate monitor to open the Axes
panel or Curves panel for that surface-monitor plot. See Sections 25.8.8
and 25.8.9 for details.
22.16.5
Monitoring Volume Integrals
At the end of each solver iteration or time step, the volume or the sum,
volume integral, volume average, mass integral, or mass average of a
field variable or function can be monitored in one or more cell zones.
You can print and plot these convergence data, and also save them in
an external file. The external file is written in the FLUENT XY plot file
format described in Section 25.8.5. The report types available are the
same as those in the Volume Integrals panel, as described in Section 26.6.
Monitoring volume integrals can be used to check for both iteration
convergence and grid independence. For example, you can monitor the
average value of a certain variable in a particular cell zone. When this
value stops changing, you can stop iterating. You can then adapt the
grid and reconverge the solution. The solution can be considered gridindependent when the average value in the cell zone stops changing between adaptions.
Overview of Defining Volume Monitors
You can use the Volume Monitors panel (Figure 22.16.6) to create volume monitors and indicate whether and when each one’s history is to
be printed, plotted, or saved. The Define Volume Monitor panel (Figure 22.16.7), opened from the Volume Monitors panel, allows you to define what each monitor tracks (i.e., the volume or the sum, integral, or
average of a field variable or function in one or more cell zones).
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Defining Volume Monitors
You will begin the volume monitor definition procedure in the Volume
Monitors panel (Figure 22.16.6).
Solve −→ Monitors −→Volume...
Figure 22.16.6: The Volume Monitors Panel
The procedure is as follows:
1. Increase the Volume Monitors value to the number of volume monitors you wish to specify. As this value is increased, additional
monitor entries in the panel will become editable. For each monitor, you will perform the following steps.
2. Enter a name for the monitor under the Name heading, and use
the Plot, Print, and Write check buttons to indicate the report(s)
you want (plot, printout, or file), as described below.
3. Indicate whether you want to update the monitor every Iteration
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down list below Every. Time Step is a valid choice only if you
are calculating unsteady flow. If you specify every Iteration, and
the Reporting Interval in the Iterate panel is greater than 1, the
monitor will be updated at every reporting interval instead of at
each iteration (e.g., for a reporting interval of 2, the monitor will
be updated after every other iteration). If you specify every Time
Step, the reporting interval will have no effect; the monitor will
always be updated after each time step.
4. Click on the Define... button to open the Define Volume Monitor
panel (Figure 22.16.7). Since this is a modal panel, the solver will
not allow you to do anything else until you perform steps 5–10,
below.
Figure 22.16.7: The Define Volume Monitor Panel
5. In the Define Volume Monitor panel, choose the integration method
for the volume monitor by selecting Volume, Sum, Volume Integral,
Volume-Average, Mass Integral, or Mass-Average in the Report Type
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drop-down list. These methods are described in Section 26.6.
6. In the Cell Zones list, choose the cell zone(s) on which you wish to
integrate.
7. Specify the variable or function to be integrated in the Report Of
drop-down list. First select the desired category in the upper dropdown list. You can then select one of the related quantities in the
lower list. (See Chapter 27 for an explanation of the variables in
the list.)
8. If you are plotting the data or writing them to a file, specify the
parameter to be used as the x-axis value (the y-axis value corresponds to the monitored data). In the X Axis drop-down list, select
Iteration, Time Step, or Flow Time as the x-axis function against
which monitored data will be plotted or written. Time Step and
Flow Time are valid choices only if you are calculating unsteady
flow. If you choose Time Step, the x axis of the plot will indicate
the time step, and if you choose Flow Time, it will indicate the
elapsed time.
9. If you are plotting the monitored data, specify the ID of the graphics window in which the plot will be drawn in the Plot Window
field. When FLUENT is iterating, the active graphics window is
temporarily set to this window to update the plot, and then returned to its previous value. Thus, each volume-monitor plot can
be maintained in a separate window that does not interfere with
other graphical postprocessing.
10. If you are writing the monitored data to a file, specify the File
Name.
11. Remember to click OK in the Volume Monitors panel after you finish
defining all volume monitors.
Printing, Plotting, and Saving Volume Integration Histories
There are three methods available for reporting the selected volume integration. To print the volume integration in the text (console) window
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after each iteration, turn on the Print option in the Volume Monitors
panel. To plot the integrated values in the graphics window indicated
in Plot Window (in the Define Volume Monitor panel), turn on the Plot
option in the Volume Monitors panel. If you want to save the values to a
file, turn on the Write option in the Volume Monitors panel and specify
the File Name in the Define Volume Monitor panel. You can enable any
combination of these options simultaneously.
! If you choose not to save the volume integration data in a file, this
information will be lost when you exit the current FLUENT session.
Plot Parameters
You can modify the attributes of the plot axes and curves used for each
volume-monitor plot. Click on the Axes... or Curves... button in the
Define Volume Monitor panel for the appropriate monitor to open the Axes
panel or Curves panel for that volume-monitor plot. See Sections 25.8.8
and 25.8.9 for details.
22.17
Animating the Solution
During the calculation, you can have FLUENT create an animation of
contours, vectors, XY plots, monitor plots (residual, statistic, force, surface, or volume), or the mesh (useful primarily for moving mesh simulations). Before you begin the calculation, you will specify and display
the variables and types of plots you want to animate, and how often you
want plots to be saved. At the specified intervals, FLUENT will display
the requested plots, and store each one. When the calculation is complete, you can play back the animation sequence, modify the view (for
grid, contour, and vector plots), if desired, and save the animation to a
series of hardcopy files or an MPEG file
Instructions for defining a solution animation sequence are provided in
Section 22.17.1. Sections 22.17.2 and 22.17.3 describe how to play back
and save the animation sequences you have created, and Section 22.17.4
describes how to read a previously-saved animation sequence into FLUENT.
