A HIGH-ORDER IMMERSED BOUNDARY METHOD ... UNSTEADY INCOMPRESSIBLE FLOW CALCULATIONS

A  HIGH-ORDER  IMMERSED  BOUNDARY  METHOD ... UNSTEADY  INCOMPRESSIBLE  FLOW  CALCULATIONS
A HIGH-ORDER IMMERSED BOUNDARY METHOD FOR
UNSTEADY INCOMPRESSIBLE FLOW CALCULATIONS
by
Mark N. Linnick
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
WITH A MAJOR IN AEROSPACE ENGINEERING
In the Graduate College
THE UNIVERSITY OF ARIZONA
200 3
UMI Number: 3119962
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entitled
Mark N. Linnirk
A High-Order Immersed Boundary Method for Unsteady
Inp.ompresslble Flow Calculations
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requirement for the Degree of
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///•
<5 J'
Date
ermannXF/ Fasel
////y/)3
Alfonscf^rtega
Date
Anat
Da
Tumin
1/ / l ^ (
/•
Date
Edward J. Kerschen
Vr
U
Date
Michael Tabor
Final approval and acceptance of this dissertation is contingent upon
the candidate's submission of the final copy of the dissertation to the
Graduate College.
I hereby certify that I have read this dissertation prepared under my
direction and recommend that it be accepted as fulfilling the dissertation
ement.
"/iq/o3
Dissertaticsn/Director
Hermann F. Fasel
Date
3
STATEMENT BY AUTHOR
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advanced degree at The University of Arizona and is deposited in the University
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Brief quotations from this dissertation are allowable without special permission,
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SIGNED:
4
Acknowledgements
I would like to thank the following people who have, in some way or another, made
this dissertation possible; Professor Hermann Fasel for supporting me during my six
years at the University of Arizona. Professors Edward Kerschen, Michael Tabor, Al­
fonso Ortega, and Anatoh Tumin for serving on my committee, and also Professor
Mosey Brio for serving on my comprehensive examination committee. Professor Mike
Gaster, under whose supervision my graduate career began at the University of Cam­
bridge. All of the great teachers and professors with whom I've had the opportunity to
study. Fellow graduate and postdoc students: Jiirgen Seidel, Hui-Liu Zhang, Andreas
Gross, Stefan Wernz, Daniel Israel, Armin Kurz, Dieter Postl, Richard Sandberg,
Pietro Valsecchi, Dominic von Terzi (special thanks for proof-reading this thesis),
Frank Husmeier, and all of the exchange students, in particular, Juliane Urban, from
the Universitat Stuttgart I've met while working in the CFD Lab. Markus Kloker, Uli
Rist, and all the members of the Transition Group at the Institut fiir Aerodynamik
und Gasdynamik for allowing me the opportunity to work with them during my stay
in Stuttgart. Finally, I would like to thank my parents, Greg and Averil Linnick, for
their support throughout the many, many years I've spent studying.
5
Table of Contents
List of Figures
8
List of Tables
14
Abstract
15
Nomenclature
16
1.
19
Introduction
1.1
2.
20
1.1.1
Structured Grid Methods
21
1.1.2
Unstructured Grid Methods
22
1.2
The Immersed Boundary Method
25
1.3
Objective and Scope of Present Research
27
Governing Equations
29
2.1
The Incompressible Navier-Stokes Equations
29
2.2
The Vorticity-Velocity Formulation
29
2.3
The Stream Function-Vorticity Formulation
31
2.3.1
Boundary Conditions for the Stream Function
32
2.3.2
Boundary Conditions for the Vorticity
33
2.3.3
Outflow Treatment
36
2.3.4
Initial Conditions
36
2.4
3.
Standard Numerical Methods for Complex Geometries
The Reynolds-Averaged Navier-Stokes Equations
Analysis of the Original Immersed Boundary Method
37
41
3.1
Using Singular Sources with Discrete Approximations
41
3.2
Formulation for the Navier-Stokes Equations
43
3.3
Model Equations with Singular Source Terms
44
6
Table of Contents- Continued
3.4
4.
5.
Application to the Navier-Stokes Equations
51
3.4.1
Numerical Implementation
51
3.4.2
Flow Control Actuators
53
3.4.3
Tollmien-Schlichting Waves
62
3.4.4
Further Discussion of Results from the Original N-S IBM ...
71
A New Approach to the Immersed Boundary Method
74
4.1
Taylor Series Analysis of Functions with Jump-Singularities
74
4.2
Derivation of Jump-Corrected Finite-Differences
76
4.3
Obtaining Jump-Corrections
78
4.3.1
Differential Equations with Singular Forcing Terms
78
4.3.2
Differential Equations without Singular Forcing Terms ....
80
Numerical Method
5.1
5.2
85
The Convection-Diffusion Equation
85
5.1.1
Time Integration
85
5.1.2
Spatial Discretization
86
5.1.3
Stability Analysis
88
5.1.4
Validation
92
The Poisson Equation
95
5.2.1
Discretization
95
5.2.2
Resulting System of Equations
98
5.2.3
Validation
104
5.3
The k-oj Transport Equations
104
5.4
Moving Immersed Boundaries
106
5.5
Validation of the Combined N-S/IBM Code
Ill
7
Table of Contents—Continued
6.
7.
Results
115
6.1
Uniform Flow Past a Circular Cylinder
115
6.2
Tollmien-Schlichting Waves in a Blasius Boundary Layer
130
6.3
Turbulent Flat Plate Boundary Layer
133
6.4
Turbulent Flow over a Step
144
6.5
Additional Examples
146
Conclusions
159
Appendix A
Computational Parameters
164
Appendix B
Finite-difference Coefficients
170
References
172
8
List of Figures
1.1
Example of a 2-D, non-orthogonal, structured grid for the computation
22
of the flow past a membrane-type flow control actuator
1.2
One possible distribution of points {xi,yi) that could be used in the un­
23
structured finite-difference stencil given by equation (1.2)
1.3
Points
in the unstructured finite-difference stencil used for the
24
numerical study given in table 1.1
1.4
An example of an irregular boundary dVli immersed inside a Cartesian
computational grid
26
2.1
Stencils used for computing wall vorticity at an immersed boundary. . .
34
3.1
The simplified stair-step representation of an immersed boundary
43
3.2
A discrete representation of the Dirac delta function from equation (3.13)
for several a / h
46
3.3
Analytical and numerical solution of equation (3.8) with a j h = 1
49
3.4
Analytical and numerical solution of equation (3.8) for several a/h. . . .
49
3.5
Error in the numerical solution of equation (3.8) for grid size N =
N — 101, and N = 201. Equation (3.13) with a / h — 2 was used as a
discrete representation of the singular source term
3.6
50
Error in the numerical solution of equation (3.8) for grid size
N = 101, and N — 201. The fourth-order function
= 51,
in Walden
(1999) was used as the discrete representation of the singular source term. 50
3.7
Truncated Fourier series for the periodic delta-function
52
3.8
2-D vibrating ribbon simulation
56
3.9
Streamwise velocity profiles u near the leading and trailing edge of a
vibrating ribbon
57
3.10 Streamwise and wall-normal disturbance velocity produced by a vibrating
ribbon
58
9
List of Figures—Continued
3.11 Wall-normal v and streamwise u disturbance velocity created by a vibrat­
ing ribbon
59
3.12 Spanwise disturbance vorticity created by the sinusoidal movement of a
wall-mounted actuator
60
3.13 Contours of constant li-disturbance velocity created by a piston actuator.
61
3.14 A flat plate immersed in a rectangular computational domain
62
3.15 Comparison of tt-velocity profiles at several downstream locations com­
puted using an immersed wall
64
3.16 Comparison of u-velocity profiles at several downstream locations com­
puted using an immersed wall
65
3.17 Comparison of cj-vorticity profiles at several downstream locations com­
puted using an immersed wall
66
3.18 Comparison of disturbance li-velocity profiles at several downstream lo­
cations computed using an immersed wall
67
3.19 Comparison of disturbance w-velocity profiles at several downstream lo­
cations computed using an immersed wall
68
3.20 Comparison of disturbance o)-vorticity profiles at several downstream lo­
cations computed using an immersed wall
69
3.21 Comparison of the growth rates of a 2-D T-S wave computed using an
immersed wall
3.22 A simple two-level AMR grid
70
71
3.23 Impulsively started circular cyhnder (a; vorticity), Reij = 2000, modelled
with an immersed boundary technique that uses singular sourcc terms. .
72
4.1
A function f { x ) with discontinuity a t x ~ X a
75
4.2
A typical solution of equation (4.16)
79
10
List of Figures—Continued
4.3
Spatial configuration of one-sided finite-difference stencils used to com­
pute jumps in equation (4.22)
81
5.1
Eigenvalues u' of equation (5.24) for the case of pure convection
90
5.2
Eigenvalues uj' of equation (5.26) for the case of pure diffusion
91
5.3
Eigenvalues u' of equation (5.28) for the case of combined convectiondiffusion
5.4
91
Numerical solution of equation (2.7) on a 65 x 65 grid, and corresponding
error £
5.5
93
The intersection of the 9-point stencil given in equation (5.35) with an
immersed boundary, and the jump-corrected stencil
5.6
97
Error in the numerical solution of equation (5.32), ||£||oo) versus multigrid
v-cycle iteration number for various grid sizes with and without immersed
boundaries
103
5.7
Numerical solution of equation (5.32), and corresponding error e
105
5.8
Error in the numerical solution of equation (5.32), ||£||oo, versus multigrid
5.9
v-cycle iteration number for various grid sizes
105
A moving immersed boundary at time t — ta, and at a later time t = tb-
107
5.10 Numerical and analytical solution to equation (5.61) using parameters
shown in table A.4
109
5.11 Uniform flow past a circular cylinder, Re^ = 40. Two figures showing the
full computational domain used for the vorticity definition/divergence-free
study of section 5.5
113
5.12 Divergence-free and vorticity definition error distribution over the upper
half of the computational domain for the finest grid
6.1
114
Grid refinement study for flow past a circular cylinder with Reo = 200 :
Cd and
C l computed on three different
grids
116
11
List of Figures—Continued
6.2
Uniform flow past a circular cylinder, Reo = 20
119
6.3
Uniform flow past a circular cylinder, Reo = 40
120
6.4
Uniform flow past a circular cylinder, Ke^ = 50
121
6.5
Nomenclature for dimensions given in table 6.2
122
6.6
Co and C i versus time t for uniform flow past a circular cylinder, Rep =
200, A = 0.023
6.7
Spanwise vorticity u>. Uniform flow past circular cylinders for Re^ = 40,
Rbd = 50 (unsteady), Re^) = 100, and Re^i = 200
6.8
126
Close-up view of the streamwise velocity u in the vicinity of the cylinder
surface for Re^ = 20 and Rep = 200
6.9
124
127
Close-up view of the normal velocity v in the vicinity of the cylinder
surface for Re^ = 20 and Re^j = 200
128
6.10 Close-up view of the spanwise vorticity uj in the vicinity of the cylinder
surface for Reip = 20 and Re^) = 200
129
6.11 Comparison of tt-velocity profiles at several downstream locations com­
puted using an immersed wall
134
6.12 Comparison of f-velocity profiles at several downstream locations com­
puted using an immersed wall
135
6.13 Comparison of a;-vorticity profiles at several downstream locations com­
puted using an immersed wall
136
6.14 Comparison of oj-vorticity profiles at several downstream locations com­
puted using the old and new IBM
137
6.15 Com.parison of disturbance t/,-velocity profiles at several downstream lo­
cations computed using an immersed wall
138
6.16 Comparison of disturbance f-velocity profiles at several downstream lo­
cations computed using an immersed wah
139
12
List of Figures—Continued
6.17 Comparison of disturbance oi-vorticity profiles at several downstream lo­
cations computed using an immersed wall
140
6.18 Comparison of disturbance u-velocity phase profiles at several downstream
locations computed using an immersed wall
141
6.19 Comparison of the growth rates of a 2-D T-S wave computed using an
immersed wall
142
6.20 RANS results for the zero pressure gradient, turbulent flat-plate boundary
layer using the model of Menter (1994): U versus y in the inner and outer
regions
145
6.21 RANS results for the zero pressure gradient, turbulent flat-plate boundary
layer:
versus
for Re^ = 5020. Present IBM results using the model
of Menter (1994) compared with a body fitted code using the k-ui model. 146
6.22 RANS results for the zero pressure gradient, turbulent fiat-plate boundary
layer. Skin friction c/ versus Re^
147
6.23 RANS results for the zero pressure gradient, turbulent flat-plate boundary
layer using the model of Menter (1994): k'^ versus y'^, and
6.24 RANS results for the turbulent wall jet :
versus y^. 148
versus y^
149
6.25 RANS results for the turbulent boundary layer : [/+ versus y^ for Ay
with limiter
149
6.26 Streamlines for turbulent fiow over a step. Shown are, from top to bottom,
streamlines, contours of vorticity, and ?7-velocity profiles
150
6.27 Turbulent kinetic energy k for turbulent flow over a step
151
6.28 Comparison of
velocity profiles at several a:-locations for turbulent flow
over a step: IBM method with SST model (—), and k-w model (— --),
compared with compressible code results (- • -)
6.29 Flow over a curved wall. Re// = 2500
151
153
13
List of Figures—Continued
6.30 Flow over a 6:1 ellipse at zero angle of attack, Rec=2400
154
6.31 Startup flow around a 6:1 ellipse (Rec = 300) at a = 30° angle of attack
that has been impulsively set into motion
155
6.32 Startup flow around a 6:1 ellipse (Rec = 300) at a = 30° angle of attack
that has been impulsively set into motion
156
6.33 Vortex street formed by flow around a 6:1 ellipse (Rec = 300) at a = 30°
angle of attack
156
6.34 Flow over a bluff body, Re//=5000
157
6.35 Computational grid for results shown in figure 6.34
158
7.1
A point requiring jump-correction that would cause problems in the present
IBM
7.2
Stretched and unstretched body-fitted grids for the computation of the
flow around a circular cylinder
7.3
162
163
Stretched and unstretched Cartesian grids for the computation of the flow
around a circular cylinder
163
14
List of Tables
1.1
Error in the 6-point unstructured finite-difference computation (9//9x. .
25
2.1
Error in the numerical computation of wall vorticity
35
3.1
Error in the numerical solution of a model ODE with singular source term
on various grids. Comparison of a Gaussian representation of 5'{x) with
that of Walden (1999)
47
5.1
5.2
5.3
5.4
5.5
5.6
5.7
6.1
6.2
6.3
A.l
A.2
A.3
A.4
A.5
A.6
A.7
A.8
Error in the numerical solution of the convection-diffusion equation (2.7)
on various grids
Error in the numerical solution of the convection-diffusion equation (2.7)
for various time-steps
Computation times for test problem of section 5.2.3
Error in the numerical solution of the Poisson equation (5.32) on various
grids
Error in the numerical solution of equation (5.61) on various grids. . . .
Error in the numerical solution of equation (5.61) for various time stepsizes At
Convergence study of the error in the divergence and curl of the computed
velocity field
Grid refinement study for fiow past a circular cylinder with Re^ = 200
and A = 0.056: Ciu and
for three different grids
Steady flow past a circular cylinder: length L of standing eddy behind
cylinder, locations a and b of the vortex centers, separation angle 6, and
drag coefficient Co for Re^ = 20 and Reo =40
Unsteady flow past a circular cylinder: period r, Strouhal number St,
drag coefficient Cd, and lift coefficient C^. Comparison of present results
with results published in the literature
Parameters for the spatial convergence study of section 5.1.4
Parameters for the temporal convergence study of section 5.1.4
Parameters for the spatial convergence study of section 5.2.3
Parameters for 1-D moving IB computations of section 5.4
Parameters for the circular cylinder computations of section 6.1
Parameters for T-S wave computation of section 6.2 using the code nst2d
of Meitz & Fasel (2000)
Parameters for T-S wave computation of section 6.2 using the immersed
boundary code nsib
Parameters for fiat plate RANS calculations of section 6.3
92
94
102
104
110
110
112
117
122
123
164
165
166
166
167
168
169
169
15
Abstract
A high-order immersed boundary method (IBM) for the computation of unsteady,
incompressible fluid flows on two-dimensional, complex domains is proposed, ana­
lyzed, developed and validated. In the IBM, the equations of interest are discretized
on a fixed Cartesian grid. As a result, domain boundaries do not always conform
to the (rectangular) computational domain boundaries. This gives rise to 'immersed
boundaries', i.e., boundaries immersed inside the computational domain. A new IBM
is proposed to remedy problems in an older existing IBM that had originally been
selected for use in numerical flow control investigations.
In particular, the older
method suffered from considerably reduced accuracy near the immersed boundary
surface where sharp jumps in the solution, i.e., jump discontinuities in the function
and/or its derivatives, were smeared out over several grid points. To avoid this be­
havior, a sharp interface method, originally developed by LeVeque & Li (1994) and
Wiegmann & Bube (2000) in the context of elliptic PDEs, is introduced where the
numerical scheme takes such discontinuities into consideration in its design. By com­
paring computed solutions to jump-singular PDEs having known analytical solutions,
the new IBM is shown to maintain the formal fourth-order accuracy, in both time
and space, of the underlying finite-difference scheme. Further validation of the new
IBM code was accomplished through its application to several two-dimensional flows,
including flow past a circular cylinder, and T-S waves in a flat plate boundary layer.
Comparison of results from the new IBM with results available in the literature found
good agreement in all cases.
16
Nomenclature
Latin Characters
D
diameter
h
grid step-size
i, j
grid indices for x , y
k
turbulence kinetic energy
L
length scale, [m]
n
curvilinear coordinate normal to a body
N
number of grid points or intervals
n
unit vector normal to a body
p
pressure
Re
Reynolds number: UooL/u
Re(7
Reynolds number based on chord C: UooC/v
Ret
Reynolds number based on eddy-viscosity Vt- UooL/i't
R^j
Reynolds number based on displacement thickness
s
curvilinear coordinate along a body
t
time
At
time step
u, V
velocity in x , y direction
u
velocity (vector)
Uoo
free-stream, or reference velocity, [m/s]
Ax, Ay
grid size m . x , y direction
X
coordinate (vector)
X, y
Cartesian coordinates
Greek Characters
a
quantity evaluated at the immersed boundary, or at a singularity
5
Dirac delta function; Kronecker delta; boundary layer thickness
£
error
Q
momentum thickness
17
A
ratio of cylinder diameter D to domain height H] wavelength
u
kinematic viscosity, [m^/s]
Ut
turbulent (eddy) viscosity
p
density, [kg/m^]
r
unit vector tangent to a body; stress tensor
ip
stream function
LU
vorticity (scalar); specific dissipation rate; circular frequency
(jH
vorticity (vector)
Modifiers
a, n
analytical, numerical
B
buffer domain
c
center
o, i
outer, inner
t
derivative with respect to time t; turbulence quantity
X, y
component or derivative in x,y direction
+, —
outside, inside immersed boundary; right, left limits
*
dimensional quantity
2
2-norm, ||a;^|l2 =
CO
free-stream; oo-norm, ||xi||oo = max|xi|
0
bold character </) : vector, tensor, or matrix
' fluctuating
component; derivative; modified quantity
'n}^ derivative of (j), d'^(p/dx'^
[(j)]a
jump in (j) at
i.e., [<1)]^ =
Frequent Abbreviations
AMR
Adaptive Mesh Refinement
CFD
Computational Fluid Dynamics
CFL
Courant-Friedrichs-Lewy
CS
Control Surface
CV
Control Volume
(pix) - lim^_^- 0(x)
18
DNS
Direct Numerical Simulation
F.D.