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22.17.1
Defining an Animation Sequence
You can use the Solution Animation panel (Figure 22.17.1) to create an
animation sequence and indicate how often each frame of the sequence
should be created. The Animation Sequence panel (Figure 22.17.2),
opened from the Solution Animation panel, allows you to define what
each sequence displays (e.g., contours or vectors of a particular variable),
where it is displayed, and how each frame is stored.
You will begin the animation sequence definition in the Solution Animation panel (Figure 22.17.1).
Solve −→ Animate −→Define...
Figure 22.17.1: The Solution Animation Panel
The procedure is as follows:
1. Increase the Animation Sequences value to the number of animation sequences you wish to specify. As this value is increased,
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additional sequence entries in the panel will become editable. For
each sequence, you will perform the following steps.
2. Enter a name for the sequence under the Name heading. This name
will be used to identify the sequence in the Playback panel, where
you can play back the animation sequences that you have defined
or read in. This name will also be used as the prefix for the file
names if you save the sequence frames to disk.
3. Indicate how often you want to create a new frame in the sequence
by setting the interval under Every and selecting Iteration or Time
Step in the drop-down list below When. (Time Step is a valid choice
only if you are calculating unsteady flow.) For example, to create
a frame every 10 time steps, you would enter 10 under Every and
select Time Step under When.
4. Click on the Define... button to open the Animation Sequence panel
(Figure 22.17.2).
Figure 22.17.2: The Animation Sequence Panel
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5. Define the Sequence Parameters in the Animation Sequence panel.
(a) Specify whether you want FLUENT to save the animation sequence frames in memory or on your computer’s hard drive
by selecting Memory or Disk under Storage Type.
!
Note that a FLUENT metafile is created for each frame in the
animation sequence. These files contain information about
the entire scene, not just the view that is displayed in the
plot. As a result, they can be quite large. By default, the
files will be stored to disk. If you do not want to use up disk
space to store them, you can instead choose to store them
in memory. Storing them in memory will, however, reduce
the amount of memory available to the solver. Note that the
playback of a sequence stored in memory will be faster than
one stored to disk.
(b) If you selected Disk under Storage Type, specify the directory
where you want to store the files in the Storage Directory field.
(This can be a relative or absolute path.)
(c) Specify the ID of the graphics window where you want the
plot to be displayed in the Window field, and click Set. (The
specified window will open, if it is not already open.)
When FLUENT is iterating, the active graphics window is set
to this window to update the plot. If you want to maintain
each animation in a separate window, specify a different Window ID for each.
6. Define the display properties for the sequence.
(a) Under Display Type in the Animation Sequence panel, choose
the type of display you want to animate by selecting Grid,
Contours, Vectors, XY Plot, or Monitor. If you choose Monitor,
you can select any of the available types of monitor plots in
the Monitor Type drop-down list: Residuals, Force, Statistics,
Surface, or Volume.
The first time that you select Contours, Vectors, or XY Plot, or
one of the monitor types if you select Monitor, FLUENT will
open the corresponding panel (e.g., the Contours panel or the
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Vectors panel) so you can modify the settings and generate
the display. To make subsequent modifications to the display
settings for any of the display types, click the Properties...
button to open the panel for the selected Display Type.
(b) Define the display in the panel for the selected Display Type
(e.g., the Contours or Solution XY Plot panel), and click Display
or Plot.
!
You must click Display or Plot to initialize the scene to be
repeated during the calculation.
See below for guidelines on defining display properties for grid,
contour, and vector displays.
7. Remember to click OK in the Solution Animation panel after you
finish defining all animation sequences.
Note that, when you click OK in the Animation Sequence panel for a
sequence, the Active button for that sequence in the Solution Animation
panel will be turned on automatically. You can choose to use a subset
of the sequences you have defined by turning off the Active button for
those that you currently do not wish to use.
Guidelines for Defining an Animation Sequence
If you are defining an animation sequence containing grid, contour, or
vector displays, note the following when you are defining the display:
• If you want to include lighting effects in the animation frames,
be sure to define the lights before you begin the calculation. See
• If you want to maintain a constant range of colors in a contour
or vector display, you can specify a range explicitly by turning
off the Auto Range option in the Contours or Vectors panel. See
Section 25.1.2 or 25.1.3 for details.
• Scene manipulations that are specified using the Scene Description
panel will not be included in the animation sequence frames. View
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modifications such as mirroring across a symmetry plan will be
included.
22.17.2
Playing an Animation Sequence
Once you have defined a sequence (as described in Section 22.17.1) and
performed a calculation, or read in a previously created animation sequence (as described in Section 22.17.4), you can play back the sequence
using the Playback panel (Figure 22.17.3).
Solve −→ Animate −→Playback...
Figure 22.17.3: The Playback Panel
Under Animation Sequences in the Playback panel, select the sequence
you want to play in the Sequences list. To play the animation once
through from start to finish, click on the “play” button under the Playback heading. (The buttons function in a way similar to those on a
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standard video cassette player. “Play” is the second button from the
right—a single triangle pointing to the right.) To play the animation
backwards once, click on the “play reverse” button (the second from the
left—a single triangle point to the left). As the animation plays, the
Frame scale shows the number of the frame that is currently displayed,
as well as its relative position in the entire animation. If, instead of playing the complete animation sequence, you want to jump to a particular
frame, move the Frame slider bar to the desired frame number, and the
frame corresponding to the new frame number will be displayed in the
graphics window.