Finite-Difference
IBM
Immersed Boundary Method
IB
Immersed Boundary
ILLU
Incomplete Line Lower Upper
LFC
Laminar Flow Control
LST
Linear Stability Theory
N-S
Navier-Stokes
ODE
Ordinary Differential Equation
PDE
Partial Differential Equation
RANS
Reynolds-Averaged Navier-Stokes
R-K
Runge-Kutta
SST
Shear-Stress Transport model of Menter (1994)
T-S
Tollmien-Schlichting
(1, 2, 3)-D
one, two, three dimensional
19
1.
Introduction
There are essentially three approaches available to the engineer for investigating prob­
lems in fluid mechanics; experimental, analytical, and numerical. Increasingly, the
numerical approach is being selected as the design tool of choice. As computing power
continues to grow, so does the demand for accurate flow simulations that, to the ex­
tent necessary, contain all of the complexity of their real-life counterparts. Of the
many factors contributing to this complexity (e.g. turbulence, chemical reactions),
the one with which the present work is concerned is geometry, that is, simulating
flows through and/or past surfaces that have complex, and possibly time-dependent,
geometrical shape. In particular, the development of the numerical method discussed
in this dissertation was originally motivated by the desire to simulate fluid flows in
flow control applications that utilize actuators of relatively complex geometry. These
actuators, in contrast to heating element actuators, for example, manipulate the flow
by introducing disturbances through motion of the actuator. An example of such an
actuator would be a wall-mounted membrane that is deflected into the flow (such as
that used in the studies of Hofmann & Herbert, 1997; Carlson & Lumley, 1996).
The goal of flow control is to modify fluid flow in such a way that the perfor­
mance of an engineering design is improved. For example, significant research has
been conducted on the use of periodic injection of momentum into a flow to delay
fluid separation from airfoil lifting surfaces (Wygnanski, 1997). Another example
can be found in laminar flow control (LFC) applications that attempt to maintain
the inherently low skin-friction associated with laminar flow by delaying transition
to turbulence (Breuer et ai, 1989; Nelson et al., 1997). A contributing factor in the
success of any flow control scheme is an understanding of the phenomena that are
used to effect the control, namely, the transient behavior of the flow over a moving
actuator.
20
To date, most numerical investigations of flow control schemes have used wall
blowing and suction as the disturbance source. Such blowing and suction actuators are
easily realized in numerical simulations through the specification of non-zero boundary
conditions on fluid velocities at surface boundaries. The present investigation seeks
to extend the range of flow control schemes that can be studied, from those using
simple blowing-and-suction type actuators, to those using wall-mounted actuators of
varying shape and complexity. The work presented in this dissertation is a first step
towards achieving the goal of an efficient numerical method that combines the ability
to handle a range of complex, time-dependent geometries together with the high wave
transport accuracy required to obtain reliable results in flow control applications. Of
course, the method is not limited to such applications, and can be used wherever
accurate flow simulations on time-dependent, multiply-connected, irregularly-shaped
domains are required.
A number of standard numerical methods capable of handling geometries of vary­
ing complexity are already available. A short description of these methods is presented
next, followed by some advantages and disadvantages of each. This review is given to
motivate the introduction of an alternative method investigated in the present work,
the immersed boundary method.
1.1 Standard Numerical Methods for Complex Geometries
Numerical methods obtain approximate solutions to partial differential equations us­
ing information at a flnite number of discrete locations inside a given domain. The
arrangement of these discrete locations, called grid points or nodes, deflnes the nu­
merical grid, of which there are essentially two diff'erent types: structured and un­
structured. The type of grid selected for use mostly depends on the application; the
generation of the grid, often a tedious and time-consuming task, is itself a topic of
many journal papers, conferences, and books.
21
1.1.1 Structured Grid Methods
Structured grids consist of families of coordinate lines, with the property that lines
of a given family do not intersect, and a line from one family intersects a line from
another family only once. These intersections define the grid points, which, because
they map one-to-one onto a Cartesian grid, can be uniquely defined by a set of indices,
e.g. [iii) in two dimensions. In two dimensions, points in the Cartesian plane {x,y)
are described by two new coordinates,
X=
via a transformation
(1.1)
y = y{^,r]).
Each coordinate line in the domain is a line of either constant ^ (the ^ family) or
constant
rj
(the
rj
family). An example of a structured grid that might be used to
compute the flow over a membrane-type actuator is shown in figure 1.1: horizontal
lines represent lines of constant ri, vertical lines, lines of constant
If finite-difference
methods are used, the original PDE, written in physical space with x and y as in­
dependent variables, must be transformed so that ^ and
rj
appear as independent
variables (barring the discussion given below in section 1.1.2). The grid in physical
space can also be viewed as a mesh of finite volume cells upon which finite volume
or finite element calculations can be directly made.
The advantage of using structured grids is that specific grid lines, ^ = constant
or rj =
constant, define the boundaries of the domain in which a solution is sought,
thus simplifying the implementation of boundary conditions. Additionally, because
the grids generated by such transformations are boundary-fitted, the computational
grid conforms naturally to the domain boundary. In CFD applications, for example,
this can be used to cluster points advantageously near a body surface in order to
capture thin boundary layers. However, it is often difficult to control the distribution
of grid points: concentrating grid points at one location in the domain produces an
unnecessary concentration somewhere else in the domain, thus increasing the cost of
the computation. Another advantage of structured grid methods is their connectivity:
22
X
Figure 1.1 Example of a 2-D, non-orthogonal, structured grid for the computation of
the flow past a membrane-type flow control actuator.
the indices of any grid point (i,j) differ from those of its neighbors by at most
(±1, ±1). This simple connectivity leads to algebraic systems of equations with very
regular structure, which in turn leads to efficient solution schemes and computer
codes.
A disadvantage of structured grid methods is that they are usually most appropri­
ate for geometrically simple solution domains. Further, in curvilinear finite-difference
schemes, calculation of the Jacobian and other geometric information used in the
transformation introduces additional errors into the numerical solution. The trans­
formation itself usually introduces many additional terms, particularly when the grid
is non-orthogonal, and mixed derivatives appear where they were absent in the orig­
inal Cartesian equation. Even for simple coordinate systems (e.g. cylindrical coordi­
nates), the coordinate transformation can significantly increase the complexity of the
equations in comparison with their Cartesian counterparts.
1.1.2
Unstructured Grid Methods
Unstructured grid methods make use of grids whose points can, in principal, be
placed anywhere in the domain without regard for neighboring points.
There is
no hmitation on the shape that the finite control volumes created by the grid are
23
<^2'y2>
(X,,Y,)
Figure 1.2 One possible distribution of points {xi,yi) that could be used in the un­
structured finite-difference stencil given by equation (1.2).
allowed to assume, nor is there a limitation on the number of neighboring grid points.
Unstructured grids are therefore well suited to handling complex, arbitrarily shaped
geometries. However, the flexibihty so obtained is made up for by the complexity
of node connectivity, which, no longer implicit in node indices, must be explicitly
specified. The systems of algebraic equations that result are no longer regular, and
the corresponding solvers therefore slower and less efficient.
Unstructured grids are most often used with finite volume and finite element
discretizations. A method that, to the present author's knowledge, has not been
presented in published literature is the use of finite-differences on unstructured grids.
With reference to figure 1.2, for example, one would like to compute a finite-difference
stencil for the partial derivative, say df/dx, of a function of two variables f{x,y) at
point {xo,yo) as a linear combination of the surrounding points
6
dx [xo,yo)
(1.2)
i=l
As in the one-dimensional case, the coefficients
Q can be determined through a Taylor
expansion (here, a two-dimensional Taylor expansion) of the /(xj, y^), resulting in the
24
2.5
,5
0.5
0
-0.:
2.5
0.5
X
Figure 1.3 Points {xi,yi) in the unstructured finite-difference stencil used for the
numerical study given in table 1.1.
system of equations for the
1
Axi
Ai/i
Ax\
AxiA^i
Ayl
1
AX 2
Ay2
Ax%
Ax2Ay2
Ayl
...
1
"
...
Axq
...
A?/6
...
Axg
. . . AxeAye
...
Ayl
.
Cl
C2
C3
C4
C5
^6 .
'0"
1
0
0
0
0
where Axi = Xi — Xq, and Ay = yi — yo- If Ax, Ay — 0{h), the neglected terms are
0{h?), and the coefficients Cj will be 0{l/h). The resulting stencil in equation (1.2)
with coefficients determined from equation (1.3) will then be
As a numerical
example, consider computing d f / d x of the function f { x , y ) = exp(x^
y ^ ) at the
point {x,y) = (1,1). The points in the stencil lie at the corners of a hexagon with
radius h, as shown in figure 1.3. Results shown in table 1.1 for three different grid
sizes h indicate that the finite-difference stencil is indeed second-order accurate.
The use of unstructured finite-differences for numerically solving simple elliptic
and parabolic PDEs met with limited success. Among the many issues remaining
to be resolved is the question of the invertibility of the matrix used to determine
the finite-difference coefficients (e.g. equation (1.3)), and the stability of an explicit
numerical scheme used for marching parabolic equations in time. With regard to
25
h
e
0.01
0.005
0.0025
2.955 X 10"^^
7.389 X 10"^
1.847 X IQ-'^
Table 1.1 Error e = f x ,a- f x ,n in the 6-point unstructured finite-difference computation
d f / d x of f{x,y) = exp(a;^ + y'^) at the point {x,y) = (1,1). The error is e = Ah"-,
with A=29.57, and n=2.0.
the first issue, it is clear that the finite-difference coefficients in the one-dimensional
case are unique. In two and higher dimensions, this will not necessarily be the case.
Indeed, in the numerical example presented above in table 1.1, the coefficient matrix
turned out to be singular. Here, the pseudoinverse solution to the system produced
coefficients that were symmetrically weighted.
1.2
The Immersed Boundary Method
Although numerically efficient, structured grid methods are limited to relatively sim­
ple geometries. On the other hand, unstructured methods can handle very complex
geometries, but are considerably less efficient. The immersed boundary method (IBM)
has been developed as an alternative meant to bridge this gap. The IBM is a nontraditional approach to numerically solving boundary-value and initial/boundaryvalue problems on domains with complex geometric boundaries. It uses an underlying
structured Cartesian grid method, thus obviating the need for grid generation. In
this section, a basic overview of the method is given.
In the immersed boundary method, the equations of interest are discretized on
a fixed Cartesian grid. As a result, the domain boundaries do not always conform
to the (rectangular) computational domain boundaries. This gives rise to 'immersed
boundaries', i.e., boundaries immersed inside the computational domain. With regard
to figure 1.4, for example, one would typically like to solve a PDE defined on the open
26
1
" il
0.8
'
/
V
0.6
!
N
\
\
n
/
/
0.4
,
\
\
/
0.2
\
s
0
0
A
0.2
0.4
0.6
0.8
1
X
Figure 1.4 An example of an irregular boundary dQi immersed inside a Cartesian
computational grid.
region ^2+ with boundary conditions on dClo, the outer boundary which conforms to
the computational boundary, and d^i, the immersed boundary which does not. The
solution in the region 17" may or may not be of interest. For example, if one would
like to simulate the interaction of a very thin, elastic membrane submerged in, and
filled with, a fluid, then the governing equations both inside (in 1]") and outside (in
Q"*") of the membrane would have to be solved. Upon approaching the limit of an
infinitely rigid shell, however, only the solution outside of the membrane would likely
be of interest. In either case, the immersed boundary dQi represents a singularity if
one considers that a particular set of governing partial differential equations apply
throughout the entire domain enclosed by
SO
q
(as in the analytical study of Sirovich,
1968, for example); field variables and/or their derivatives will likely be discontinuous
across the immersed boundary.
The immersed boundary method determines a solution at every grid-point within
the domain enclosed by dVlo, both inside and outside the immersed boundary. The
27
equations to be numerically solved are discretized on a rectangular Cartesian grid
which is allowed to pass through dQ.i, as shown in figure 1.4. Several methods for
handling the singularity at the immersed boundary have been proposed in the past.
One of the original methods is discussed in section 3.1; more recent methods, in
particular, the one used in the present work, are introduced in chapter 4.
1.3
Objective and Scope of Present Research
As mentioned earlier, the research presented in this dissertation was originally moti­
vated by the desire to accurately simulate flow control strategies that use actuators
of complex geometry. Past and ongoing flow control investigations of transitional and
turbulent flows by Fasel and coworkers (amongst whom the present author is counted)
using the computational approach have required high-order numerical methods^ lim­
iting investigations to flow geometries with surfaces that conform to relatively simple
coordinate systems, e.g. flow over a flat plate or backward-facing step (Cartesian
coordinates), flow between concentric cylinders or past the tail-end of a cylindrical
bluff body (cylindrical coordinates). For fundamental studies, explicitly limiting in­
vestigations to such geometries is often desired and well justified. Wernz (2001), for
example, motivates his investigation of forced transitional and turbulent wall jets
with several engineering applications including separation control over airfoils and
turbine-blade film cooling, where the geometry could be quite difficult to simulate;
the actual geometry investigated by Wernz (2001) is a fiat plate. The next step in
the progression towards an end application of his research would, for example, be the
addition of curvature, a more challenging, yet still manageable computational task
(at high order). One could imagine the end of the progression, which began with tlie
fiat plate, as a simulation, for example, of an entire 3-D airfoil with fiaps, and time^ High-order here means greater than or equal to fourth order (> C(/i^)) - the resolution required
to correctly model the physics of fluids in flow control and/or transition studies using lower order
methods is, in most cases, likely to be prohibitively large.
28
dependent flow control actuator for forcing. Unfortunately, geometrical complexity
at high numerical order and computational efficiency become requirements that are
difficult to satisfy simultaneously. The objective of the present research, then, is to
bridge this gap by
i. Developing a temporally and spatially fourth order accurate code for the simula­
tion of unsteady fluid flow over and/or through surfaces of complex geometrical
shape.
ii. Taking advantage of the computational efficiency of structured, Cartesian com­
putational grids.
To this end, a high-order immersed boundary method for the computation of un­
steady, incompressible fluid flows on two-dimensional, complex domains is proposed,
analyzed, developed and validated. The equations that govern incompressible fluid
motion, and to which the IBM will be applied, are discussed in the next chapter of
this dissertation. The idea of the IBM is not a new one, and in chapter 3, older
immersed boundary methods are presented, in particular, the one originally selected
for flow control actuator investigations by the present author. This particular IBM
is analyzed to reveal its deficiencies, which it shares in common with similar ap­
proaches discussed in this chapter, and to motivate the development of a new IBM
discussed in chapter 4. In chapter 5, the actual numerical method used in the present
investigation to numerically solve parabolic and elliptic equations is described. Here,
numerical experiments using equations with known analytical solutions are carried
out to show that the formal accuracy of the new IBM is actually achieved. Further
validation of the new IBM is carried out in chapter 6 in its application to physically
relevant incompressible fluid flows, including uniform flow past a circular cylinder and
Tollmien-Schlichting waves in a boundary layer. Finally, conclusions and recommen­
dations for future work are discussed in chapter 7.
29
2.
Governing Equations
2.1 The Incompressible Navier-Stokes Equations
An excellent mathematical model of incompressible viscous fluid flow is found in the
(incompressible) Navier-Stokes equations
(3ll
.
-—f
1 r—rO
^ +(u.V)u = -Vp+pjVu
V-u = 0
(2.1)
(2.2)
where u(x, t ) i s t h e f l u i d v e l o c i t y , p (x, t) is the pressure, and Re = UooL/v is the
Reynolds number based on velocity scale Uoo, length scale L and kinematic viscosity
y. As shown in section 3.2, an optional body-force term F(x, i) may be added to
the right-hand side of equation (2.1). Methods that obtain numerical solutions to
the incompressible Navier-Stokes equations by working directly with equation (2.1),
while enforcing equation (2.2), are said to be using the primitive variable formulation
of these equations.
2.2
The Vorticity-Velocity Formulation
The difficulty in finding boundary conditions for pressure in a primitive variable
formulation of the Navier-Stokes equations led researchers to seek alternative formu­
lations that would allow them to get rid of the pressure. The vorticity-velocity for­
mulation (Fasel, 1976; Stella & Guj, 1989; Daube, 1992; Quartapelle, 1993; Wu
et ai,
1995; Giannattasio & Napolitano, 1996; Clercx, 1997; Huang & Li, 1997; Mdhring
& Mohring, 1998; Meitz Sz Fasel, 2000) is one such example in which pressure is
eliminated by taking the curl of equation (2.1), leaving velocity u, and the vorticity,
a; = V
X
u, as the dependent variables.
The vorticity-velocity formulation was originally selected for the investigation of
the new IBM developed in this dissertation, in particular because it is valid in both
30
two and three dimensions.
However, a successful implementation of the method,
even in the absence of immersed boundaries, was never achieved^ It is believed that
the problems stemmed from replacing the original Cauchy-Riemann type vorticityvelocity formulation
V - u = 0, V x u = a;,
(2.3)
with one that makes use of a Poisson equation for the velocity u,
V2u=-Vxa;.
(2.4)
Because numerical techniques for solving equation (2.4) are much more developed
than those for solving equation (2.3), almost all vorticity-velocity formulations make
use of the latter. However, solutions to the derived equation (2.4) will not necessar­
ily be solutions to the more primitive equations (2.3) if proper boundary conditions
are not explicitly enforced. Indeed, improper boundary conditions yield a velocity
field that is not divergence-free, and a "vorticity" field that is not the curl of the
velocity field. A review of published hterature, some of which has been cited above,
indicates that the issue of proper numerical boundary conditions for the vorticityvelocity formulation is still unresolved, particularly for finite-difference approaches
that use nonstaggered grids. Daube (1992), for example, uses the vorticity-velocity
formulation together with a finite-difference method to compute the 2-D incompress­
ible flow inside a lid-driven cavity. He applies an influence matrix technique to enforce
what, from an analytical standpoint, appear to be proper boundary conditions in the
sense that solutions to equation (2.4) will also be solutions to equation (2.3). How­
ever, he found that "[every] attempt to use a nonstaggered grid ... led to either the
blowing up of the computations or to nonphysical flows ..." Other investigators, in­
cluding Fasel (1976), Fasel
et al.
(1990), and Meitz & Fasel (2000), on the other hand,
have reported on nonstaggered-grid implementations that can be used to successfully
^The zero-pressure gradient, fiat-plate boundary layer with T-S waves (see section 6.2), and the
unsteady lid-driven cavity (a standard benchmark problem, see Guj & Stella, 1988; Pozrikidis, 1997)
were investigated using a body-fitted version of the code.
31
reproduce the physics of (laminar-turbulent) transitional flows (the issues mentioned
above, in particular, whether or not equation (2.3) will be satisfied in the limit of
vanishing grid-size, has not been discussed by these investigators). This latter work,
however, is formulated in such a way that it is not directly applicable to the more
general geometries of interest in the present IBM investigation.
2.3
The Stream Function-Vorticity Formulation
The unresolved issues in the vorticity-velocity formulation mentioned above led to
its being abandoned in favor of the stream function-vorticity formulation (a detailed
description of the formulation can be found in Roache, 1972; Quartapelle, 1993).
This formulation is only valid in two dimensions, although it can be generalized to
3-D by making use of a vector potential (resulting in a formulation with which it is
relatively inconvenient to work). In the stream function-vorticity formulation, the
variables u and p are replaced by two scalar variables, the vorticity uj and the stream
function -0, which will be defined below. The advantage of this formulation is that
the incompressibility constraint, i.e., equation (2.2), is automatically enforced.
The vorticity
LU
is defined here as^
_ du
dy
dv
dx'
^
and the stream function ijj such that
dip
u= — , v=
ay
di>
.
ax
, .
2.6)
It is clear that V • u = 0 with this definition. An additional consequence is that
contours of •0=constant define instantaneous streamlines in the flow.
By taking the curl of the momentum equation (2.1), a transport equation for the
vorticity
UJ
is obtained:
duj
dt
d(uu})
dx
d{vuj)
dy
^the standard definition \s to = d v / d x — d u / d y
1 /d ' ^ o j
Re
d'^oj \
dy'^ )
32
From the definitions of vorticity and the stream function, the following Poisson equa­
tion for the stream function is obtained:
=
(2-8)
Equations (2.7) and (2.8), together with boundary and initial conditions, are the two
main equations to which the new IBM discussed in the present work are to be applied.
2.3.1
Boundary Conditions for the Stream Function
Let n denote the outward-pointing unit vector normal to the body S ,
T
the tangen­
tial unit vector, and s the curvilinear coordinate along the body (counter-clockwise
orientation). Given the velocity of the boundary S as
Us {s,t),
the corresponding
boundary conditions on ip are
dtp
ds s
dip
dn 5
{s,t),
=
n-us
=
-T - U s { s , t ) .
(2.9)
(2.10)
Equation (2.9) can be integrated over S to yield an equivalent Dirichlet boundary
condition
so that one has instead
tp\g = a{s,t),
dip
^ -T-us{s,t).
dn
(2.11)
(2.12)
There are more boundary conditions on i p than required by equation (2.8), so bound­
ary conditions on
LJ
must be specified such that both constraints on i p can be satisfied.
In the present work, only equation (2.11) is explicitly used in the finite-difference
scheme; the second constraint, equation (2.12), is enforced by proper selection of the
wall vorticity.
In multiply connected domains, the constant of integration for equation (2.11) is
determined by enforcing the condition that the pressure is single valued,
Vp • d s — 0 ,
(2-13)
33
where the integral is taken around a closed loop enclosing the body. We see that by
Stokes' theorem,
j\/p-ds = j ) V x V p d A = 0,
(2.14)
this condition is automatically satisfied on simply-connected domains. To obtain Vp
for equation (2.13), the momentum equation in the form
-Vp = ^ + u • Vu ot
Re
(2.15)
is used.
One additional complication that arises in multiply connected domains is when
the velocity normal to the immersed boundary surface is non-zero. Taking ?/; as a
function of arc length s on the immersed boundary,
tp{s) = ^p{Sl) + [ Vn ds,
J Si
where
is the velocity normal to the body surface. Clearly, if
(2.16)
7^ 0 over the entire
surface, or is not specified such that the net volume flux through the body surface
is zero, then integrating equation (2.16) around the body all the way back to si wiU
yield a jump in •0(s)- To handle this situation, a branch cut is introduced into the
solution in a manner similar to the way it is done in the theory of complex variables.
Across the branch cut, the value of 'il){x,y,t) jumps by an amount determined by
integrating equation (2.16) full-circle. All derivatives of ip{x,y,t), however, will be
continuous across the branch cut.
2.3.2
Boundary Conditions for the Vorticity
In the present work, the wall vorticity uj is computed from its definition, equa­
tion (2.5).
Where allowed by the intersection, explicit finite-differences are used
to compute Uy and v^. For example, referring to figure 2.1, one can see that at the
intersection denoted by the symbol •, one can easily compute v^, but the computation
34
dQ.
I
\
\'
Figure 2.1 Stencils used for computing wall vorticity at an immersed boundary: first
d e r i v a t i v e i n y ( o ) , an d f i r s t d e r i v a t i v e i n x , ( • ) .
of Uy would require a 2-D finite-difTerence stencil. For the intersection denoted by o,
the reverse is true.
The following strategy was developed to avoid the use of 2-D stencils in the cases
mentioned above. We consider the case of computing
at the intersection denoted
by the symbol o. If the tangent vector makes an angle cp with the x axis such that
7''/4 < \4>\ < 37r/4, then
at o-intersections are computed by interpolating nearest
neighbor a-intersection values along the arc defining the immersed boundary^. For
the angles (j) not satisfying this inequality, the known derivative Vg computed along
the arc length s, and Vy computed using an explicit finite-difference stencil are used
to obtain
Xs
(2A7)
where the tangent vector t = { x s , y s ) - A similar strategy is used to obtain Uy at
•-intersections.
To numerically validate this strategy for computing wall vorticity, the values of u
all angles are given assuming a ±7r branch-cut in the arg function
35
N
Iklloo
Iklh
129
257
513
1.3946 X 10"^
8.6255 x 10^^
5.4499 x 10~®
3.9883 x 10"®
2.0887 x 10"^
1.6603 x 10"®
Table 2.1 Error {e = LOa — uJn) in the numerical computation of wall vorticity on grids
of size NxN {or step-size h= 1/{N — 1)). The error is e = Ah^, with A=3.7251 x 10^,
n=4.0 for ||£l|oo, and A=7.9925 x 10^, n=4.0 for 1|£||2-
and V are set to analytical functions
u { x , y ) = sin(a;x) sin(a;y),
{2AS)
v{x, y) = cos{uJx) cos{ujy).
(2.19)
on an equidistant grid x , y E [0,1] with a circular immersed boundary of radius
r = 0.1 centered at {x,y) = (0.5,0.5). Fourth-order finite-difference stencils are used,
where allowed by the grid/immersed boundary intersection, to compute Uy and
interpolation along the immersed boundary is also done to fourth order. As seen in
table 2.1, the resulting wall vorticities are computed to fourth-order accuracy.
One can imagine several cases where the simple strategy described above may
produce less accurate results, or even fail completely. This may happen, for example,
in locations where there is a rapid change in surface geometry and there are an
insufficient number of points available for interpolation. These geometries have been
explicitly avoided in the present work. It was felt that the development of a more
general strategy, while more difficult to implement than the simple method described
above, would still be feasible.
36
2.3.3
Outflow Treatment
Depending on the problem investigated, any boundary, z = 1, z = n^, J = 1, or
j = riy, can be an outflow boundary'^. Upstream of outflow boundaries, the bufl^er
domain technique of Meitz & Fasel (2000) is used to ramp the vorticity down to
some prescribed value Ur- For example, if the i = Ux domain boundary is an outflow
boundary, then the vorticity from i = ii to i = i2 will be modified as
w = c(Ou; + (1 -
(2.20)
where the function c(^) is
c(e) = e-^'/l°(l-^^°)^
(2.21)
and
e =
(2.22)
^2 - H
Prom i = i 2 t o i = U x ,
U — Ur-
(2.23)
This ramping of the vorticity is performed after every time integration step, and at
every stage of the Runge-Kutta scheme. The ramp vorticity uj^ is problem dependent,
and requires some creativity on the part of the numerical investigator. The outflow is
always placed far enough away that any upstream influence caused by this artificial
outflow condition is minimal.
2.3.4
Initial Conditions
Two main types of flows are considered in the present investigation: wall-bounded
flows, where the wall runs from the inflow to the outflow boundary, and flows around
bluff bodies situated in a uniform free-stream. In general, some inventiveness is again
required to find a good initial condition for a particular calculation. For example,
"^Grid points of the rectangular, Cartesian, computational box are denoted by
where i = 1... rij,; for y, j and Uy are used
i in the .x-direction,
37
for flows over a flat plate, the inflow boundary layer profile can be copied parallel
to the plate and used as an initial condition. The resulting flow, then, has at every
X
location a boundary layer profile that is the same as the inflow boundary layer
profile. For flows around bluff bodies, an inviscid flow solution can be specified as
an initial condition. However, special care is needed here, as suddenly imposing noslip boundaries on the inviscid solution may cause the numerical solution to blow-up.
To circumvent this problem, first-order differences are used to compute wall vorticity
during the initial start-up phase. Shortly thereafter, however, higher-order differences
are switched on for the remainder of the calculation.
2.4
The Reynolds-Averaged Navier-Stokes Equations
As most fiuid fiows of engineering interest are turbulent, a Reynolds-averaged NavierStokes (RANS) turbulence model has been implemented in the immersed boundary
code. Some form of turbulence model is required whenever one wishes to carryout numerical simulations of a turbulent flow, but does not possess the computer
power necessary to capture all relevant length and time scales in the flow. Usually,
obtaining a complete time history of the flow over all spatial coordinates is not even of
interest, and instead only average quantities, such as the mean skin friction or mean
heat transfer rate, are sought. A turbulence model is a mathematical model used to
approximate the physics of turbulence, requiring the minimum amount of temporal
and spatial resolution to obtain the necessary information about a given flow.
The three-dimensional, time dependent nature of turbulence means that a com­
plete description of a turbulent flow would require an enormous amount of informa­
tion. Such information can be obtained through direct numerical simulation (DNS),
where all relevant length and time scales in the flow are captured, and no modelling
assumptions, outside of those inherent in the full Navier-Stokes equations, are used.
However, present-day computing power is still far from sufficient, and DNS computa­
38
tions are limited to low Reynolds numbers (relative to those of engineering interest).
In comparison to DNS, RANS simulations are found at the low end of the temporalspatial resolution spectrum. The RANS approach decomposes the instantaneous ve­
locity, Ui(x, t) (the ith component of u (x, t)), as the sum of a mean, J7i(x, t), and a
fluctuating part,
m-(x, t),
Mj(x,t) = ?7i(x,t)-f-w-(x,t),
(2.24)
where the temporal average
1
?7,(x,t) ='a,(x,t) = - y
u,{y.,t)dt
(2.25)
is used to remove the turbulent fluctuations. For flows with a time-dependent mean,
the averaging interval T must be much greater than the time scale Ti associated with
the fluctuating velocity component, but much less than T2, that associated with the
time variation of the mean, i.e., Ti
T -C T2 (Wilcox, 1998). Applying the averaging
operator to the incompressible Navier-Stokes equations given in section 2.1 yields the
RANS equations
dUi
dP
dx,
1
d
fdUA
d
,
0.
(2.27)
The term —u[u'^ is known as the Reynolds-stress tensor r^. It is the goal of turbulence
modelling to determine an expression for
such that the system of RANS equations
is closed, and hence can be used to solve for the mean-flow quantities [/j and P. For
the present study, closure is obtained using a two-equation eddy-viscosity turbulence
model of Menter (1994): the Shear-Stress Transport (SST) model.
In the SST model, Tjj is determined as
1
fdU,
dUi \
2 , ,
39
where Ret (Rej = UooL/i/*, ul is the dimensional eddy-viscosity) is a function of A:,
the turbulence kinetic energy, and uo, the so-called specific dissipation rate'^; for the
k-LO model, the relation is Vt = k/u. The quantities k and to are determined by solving
two additional transport equations
dk ^
ot
dk
_
OXJ
dU,
^ <9
O'k \ d k
Ret ) d X j
Y 1
\Re
(2.29)
(^LO\ duJ
d
\l
d x ^ .1
OXJ
1 dk du
2(1 — Fi)a^^2 — 7^ 7^ >
oj d x j d x j
Rcj / d x j J
(2.30)
where F i = F i { x , y , t ) is a blending function such that constants (cjfc,
/3, /?*, 7) are
determined as
0 = Fi(/)i(1 - FI)02,
where (pi represents any of
(2.31)
P i , P i , 1 i ) from the k - u ) turbulence model, and,
similarly, ^2 represents any of {o'k2,'^uj2, P2, ^2^12) from the standard k - e model; the
numerical values of both sets are given in Menter (1994).
At a soUd surface, the boundary condition for A; is A: = 0. Following Menter (1994),
the boundary condition for u is taken as
ReA(Ayi)''
where A^i is the distance to the next point away from the wall. Other required bound­
ary conditions, such as inflow conditions, as well as initial conditions, are discussed
later on a case-by-case basis alongside corresponding results.
For RANS calculations using the stream function-vorticity formulation, one must
add —V
X
(V • r) to the right-hand side of the vorticity transport equation (2.7)
where
- V x ( V .t) = | dy \ dx
dy J
dx \ dx
®Not to be confused with vorticity, which is also denoted as
section, mean vorticity will be denoted by il.
+^),
dy J
(2^33)
u) outside of this section. In this
which can be simphfied to
/dvt dQ
\dx
dy"^
dx
dx"^ y \
d'^vt d U
dxdy dx
W\
^
y
dut dfl \
dy
dy J
^
\ dx"^
using the continuity equation and the definition of vorticity.
dy"^
41
3.
Analysis of the Original Immersed Boundary Method
This chapter introduces the immersed boundary method originally selected to in­
vestigate fluid flows manipulated by flow control actuators. An abridged review of
previous IBM research is given, followed by a presentation of the theory behind, and
results from, the method. This theory is used to explain why results obtained from
the original IBM were less than satisfactory, and, further, how the immersed bound­
ary method can be improved to yield the IBM developed in the present work, and
described in chapters 4 and 5.
3.1
Using Singular Sources with Discrete Approximations
In the IBM, dependent variables and/or their derivatives change discontinuously
across the immersed boundary interface. In this section, formulations of the im­
mersed boundary method are discussed where the jump-singularity at the immersed
boundary is represented, either explicitly or implicitly, by a singular forcing term in
the equations governing fluid motion.
The earliest IBM approach is probably that of Viecelli (Viecelli, 1969, 1971),
who proposed a numerical method for computing inviscid, incompressible flow with
arbitrarily-shaped curved boundaries using a Cartesian grid.
In his calculations,
the pressure at the immersed boundary is iteratively modified until fluid particles
move along the tangent to the boundary. The original immersed boundary method,
however, is usually attributed to the work of Peskin (early work includes Peskin,
1972, 1977; Peskin & McQueen, 1980, 1989) who used the method to investigate
flow patterns through heart valves and in a beating heart. In Peskin's case, the
geometry of the immersed boundary is determined as part of the overall solution,
being dependent upon material properties and the surrounding viscous fluid flow.
More recent investigations and applications of Peskin's immersed boundary method
42
that utihze some type of discrete approximation for the forcing term include the
work of Beyer & LeVeque (1992), who investigate the immersed boundary method by
studying a one-dimensional model equation
ut = u^^+c{t)5{x-a{t)).
(3.1)
They were interested, in particular, in accuracy issues related to the choice of the
representation for the discrete delta function. Goldstein et al. (1993) use Peskin's
IBM for geometries where the boundary is prescribed, and not computed as part of
the flow solution. A feedback scheme is used to iteratively determine the magnitude
of the force required to obtain a desired velocity on the immersed boundary. An
implementation of this method is discussed below in section 3.4. Saiki & Biringen
(1996) have implemented the method of Goldstein et al. (1993) to compute the flow
over circular cylinders for a range of Reynolds numbers. Considering the relative
simplicity of this model/method, good agreement with both experimental results and
other calculations was found. The parameters investigated (length of the separation
bubble, location of vortex centers, Strouhal number, etc.), however, may not be par­
ticularly sensitive to near-wall accuracy. Indeed, close inspection of some of the plots
given (e.g., figures 5 and 6 of Goldstein et al, 1993, and figure 2 of Saiki & Biringen,
1996) show considerable noise in the vicinity of the immersed boundary. Because
of the importance of capturing near-wall behavior correctly, the prediction of T-S
wave amplification rate is proposed in section 6.2 as a more challenging test for any
immersed boundary method.
Fadlun e t a l . (2000) discuss an approach that they claim to be more direct than
the iterative method of Goldstein et al. (1993). The simplest possibility is that the
immersed boundary is represented by a stair-step approximation, as shown, for exam­
ple, in figure 3.1. The velocity at points closest to the immersed boundary is simply
set at every time-step, and one can think of this as applying an equivalent forcing
term to the Navier-Stokes equations. To avoid a stair-step approximation, interpo-
43
Figure 3.1 The simplified stair-step representation of an immersed boundary.
lation is used to set the velocity at points closest to the boundary, if the boundary
would have the given desired velocity.
In the next section, the formulation of the immersed boundary method used in
several of the studies mentioned above, and in the implementation discussed in sec­
tion 3.4 below, is given. The remaining sections of the chapter will be devoted to
evaluating its performance.
3.2
Formulation for the Navier-Stokes Equations
The momentum equation (2.1) of the incompressible Navier-Stokes equations is writ­
ten with an external forcing term F
1
^ -1- u • Vu = F(x, t ) - V p + — V^u,
(3.2)
where the singular force F has support only on the immersed boundary dQi{t)
f(r, s, t)(5(x — X(r, s, t ) ) d S .
(3.3)
' K { r , s , t ) is a parametric representation of the immersed boundary, and 5(x) is the
three-dimensional Dirac delta function.
As mentioned above, the location of the
immersed boundary is either determined as part of the overall solution, or specified
explicitly. If determined as part of the solution, then one must solve
5X(r, 5,t)
, ,
=u(X(r,s,t),t).
(3.4)
44
The term f (r, s,if:) in equation (3.3) is the force strength, and is usually determined
during the course of the calculation. In the case of immersed membranes, for example,
f is determined from the membrane surface tension, which in turn is a function of the
membrane geometry. For the case in which the location of the immersed boundary is
explicitly specified, some other procedure must be used to determine f. Usually, the
additional constraint
u(x,t)|x=x = u(X, t)
(3.5)
is imposed where u(X, t ) is the known velocity of the immersed boundary. For exam­
ple, with u(X, t) = 0, Goldstein et al. (1993) allow the force to adapt itself iteratively
to the local flow field through the use of a feedback loop
f(X(r, s, t), t) = Q; f u ( X { r , s , i ) , i ) d i + P u { X . { r , s , t ) , t ) ,
Jo
(3.6)
where a and (3 are negative constants which determine the amount of control.
In the numerical implementation, the integral in equation (3.3) is usually written
as a sum of discrete forces
F(x,t) =
5,(x - Xfc)
(3.7)
k
where
is determined point-wise from equation (3.6), and Sh{x.) is a discrete represen­
tation of the three-dimensional Dirac delta function. Higher-order discretizations of
equation (3.3) have not been considered; the representation of 5{x) by Sh{x), however,
has been investigated, and is the subject of the following section.