! For smoother animations, turn on the Double Buffering option in the
Display Options panel (see Section 25.2.7). This will reduce screen flicker
Additional options for playing back animations are described below.
Modifying the View
If you want to replay the animation sequence with a different view of the
scene, you can use your mouse to modify (e.g., translate, rotate, zoom)
it in the graphics window where the animation is displayed. Note that
any changes you make to the view for an animation sequence will be lost
when you select a new sequence (or reselect the current sequence) in the
Sequences list.
Modifying the Playback Speed
Different computers will play the animation sequence at different speeds,
depending on the complexity of the scene and the type of hardware used
for graphics. You may want to slow down the playback speed for optimal
viewing. Move the Replay Speed slider bar to the left to reduce the
playback speed (and to the right to increase it).
Playing Back an Excerpt
You may sometimes want to play only one portion of a long animation
sequence. To do this, you can modify the Start Frame and the End Frame
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frames, but you want to play only frames 20 to 35, you can set Start
Frame to 20 and End Frame to 35. When you play the animation, it will
start at frame 20 and finish at frame 35.
“Fast-Forwarding” the Animation
You can “fast-forward” or “fast-reverse” the animation by skipping some
of the frames during playback. To fast-forward the animation, you will
need to set the Increment and click on the fast-forward button (the last
button on the right—two triangles pointing to the right). If, for example,
you click on the fast-forward button, the animation will show frames 1,
3, 5, 7, 9, 11, 13 and 15. Clicking on the fast-reverse button (the first
button on the left—two triangles pointing to the left) will show frames
15, 13, 11,...1.
Continuous Animation
If you want the playback of the animation to repeat continuously, there
are two options available. To continuously play the animation from beginning to end (or from end to beginning, if you use one of the reverse
play buttons), select Auto Repeat in the Playback Mode drop-down list.
To play the animation back and forth continuously, reversing the playback direction each time, select Auto Reverse in the Playback Mode dropdown list.
To turn off the continuous playback, select Play Once in the Playback
Mode list. This is the default setting.
Stopping the Animation
To stop the animation during playback, click on the “stop” button (the
square in the middle of the playback control buttons). If your animation
contains very complicated scenes, there may be a slight delay before the
animation stops.
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Advancing the Animation Frame by Frame
To advance the animation manually frame by frame, use the third button
from the right (a vertical bar with a triangle pointing to the right). Each
time you click on this button, the next frame will be displayed in the
graphics window. To reverse the animation frame by frame, use the
third button from the left (a left-pointing triangle with a vertical bar).
Frame-by-frame playback allows you to freeze the animation at points
that are of particular interest.
Deleting an Animation Sequence
If you want to remove one of the sequences that you have created or read
in, select it in the Sequences list and click on the Delete button. If you
want to delete all sequences, click on the Delete All button.
! Note that if you delete a sequence that has not yet been saved to disk (i.e.,
if you selected Memory under Storage Type in the Animation Sequence
panel), it will be removed from memory permanently. If you want to
keep any animation sequences that are stored only in memory, you should
be sure to save them (as described in Section 22.17.3) before you delete
them from the Sequences list or exit FLUENT.
22.17.3
Saving an Animation Sequence
Once you have created an animation sequence, you can save it in any of
the following formats:
• Solution animation file containing the FLUENT metafiles
• Hardcopy files, each containing a frame of the animation sequence
• MPEG file containing each frame of the animation sequence
Note that, if you are saving hardcopy files or an MPEG file, you can
modify the view (e.g., translate, rotate, zoom) in the graphics window
where the animation is displayed, and save the modified view instead of
the original view.
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Solution Animation File
If you selected Disk under Storage Type in the Animation Sequence panel,
then FLUENT will save the solution animation file for you automatically. It will be saved in the specified Storage Directory, and its name
will be the Name you specified for the sequence, with a .cxa extension
(e.g., pressure-contour.cxa). In addition to the .cxa file, FLUENT
will also save a metafile with a .hmf extension for each frame (e.g.,
pressure-contour 2.hmf). The .cxa file contains a list of the associated .hmf files, and tells FLUENT the order in which to display them.
If you selected Memory under Storage Type, then the solution animation
file (.cxa) and the associated metafiles (.hmf) will be lost when you exit
from FLUENT, unless you save them as described below.
You can save the animation sequence to a file that can be read back into
FLUENT (see Section 22.17.4) when you want to replay the animation.
As noted in Section 22.17.4, the solution animation file can be used for
playback in FLUENT independent of the case and data files that were
used to generate it.
To save a solution animation file (and the associated metafiles), select
Animation Frames in the Write/Record Format drop-down list in the Playback panel, and click on the Write button. FLUENT will save a .cxa
file, as well as a .hmf file for each frame of the animation sequence.
The filename for the .cxa file will be the specified sequence Name (e.g.,
pressure-contour.cxa), and the filenames for the metafiles will consist of the specified sequence Name followed by a frame number (e.g.,
pressure-contour 2.hmf). All of the files (.cxa and .hmf) will be
saved in the current working directory.
Hardcopy File
You can also generate a hardcopy file for each frame in the animation
sequence. This feature allows you to save your sequence frames to hardcopy files used by an external animation program such as ImageMagick.
As noted above, you can modify the view in the graphics window before
you save the hardcopy files.
To save the animation as a series of hardcopy files, follow these steps:
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1. Select Hardcopy Frames in the Write/Record Format drop-down list
in the Playback panel.
2. If necessary, click on the Hardcopy Options... button to open the
Graphics Hardcopy panel and set the appropriate parameters for
saving the hardcopy files. (If you are saving hardcopy files for use
with ImageMagick, for example, you may want to select the window
dump format. See Section 3.12.1 for details.) Click Apply in the
Graphics Hardcopy panel to save your modified settings.