3.3
Model Equations with Singular Source Terms
The model ODE
u" — uj'^u ~ a5'{x — a), x E [0,1],
(3-8)
is used to investigate several methods for numerically approximating the solution to
differential equations that have singular source terms. The function S^^\x) is the nth
45
derivative of the Dirac delta function. The singular source term in equation (3.8) is
chosen as 6'{x) to produce a discontinuity in the function value u at the singularity,
i.e., [u]a = \i'm^^oc+u{x) —
u{x) ^ 0. Later it will be seen that the IBM
applied to incompressible flows produces a vorticity uj that has a similar non-zero jump
at the immersed boundary (or in primitive variables, a pressure p with a non-zero
jump at the immersed boundary). In what follows, the parameters in equation (3.8)
are set (essentially arbitrarily) to the particular values a = 2, a = 1/3, and u — 2.
As posed, the analytical solution u{x) to equation (3.8) can be shown to be
X <a
(3.9)
X > a.
As the method will be used in what comes later, the extra effort is made here to use
fourth-order compact finite-differences (Lele, 1992) to discretize the left-hand side of
equation (3.8) on an equidistant grid of size h {h = 1/{N —1), where N is the number
of grid points)
-l- Liu'l -1"
-|- RiUi -|-
(3.10)
where Lj and Ri are the compact difference stencil coefficients that are functions
of h. This discretization, together with equation (3.8) results in a linear system of
equations for the Ui
(-^2 — 1 ^
R i —\ )^i — 1
"t" (-^z ^
1^
Ri)^i
-{L,_ia5'^{xi-i - a ) + LiaS'f^{xi - a ) +
Ri+ 1 )^? +1
a S'^{xi+i - a ) ) ,
(3.11)
where S'f^{x) is a discrete representation of the effect of the singular source term. Some
representations that have been considered in past immersed boundary implementa­
tions include
(cos(7rx/(2/i)) + l)/(4/?,) lx| < 2 h
0
otherwise
(3.12)
which was used by Peskin (1977) and a Gaussian function
1
e
(3.13)
46
1.4
1.2
Ql
-5
-4
-3
-2
-1
2
0
3
4
5
x/h
Figure 3.2 A discrete representation of the Dirac delta function from equation (3.13)
for several a/h, where h is the computational grid size.
used by Goldstein e t a l . (1993).
This latter representation, shown in figure 3.2,
behaves as one would expect a good approximation of a delta function to behave:
as a ^ 0, the function becomes taller and narrower, yet the area under the curve
remains constant. Indeed, one finds that as a generalized hmit
S ( x ) = lim —7=— e
<T^0 ^TTCr
X
(3.14)
Many other discrete representations have been proposed in the literature, and the
first question that one might ask is, which representation provides the most accu­
rate result? If the singular source terms are treated in the right way, Walden (1999)
has shown that one can obtain any desired order of accuracy, but only away from
the singularities. Specifically, using nth order finite-differences to discretize a given
differential equation, an nth order point-wise convergence of the numerical approxi­
mation away from the singularities will be obtained if an nth order approximation of
is used. The function
(3.15)
47
is defined by Walden to be an nth order approximation of
— a)
if it satisfies
^ ^ J " d ( i - a ) ( t - d f = Sk-s, V a e R , k = 0 , . . . , N + s - 1,
SI
(3.16)
i
where 6k~s is the Kronecker delta. Walden does not motivate equation (3.16), but
one might suspect that it is a discrete representation of the analytical statement
OO
(/)(x)5(")(x)
/
dx =
(-1)"0(")(O).
(3.17)
•OO
The analysis aside, numerical solutions to equation (3.8) clearly reflect the effect
of selecting proper discrete representations for the singular source term. Figure 3.3
compares one numerical solution, computed using the representation given in equation
(3.13), with its corresponding analytical solution. The effect of the selected singular
source representation is to smear out what should be a sharp discontinuity in the
solution in the vicinity of the singularity located
a,t x — a.
As a is made larger, the
region that is affected grows correspondingly, as seen in figure 3.4. The data given
in table 3.1 and in figures 3.5 and 3.6 indicate the lower order convergence of this
representation compared with that given by Walden, which is seen to converge to the
expected fourth order (outside of some neighborhood of the singularity). In closing,
it is noted that the results using equation (3.13) do depend on the choice of a / h .
N
llelloo (Gaussian)
||£||oo (Walden, 1999)
51
101
201
1.2770 X
3.1906 X 10 -4
7.9752 X 10^-^
2.2353 x 10~®
2.9618 x 10-^
8.6236 x 10""
Table 3.1 Error [ e = U a — u ^ ) in the numerical solution of equation (3.8) on grids
of size N (or step-size h = 1/(A^ — 1)). The error is ||e||oo =
with 71=3.1991,
and n =2.00 for the Gaussian representation, equation (3.13) with a/h = 2, and
A=0.1863, and n=4.01 for the fourth-order representation given in Walden (1999).
The range x G [0.2, .5] in the vicinity of the singularity has been excluded from the
analysis.
48
Although an optimal value exists, the convergence of the numerical solution does not
improve significantly.
With the first question answered, the second question one might ask is, does one
want to discretize the delta function at all? That is, could the effect of the singular
forcing term be introduced in some other way? Before proceeding to answer this
question, it is first interesting to look at the performance of a complete immersed
boundary Navier-Stokes code in light of the results presented in this section. This
first code utilizes the Gaussian representation from equation (3.13) to represent the
discrete singular forcing terms in the immersed boundary formulation. The represen­
tations from Walden could not be run without the computation becoming unstable.
49
0.2
0.6
0.4
0.8
x
Figure 3.3 Analytical (—) and numerical (o) solution of equation (3.8). N = 101
points were used to compute the numerical solution, and equation (3.13) with a / h — 1
was used as a discrete representation of the singular source term.
2.5
3
0.5
•0.5
-1 >-
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x
Figure 3.4 Analytical (—) and numerical (symbols) solution of equation (3.8). N =
101 points were used to compute the numerical solution, and equation (3.13) with
a/h = 1 (o), ojh — 2 (o), and a/h — A (*) was used as a discrete representation of
the singular source term.
50
<jJ
-4
.-8
0.2
0.4
0.6
0.8
x
Figure 3.5 Error in the numerical solution of equation (3.8) for grid size
= 51 ( - ),
N = 101 (
), and N = 201 (- • -). Equation (3.13) with a / h = 2 was used as a
discrete representation of the singular source term.
-2
,-4
-10
,-12
.-14
0.2
0.4
0.6
0.8
x
Figure 3.6 Error in the numerical solution of equation (3.8) for grid size
= 51 (—),
N — 101 (
), and N = 201 (- • -). The fourth-order function 5 ' ^ { x ) in Walden
(1999) was used as the discrete representation of the singular source term. The data
in table 3.1 indicates that fourth order convergence of the error is achieved (away
from the neighborhood of the singularity).
51
3.4
Application to the Navier-Stokes Equations
The Immersed Boundary scheme described in section 3.1 was implemented in an ex­
isting Navier-Stokes code, written in vorticity-velocity formulation (see Meitz, 1996;
Meitz & Fasel, 2000, for details). This code utilizes a fourth-order Runge-Kutta
scheme for time integration, fourth-order compact finite-differences for spatial deriva­
tives in the streamwise x and wall-normal y directions, and a pseudo-spectral ap­
proach for the spanwise 2 direction. In the following, the numerical implementation
is first discussed, followed by its application to flow control actuators and a T-S wave
simulation. Finally, a discussion of the method's shortcomings is presented.
3.4.1
Numerical Implementation
For the pseudo-spectral implementation, the forcing term F(X,i) in equation (3.3) is
expanded as a truncated Fourier series
¥{x,y,z,t) =
F fx ' h t ) I
°
h
N
•
^
Ffc,(x,?/,t)sin(7fc2:) ^ ¥ k c { x , y , t ) c o s { - i k z ) ,
(3.18)
k=\
where
^ks{x,y,t)
^kc{x,y,t)
and j k = k
Y [
sin(7fcZ)
cos(7fcZ)
S{x - X)S{y - Y ) d S ,
(3.19)
j X z is the wave number, X = ( X , Y , Z ) = (X(r, s , t ) , Y (r, s, t ) , Z { r , s , t ) )
is a parametric (r, s) representation of the immersed boundary surface S , and dS =
\\dX/dr
X
dX/ds\ \ dr ds. The Fourier series representation of such a localized forc­
ing is expected to generate considerable ringing in the numerical solution, a result
epitomized by the representation of the Dirac delta-function shown in figure 3.7.
There are three transport equations for the three vorticity components in spectral
52
n=16
to
10
-10
-0.5 -0.4 -0.3 -0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
x/X
Figure 3.7 Truncated Fourier series for the periodic delta-function
with A'' = 4, 8, and 16 terms.
S{x — n \ )
space
d f lX
fc®
dt
dVty
dt
dt
where Ak, B^, and
dA,
1
i ^ k C k f . + 777 fcs
+ Gx ki 1
dy
Re
dAf.s
1
T IkBki + ^"^k^yki +
dx
dB,
dCks
1
^"k^zki + G:^ki
dx
dy
Re
(3.20)
(3.21)
(3.22)
are the Fourier coefficients for the non-linear terms
and the Laplacian operator
a = V Ux — ucUy,
(3.23)
b = w ujy — V u j z ,
(3.24)
C = ULUz — w U t ,
(3.25)
is transformed into
^2
q2
The forcing terms, G^k, Gyk, and Gzk, are obtained by taking minus the curl of
53
F(X, t ) resulting in
Gkxsix,y,t)
Gkxc{x,y,t) ^
cos(7fcZ)
sin(7fcZ)
Jdftiit)
fz
G k y s( ^ ) y1
Gkycix,y,t)
fz
Jd0 . i {t)
Ikfc
Gk
z osip^lU
— n, ^
\
I o )
/
Ghzci^iUji) _
I
^zJdfli{t)
fx
5{x^X)6{y-Y)-
sin(7fcZ)
S{x-X)S'{y-Y)dS,
cos(7fcZ) _
sin(7fcZ)
_ cos(7fcZ)
S'{x-X)S{y-Y)±
cos(7fcZ)
sin(7fcZ)
S{x-X)5{y-Y)dS,
sin(7fcZ)
cos(7a;Z)
S{x-X)5'{y-Y)-
sin(7fcZ)
cos h k Z )
S'{x-X)S{y-Y)dS.
(3.27)
(3.28)
(3.29)
In practice, the surface integrals in the above equations are evaluated as discrete sums,
to equation (3.7). Additionally, the feedback loop given in equation (3.6) was
similar to
used to compute the magnitude
3.4.2
f = {fx, fy, fz) of each of the discrete forcing terms.
Flow Control Actuators
This section describes results obtained using the original immersed boundary method
to simulate the response of a fluid to moving flow control actuators. The results
are presented to give an indication of the type of simulations that motivated the
development of the new IBM. The actuators modelled are mounted on, above, or
below the surface of a flat plate, upon which a zero-pressure-gradient boundary layer
has formed. Disturbances within the boundary layer are produced by motion of the
actuators.
The first actuator investigated is a vibrating ribbon actuator. Following the de­
velopment of the linear stability theory by Tollmien and Schlichting, experimental
investigations were carried out to test the theory (see also the discussion in sec­
tion 6.2). Many of these original experiments, beginning in the early 1940's with
those of Schubauer & Skramstad (1948), made use of a vibrating ribbon to excite
54
2-D boundary layer oscillations. The ribbon was held under tension across the span
of a flat plate, and made to oscillate by running an AC current through the ribbon
which was immersed in a magnetic field. This technique was found to be capable of
producing the desired 2-D boundary layer disturbances.
A numerical simulation of a vibrating ribbon is described in this section. The
ribbon is modelled using the IBM with singular forcing, allowing it to move freely in
the computational domain; its location is not constrained to grid points. To simu­
late the motion of the ribbon actuator, discrete volume forces are distributed evenly
along the ribbon surface; the flat plate wall itself corresponds to the computational
domain, and is not simulated using volume forces. The forces are moved through the
computational grid, and their magnitude is adjusted according to equation (3.6) so
that the no-slip, no-penetration condition is enforced at each time-step.
The ribbon, which is about 1/4 of a T-S wavelength long, is placed at approxi­
mately 5.6% S from the wall and moves sinusoidally. The near-field disturbance caused
by the ribbon is shown in figure 3.8 after a periodic state has been reached. The fiow
field appears as one would expect; as the ribbon moves upward (downward), fluid
moves inward (outward), under the ribbon; at the top or bottom of the ribbon stroke,
a low Reynolds number flow regime exists. Streamwise velocity proflles corresponding
to flgure 3.8 are provided in flgure 3.9 to show more detail.
The global behavior of the disturbance created by the ribbon is shown in flg­
ure 3.10. It is clear from this flgure that the frequency of the ribbon is such that
amplifying waves are produced over a particular range of downstream distances. The
disturbance distributions in the wall-normal direction shown in figure 3.11 compare
well with the predictions of linear stability theory.
The second actuator investigated is a 2-D membrane-type actuator attached to
the flat plate wall. The height h of the actuator above the wall is described by
h ( x , t) = — (1 — cos(27r/t)) cos^ ( —
2
\
w
J
, \x — xd < w/2,
(3.30)
55
where Xc is the coordinate of the center of the actuator, and
w
is the total width of
the actuator.
The spanwise disturbance vorticity created by the sinusoidal motion of the actu­
ator is shown in figure 3.12. Shown is a time history of the actuator movement; the
actuator at mid-stroke moving upward, the actuator at top stroke and stationary, and
finally at mid-stroke moving downward.
The final case is that of a piston-type actuator, where the piston moves in and
out of the flat plate, travelling in a hole drilled into the plate. Figure 3.13 depicts
one of the few 3-D simulations that were run using the original IBM. Shown in the
figure are contours of constant disturbance M-velocity created by the motion of the
piston. Three-dimensional simulations using the old immersed boundary method were
considerably less successful than their 2-D counterparts, the main reason being that
3-D simulations, which utilize a pseudo-spectral approach, required an unreasonably
high spanwise resolution to obtain satisfactory results.
56
center stroke, moving upward
top stroke
center stroke, moving downward
bottom stroke
Figure 3.8 Sinusoidal motion of a 2-D vibrating ribbon simulated inside the compu­
tational domain. Shown are total-velocity vectors.
57
0.0015
0.001
center stroke, up
top stroke
center stroke, down
bottom stroke
0.0005
-0.1
-0.05
0
0.05
U
0.1
0.15
0.2
0.15
0.2
0.0015
0.001
center stroke, up
top stroke
center stroke, down
bottom stroke
0.0005
0 L-0.1
-0.05
0.05
U
Figure 3.9 Streamwise velocity profiles u near the leading edge of the ribbon (upper
graph) and the trailing edge (lower graph).
Figure 3.10 Streamwise (top) and wall-normal (bottom) disturbance velocity pro
duced by a vibrating ribbon. Uoo = 30.0 m/s,
= 1.5 x 10"^ m^/s, Rer = 10®
F =150 X 10-® or f* = 14.3 Hz.
59
0.6
>
0.4
Orr-Sommerfeld solution
Navier-Stokes solution
0.2
'I'
0.8 - 'I)
if-'
-$
(h
V
0.6 -T
0
Orr-Sommerfeld solution
\
O Navier-Stokes solution
h
\
V
0.4
0.2
•"'O-.-Q.
-
0
10
11
12
Figure 3.11 Wall-normal v (upper graph) and streamwi«e u (lower graph) disturbance
velocity. Navier-Stokes solution (fundamental) using the IBM to simulate a vibrating
ribbon (o) compared with Orr-Sommerfeld solution (—). Uoo = 30.0 m/s, = 1.5 x
10-^ Rec = 10^ F =150 x 10"® or f* = 14.3 Uz or u =
= 0.099735,
= 666.0,
a = a*Si = (0.266399, -0.002955).
^ °a94
096
oXb
T])0
i~04
1^6
ToS
1.10
1.12
X
half stroke, moving upward
I
^
-1,0
m
0,0
1.0
"lO^
top stroke, stationary
•102
half stroke, moving downward
Figure 3.12 Spanwise disturbance vorticity created by the sinusoidal movement of a
wall-mounted actuator. Actuator height is 5.6% of boundary layer thickness 5.
61
0.6
0.5
0.4
0.06%
0.3
A12%
0.2
0.1
-1.4%
-3
1
-2
0
1
2
3
506
508
z (mm)
W.001,
-0.0187%
-2
-3^
496
-0.00937%
\\
498
500
X
502
(mm)
504
0.6
0.5
0.4
0.3
0.2%
0.2
/
-0.7%
0.1%
0.1
-1.4%
496
498
500
502
X (mm)
504
506
508
Figure 3.13 Contours of constant li-disturbance velocity created by a piston actuator.
Actuator is located in a wall over which a zero-pressure gradient boundary layer has
developed.
62
3.4.3
Tollmien-Schlichting Waves
The main deficiency of the IBM mentioned earher, namely, that it smears out jump
discontinuities at the immersed boundary, is highlighted in the test case to he pre­
sented, which is shown schematically in figure 3.14. A zero-pressure gradient, flatplate boundary layer develops on a virtual wall, offset parallel to the computational
wall, and modelled using an immersed boundary. Initially, the boundary layer is
undisturbed, and appears as shown in figures 3.15 through 3.17. For comparison, an
identical body-fitted grid calculation was computed without the immersed boundary.
The u and v velocity profiles computed with the immersed boundary agree relatively
well with the body-fitted results. The spanwise vorticity uj profiles, however, exhibit
smoothing in the vicinity of the singular source terms instead of a sharp discontinuity.
This is the same behavior that was seen in section 3.3.
Next, the steady boundary layer is forced using a dipole body force near the
inflow, as seen in figure 3.14. The frequency was selected to be in the range to which
the undisturbed boundary layer is linearly unstable. The growth of the resulting
disturbance can be compared with the prediction of linear stability theory, as obtained
from the Orr-Sommerfeld equation, as well as with the body-fitted code results. The
disturbance u, v, and
UJ
profiles, shown in figures 3.18 through 3.20, exhibit similar
behavior as their steady counterparts. As seen in figure 3.21, the amplitude growth
body forcing
immersed wall
computational wall
buffer zone
//////////
Figure 3.14 A flat plate immersed in a rectangular computational domain.
63
predicted from the immersed boundary computation agree reasonably well with the
LST and body-fitted results.
In order to obtain these results however, the time
step, given non-dimensionally in the figure as a CFL number {UooAt/Ax), must be
roughly ten times smaller than is required for stability of time integration (note that
Goldstein et al, 1993, also encounter this restriction). One explanation might be
that the feedback control given by equation (3.6) acts a type of iterative solver: the
magnitudes
of the discrete singular forces in equation (3.7) are coupled and need
to be determined simultaneously. If the numerical scheme is to be stable, the flow
field cannot be allowed to change significantly before the 'iteration' comes close to
'converging' to the proper force magnitudes. A type of infiuence matrix technique
(Daube, 1992) that takes the force coupling into account would be required if the
feedback control loop is to be eliminated.
64
0.8
0.6
0.4
0.2
0
0.01
0.02
0.03
0.04
y
0.4
0.35
0.3
0.25
3
0.2
0.15
0.05
-0.05
-4
-2
y
X 10-3
Figure 3.15 Comparison of w-velocity profiles at several downstream locations: x =
1.213 {Rs, = 600) (o), X = 2.158
= 800) (o), and x = 3.375
= 1000) (*).
Symbols indicate values computed using an immersed wall at y = 0, and solid lines
the body-fitted code.
65
X
-3
10
2.5
>
0.5
-0.5
0.01
0.02
0.04
0.03
y
5
X
.-4
10
4
3
2
1
0
1
-4
-2
0
2
4
6
Figure 3.16 Comparison of f-velocity profiles at several downstream locations: x =
1.213 {Rs, = 600) (o), X = 2.158 (% = 800) (o), and x = 3.375
= 1000) (*).
Symbols indicate values computed using an immersed wall at y = 0, and solid lines
the body-fitted code.
66
100
lOOr
O
o.
80
OOOO <> 0 0
60
;
6*
40
0
*
it * « «
20
0 -
-4
-2
0
2
y
-3
X 10
Figure 3.17 Comparison of cu-vorticity profiles at several downstream locations: x =
1.213 (R5, = 600) (o), X = 2.158 (% = 800) (o), and x = 3.375 (% = 1000) (*).
Symbols indicate values computed using an immersed wall at |/ = 0, and solid lines
the body-fitted code.
67
1.5
X
-3
10
1
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
y
X
10
0.5
-5
-2.5
2.5
y
X 10 -3
Figure 3.18 Comparison of disturbance u-velocity profiles (fundamental) at several
downstream locations: x = 1.213 (R^j = 600) (o), x = 2.158
= 800) (o), and
X = 3.375 (R^j = 1000) (*). Symbols indicate values computed using an immersed
wall at y = 0, and solid lines the body-fitted code.
68
X 10
>3
0.01
0.02
0.03
0.04
0.05
0.06
y
>
0.5
-5
-2.5
2.5
y
X 10-3
Figure 3.19 Comparison of disturbance velocity profiles (fundamental) at several
downstream locations: x = 1.213 (R^^ = 600) (o), x = 2.158
= 800) (o), and
X = 3.375 (R^i = 1000) (*). Symbols indicate values computed using an immersed
wall at y = 0, and solid lines the body-fitted code.
69
0.2
0.15
S" 0.1
0.05
0
0
0.01
0.02
0.03
0.04
0.05
0.06
y
1.5
1
0.5
0
-5
-2.5
0
y
2.5
5
X 10-3
Figure 3.20 Comparison of disturbance (x;-vorticity profiles (fundamental) at several
downstream locations: x — 1.213 (R^j = 600) (o), x = 2.158 (R^j = 800) (o), and
X — 3.375 (R^j = 1000) (*). Symbols indicate values computed using an immersed
wall at y = 0, and solid lines the body-fitted code.
70
3
)
1
2.5
8.93E-2
/
2
/
/
O
/
< 1.5
/
<
4.46E-2
/
-
- • •
. . . . .
^
"
-_
/
.
/
<2.23E-2 •
/
X'
0.5
0
-0.5
,
600
700
800
900
1000
R.
Figure 3.21 Comparison of the growth rates of a 2-D T-S wave {A = inner maximum
of M(i,o),) for frequency F =1.4 x 10""^: immersed wall and various CFL numbers (• -), standard DNS (
), and LST (—). Note: C¥\j=Uoo^t/and Ax is held
constant here.
71
3.4.4
Further Discussion of Results from the Original N-S IBM
In some situations, one might be content to accept solutions that do not have sharp
interfaces at the immersed boundary, as well as the extra overhead of running with
a reduced CFL number, considering the increased fiexibihty in geometry allowed by
the IBM. For example, figure 3.23 shows a simulation of the interaction between the
wake of a circular cylinder, and a flat plate boundary layer. Despite the noise in the
vicinity of the immersed boundary, the wake looks quite reasonable. In cases where
near-wall accuracy is important, however, one may have to resort to a considerable
resolution increase in the vicinity of the immersed boundary. This increased resolution
is usually only required locally, and is most efficient when coupled with adaptive mesh
refinement (AMR, see Berger & Colella, 1989, for example). In the AMR approach,
particular regions of a computational domain consist of nested rectangular patches
that have a finer grid spacing than the underlying global coarse grid. The finer grid
is used where the error in the numerical solution on the coarser grid would otherwise
be too high. A simple example using two levels of grid nesting is shown in figure 3.22.
More complicated grids using higher levels of nesting and more compficated fine-grid
distributions can be found in the literature.
0.8
0.6
0.4
0.2
qI I I I I I I ' I I I I I ' I I I I I I0
0.2
0.4
0.6
0.8
1
Figure 3.22 A simple two-level AMR grid.
72
Figure 3.23 Impulsively started circular cylinder {uj vorticity), Ksd = 2000, modelled
with an immersed boundary technique that uses singular source terms. The cylinder
is placed above a flat plate, whose disturbed boundary layer is seen in the bottom
figure.
73
An investigation of the Peskin IBM coupled with AMR was carried out by Roma
et al. (1999), who found that enhanced accuracy for the IBM could be obtained
by locally covering the immersed boundary with a sequence of nested, increasingly
finer grids. His results, however, remain formally only first-order accurate in space.
Temporal accuracy, on the other hand, is not investigated by Roma. A study carried
out by the present author showed that the interpolation procedure required by a
finite-difference AMR code reduces the temporal accuracy by one order.
74
4.
A New Approach to the Immersed Boundary Method
Earlier the question was asked as to whether, in the numerical solution of equations
with singular forcing terms, one should attempt to discretize the singular forcing
term. For problems where information is not needed in the neighborhood of the
singularity, a proper numerical discretization will produce accurate results outside of
this neighborhood. The immersed boundary method, however, requires information at
the immersed surface, i.e., at the location of the singularity itself, e.g., equation (3.6).
Further, one is often interested in the value of surface quantities, such as temperature,
skin friction, heat flux, etc., which require sufficient grid resolution in the IBM to
capture them (using, perhaps, AMR). As the old IBM tends to smear out the solution
in the neighborhood of the singularity, what is required for accurate results at the
immersed boundary is a sharp interface method designed to acknowledge the presence
of jump discontinuities, and accurately reproduce them. A new approach to the IBM
that does just this is introduced in the following sections. The new approach is
based on the work of Wiegmann & Bube (2000) who investigated the numerical
solution of linear elliptic problems using second-order, 0{h^), finite-differences. In
this dissertation, their work is extended to solve the unsteady Navier-Stokes equations
using fourth-order, 0{h^), finite-difference methods.
4.1 Taylor Series Analysis of Functions with Jump-Singularities
The basic idea behind the new IBM approach, first discussed by LeVeque & Li (1994),
and later clarified by Wiegmann &: Bube (2000) \ is to recognize that the standard
Taylor series upon which finite-difference stencils are based is not valid through a
singularity, and that finite-differences at the interface of the immersed boundary
^Wiegmann & Bube (2000) introduced the idea of the Exphcit-Jump Immersed Interface Method
(EJIIM).
75
h
i
\
1
h
Figure 4.1 A function /(x) with discontinuity at x =
must be corrected in order to maintain formal accuracy of the underlying numerical
scheme. With regard to figure 4.1, for example, one would like to write a Taylor series
at point Xi to evaluate the function f{x) at point Xj+i. Assume that /(x) is analytic^
everywhere in the domain D = {x | Xi_i < x < Xi+i} except at the point Xq (and
only at this point, in all of what follows) where it has a jump discontinuity in the
function value itself and/or higher derivatives. If Xi < Xa, the standard Taylor series
cannot proceed through
to correctly predict /(xj+i) unless a correction term
is
added:
= f { X i ) + f ' { X i ) h + /"(Xj)— +
where
+ . . . + J q,
(4-1)
+... ,
(4,2)
is
J. = [/la + l/'lafc+ + ^lf"Uh*f +
h = Xj+i — Xj, and h~^ = Xj+i — Xq,. The term [0]q, represents the jump in the value
of 0 at X = Xa, that is
[(t>]a= lim^ 0(x) — hm 0(x),
(4.3)
so that [/]„ represents the jump in the function value at x = X q , [ f ' ] a the jump in
the value of the first derivative of the function, and so on. The case Xj = x^ requires
real function is analytic if for each a e D it has a power series expansion
valid in some neighborhood of a.
f{a+h) =
"n/i"
76
one to decide whether the terms f{xi), f'{xi), etc. are to be defined as .left or right
limits, this determining in turn whether or not a correction to the Taylor series will
be required. Finally, for the case
< Xi, no correction is required to predict /(xj+i).
Similar arguments can be made for finding /(a;j_i) in terms of a Taylor expansion
about point Xj, the result being that
/(Xi_i) = f{Xi) - f{xi)h+
where
+ . . .+ Ja,
(4.4)
is
J . = - I f ] . + if'Uh- - ^ \ r u h - Y +
^.
(4.5)
h = Xi — Xj_i, and h~ = Xa — Xi-i. As above, Jq, is nonzero only when a correction
is required.
Equations (4.1) and (4.4) are termed corrected Taylor series. The corrected Taylor
series will be used in the next section to correct finite-difference stencils that have
been obtained using the standard Taylor series.
4.2
Derivation of Jump-Corrected Finite-Differences
In this section, the case Xi < Xa discussed above is considered. Using the correction
term Jq,, one can now modify any type of standard finite-difference stencil to obtain
its jump-corrected counterpart. The jump-corrected stencil will maintain the order
of accuracy of the original stencil when the stencil passes through a jump-singularity
of the function to which it is applied.
Consider, for example, a finite-difference stencil for the computation of a second
derivative
-H Lif- ^
-h Rifi + Ri+ifi+i -(-
C(/i^),
(4.6)
77
where h is some measure of the local grid-size^, e.g., (xj+i — Xi^i)l2. If the function
f{x) is analytic in the entire domain Xi^i < x < Xj+i, then the stencil given in
equation (4.6) will be accurate up to the truncation error built into the approximation
it represents. For equation (4.6), this error is proportional to h'^.
Next, consider the case discussed above in section 4.1, and depicted in figure 4.1.
If the stencil given in equation (4.6) is applied to this case, a large error, possibly as
high as 0(/i~^), will result. This fact, and simultaneously, the remedy, can be seen by
witnessing what would happen if the terms fk and
in equation (4.6) were to be
expanded about some point x* < x^ using the corrected Taylor series approximation
given by equation (4.1). Clearly, because it was designed to do so, all derivatives
n = 0,1,... 4 will drop out of equation (4.1), leaving the derivatives starting
at
This can be seen in the truncation error in equation (4.6). Addition­
ally, because of the singularity at
Xq
, the jump-correction terms
will also remain,
transforming equation (4.6) into
Li-\-lJa2
Ri+lJaO
d'f
^^,5
om,
(4.7)
where
= [z™]. +
Ja2 = I/'^'lc +
+ • • • •
(4^8)
+ ... ^
(4,9)
One can now see why the error using the uncorrected stencil given in equation (4.6)
will be
the coefficient
which is
multiplies the 0{1) term
in the remainder equation (4.7). If, instead, the function were continuous ([/^°^]a = 0),
but the first derivative were discontinuous
^ 0), then this error would be
reduced to 0 { h ~ ^ ) . Finally, if all derivatives up to and including
were continuous,
then the original 0{h'^) accuracy of the stencil would be maintained.
^The existence of stretched grids is allowed for in the present discussion. However, all references
to formal accuracy are stated for the equidistant-grid case; stretched grids will have a somewhat
lower accuracy not exceeding one order less.
78
To avoid the situation altogether, it is now clear how the stencil in equation (4.6)
must be modified: to the right-hand side of this equation, the term Lj+i Ja2 — -Ri+i-^ao
must be added
= Ri-lfi-l + Rifi + Ri+lfi+l + {Li+lJa2 — Ri+lJao), (4-10)
where Jao and Ja2 are truncated by taking only enough terms to maintain 0{h'^)
= [/'")«+ [/'"l«A+ + ^l/''"l<,(ft+)' + i[/'"l»('»^)'-
(4.12)
Now, when the terms in equation (4.10) are expanded using the corrected Taylor
series, the jump terms will cancel, leaving only an C(/i^) remainder term. The ques­
tion remains: how does one obtain the jumps in the function and derivatives at Xq
that are required for the correction scheme just outlined? An answer to this question
will be given in the next section.
4.3
4.3.1
Obtaining Jump-Corrections
Differential Equations with Singular Forcing Terms
As discussed in section 3.1, the IBM originally developed by Peskin (1972) used a
singular volume-force term F(x, t) added to the right-hand side of the incompressible
Navier-Stokes equations. As a simplified model of this method, consider the ODE
u" — uj'^u = 0, Xi < X < X2,
(4-13)
u{xi) = til, u{x2) = U2.
(4-14)
with boundary conditions
In the spirit of the immersed boundary method, one would like to specify an additional
"immersed boundary condition" at x^, say u{xa) — Ua, where x^ is located somewhere
79
5
4
3
0
-1
0
-0
0.2
0.6
0.4
0.8
X
Figure 4.2 A typical solution of equation (4.16).
in the domain Xi <
< X2- This immersed boundary condition can be satisfied
through the addition of a singular forcing term to equation (4.13)
u" — uj'^u = aS{x — Xa)
(4.15)
and the proper choice of a = a{ua), the magnitude of the singular forcing term. The
solution of this equation will be continuous across
but the first derivative will
most likely not be. To continue the discussion of the case where the function itself is
discontinuous across
the derivative of the Dirac delta function, S'{x)^ will instead
be used as a singular forcing term
u" — UpU = a5'{x —
Xa).
(4.16)
A typical solution of equation (4.16) is given in figure 4.2. As discussed above in
section 4.2, any numerical method attempting to solve equation (4.16) will have to
deal with the jumps in the function and derivatives present at
The correction
scheme makes use of these jumps, and hence they must be known. In the case of
ODE's or PDE's with singular forcing terms, all jumps can be obtained without prior
knowledge of the actual solution; the jumps depend solely on the magnitude of the
80
singular forcing terms and the geometry of the immersed boundary. For the case of
the Navier-Stokes equations, the derivation is somewhat involved (see Li & Lai, 2001);
for equation (4.15), however, it is relatively straightforward to show this. Integrating
this equation over a small region enclosing the singularity at
rxa+e
/
u" —
JxQ—e
rxa+e
dx —
ad'(x — Xa) dx,
(4.17)
JXa—^
and then taking the limit as e —> 0 gives
u'(a;+) - u ' { x ~ ) = [v!]oc = 0.
(4.18)
u{x'^) - u{x~) = [u]a = a.
(4.19)
Similarly, one finds
Using these two conditions, and by taking derivatives of the original ODE, equa­
tion (4.15), the entire series of jump conditions can be found
[u]a
=
a,
[u']a
=
0,
[u"]a
=
K']„
=
i0'^[u]a
=
=
0,
With these jumps now known, any finite-difference stencil used to solve equation (4.16)
numerically can be corrected using the method described above in section 4.2.
4.3.2
Differential Equations without Singular Forcing Terms
The problem of interest in this section is that of numerically solving PDE's on irregular
domains using Cartesian grids where singular forcing terms are not inherently present
in the problem. The singular-forcing function method described above in section 4.3.1
can of course be applied on irregular domains. However, some type of method (e.g.
81
F.D.
Stencil
Stencil
+
Figure 4.3 Spatial configuration of one-sided finite-difference stencils used to compute
jumps in equation (4.22). The symbol • indicates points used in the stencil, whereas
the symbol o indicates points not used in the stencil.
an iterative method) must be used to determine the magnitude of the singular forcing
terms that enforce the desired immersed boundary conditions. For problems where a
singular forcing term is not inherently present, it is probably best to use a more direct
approach that avoids the introduction of one. How this can be done is the topic of
the present section.
In contrast to the case involving singular forcing terms, the jumps in the function
and its derivatives cannot be found independently of the problem solution. Recogniz­
ing this, one is then forced to use the solution itself to obtain the jumps. Numerically,
this is accomplished through the use of one-sided finite-differences. The accuracy and
spatial configuration of the one-sided stencils must be chosen with some care, as the
following example will illustrate. The issue of accuracy is presented first, followed
by a discussion of spatial configuration, that is, which points should and should not
be used in the one-sided stencils. With reference to figure 4.3, the second derivative
of a function f{x) at point Xi is computed using an explicit finite-difference
with
82
j ump-correction
fi = Rxxi-ifi-l + Rxxifi + Rxxi+ifi+l ~ Rxxi+iJaO^
(4-21)
where
(422)
0{h^)
0{h^)
0{h)
The expressions (4.21) and (4.22) together form an 0{h'^) expression for //' in the
presence of a jump singularity at the point x = x^- The order symbols below each of
the terms in equation (4.22) indicate the order of the one-sided finite-difference stencil
which should be used to approximate this term in order to ensure 0{h?) accuracy
in equation (4.21). For example, the
quantity, and by
term is multiplied by Rxxi+D an 0{h~^)
which together make a term 0{h^). If
is represented
by a stencil with truncation error 0 { h ) , then the overall error contribution from the
entire term 1/3!
to equation (4.21) will be 0{h?).
The objection might be raised, justifiably, that if our overall solution is 0{h?)
accurate, then one will not be able to obtain the term
to the required 0{h^)
(although it need not be, it will be assumed that [/^°^]a is always known). Indeed, in
the present example, one order may well be lost locally. The effect of this phenomenon
on the errors ||£||co and ||£||2 is investigated experimentally in sections 5.1.4 and 5.2.3.
For now, the naive approach of discretizing the jump
using an 0[h^) stencil
will be used.
In order to maintain the specified order of accuracy, each of the one-sided stencils
used to compute the jumps in (4.22) must contain four points. Again referring to
figure 4.3, the spatial configuration of these four points is such that the jumps are
computed as
=
(4-23)
83
where
(4.24)
(4.25)
The + and — superscripts on
indicate respectively right and left limits at
Xq
. Note
that the points Xj and Xj+i have intentionally not been used, and that because of this,
many problems, in particular numerical stability, have been avoided. Consider, for
example, the use of an explicit time-marching scheme for solving the 2-D diffusion
problem
(4.26)
Had these points been used, the scheme would have become unstable as the singularity
approached either of them, because for a given time-step At, the criterion that
(4.27)
would have been violated { K is some constant that depends on the numerical scheme).
Admitting arbitrarily shaped immersed boundaries, or further, moving immersed
boundaries, it is clear that the singularities must be allowed to move arbitrarily
close to, or even lie on top of, grid points. The spatial configuration of the stencils
described in this section allows just that.
One additional note is in order before leaving this section. In conjunction with
the use of compact finite-difference stencils (see section 5.1.2), one might consider
computing jump-corrections making use of the derivatives which are being sought.
That is, if solving for a compact first derivative, one might consider computing the
jump given in equation (4.24) as
+ ^rii+ifUa + ^rii+ifi+i
(4.28)
84
where, in accordance with the compact finite-difference method of computing nu­
merical derivatives, terms //+2) /i'+3' etc. are unknown, and must be moved to the
left-hand side of the compact difference equation
V ;
;
i
:
ff(\
fs
n
•./ \ ' - j
-R12 ^13 i?l4
R^-21 R22 R23 R24
R'31
.