!
Do not click on the Save... button in the Graphics Hardcopy panel.
You will save the hardcopy files from the Playback panel in the next
step.
3. In the Playback panel, click on the Write button. FLUENT will
replay the animation, saving each frame to a separate file. The
filenames will consist of the specified sequence Name followed by
a frame number (e.g., pressure-contour 2.ps), and they will all
be saved in the current working directory.
MPEG File
It is also possible to save all of the frames of the animation sequence in
an MPEG file, which can be viewed using an MPEG decoder such as
mpeg play. Saving the entire animation to an MPEG file will require less
disk space than storing individual window dump files (using the hardcopy
method), but the MPEG file will yield lower-quality images.
As noted above, you can modify the view in the graphics window before
you save the MPEG file.
To save the animation to an MPEG file, follow these steps:
1. Select MPEG in the Write/Record Format drop-down list in the
Playback panel.
2. Click on the Write button.
FLUENT will replay the animation and save each frame to a separate scratch file, and then it will combine all the files into a single
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MPEG file. The name of the MPEG file will be the specified sequence Name with a .mpg extension (e.g., pressure-contour.mpg),
and it will be saved in the current working directory.
22.17.4
If you have saved an animation sequence to a solution animation file (as
described in Section 22.17.3), you can read that file back in at a later
time (or in a different session) and play the animation. Note that you
can read a solution animation file into any FLUENT session; you do not
need to read in the corresponding case and data files. In fact, you do not
need to read in any case and data files at all before you read a solution
animation file into FLUENT.
To read a solution animation file, click on the Read... button in the
Playback panel. In the resulting Select File dialog box, specify the name
of the file to be read.
22.18
Executing Commands During the Calculation
As described in Sections 22.16 and 22.17, respectively, you can report
and monitor various quantities (e.g., residuals, force coefficients) and create animations of the solution while the solver is performing calculations.
FLUENT also includes a feature that allows you to define your own command(s) to be executed during the calculation at specified intervals. For
example, you can ask FLUENT to export data to a third-party package
such as FIELDVIEW after every time step. You will specify a series of
text commands or use the GUI to define the steps to be performed.
! If you want to save case or data files at intervals during the calculation, you must use the Autosave Case/Data panel (opened with the
File/Write/Autosave... menu item). See Section 3.3.4 for details.
22.18.1
Specifying the Commands to be Executed
You will indicate the command(s) that you want the solver to execute
at specified intervals during the calculation using the Execute Commands
panel (Figure 22.18.1).
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22.18 Executing Commands During the Calculation
Solve −→Execute Commands...
Figure 22.18.1: The Execute Commands Panel
The procedure is as follows:
1. Increase the Defined Commands value to the number of commands
you wish to specify. As this value is increased, additional command
entries will become editable. For each command, you will perform
the following steps.
2. Turn on the On check button next to the command if you want
it to be executed during the calculation. You may define multiple
commands and choose to use only a subset of them by turning off
the check button for those that you do not wish to use.
3. Enter a name for the command under the Name heading.
4. Indicate how often you want the command to be executed by setting the interval under Every and selecting Iteration or Time Step in
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the drop-down list below When. (Time Step is a valid choice only
if you are calculating unsteady flow.) For example, to execute the
command every 10 iterations, you would enter 10 under Every and
select Iteration under When.
!
If you specify an interval in iterations, be sure to keep the Reporting
Interval in the Iterate panel at its default value of 1.
5. Define the command by entering a series of text commands in the
Command field, or by entering the name of a command macro you
have defined (or will define) as described in Section 22.18.2.
!
If the command to be executed involves saving a file, see Section 22.18.3 for important information.
22.18.2
Defining Macros
Macros that you define for automatic execution during the calculation
can also be used interactively by you during the problem setup or postprocessing. For example, if you define a macro that performs a certain
type of adaption after each iteration, you can also use the macro to
Definition of a macro is accomplished as follows:
1. In the Execute Commands panel, click on the Define Macro... button
to open the Define Macro panel (Figure 22.18.2). Since this is a
“modal” panel, the solver will not allow you to do anything else
until you perform step 2, below.
2. In the Define Macro panel, specify a Name for the macro (e.g.,
adapt1) and click on OK. (The Define Macro... button in the Execute Commands panel will become the End Macro button.)
3. Perform the steps that you want the macro to perform. For example, if you want the macro to perform gradient adaption, open the
!
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If the command to be executed involves saving a file, see Section 22.18.3 for important information.
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22.18 Executing Commands During the Calculation
Figure 22.18.2: The Define Macro Panel
4. When you have completed the steps you wish the macro to perform,
click on the End Macro button in the Execute Commands panel.
As noted above, once you have defined a macro for execution during the
calculation, you can use it at any time. If you defined the macro called
in the console (text) window to perform this adaption. This macro is
independent of any text menus, so you need not move to a different text
menu to use it. Macros can be saved to and read from files. To save all
macros that are currently defined, use the file/write-macros text command. To read all the macros in a macro file, use the file/read-macros
text command.
! A macro, like a journal file, is a simple record/playback function. It
will therefore know nothing about the state in which it was recorded or
the state in which it is being played back. You must be careful that all
surfaces, variables, etc. that are used by the macro have been properly
defined when you (or FLUENT) invoke the macro.
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22.18.3
Saving Files During the Calculation
If the command to be executed during the calculation involves saving a
file, you must include a special character in the file name when you enter
it in the Select File dialog box so that the solver will know to assign a
new name to each file it saves. You can number the files by iteration
number or by time step. (These special characters for numbering files
are also useful when you are saving window dumps for use in animations,
as described in Section 3.12.1.) See Section 3.1.7 for details about these
special characters for filenames.
! If you want to save case or data files at intervals during the calculation, you must use the Autosave Case/Data panel (opened with the
File/Write/Autosave... menu item). See Section 3.3.4 for details.
22.19
Convergence and Stability
Convergence can be hindered by a number of factors. Large numbers
of computational cells, overly conservative under-relaxation factors, and
complex flow physics are often the main causes. Sometimes it is difficult
to know whether you have a converged solution. In the following sections,
some of the numerical controls and modeling techniques that can be
exercised to enhance convergence and maintain stability are examined.
• Section 22.19.1: Judging Convergence
• Section 22.19.2: Step-by-Step Solution Processes
• Section 22.19.3: Modifying Algebraic Multigrid Parameters
• Section 22.19.4: Modifying FAS Multigrid Parameters
• Section 22.19.5: Modifying Multi-Stage Time-Stepping Parameters
You should also refer to Sections 22.7 and 22.8 for information about how
the choice of discretization scheme or (for the segregated solver) pressurevelocity coupling scheme can affect convergence. Manipulation of underrelaxation parameters and multigrid settings to enhance convergence is
discussed in Sections 22.9 and 22.19.3.
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22.19.1
Judging Convergence
There are no universal metrics for judging convergence. Residual definitions that are useful for one class of problem are sometimes misleading
for other classes of problems. Therefore it is a good idea to judge convergence not only by examining residual levels, but also by monitoring
relevant integrated quantities such as drag or heat transfer coefficient.
For most problems, the default convergence criterion in FLUENT is sufficient. This criterion requires that the scaled residuals defined by Equation 22.16-4 or 22.16-9 decrease to 10−3 for all equations except the
energy and P-1 equations, for which the criterion is 10−6 .
Sometimes, however, this criterion may not be appropriate. Typical
situations are listed below.
• If you make a good initial guess of the flow field, the initial continuity residual may be very small leading to a large scaled residual
for the continuity equation. In such a situation it is useful to examine the unscaled residual and compare it with an appropriate
scale, such as the mass flow rate at the inlet.
• For some equations, such as for turbulence quantities, a poor initial guess may result in high scale factors. In such cases, scaled
residuals will start low, increase as non-linear sources build up,
and eventually decrease. It is therefore good practice to judge convergence not just from the value of the residual itself, but from
its behavior. You should ensure that the residual continues to decrease (or remain low) for several iterations (say 50 or more) before
concluding that the solution has converged.
Another popular approach to judging convergence is to require that the
unscaled residuals drop by three orders of magnitude. FLUENT provides
residual normalization for this purpose, as discussed in Sections 22.16.1
and 22.16.1. In this approach the convergence criterion is that the normalized unscaled residuals should drop to 10−3 . However, this requirement may not be appropriate in many cases:
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• If you have provided a very good initial guess, the residuals may
not drop three orders of magnitude. In a nearly-isothermal flow, for
example, energy residuals may not drop three orders if the initial
guess of temperature is very close to the final solution.
• If the governing equation contains non-linear source terms which
are zero at the beginning of the calculation and build up slowly
during computation, the residuals may not drop three orders of
magnitude. In the case of natural convection in an enclosure, for
example, initial momentum residuals may be very close to zero
because the initial uniform temperature guess does not generate
buoyancy. In such a case, the initial nearly-zero residual is not a
good scale for the residual.
• If the variable of interest is nearly zero everywhere, the residuals
may not drop three orders of magnitude. In fully-developed flow in
a pipe, for example, the cross-sectional velocities are zero. If these
velocities have been initialized to zero, initial (and final) residuals
are both close to zero, and a three-order drop cannot be expected.
In such cases, it is wise to monitor integrated quantities, such as drag
or overall heat transfer coefficient, before concluding that the solution
has converged. It may also be useful to examine the un-normalized
unscaled residual, and determine if the residual is small compared to
some appropriate scale. Alternatively, the scaled residual defined by
Equation 22.16-4 or 22.16-9 (the default) may be considered.
Conversely, it is possible that if the initial guess is very bad, the initial
residuals are so large that a three-order drop in residual does not guarantee convergence. This is specially true for k and equations where good
initial guesses are difficult. Here again it is useful to examine overall integrated quantities that you are particularly interested in. If the solution
is unconverged, you may drop the convergence tolerance, as described in
Section 22.16.1.
22.19.2
Step-by-Step Solution Processes
One important technique for speeding convergence for complex problems
is to tackle the problem one step at a time. When modeling a problem
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with heat transfer, you can begin with the calculation of the isothermal
flow. To solve turbulent flow, you might start with the calculation of
laminar flow. When modeling a reacting flow, you can begin by computing a partially converged solution to the non-reacting flow, possibly
including the species mixing. When modeling a discrete phase, such as
fuel evaporating from droplets, it is a good idea to solve the gas-phase
flow field first. Such solutions generally serve as a good starting point
for the calculation of the more complex problems. These step-by-step
techniques involve using the Solution Controls panel to turn equations on
and off.
Selecting a Subset of the Solution Equations
FLUENT automatically solves each equation that is turned on using the
Models family of panels. If you specify in the Viscous Model panel that
the flow is turbulent, equations for conservation of turbulence quantities
are turned on. If you specify in the Energy panel that FLUENT should
enable energy, the energy equation is activated. Convergence can be
sped up by focusing the computational effort on the equations of primary
importance. The Equations list in the Solution Controls panel allows you
to turn individual equations on or off temporarily.
Solve −→ Controls −→Solution...