R33 R34
R.-41 -R42 R43 i?44
/ r41
A
\
fh\
f2
h
h
CO
to
/Lii L12 -^13 Lii • • •\
L21 1/22 -^23 -^24
L^l L32 -^33 L^A
L41 L42 I/43 L44
7
(4.29)
\'' J
This strategy would be useful for either increasing the order of the stencil for a fixed
number of points, or reducing the width of the stencil for a fixed formal accuracy. The
strategy, however, did not prove to be as effective as using explicit finite-difference
stencils to compute the jump-corrections: the error near the boundary was found to
be one or two orders of magnitude larger when the sought derivatives were included in
the jump-correction terms, as opposed to the case when only known function values
were used (as is the case in equations (4.24) and (4.25)).
85
5.
Numerical Method
The solution of the full equations of motion can essentially be split into two main
tasks: time integration of a parabolic convection-diffusion equation, the vorticity
transport equation (2.7), and the solution of an elliptic poisson equation, the stream
function equation (2.8). If a turbulence model is used, then equations (2.29) and
(2.30) must also be integrated in time, and are handled in almost the same manner
as the vorticity transport equation. The nature of the equation, whether parabolic or
elliptic, determines how it is to be treated numerically. In the following sections, the
numerical method of solution for each equation is discussed separately, with special
emphasis on how the immersed boundaries are handled. To begin the discussion,
only stationary immersed boundaries are considered; moving immersed boundaries
are treated in section 5.4.
5.1 The Convection-Diffusion Equation
5.1.1
Time Integration
For time integration, either a second-order predictor corrector or a fourth-order
Runge-Kutta method is used. Both methods are exphcit in time, and hence un­
stable for time steps At larger than some limiting value. The convection-diffusion
equation is written generically as
(5.1)
The time integration scheme advances the solution cf) of equation (5.1) from time
to tn+\ = tn + At. For the predictor-corrector
(/>! =
(f^n+l
=
4>n + Atf{(l)n),
+ -^(/(0n) + /((J^*!))
(5.2)
(5.3)
86
and for the fourth-order Runge-Kutta
<i>i = <l'n +
(5-4)
02 =
(5^5)
03 = 0n + Ai/(02))
(5-6)
0n +l
0n + -^(/(0n) + 2 /((/)i) H- 2/(02) -h /(03))6
=
(5-7)
One intermediate variable can be ehminated by rewriting the scheme as
01 = 0n + ~^/(0n)5
(5-8)
02
= 0n + ^ / ( 0 l ) ; 01 = 01 + 202,
(5.9)
02
= 0n + ^^/(02); 01 = 2 (~0n + 01 + 02))
(5.10)
=
(5.11)
0n +l
01 + -^/(02)-
This latter form of the time integration scheme was used in the present code.
5.1.2
Spatial Discretization
A three-point-wide, fourth-order compact finite-difFerence scheme (Lele, 1992; Kloker,
1998) is used for the discretization of spatial first,
r1
f ( l ) I /• 1 f ( l ) I r 1
Ri-lfi-1
r2
e{2)
, r 2 r ( 2 ) _i_ r 2
J-'i-lJi-l + -L'iJi
and second, f^'^\ derivatives
/•(!)_
+ Rifi + Ri+lfi+1 + {L]Jal — RjJao),
^(2)
+-^i+l/i+i —
RlJ^-i + Rffi +
+ {Lpc.2 - RjJao),
(5.12)
87
where I — i -\- 1 \i the jump singularity occurs for Xi < Xa <
In this case one
has /i"*" = Xi+i — Xa and
jI/''"l»(ft+)* + ^l/'''l.(A^)=.
{5.13)
{5.14)
J„ 2 = |/">l„ + l/Wl„A+ + ^l/''"l„(ft+)' + i[/%(A+)'.
(5.15)
If the jump singularity occurs for Xi_i < x ^ < X i , then I — i — 1 . In this case one
has h~ — Xa — Xi^i and
Jo.2 =
^lf'X(h-)' + lilf'X(h~)',
(8.16)
5l/®l.{''")'.
(5.17)
+ l/'='l«ft--i[/''"WA-)'+i[/®)o{''-)'.
(5.18)
When X a falls exactly on a grid-point, the decision must be made as to whether the
function value here is set to the left or right limit. The stencil is then corrected or
not corrected accordingly. For example, if x ^ = X i , and /(xj) = lim^^^+ f { x ) , then
I = i — I . However, if X a — Xj_i and /(xj-i) = lim
+
f { x ) , then no correction
i —1
will be required. The other cases can be handled similarly, and if done consistently,
pose no particular problems.
All jumps are discretized at the immersed boundary using one-sided finite-differences
of an order such that the stencils in equation (5.12) maintain their formal (fourthorder) accuracy if / were known to one order higher (fifth). The term
for
88
example, is discretized to 0{h^),
to 0(/i^), etc. (see the discussion in section
4.3.2), requiring a total of six points in each case.
5.1.3 Stability Analysis
In this section, an eigenvalue analysis of the schemes presented above in section 5.1.2 is
carried out in order to investigate whether or not they will be numerically stable when
solving time-dependent PDEs. The case of pure convection with constant convection
velocity c,
ot
(5.19)
ox
on the domain x € [0,1] is considered first.
The immersed boundary is fixed at x = X a , and for x < X a , the solution is
identically set to zero, cf) = 0. The domain [0,1] is divided into N equal intervals
of step-size h so that Xi — ih, i = 0,1,N, and (pi = (j){xi). It is assumed that,
for the compact differencing, the derivatives (px are known at the boundary points
z = 0 and i = N. Equation (5.19) is rewritten as a system of equations for the
(pi e $,^ = 0,...,A^-|-1,
at
ox
Whereas equations i = 0 , . . . , N are arranged to be in one-to-one correspondence to
the grid points Xj, equation i — N + I represents equation (5.19) at the immersed
boundary point x^The compact difference approximation for
=
IJX
is
+
(5.21)
and Rj; are sparse matrices that hold the compact difference coefficients, and b^:
is a column vector containing the values of
the corresponding rows of
at z = 0, j = A'' and i = N -\-l (so that
and R^^ are filled with 0, except for the {i,i) entry of
89
Lx which is 1). Note, however, that
is written such that there is no dependence
i = 0, • • • , N , on
w h a t s o e v e r of
Equation (5.20) is pre-multipUed by L^. so that it can be rewritten in the semidiscretized form
ot
=-c(R,$ + b,).
(5.22)
Seeking normal mode solutions of the form $ oc exp(c<;t) to the homogeneous part of
the above equation results in
=
where
(5.23)
h
= /iR^- Rearranging this equation yields the generalized eigenvalue prob­
lem
(a;'L, + R,)$ = 0
(5.24)
for eigenvalue uj' = hujic. Stable numerical solutions will require that the real part
of U' lie in the left half of the complex
In general, the eigenvalue
UJ'
UJ'
plane.
is dependent upon the grid size, the interior and
boundary differencing schemes, and the location of the immersed boundary. As such,
a general analysis cannot be given. Instead, the specific case of
= 50 is presented
for an immersed boundary located half-way between grid points Xie and Xjy,
=
1/2 (xie + X17), and also at a location that is almost on top of grid point i = 16,
Xa = 3:16 -I- 1
x
10"^. The result, shown in figure 5.1, indicates that the present
differencing scheme, when applied to the pure convection equation (5.19), will not be
stable, as eigenvalues
UO'
are found in the right half of the
UJ'
plane. This result is
borne out in numerical solutions of equation (5.19).
A similar analysis (with similar notation) of the pure diffusion equation
dcf)
d'^cj)
yields the system
{uj' l x x - Rxx)^ = 0
(5.26)
90
2
1.5
1
0.5
5. 0
E
-0.5
-1
-1.5
-2
-0.06 -0.04 -0.02
0
0.02 0.04
Re (0)')
0.06
0.08
0.1
Figure 5.1 Eigenvalues uj ' of equation (5.24) for the case of pure convection, with
= 50 for Xa = 1/2 (xi6 + xn) = 0.31, (o), and xa = xiq + 1 x 10"^ = 0.3000001,
(x).
for eigenvalues lo' = h^uj/u. The result, shown in figure 5.2, indicates that the im­
mersed boundary scheme is numerically stable for the pure diffusion equation.
Finally, an analysis of the combined convection-diffusion equation
84)
dcf)
yields the system
{u;'l + ReL-'K, -
=0
(5.28)
for eigenvalues uj' = h ^ u j / u , where Re = c h j u is the cell Reynolds number. The
results, shown in figure 5.3, indicate that the numerical scheme will be stable for cell
Reynolds numbers less than some limiting value that depends upon the location of
the immersed boundary. For example, for the results shown in figure 5.3, the largest
cell Reynolds number for which the numerical scheme is stable appears to be greater
for the case where the immersed boundary is located half-way between grid points,
compared with the case where the immersed boundary is almost right on top of a
grid point.
91
:• O • : • • • -
X
(d-;® ^ 0
<XX9>OS38S8a
o •- •
•2.5'
-8
'
-7
'
-6
'
-5
'
^^'
-4
-3
-2
-1
Re (lo')
0
Figure 5.2 Eigenvalues LO' of equation (5.26) for the case of pure diffusion, with N' = 50
for Xa = 1/2 {xiQ + Xn) — 0.31, (o), and Xa — Xi& + 1 x 10^^ = 0.3000001, (x).
0 (DECOOeg^OOOOO'OiOOOXICCCglMM I
Re ((0 )
Re (CO')
Figure 5.3 Eigenvalues UJ' of equation (5.28) for the case of combined convectiondiffusion, with A'' = 50 for x^ — l/2(a:i6 + xii) = 0.31 and cell Reynolds numbers
Re = 10, (o), 100, (x), and 200, (*), left, and
= Xi^ + 1 x 10"^ = 0.3000001 and
cell Reynolds numbers Re = 0.1, (o), 1, (x), and 10, (*) right.
92
5.1.4
Validation
In this section, the vorticity-transport equation (2.7) is solved with u and v specified:
u { x , y, t ) =
sin(/?x) sin(/3y),
(5.29)
v{x, y,t) =
cos(/3x) cos(/5y).
(5.30)
A forcing term is added to the right-hand side of equation (2.7), and boundary con­
ditions are imposed such that the analytical solution is known to be
u j { x , y , t ) = 2/3e~"^ sin(/3x) cos(/3ty).
(5.31)
By comparing the numerical solution of equation (2.7) with the above analytical so­
lution, the spatial and temporal convergence properties of the immersed boundary
scheme, as applied to the convection-diffusion equation, can be experimentally deter­
mined. The numerical solution is computed on the square domain x,y ^ [0,1], and
the immersed boundary is a circle of radius r = 0.1 centered at {x,y) = (0.5,0.5).
Fourth-order stencils are used for the spatial discretization, with fourth-order correc­
tions on the immersed boundary, and a second-order Runge-Kutta method is used for
time integration. A typical result is shown in figure 5.4. Note the sharp interface near
the immersed boundary, and the corresponding smoothness of the error distribution.
Table 5.1 presents the results of the spatial convergence study, and the corre­
sponding computational parameters are given in table A.l. The numerical solution
N
||g|U
llfib
65
129
257
3.113409 X 10"®
1.949443 x 10^^
1.210198 x 10"®
1.101375 x 10"®
6.972817 x 10"®
4.368400 x 10"^
Table 5.1 Error (e = uja—ujn) in the numerical solution of equation (2.7) on grids of size
N X N { o i s t e p - s i z e A x = 1 / { N — 1 ) ) w i t h f i x e d A t . T h e e r r o r i s II^Hoo = ^
+ B,
with /1=51.49, n=4.00, and 5=—1.1875 x 10~^°; lle||2 = A Ax" + B, with A=17.03,
and n=3.98, and
5.2543 x 10"^^.
93
Figure 5.4 Numerical solution of equation (2.7) on a 65 x 65 grid (top), and corre­
sponding error e =
(bottom).
94
At
4x
2 x
1 x
5x
10-5
10"®
10-5
10"®
||£||oo
4.810835 x 10"®
4.807736 x 10"®
4.806962 x 10"®
4.806768 x 10"®
Iklh
1.695369 x
1.694273 x
1.693999 x
1.693930 x
10"®
10^®
10"®
10"®
Table 5.2 Error (£ = /„ — /„) in the numerical solution of equation (2.7) on grids
of size 61 x 61 for various time-steps At. The error is ll^lloo =
+ B, with
yl=2.61325, and n=2.0, and 5=4.8067 x 10"®; ||£||2 = A At" + B , with ^=0.92521,
and n=2.0, and 5=1.6939 x 10~®.
to equation (2.7) is computed on several grids with the time-step held fixed. The re­
sulting numerical solution is compared with the analytical solution, equation (5.31),
and the maximum absolute error is found as a function of Ax : ||e||oo = A Ax"' -I- B.
From the data in table 5.1, the exponent n is found to be 4.00; for ll£l|2, the exponent
n is 3.98. Here, one can see that the loss of accuracy about which had been specu­
lated in section 4.3.2 has not been realized. There it was noted that the immersed
boundary scheme would require that the numerical solution be known to one order
higher than was available.
The temporal convergence study is carried out in a similar fashion. The numerical
solution to equation (2.7) is computed for several different time-steps, this time with
the spatial grid held fixed. Table 5.2 presents the results of the temporal convergence
study, and the corresponding computational parameters are given in table A.2. The
resulting numerical solutions are again compared with the analytical solution, and
the maximum absolute error is found as a function of At : ||£||oo = ^ At" -f B. From
the data in table 5.2, the exponent n is found to be 2.0; for ||£||2, n = 2.0 is also
found. Thus, second-order temporal accuracy is maintained.
95
5.2
The Poisson Equation
The Poisson equation (2.8) for the solution of the stream function ijj is of the form
V'^f{x,y) = p{x,y).
(5.32)
Both Dirichlet, f \ g n ^ = a i x , y ) , and Robin, d f / d n \ g Q ^ = a { x , y ) f { x , y ) + b { x , y ) ,
boundary conditions on dflo will be considered. On the immersed boundary,
only
Dirichlet boundary conditions are considered. In this section, the numerical solution
of the Poisson equation, with and without immersed boundaries, is discussed.
5.2.1
Discretization
The discretization of equation (5.32) is based on two 1-D, fourth-order compact finitedifference stencils^ for second derivatives in x and y:
(5.33)
First, the discretization of equation (5.32) is shown for the case when no jumpcorrections are needed.
Using the two expressions in equation (5.33), together with equation (5.32), the
2-D compact stencils at point (z, j) inside the computational domain are derived as
follows:
^Index notation, employing the standard summation convention, is used throughout with respect
to i and j.
96
where
xxi ^ y y j
(5.34)
)
resulting in a nine-point 2-D stencil
(
Riz
i?12
i?ll
R23
R33 \
Rz2
/ Pi3
R22
R21
R31
\ Pll
1•I
)
P23
P33 \
P12 P22 P32
P21
P31
j
J
)
(5.35)
where the operator (•) is defined such that corresponding elements are to be mul­
tiplied, and the result summed to form a scalar quantity. A similar derivation can
be carried out to determine the 2-D stencil when near a boundary with a prescribed
Robin boundary condition. In this case, one (or both) of the stencils in equation (5.33)
are replaced with
•yj J y j '
where / = 1 or / =
(5.36)
and J — 1 or J — Uy.
When, as shown in figure 5.5 (a), the nine-point 2-D stencil derived above inter­
sects an immersed boundary, jump-corrections to the 1-D stencils in equation (5.33)
must be made:
(5.37)
(5.38)
where
(5.39)
(5.40)
and
(5.41)
97
a""
d aI
d a .I
N
Q"
\'
Q"
\
\
(a)
(b)
Figure 5.5 The intersection of the 9-point stencil (o) given in equation (5.35) with an
immersed boundary (a), and (b), the jump-corrected stencil, equation (5.42), with
jump-correction terms (•). The points where (o) and (•) overlap are represented by
(-)•
and similarly for J (see section 4.1 for a discussion of what happens when
lies on
a grid point). The jumps JaOx, etc., are defined as in section 5.1.2. The resulting 2-D
discretization of the Poisson equation with jump-corrections becomes
{.RxXiLyy^
-f-
LxXiRyyj) fij
L/xXfLyy^ Pij
+
(^Lyy.JcyxIj
~l~
>^01/Jj )1
(5.42)
which can be derived as was done for the stencil without jumps. An example stencil is
shown in figure 5.5 (b), where 6-point, one-sided explicit finite-differences have been
used to compute the required jump-corrections.
98
5.2.2
Resulting System of Equations
The discretization of the Poisson equation without immersed boundaries leads to a
system of equations Ax = b where the matrix A has the following structure
A =
L2
0
0
0
[/i 0
B2 U2
-L3 B 3
0
0
0
0
0
0
Us
0
0
0
0
0
0
Ln^-I
0
Bn^^i
Ln,
Un^-i
Bn,
with tridiagonal blocks Lj, B i , and Ui of size UyXriy, and where i = 1 . . .
(5.43)
sweeping
in j. This system Ax = b is solved using a multigrid technique together with an ILLU
(Incomplete Line LU decomposition) relaxation method (Sonneveld et ai, 1985). The
ILLU has very good convergence properties, even for highly stretched grids. Standard
relaxation methods, for example successive over relaxation (SOR), successive line
over relaxation (SLOR), or Gauss-Seidel, which perform well on equidistant grids
will perform poorly on stretched grids (de Zeeuw, 1997; Botta & Wubs, 1993). The
ILLU decomposition and relaxation procedure will be described first, followed by
a discussion of the multigrid method. As the solution procedure without immersed
boundaries requires only minor modifications to take them into account, the discussion
of the ILLU procedure will begin assuming that no immersed boundaries are present.
Later, the extensions required to include immersed boundaries will be discussed.
In the ILLU decomposition, one attempts to find a matrix D such that
(5.44)
A = (L + D)D-^(D + U)
where
0
L =
La' 2
0
0
0
0
0
0
0 0
L3 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Ln, 0
(5.45)
99
0 Ui
0 0
0 0
U=
0
0
0
0
0 0
U2 0 0
0 Us 0
5
0
0
0
0
0
0
0
0
0 Un.-1
0
0
and
'
D=
D, 0
0 D2
0
0
0
0
0
0
0 0
0 0
Ds 0
0
0
0
0
(5.46)
0
0
0
0
0
0
-1
(5.47)
0
0
Equation (5.44) can be expanded as
A = L + D + U + LD -^U
where LD
(5.48)
is the block-diagonal matrix
'0
0
0
LD^U =
0
L2D1
0
0
0
0
0 0
0
0 0
LsD^'U2 0 0
0
0
0
0
(5.49)
0 0 ^rix Dun­
From equations (5.48) and (5.49) it can be seen that E can
Di = Bi, D i ^ Bi~ LiD^\Ui-i, i = 2,3,... ,n
The matrix
(5.50)
^ is in general full, so the approximation of taking only the tridiagonal
part is made, resulting in algorithm (5.50) being modified as
D i = E l , b i = B i - ix\di&g{LiD~\Ui^i), i = 2 , 3 , . . . ,n^.