A typical example is the computation of a flow with heat transfer. Initially, you will define the full problem scope, including the thermal
boundary conditions and temperature-dependent flow properties. Following the problem setup, you will use the Solution Controls panel to
temporarily turn off the energy equation. You can then compute an
isothermal flow field, remembering to set a reasonable initial value for
the temperature of the fluid.
! This is possible only for the segregated solver; the coupled solvers solve
the energy equation together with the flow equations in a coupled manner, so you cannot turn off the energy equation as described above.
When the isothermal flow is reasonably well converged, you can turn
the energy equation back on. You can actually turn off the momentum
and continuity equations while the initial energy field is being computed.
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When the energy field begins to converge well, you can turn the momentum and continuity equations back on so that the flow pattern can adjust
to the new temperature field. The temperature will couple back into the
flow solution by its impact on fluid properties such as density and viscosity. The temperature field will have no effect on the flow field if the
fluid properties (e.g., density, viscosity) do not vary with temperature.
In such cases, you can compute the energy field without turning the flow
equations back on again.
! If you have specified temperature-dependent flow properties, you should
be sure that a realistic value has been set for temperature throughout
the domain before disabling calculation of the energy equation. If an
unrealistic temperature value is used, the flow properties dependent on
temperature will also be unrealistic, and the flow field will be adversely
affected. Instructions for initializing the temperature field or patching a
temperature field onto an existing solution are provided in Section 22.13.
Turning Reactions On and Off
To solve a species mixing problem prior to solving a reacting flow, you
should set up the problem including all of the reaction information, and
save the complete case file. To turn off the reaction so that only the
species mixing problem can be solved, you can use the Species Model
panel to turn off the Volumetric Reactions.
Define −→ Models −→Species...
Once the species mixing problem has partially converged, you can return
to the Species Model panel and turn the Volumetric Reactions option on
again. You can then resume the calculation starting from the partially
converged data.
For combustion problems you may want to patch a hot temperature in
the vicinity of the anticipated reactions before you restart the calculation.
See Section 22.13.2 for information about patching an initial value for a
flow variable.
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22.19 Convergence and Stability
22.19.3
Modifying Algebraic Multigrid Parameters
The default algebraic multigrid settings are appropriate for nearly all
problems, but in rare cases you may need to make minor adjustments.
This section describes how to analyze the multigrid solver’s performance
to determine which parameters should be modified. It also provides examples of suggested settings for particular types of problems and explains
how to set the multigrid parameters.
Analyzing the Algebraic Multigrid Solver
As mentioned earlier, in most cases the multigrid solver will not require
any special attention from you. If, however, you have convergence difficulties or you want to minimize the overall solution time by using more
aggressive settings, you can monitor the multigrid solver and modify
the parameters to improve its performance. (The instructions below assume that you have already begun calculations, since there is no need to
monitor the solver if you do not fit into one of the two categories above.)
To determine whether your convergence difficulties can be alleviated by
modifying the multigrid settings, you will check if the requested residual
reduction is obtained on each grid level. To minimize solution time, you
will check to see if switching to a more powerful cycle will result in overall
reduction of work by the solver.
Monitoring the Algebraic Multigrid Solver
The steps for monitoring the solver are as follows:
1. Set multigrid Verbosity to 1 or 2 in the Multigrid Controls panel.
Solve −→ Controls −→Multigrid...
2. Request a single iteration using the Iterate panel.
Solve −→Iterate...
If you set the verbosity to 2, the information printed in the console (text)
window for each equation will include the following:
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• equation name
• equation tolerance (computed by the solver using a normalization
of the source vector)
• residual value after each fixed multigrid cycle or fine relaxation for
the flex cycle
• number of equations in each multigrid level, with the zeroth level
being the original (finest-level) system of equations
Note that the residual printed at cycle or relaxation 0 is the initial residual before any multigrid cycles are performed.
When verbosity is set to 1, only the equation name, tolerance, and residuals are printed.
A portion of a sample printout is shown below:
pressure correction equation:
tol. 1.2668e-05
0 2.5336e+00
1 4.9778e-01
2 2.5863e-01
3 1.9387e-01
multigrid levels:
0
918
1
426
2
205
3
97
4
45
5
21
6
10
7
4
Interpreting the Algebraic Multigrid Report
By default, the flex cycle is used for all equations except pressure correction, which uses a V cycle. Typically, for a flex cycle only a few (5–10)
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relaxations will be performed at the finest level and no coarse levels will
be visited. In some cases one or two coarse levels may be visited. If
the maximum number of fine level relaxations is not sufficient, you may
want to increase the maximum number (as described in Section 22.19.3)
or switch to a V cycle (as described in Section 22.19.3).
For pressure correction, a V cycle is used by default. If the maximum
number of cycles (30 by default) is not sufficient, you can switch to a W
cycle (using the Multigrid Controls panel, as described in Section 22.19.3).
Note that for the parallel solver, efficiency may deteriorate with a W
cycle. If you are using the parallel solver, you can try increasing the
maximum number of cycles by increasing the value of Max Cycles in the
Multigrid Controls panel under Fixed Cycle Parameters.
Solve −→ Controls −→Multigrid...
Changing the Maximum Number of Relaxations
To change the maximum number of relaxations, increase or decrease the
value of Max Fine Relaxations or Max Coarse Relaxations in the Multigrid
Controls panel (Figure 22.19.1) under Flexible Cycle Parameters.
Solve −→ Controls −→Multigrid...
Specifying the Multigrid Cycle Type
By default, the V cycle is used for the pressure equation and the flex
cycle is used for all other equations. (See Section 22.5.2 for a description
of these cycles.) To change the cycle type for an equation, you will use
the top portion of the Multigrid Controls panel (Figure 22.19.1).
For each equation, you can choose Flexible, V-Cycle, W-Cycle, or F-Cycle
Setting the Termination and Residual Reduction Parameters
When you use the flex cycle for an equation, you can control the multigrid
performance by modifying the Termination and/or Restriction criteria for
that equation at the top of the Multigrid Controls panel (Figure 22.19.1).
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Figure 22.19.1: The Multigrid Controls Panel
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Solve −→ Controls −→Multigrid...
The Restriction criterion is the residual reduction tolerance, β in Equation 22.5-14. This parameter dictates when a coarser grid level must be
visited (due to insufficient improvement in the solution on the current
level). With a larger value of β, coarse levels will be visited less often
(and vice versa). The Termination criterion, α in Equation 22.5-15, governs when the solver should return to a finer grid level (i.e., when the
residuals have improved sufficiently on the current level).
For the V, W, or F cycle, the Termination criterion determines whether
or not another cycle should be performed on the finest (original) level. If
the current residual on the finest level does not satisfy Equation 22.5-15,
and the maximum number of cycles has not been performed, FLUENT
will perform another multigrid cycle. (The Restriction parameter is not
used by the V, W, and F cycles.)
There are several additional parameters that control the algebraic multigrid solver, but there will usually be no need to modify them. These
additional parameters are all contained in the Multigrid Controls panel
(Figure 22.19.1).
Solve −→ Controls −→Multigrid...
Coarsening Parameters
For all multigrid cycle types, you can control the maximum number of
coarse levels (Max Coarse Levels under Coarsening Parameters) that will
be built by the multigrid solver. Sets of coarser simultaneous equations
are built until the maximum number of levels has been created, or the
coarsest level has only 3 equations. Each level has about half as many
unknowns as the previous level, so coarsening until there are only a few
cells left will require about as much total coarse-level coefficient storage
as was required on the fine mesh. Reducing the maximum coarse levels
will reduce the memory requirements, but may require more iterations
to achieve a converged solution. Setting Max Coarse Levels to 0 turns off
the algebraic multigrid solver.
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Another coarsening parameter you can control is the increase in coarseness on successive levels. The Coarsen By parameter specifies the number
of fine grid cells that will be grouped together to create a coarse grid cell.
The algorithm groups each cell with its closest neighbor, then groups the
cell and its closest neighbor with the neighbor’s closest neighbor, continuing until the desired coarsening is achieved. Typical values are in
the range from 2 to 10, with the default value of 2 giving the best performance, but also the greatest memory use. You should not adjust
this parameter unless you need to reduce the memory required to run a
problem.
Fixed Cycle Parameters
For the fixed (V, W, and F) multigrid cycles, you can control the number
of pre- and post-relaxations (β1 and β3 in Section 22.5.2). Pre Sweeps sets
the number of relaxations to perform before moving to a coarser level.
Post Sweeps sets the number to be performed after coarser level corrections have been applied. Normally one post-relaxation is performed and
no pre-relaxations are done (i.e., β3 = 1 and β1 = 0), but in rare cases,
you may need to increase the value of β1 to 1 or 2.
Returning to the Default Multigrid Parameters
If you change the multigrid parameters, but you then want to return
to FLUENT’s default settings, you can click on the Default button in
the Multigrid Controls panel. FLUENT will change all settings to the
defaults, and the Default button will become the Reset button. To get
your settings back again, you can click on the Reset button.
22.19.4
Setting FAS Multigrid Parameters
For most calculations, you will not need to modify any FAS multigrid parameters once you have set the number of coarse grid levels. If, however,
you encounter convergence difficulties, you may consider the following
suggested procedures.
! Recall that FAS multigrid is used only by the coupled explicit solver.
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Combating Convergence Trouble
Some problems may approach convergence steadily at first, but then the
residuals will level off and the solution will “get stuck.” In some cases
(e.g., long thin ducts), this convergence trouble may be due to multigrid’s
slow propagation of pressure information through the domain. In such
cases, you should turn off multigrid by setting Multigrid Levels to 0 in
the Solution Controls panel (under Solver Parameters).
Solve −→ Controls −→Solution...
“Industrial-Strength” FAS Multigrid
In some cases, you may find that your problem is converging, but at
an extremely slow rate. Such problems can often benefit from a more
aggressive form of multigrid, which will speed up the propagation of the
solution corrections. For such problems, you can try the “industrialstrength” multigrid settings.
! These settings are very aggressive and assume that the solution information passed through the multigrid levels is somewhat accurate. For
this reason, you should only attempt the procedure described here after you have performed enough iterations that the solution is off to a
good start. Using “industrial-strength” multigrid too early in the calculation process—when the solution is far from correct—will not help
convergence and may cause the calculation to become unstable, as very
incorrect values are propagated quickly to the original grid. Note also
that while these multigrid settings will usually reduce the total number
of iterations required to reach convergence, they will greatly increase
the computation time for each multigrid cycle. Thus the solver will be
performing fewer but longer iterations.
The strategy employed is as follows:
• Increase the number of iterations performed on each grid level before proceeding to a coarser level
• Increase the number of iterations performed on each grid level after
returning from a coarser level
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• Allow full correction transfer from one level to the next finer level,
instead of transferring reduced values of the corrections
• Do not smooth the interpolated corrections when they are transferred from a coarser grid to a finer grid
You can set all of the parameters for this strategy under FAS Multigrid Controls in the Multigrid Controls panel (Figure 22.19.2) and then
continue the calculation.
Solve −→ Controls −→Multigrid...