(5.51)
The ILLU decomposition of A is now defined to be
A = (L + D)D~^(D + U)+E,
(5.52)
100
where E is the error matrix. With this decomposition, the single-grid iterative method
for solving Ax = b with starting guess x" becomes:
r = b - Ax"
(L + D)c = r
(D + U)e = Dc
x"+i = x" + e
In the present derivation of the ILLU algorithm, a considerable amount of detail
has been left out. An efficient implementation of the ILLU algorithm requires this
additional information, and the interested reader is referred to Sonneveld et al. (1985).
The multigrid components of the Poisson solver will be discussed next (a good
introduction to multigrid techniques can be found in Briggs, 1987). The solver is
based on a standard V-cycle, as a good initial guess solution is always available.
Specifically, when solving equation (2.8), -0""^ from the previous time-step is taken
as a starting guess to compute
at the present time. A template for the V-cycle
scheme is given by
^ HgV(/,p'')
1. Relax NPRE times on
with initial guess /'^ (pre-smoothing relaxation).
2. Iff]'^ is coarsest grid, go to 4 with NPST=NCG, else,
^
/f(p'^-AV'')
f" 4- 0
/2. ^ MgV(/2^p^'^)
3. Add correction term
4. Relax NPST times on
+
— p'^ (post-smoothing relaxation).
101
The nomenclature used above requires some explanation. The superscript h denotes
a grid Q'' one level finer than grid
Grid
is formed here by the standard
coarsening strategy of taking every second grid point of fi''.
The operator
an interpolation operator which takes a quantity from grid
to grid
operator
is
and the
is a restriction operator, which performs the reverse operation. Coarse
to fine grid interpolation
is based on linear interpolation
fi =
hi + h2
and restriction
hi + h2
is based on a second-order filter
^ o2{hi
fh +
1 . hh2)
\
2
o2{hi
fh +
_L hh2)
(5.54)
where h2 = Xi+i—Xi, and hi = Xi — Xi-i. On equidistant grids, the restriction operator
collapses to the standard full restriction operator. Two-dimensional interpolations
and restrictions are based on two applications of the one-dimensional versions, i.e.,
equations (5.53) and (5.54).
The extensions required to included immersed boundaries will now be given. As
discussed in section 5.2, the jump-corrections to the 9-point compact discretization
are solution dependent. This introduces terms into locations (row m, column n) in
the matrix A which, unlike the tridiagonal matrices Lj, i?,, and Ui above, are not at
all regular^. To handle these irregular entries, the following strategy was developed:
A is ILLU decomposed, and ILLUSolve(r) is defined as a function returning the
solution e of
(L + D)D-^(D + U)e = r.
The relaxation procedure then becomes
1. r = b — Ax"
2. e ^ ILLUSolvefr)
The matrix A always denotes a discretization of the Poisson equation without jump-corrections.
102
3. r* <— CiY + C2JumpCorrect(e)
4. e* ^ ILLUSolve(r*)
5.
^ x" + e*
where ci = 0.8 and C2 = 0.2, and the function JumpCorrect adds all jump-correction
terms to the right-hand side of Ax = b given a guess solution x". Specifically, in the
presence of immersed boundaries, to A is added a matrix A' and (A -|- A') x = b is
solved to obtain the jump-corrected solution x to the Poisson equation. The function
JumpCorrect (x"), then, returns the column vector — A'x".
The procedure just outlined was empirically determined, and was found to con­
verge in all cases in which it was employed to solve the Poisson equation with im­
mersed boundaries. It is a convenient algorithm that allows the ILLU decomposition
to be performed on the regular (block tridiagonal) matrix A, yet it easily accommo­
dates the irregularly located entries introduced by immersed boundaries.
As a final note, an estimate of the time penalty associated with adding immersed
boundaries is given. The modified relaxation procedure described above requires one
additional call more to ILLUSolve than the relaxation procedure without immersed
boundaries.
We can estimate, therefore, that a solution of the Poisson equation
with immersed boundaries will probably be about twice as expensive to compute,
if the convergence rate of the multigrid is not affected by the modified relaxation
procedure. Table 5.3 presents run times to compute 200 multigrid V-cycles for the
Grid Size ^
time, Mg without IB
time, Mg with IB
% increase
65
x
65
129
x
129
0.85 s
2.0 s
6.7 s
11.3 s
135%
69.0%
257
x
257
36.0 s
57.2 s
59.0%
Table 5.3 Computation times for test problem of section 5.2.3 using 200 multigrid
V-cycles and various equidistant grids. A maximum of 4 grids is allowed (NGMAX=:4),
NPRE=2, NP0ST=1, and NCG=4.
103
,0
10'
• •
10
0
5
10
Qp^g^af ggouf^Qogf
15
20
25
30
iteration
Figure 5.6 Error in the numerical solution of equation (5.32), ||£||oo, versus multigrid
v-cycle iteration number for various grid sizes with (—) and without (• • •) immersed
boundaries : 65 x 65 (o), 129 x 129 (*), and 257 x 257 (o, •). A maximum of 4 grids
(NGMAX=4) is allowed (except for • where NGMAX=8), NPRE=2, NP0ST=1, and NCG=4.
problem described in section 5.2.3 with and without immersed boundaries. For the
problem sizes of interest, the immersed boundary multigrid solver is seen to require
around 60% to 70% more run time than the solver without immersed boundaries. As
seen in figure 5.6, the convergence rate of the solver is not adversely affected by the
presence of the immersed boundary. The slight deterioration in the convergence rate
at the tenth iteration for the 257 x 257 grid (without immersed boundaries) is due to
the fact that a maximum of 4 grids was allowed (NGMAX=4). Increasing the number of
allowed grids to 8 eliminated this deterioration, which can also be seen in figure 5.6.
104
N
Iklloo
Iklh
41
81
161
3.851207 X 10"^
2.317782 x 10"^
1.428926 x 10"^
1.411451 x 10"^
8.371542 x 10"^
5.175142 x 10"^°
Table 5.4 Error (£ = /„ — /„) in the numerical solution of equation (5.32) on grids of
size N X N (or step-size h = 1/{N — 1)). The error is ||£||oo = Ah^, with A=1.126,
and n=4.04; ||£:||2 =
with A=0.425, and n=4.05.
5.2.3
Validation
The Poisson equation (5.32) is solved numerically, with the right-hand side source
term p{x,y) and boundary conditions chosen such that the analytical solution is
f{x,y) — { —l/Lo)sm{uix)cos{ujy).
(5.55)
The computational parameters used are given in table A.3.
A computed solution and the corresponding error is shown in figure 5.7. We again
note the sharp interface in the solution obtained, and the relative smoothness of the
error distribution in the vicinity of the immersed boundary. Table 5.4 presents the
results of the numerical experiment. The solution and the corresponding error is
computed on several grids, and the errors ||£||cx) and ||e||2 tabulated as a function
of grid size h. We find the error is ||£||oo = Ah^, with n=4.05, and ||e||2 = Ah,"',
with n=4.07. Both measures show fourth-order convergence. Figure 5.8 shows the
corresponding convergence history of the multigrid solver.
5.3 The k-uj Transport Equations
The inclusion of the turbulence model introduced in section 2.4 requires that two
additional transport equations be solved: one for the turbulence kinetic energy /c,
equation (2.29), and one for the specific dissipation rate u, equation (2.30). The
treatment of these two equations is very similar to the treatment of the vorticity
105
Figure 5.7 Numerical solution of equation (5.32), left, and corresponding error e =
right.
t$ - • •
%
%
-
^
° $
ot
O
j O O O O O O O O O O O O O O C )
^ O O O O O O O O O O O O o
* * * * * * * * * * *
iteration
Figure 5.8 Error in the numerical solution of equation (5.32), ||e|loo, versus multigrid
v-cycle iteration number for various grid sizes: 41 x 41 (o), 81 x 81 (o), and 161 x 161
(*)•
106
transport equation. Solutions to the transport equations for k and u) (here again,
u> denotes specific dissipation rate), are numerically determined using the spatial
and (explicit) temporal discretization schemes given above in section 5.1. The only
modification that was required for stability of the explicit time-marching scheme was
that the derivatives dk/dx, dk/dy, du/dx, du/dy had to be computed using firstorder upwind derivatives when used as convective terms. For example, u dk/dx would
require that dk/dx at point {i,j) be computed as {kij — ki^ij)/Axi if Uij > 0, or
{ki+ij — kij)/Axi+i if Uij < 0. Jump-corrections to first order are used:
=
+ i (-[/]„+ 1/'U-)
(5.56)
for backward differences where the immersed boundary lies between grid points i and
z — 1, and
n=
^ (l/l« + [flh*)
(5-57)
for forward differences where the immersed boundary lies between grid points i and
i +\. All other derivatives required by the RANS model were computed using fourthorder compact stencils.
The mixed derivative d'^Ut/dydx was computed using two applications of a first
d e r ivative, first with respect to the x direction, then the y. The value of dvt/dx on
the immersed boundary required for jump-correcting the y derivative was determined
in the same manner as was the wall vorticity (see section 2.3.2).
5.4
Moving Immersed Boundaries
Most of what has been discussed so far for stationary immersed boundaries is also
applicable to moving immersed boundaries. The only special problem that arises
with regard to moving immersed boundaries is when, with regard to figure 5.9, points
such as the one labelled "5b" are uncovered: at time ta, point 5b is inside the im­
mersed boundary; at time tb, it has emerged into the fluid, and is now outside of
107
MC
3
*
1
05a
^
5b
y
K',
4
\
\
\
\
\
and at a later time
Figure 5.9 A moving immersed boundary at time t = t a
t = tb, i
).
the immersed boundary. In the stream function-vorticity formulation, the problem is
to determine a value for the vorticity u) at this point. One approach that has been
used by Udaykumar et al. (2001), for example, is to interpolate from surrounding
points such as grid points 1 through 4, and the immersed boundary point
7. In the
stream function-vorticity formulation, however, the wall vorticity is not known at the
time when point 5b is uncovered, hence u at this point can only be determined by
extrapolation, a numerical method that is often ill-behaved.
The following strategy was developed to handle freshly cleared points, such as
point 5b from the discussion above: to obtain the unknown value of u at point 5b,
a Lagrangian standpoint is adopted in which the unknown value of u is determined
convectively. With D[-)/Dt denoting the convective derivative
-f-u • V(-), one
has from equation (2.7)
Doo
1 _9
— = —
Dt
Re
5
.
5
,
^
>
8
'
As a first step in an explicit predictor-corrector type time integration scheme, the
coordinates of (the Lagrangian) point 5a, Xs^, which is convected to (the Eulerian)
point 5b must be determined:
X5b = X5a + AtUsa,
(5.59)
where A t — tf, — t a , and U5a is the velocity of point 5a at time t a - Once point 5a is
108
determined, the value of lo at point 5b can be determined as
DUJ
Dt
<^56 —
(5.60)
5a
where, again, the derivative D u j / D t is computed at time level ta- Any values required
at point 5a, such as
cjsq
, must be obtained by interpolation from neighboring points.
This is not a problem since all required information is available at time taTo test the strategy, the 1-D convection-diffusion equation for (j){x,t),
84)
/
d'^4>
/r-
with /^=constant, and time dependent convection velocity
u{t) =
AQUJQ
sin(a;o^)
(5.62)
was used. The location of the immersed boundary Xa was varied as
Xa{t) =
X a o -
Ao cos{ujot).
(5.63)
With proper initial and boundary conditions, an exact solution of equation (5.61) can
be shown to be
(l){x, t) =
sin(A;[x + Aq cos(a;ot)]).
(5.64)
The solution with x < x ^ i s considered to be "inside" the immersed boundary, and is
set to zero. Second order approximations in both space and time are used to compute
the numerical solution.
Table 5.5 presents the results of the spatial convergence study, and the corre­
sponding computational parameters are given in table A.4. The numerical solution
to equation (5.61), several time-steps of which are shown in figure 5.10, is computed
on several grids with the time-step held fixed. The resulting numerical solution is
compared with the analytical solution, equation (5.64), and the maximum absolute
error is found as a function of Ax ; ||£||oo = A Ax". From the data in table 5.5, the
109
0.5
0.5
-o-
Qoooooc
-0.5
-0.5
0.2
0.4
0.6
X
t = 0/4T
0.2
0.8
0.5
0.5
-0.5
-0.5
0.2
0.4
X
t = l/4T
0.8
0.2
0.4
X
t^2i4T
0.4
0.6
X
t = 3/4T
0.8
Figure 5.10 Numerical (o) and analytical (—) solution to equation (5.61) using pa­
rameters shown in table A.4. The immersed boundary, whose spatial envelope is
denoted in the figure by ( x — x ) , moves according to equation (5.63) with period T .
no
N
||g||oo
Iklh
41
81
161
4.632733 x 10"^
1.180837 x 10"^
2.787453 x 10"^
2.297667 x 10"^
5.784433 x lO"'^
1.368646 x 10^^
Table 5.5 Error [ e = f a — f n )
the numerical solution of equation (5.61) on grids of
size N X N (or step-size h = 1/{N — 1)) at t = 0.1 with At —8.3333 x 10~^. The error
is ll^lloo = A/i", with A=8.307017, and n=2.027; ||e||2 = A/i", with A=4.221318, and
n=2.035.
At
2.500 x
1.667 x
1.250 x
1.000 x
6.250 x
ll^lloo
10-^
10"^
10"^
10-^
10-®
4.448002 x
4.562156 x
4.603101 x
4.622262 x
4.643223 x
||£||2
10"^
10-^
10-3
IQ-^
IQ-^
2.213553 x 10-3
2.264745 x 10-^
2.283621 x 10-3
2.292595 x IQ-^
2.302743 x i q - ^
Table 5.6 Error (e = /„ — /„) in the numerical solution of equation (5.61) for various
time step-sizes At at t = 0.1 with
= 41 grid points in x. The error is ||£||oo =
A Ar+5, with A=-2.13422x 103, 5=4.6574x10-3, andn=1.946; II^Hz = AAt'^ + B,
with /1=-519.197, 5=2.3100 x 10-3, and n=1.869.
exponent n is found to be 2.0; for ||£||2, the exponent n is also 2.0. Thus, second-order
spatial accuracy is obtained.
The temporal convergence study is carried out in a similar fashion, and table 5.6
presents the results. The numerical solutions are again compared with the analytical
solution, and the maximum absolute error is found as a function of At : ||e ||oo
=
A At"- + B. Prom the data in table 5.6, the exponent n is found to be 1.9; for ||£||2,
n = 1.9 is also found. This small reduction in temporal accuracy, i.e. n = 1.9
instead of n = 2.0, may be symptomatic of a degradation of temporal accuracy by
interpolation, similar to the problem discussed in section 3.4.4 with regard to AMR.
In this dissertation, the investigation of moving immersed boundaries using the
new IBM was limited to 1-D examples. Tests in two and three dimensions will need
to be carried out before the robustness of the moving immersed boundary strategy
Ill
discussed above can be determined. In this regard, it is noted that the CFL criterion
imposed on expUcit time integration schemes will limit the movement of the immersed
boundary to at most one grid-point per time-step. Additionally, higher order schemes
may also prove to be challenging. Nevertheless, the preliminary results presented here
are very encouraging.
5.5
Validation of the Combined N-S/IBM Code
In later sections, the immersed boundary code is used to compute flows for which
comparison with published results, both experimental and numerical, can be made.
In this section, a check is made to verify the zero-divergence of the computed velocity
fields, and that the definition of the vorticity, the vorticity-velocity relation given in
equation (2.5), is satisfied. Each criterion must be satisfied in the limit of vanishing
grid size if one is to claim to have a numerical solution that converges to a solution of
the incompressible Navier-Stokes equations. As discussed in section 2.2, if boundary
conditions are not properly specified, one may obtain a numerical solution whose
velocity field is not divergence-free, and/or whose vorticity is not the curl of the
velocity.
The test case selected is that of steady, uniform fiow past a circular cylinder with
R cd = 40. The computed stream function and vorticity are shown in figure 5.11.
Figure 5.11 also depicts the entire computational domain used. The domain is con­
siderably smaller than that required to accurately simulate a uniform flow in an
unbounded domain (see section 6.1), but is sufficient for the convergence study of
interest in this section. Three different equidistant grids with Ax = Ay = h are used,
halving the grid size for each subsequently coarser grid.
The divergence and curl of the computed velocity fields are computed numerically
using the same routines that have been used to advance the vorticity from the vor­
ticity transport equation. The results of the study are presented in table 5.7 and in
112
h
||V-u ||oo
l|V-u|l2
0.05
0.025
0.0125
1.268327 X 10-^
1.547833 X 10-2
1.578707 X 10-3
1.902672 X 10-3
2.182507 X 10-'^
1.229118 X 10-^
h
lieu - k • (V X u )||oo
||cu-k-(vxu )||2
0.05
0.025
0.0125
2.942548 x 10"^
1.854906 x lO'^
1.169890 x lO^^
4.793289 x 10"^
2.678087 x 10"^
1.951912 x 10"®
Table 5.7 Convergence study of the error in the divergence and curl of the computed
velocity field on grids of size h. The error is e = AK^, with A=1.709 x 10^, and
n=3.16 for ||V • u|loo; A=1.156 x 10^, and ?t.=3.64 for ||V • u||2; A=4.532 x 10^, and
n=4.00 for ||c<; — k • (V x u)||oo; A=6.706 x 10^, and n=3.97 for ||c<; — k • (V x u)||2.
figure 5.12. From the data presented in the table, and the numerical investigations
presented in sections 5.2 and 5.1, one can conclude that the immersed boundary code
produces accurate results that are consistent with the Navier-Stokes equations.
One might wonder why the divergence error shown in figure 5.12 is not everywhere
zero, given that the definition of the stream function should identically enforce zero
divergence.
The reason has to do with the fact that the Robin condition on ^|J,
equation (2.12), is enforced by computing wall vorticity from its definition. In this
way, the Robin condition is only approximately enforced, i.e., the normal derivative of
ip at the immersed surface is only equal to the prescribed boundary velocities (which
are zero here) to 0{h'^). Indeed, the largest divergence error seen in figure 5.12 occurs
at the boundary.
113
spanwise vorticity 00
stream function
ip
Figure 5.11 Uniform flow past a circular cylinder, Re^ = 40. Two figures showing
the full computational domain used for the vorticity definition/divergence-free study
of section 5.5. The dashed line (
) indicates the start of the outflow buffer, xb -
114
Figure 5.12 Divergence-free, top, and vorticity definition error, bottom, distribution
over the upper half of the computational domain for the finest grid (every fourth
point is shown for clarity).
115
6.
Results
A FORTRAN 90 code was written to implement the new immersed boundary method
discussed in chapters 4 and 5. In chapter 5, tests performed using this code showed
that it is not only numerically stable, but also convergent, i.e. the numerical solution
converges to the exact analytical solution to the order of the underlying numerical
scheme. To further validate the code in its application to unsteady incompressible
flows, several test cases are presented in this chapter. Following these quantitative
studies that begin in section 6.1, additional qualitative results are given in section 6.5
to show the range of geometries that can be handled by the code.