Increasing the number of iterations performed on each grid level before
proceeding to a coarser level (the value of β1 described in Section 22.5.2)
will improve the solution passed from each finer grid level to the next
coarser grid level. Try increasing the value of Pre Sweeps (under FAS
Multigrid Controls, not under Algebraic Multigrid Controls) to 10.
Increasing the number of iterations performed on each level after returning from a coarser level will improve the corrections passed from each
coarser grid level to the next finer grid level. Errors introduced on the
coarser grid levels can therefore be reduced before they are passed further
up the grid hierarchy to the original grid. Try increasing the value of
Post Sweeps (under FAS Multigrid Controls, not under Algebraic Multigrid
Controls) to 10.
By default, the full values of the multigrid corrections are not transferred
from a coarser grid to a finer grid; only 60% of the value is transferred.
This prevents large errors from transferring quickly up to the original
grid and causing the calculation to become unstable. It also prevents a
“good” solution from propagating quickly to the original grid, however.
By increasing the Correction Reduction to 1, you can transfer the full
values from coarser to finer grid levels, speeding the propagation of the
solution and, usually, the convergence as well.
When the corrections on a coarse grid are passed back to the next finer
grid level, the values are, by default, interpolated and then smoothed.
Disabling the smoothing so that the actual value in a coarse grid cell is
assigned to the fine grid cells that comprise it can also aid convergence.
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22.19 Convergence and Stability
Figure 22.19.2: The Multigrid Controls Panel
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Using the Solver
To turn off smoothing, set Correction Smoothing to 0. Large discontinuities between cells will be smoothed out implicitly as a result of the
There are several additional parameters that control the multigrid solver,
but there will usually be no need to modify them. These additional
parameters are all contained in the Multigrid Controls panel.
Solve −→ Controls −→Multigrid...
Specifying the Multigrid Cycle Type
By default, the V cycle is used for the flow equations. (See Section 22.5.2
for a description of the available cycles.) To change to a W cycle, you
can select it in the drop-down list next to Flow at the top of the Multigrid
Controls panel (Figure 22.19.2).
Reducing the Time Step for Coarse Grid Levels
The Courant Number Reduction (at the bottom of the Multigrid Controls
panel) sets the factor by which to reduce the Courant number for coarse
grid levels (i.e., every level except the finest). Some reduction of time
step (such as the default 0.9) is typically required because the stability
limit cannot be determined as precisely on the irregularly shaped coarser
grid cells.
22.19.5
Modifying Multi-Stage Time-Stepping Parameters
The most common parameter you will change to control the multi-stage
time-stepping scheme is the Courant number. Instructions for modifying
the Courant number are presented in Section 22.10, and procedures for
modifying other less commonly changed parameters are provided in this
section.
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22.19 Convergence and Stability
Using Residual Smoothing to Increase the Courant Number
Implicit residual smoothing (or averaging) is a technique that can be
used to reduce the time step restriction of the solver, thereby allowing the
Courant number to be increased. The implicit smoothing is implemented
with an iterative Jacobi method, as described in Section 22.4.3. You can
control residual smoothing in the Solution Controls panel.
Solve −→ Controls −→Solution...
By default, the number of Iterations for Residual Smoothing is set to
zero, indicating that residual smoothing is disabled. If you increase the
Iterations counter to 1 or more, you can enter the Smoothing Factor. A
smoothing factor of 0.5 with 2 passes of the Jacobi smoother is usually
adequate to allow the Courant number to be doubled.
Changing the Multi-Stage Scheme
It is possible to make several changes to the multi-stage time-stepping
scheme itself. You can change the number of stages and set a new multistage coefficient for each stage. You can also control whether or not
dissipation and viscous stresses are updated at each stage. These changes
are made in the Multi-Stage Parameters panel (Figure 22.19.3).
Solve −→ Controls −→Multi-Stage...
! You should not attempt to make changes to FLUENT’s multi-stage scheme
unless you are very familiar with multi-stage schemes and are interested
in trying a different scheme found in the literature.
Changing the Coefficients and Number of Stages
By default, the FLUENT multi-stage scheme uses 5 stages with coefficients of 0.25, 0.166666, 0.375, 0.5, and 1.0 for the first through fifth
stages, respectively. You can decrease the number of stages using the
arrow buttons for Number of Stages in the Multi-Stage Parameters panel.
(If you want to increase the number of stages, you will need to use the
text-interface command solve/set/multi-stage.) For each stage, you
can modify the Coefficient. Coefficients must be greater than 0 and less
than 1. The final stage should always have a coefficient of 1.
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Using the Solver
Figure 22.19.3: The Multi-Stage Parameters Panel
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22.19 Convergence and Stability
Controlling Updates to Dissipation and Viscous Stresses
For each stage, you can indicate whether or not artificial dissipation
and viscous stresses are evaluated. If a Dissipation box is selected for a
particular stage, artificial dissipation will be updated on that stage. If
not selected, artificial dissipation will remain “frozen” at the value of
the previous stage. If a Viscous box is selected for a particular stage,
viscous stresses will be updated on that stage. If not selected, viscous
stresses will remain “frozen” at the value of the previous stage. Viscous
stresses should always be computed on the first stage, and successive
evaluations will increase the “robustness” of the solution process, but
will also increase the expense (i.e., increase the CPU time per iteration).
For steady problems, the final solution is independent of the stages on
which viscous stresses are updated.
Resetting the Multi-Stage Parameters
If you change the multi-stage parameters, but you then want to return to
the default scheme set by FLUENT, you can click on the Default button
in the Multi-Stage Parameters panel. FLUENT will change the values to
the defaults and the Default button will become the Reset button. To
get your values back again, you can click on the Reset button.
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Using the Solver
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