6.1
Uniform Flow Past a Circular Cylinder
In this section, the immersed boundary code is used to compute the 2-D flow around
a circular cylinder placed in a uniform free-stream. The salient features of the com­
puted flow fields are compared with results, both experimental and computational,
available in the hterature. Reynolds numbers (Rej) = UooD/u) in the range Re^ = 20
to Re£) = 200 are considered, spanning the steady, steady-to-unsteady transitional,
and unsteady flow regimes. Experimentally, it has been found that the well-known
phenomenon of periodic vortex shedding first appears for Reynolds numbers around
40 to 49 (Tritton, 1959; Belov et al, 1995); for smaller Reynolds numbers, the flow
is found to be steady.
The boundary conditions used in the computations will be discussed first.
At
the inflow, a uniform flow of i/j{xi,y) = y, or, equivalently, u = 1, is specified. The
upper and lower free-stream boundaries are specified as no-penetration, shear-free
walls, 'ip{x,yi) = yi, ipix^y^) = y2, and a; = 0 [uy = 0 and v = 0). The location of
these boundaries can have a significant impact on the calculation if not placed far
enough away. The parameter A, the ratio of the cylinder diameter D to the domain
116
,5
1
1.45
0.5
.4
0
1.35
-0.5
.3
0
15
•1
0
Figure 6.1 Grid refinement study for flow past a circular cylinder with Rep = 200 and
A = 0.056 : C d , left, and C l , right, versus time for three different grids; 641 x 281
(—)> 321 x 141 (
), 181 x 97 (
).
height H — y2 — y i , is introduced as a measure and made small enough to minimize
the influence of the free-stream boundaries. Grid stretching in the ^-direction allows
this to be accomplished with relatively few points. At the outflow, the buffer domain
technique discussed in section 2.3.3 is used to ramp the vorticity down to zero. A
uniform flow at the outflow, 'tp{x2,y) = y, can then be specifled.^
Ramping the
vorticity down to zero using the buffer domain technique was chosen for its robustness
in avoiding outflow problems, such as the reflection at the outflow of downstreamconvected disturbances that cause upstream contamination. Alternatively, one might
have considered using a convective condition on the vorticity at the outflow
Du>
^ = 0.
. .
(6.1)
Whichever method is used, the most important thing is to place the outflow far enough
away from the upstream domain of interest such that the results here are negligibly
influenced by the artificial outflow treatment. Computational efficiency plays only a
secondary role in the selection of an outflow treatment.
The computational parameters used for the investigations described in this section
^In hindsight, ipxx{x2Ty) = 0 at the outflow might have been somewhat "softer".
117
tlx ^
641x281
321x141
181x97
^l
1.370±0.046^
1.368±0.045^
1.383±0.055t
±0.701
±0.701
±0.715
Table 6.1 Grid refinement study for flow past a circular cylinder with Rep = 200 and
A = 0.056: C d and C l for three different grids. Note: f approximate (see text).
are given in table A.5. A grid refinement study for the Re^i = 200 case was carried
out by reducing (in both x and y) the number of grid points cited in this table from
641 x 281 (grid [1]), to 321 x 141 (grid [2]) and 181 x 97 (grid [3]). Note however,
that whereas grid [2] was obtained by taking every second grid point from grid [1],
grid [3] was formed by taking every second point from grid [2], except in the vicinity
the cylinder (x, y) = (xc, yc) ± (0.5, 0.5). The coarsening strategy had originally been
carried out everywhere, but the resolution of the resulting grid [3] was insufficient for
the immersed boundary method to perform properly. On the other hand, refining
grid [1] further would have required considerably longer run times to carry out the
computations. The grid [3] used here was considered to be a reasonable compromise.
Results of the convergence study are given in table 6.1 and figure 6.1, from which
sufficient convergence of the numerical solution can be concluded.
Results for the steady regime of Reynolds numbers are shown in figure 6.2 through
6.4. The attached, steady, symmetric standing eddy behind the cylinder is seen to
grow in length as Re^ increases, and the vorticity generated at the cylinder surface
is less able to penetrate the oncoming free-stream. Table 6.2 presents a comparison
of the present results with those published in the literature. For all quantities of
interest, excellent agreement is found within the scatter of the data.
The lift, C l = F y / p U ^ D ) , and drag, C d = F x / { ^ p U ^ D ) , coefficients per unit
span have been computed using a control volume analysis
1 { Cd , Cl ) = - ^ [ n d V - f u u d S - f p d S + [ d S • " E
2
at JQY
JCS
JCS
JCS
(6.2)
118
where the control surface (CS) (and hence the control volume (CV)) is an arbitrary
rectangle containing the cylinder.
and Fy are respectively the drag and lift forces
on the cylinder, p is the pressure, and S is the (deviatoric) stress tensor. Components
Txx = ( 2 / R e ) d u j d x , Tyy = ( 2 / R e ) d v / d y , a n d Ty^ = r ^ y = ( 1 / R e ) [ d u / d y + d v / d x )
of S are needed. The pressure p can be obtained by solving the momentum equa­
tion (2.1) for the pressure gradient, and integrating the pressure gradient along the
rectangle defining the control surface. The pressure datum can be arbitrarily set to
zero at one corner of the control surface.
119
spanwise vorticity to
stream function ip
Figure 6.2 Uniform flow past a circular cylinder, Re^j = 20. Contours are exponen­
tially spaced.
-1.5
8
9
10
11
12
13
14
X
spanwise vorticity uj
X
stream function tp
Figure 6.3 Uniform flow past a circular cylinder, Reo = 40. Contours are exponen­
tially spaced.
121
8
9
10
11
12
13
14
X
spanwise vorticity u
X
stream function •0
Figure 6.4 Uniform flow past a circular cylinder, Rep = 50 (steady). Contours are
exponentially spaced.
122
X
Figure 6.5 Nomenclature for dimensions given in table 6.2.
Rep = 20
Fornberg (1980)
Dennis &: Chang (1970)
Coutanceau & Bouard (1977)*
Tritton (1959)*
present, A = 0.056
present, A = 0.023
Fornberg (1980)
Dennis &: Chang (1970)
Coutanceau &: Bouard (1977)*
Tritton (1959)*
present, A = 0.056
present, A = 0.023
L
0.91
0.94
0.93
a
b
-
-
0.33
0.46
6
45.7°
43.7°
45.0°
-
-
-
-
-
-
0.93
0.93
0.36
0.36
L
2.24
2.35
2.13
a
b
-
-
-
~
0.76
0.59
6
55.6°
53.8°
53.8°
-
-
-
-
2.23
2.28
0.71
0.72
0.59
0.60
53.4°
53.6°
0.43 43.9°
0.43 43.5°
Rep = 40
Cp
2.00
2.05
-
2.09
2.16
2.06
Cp
1.50
1.52
-
1.59
1.61
1.54
Table 6.2 Steady flow past a circular cylinder: length L of standing eddy behind cylin­
der, locations a and b of the vortex centers, separation angle 6, and drag coefficient
Cp for Rep = 20 and Rep = 40. See figure 6.5 for dimension nomenclature. Note:
(1) (*) denotes experimental results. (2) The results of Coutanceau & Bouard (1977)
cited here are for A = 0, obtained via extrapolation.
123
Berger & Wille (1972)*
Liu e t a l . (1998)
present, A = 0.056
present, A = 0.023
r
5.88-6.25
6.06
5.90
6.02
Berger &: Wille (1972)*
Belov e t al. (1995)
Rogers, Kwak (1990)'''
Miyake e t a l . (1992)'''
Liu e t al. (1998)
present, A = 0.056
present, A = 0.023
r
5.26-5.56
5.18
5.41
5.10
5.21
5.02
5.07
RGD = 100
S t = 1/r
CD
0.16-0.17
0.165
1.35±0.012
0.169
1.38±0.010^
1.34±0.009
0.166
RGD = 200
St = 1/T
CD
0.18-0.19
1.19±0.042
0.193
1.23±0.050
0.185
1.34±0.043
0.196
0.192
1.31±0.049
1.37±0.046t
0.199
1.34±0.044
0.197
-
-
CL
-
±0.339
±0.337
±0.333
CL
-
±0.64
±0.65
±0.67
±0.69
±0.70
±0.69
Table 6.3 Unsteady flow past a circular cylinder; period r, Strouhal number S t , drag
coefficient C d , and lift coefficient C^. Comparison of present results with results
pubUshed in the literature. Note: (1) (*) denotes experimental results. (2) f in Belov
et al. (1995). (3)|approximate (see text). (4) St determined from time variation of
C l.
As noted above, the case Reo = 50 lies at the upper range of the 'transition' regime
between steady and unsteady flow. This case, shown in figure 6.4, was allowed to run
for a considerably long time, but remained steady for the course of the calculation.
Given enough time, the calculation may have eventually become unsteady on its
own due to round-off or other sources of numerical noise. Instead, a small pulse
was introduced into the recirculation zone behind the cylinder, and the calculation
continued. As seen in figure 6.7, the flow becomes unsteady with the formation of
a vortex street behind the cylinder. By predicting a steady/unsteady 'transition'
near Rej^ = 50, further proof has been obtained that the immersed boundary code is
capable of accurately modelling the physics of the flow.
Flows with higher Reynolds numbers are also shown in figure 6.7. Here the flow
becomes unsteady on its own, that is, without requiring any external disturbance
124
(a)
(b)
Figure 6.6 (a) Co and, (b), Cl versus time t for uniform flow past a circular cylinder,
Rec = 200, A = 0.023.
to trigger the instabihty. The computations are started with an inviscid solution as
initial condition, which is essentially equivalent to impulsively starting the cyhnder
from rest. A symmetric recirculating eddy then forms behind the cylinder, and grows
in length until the point in time at which unsteady vortex shedding of frequency / (or
period r) sets in. Table 6.3 compares the Strouhal number St = fD/Uoo, lift coeffi­
cient C l , and drag coefficient Co obtained in the present study with results published
in the literature. As for the steady flows, good agreement is found. For domains with
A = 0.056, the drag coefficient was found to exhibit behavior that was somewhat
irregular, as seen for example in figure 6.1. The deviation was more pronounced for
lower Reynolds numbers. The hft coefficient, however, was almost unaffected. Where
irregularities occurred, data values cited have been labelled approximate, and were
obtained as the mean and first harmonic of a Fourier analysis of the time series. For
domains with A = 0.023, this irregularity disappeared, and both the lift and drag
coefficients exhibited regular sinusoidal behavior, as in figure 6.6.
Close-up views of the solution in the vicinity of the immersed boundary are shown
in figures 6.8 through 6.10 for Re^ = 20 and Rep = 200. The solutions are presented
such that their behavior near the immersed boundaries can be seen. In all cases, the
125
sharp interface near the immersed boundary is evident, especially for the vorticity
seen in figure 6.10.
126
Figure 6.7 Spanwise vorticity uj. Uniform flow past circular cylinders for, top to
bottom, Reo = 40, Re^) = 50 (unsteady), Re^i = 100, and Reo = 200. Contours are
exponentially spaced.
127
0
9
RgjP = 20
0
9
R,e£) = 200
Figure 6.8 Close-up view of the streamwise velocity u in the vicinity of the cylinder
surface for Re^ = 20 and Re^ = 200. Exponentially spaced contour levels are given
below the mesh plot. For clarity, every second grid-point shown in mesh plots.
128
Re£) — 200
Figure 6.9 Close-up view of the normal velocity v in the vicinity of the cylinder surface
for Reo — 20 and Re^ = 200. Exponentially spaced contour levels are given below
the mesh plot. For clarity, every second grid-point shown in mesh plots.
129
0
9
Re£) = 20
0
9
Re/) — 200
Figure 6.10 Close-up view of the spanwise vorticity uj in the vicinity of the cyhnder
surface for Reo = 20 and Rej^ = 200. Exponentially spaced contour levels are given
below the mesh plot. For clarity, every second grid-point shown in mesh plots.
130
6.2
Tollmien-Schlichting Waves in a Blasius Boundary Layer
In this section, the phenomenon of Tollmien-Schlichting waves in a zero pressuregradient, flat-plate boundary layer is revisited. The results are computed with both
a body-fitted code (the temporally and spatially fourth-order accurate code nst2d of
Meitz & Fasel, 2000) and the new jump-corrected immersed boundary scheme.
The setup is similar to that shown in figure 3.14 where the immersed boundary
runs from the inflow to the outflow boundary. The immersed boundary does not
lie on top of grid points, but rather is situated half-way between them. Tests were
conducted with the immersed wall as close as 1 x 10"® non-dimensional units away
from a horizontal row of grid-points. The resulting solution was, for all practical
purposes, identical to the case where the immersed wall was situated half-way between
grid-points. The free-stream is placed roughly 11 boundary layer thicknesses away
from the wall, and the boundary condition here is taken as ipy = 1 in equation (2.8).
The outflow condition is set, somewhat less than optimally^, to ipxx = 0, so that one
may obtain Tp here by solving the ODE ipyy — u. This is equivalent to setting
= 0,
an approximation that is only vahd for Re^; » 1. However, the outflow is placed
far enough away so that the adverse upstream-infiuence is minimal. Computational
parameters are given in tables A.6 and A.7.
As a first step, the undisturbed flat-plate boundary layer is computed, starting
with the Blasius similarity solution as initial condition and converging to a steadystate. Figures 6.11 through 6.13 compare the resulting u, v, and uj profiles at various
x-locations along the plate. Excellent agreement between the body-fitted and im­
mersed boundary code is found. As expected, the sharp interface in u at the wall
is captured, in contrast to the results shown in figure 3.17. Part of figure 3.17 is
reproduced in figure 6.14 to show the contrast.
Next, small, time-harmonic disturbances are introduced into the steady boundary
'^''Pxxx — 0, or, equivalently, v^x = 0, would have been a better approximation.
131
layer at a given upstream location near the inflow. In the body-fitted code, the
disturbances are generated by a narrow blowing-and-suction strip at the wall between
X = Xi a n d x = X2
v^{x,t) = Avs{^{x))
(6.3)
where
xi<x<x,:
X C < X < X 2 :
v M ^ ) ) = ^(729e^ -1701^^ + 972^^),
V s i
^(x) =
Xc
( { x ) ) = -^(729^5 - 1701^^ + 972f), ^ ( x ) -
(6.4)
(6.5)
For the immersed boundary code, it turned out to be easier to add a volume forcing
term to the ^/-momentum equation^
f { x , y , t ) = A s m { L o t ) e ~ ( ^ ^ Vs{C{x)).
(6.6)
An equivalent forcing term for the vorticity transport equation (2.7) is obtained as
df/dx.
If the disturbances introduced are small enough, their behavior is, to a good ap­
proximation, linear, and the resulting velocity and vorticity perturbations can be
compared with the results from hnear stability theory (LST, see Schlichting, 1979;
Mack, 1984). LST predicts a range of unstable frequencies for which wave-like per­
turbations of a laminar boundary layer will propagate and exponentially grow. These
wave-like perturbations are known as Tollmien-Schlichting waves (T-S waves).
The results of the time-harmonic forcing are shown in figures 6.15 through 6.19.
Figures 6.15 through 6.17 depict the amplitude envelopes of the u, v and to dis­
turbance quantities. Assuming linearity, these quantities are obtained as the first
harmonic of a Fourier series with the fundamental frequency equal to the forcing fre­
quency. Numerically, these values are computed with an FFT applied to the data
^Adding a non-zero i;-velocity at the immersed wall would have worked just as well, but would
have required more programming.
132
time series. The agreement between the body-fitted results and the immersed bound­
ary results is again excellent. Once more it is noted that the sharp interface in a; is
captured perfectly by the immersed boundary scheme. A comparison of the phase
distribution for u is shown in figure 6.18 for reference, where excellent agreement is
also found. Finally, figure 6.19 shows the spatial development of the inner maximum
of the disturbance «-velocity amplitude envelope, along with the prediction of LST.
The results agree favorably, and in addition, agree well with those of Fasel & Konzelmann (1990). For comparison, results from the old IBM are also shown, to indicate
the improvement that has been obtained using the new IBM. For similar resolution,
the new IBM is seen to be considerably more accurate than the old IBM. It may be
possible to improve the results of the old IBM somewhat by increasing grid resolution
in the vicinity of the immersed wall, however this was not investigated. More inter­
esting is the fact that the new IBM appears to predict T-S wave growth to the same
accuracy as the body-fitted N-S code. Because the growth of the small amplitude
disturbances depends on the fine details of the instantaneous gradients of the u and
V
velocity components near the wall, the growth rate is a very sensitive indicator for
the near-wall accuracy of the simulation. Insufficient near-wall accuracy can dete­
riorate the solution considerably, and, in the worst case, change the flow behavior
from unstable to stable or vice versa. This sensitivity to near-wall accuracy makes
the accurate prediction of T-S waves a crucial test for any IBM method. Based upon
its performance in predicting T-S wave behavior, it can be expected that the new
IBM could be used to carry out accurate fiow control simulations, in particular those
schemes that take advantage of T-S-type instabilities.
One might consider repeating this test with the flat plate at some angle ^ 0 with
respect to the underlying computational box. This would preclude any unforseen
fortuitous agreement that might come from the plate being aligned with the under­
lying computational grid. Unfortunately, the configuration of the angled plate would
produce jump-corrections that require points outside of the computational domain
133
near the inflow. With some additional modifications to the IBM code, however, the
calculation could be carried out.
6.3 Turbulent Flat Plate Boundary Layer
In this section, the immersed boundary code with the RANS turbulence model is used
to compute the mean turbulent flow over a flat plate with zero pressure gradient. The
setup is again similar to that shown in figure 3.14 where the immersed boundary runs
from the inflow to the outflow boundary. At the inflow, mean turbulent profiles,
obtained from a boundary layer solver (i.e., a code designed to solve the turbulent
boundary layer equations; see Wernz, 2001, for details), are prescribed for all depen­
dent variables. These inflow profiles are then copied parallel to the immersed wall,
as discussed in section 2.3.4, and used as initial conditions for the entire flow.'^ The
validation of the immersed boundary code, with the RANS turbulence model, is then
to see whether or not the downstream behavior of the flow is correctly predicted. In
presenting the results, the following non-dimensional quantities will be useful:
t/+ = —, y+ = ReUrV,
U -t
(6.7)
where Ur is the so-called wall-friction velocity
Ut = \/^,
and
(6.8)
is the (non-dimensional) wall shear-stress (r^^ = T*/{pU^)).
For turbulent fiows, both internal and external, three distinct regions are discernable in the flow near a wall: an inner, outer, and overlap layer (White, 1991). In the
inner layer, dimensional analysis yields
U* = /(!/+),
(6.9)
"^That is, d(t>ldx = 0 for any dependent variable (p a,t t = 0. Of course, a better initial condition
could have been obtained by using the spatially-developing results from the boundary layer solver.
The initial condition actually used here, namely, the inflow copied parallel to the wall, was selected
to test the robustness of the IBM/RANS code.
134
» »» $ »
0.8
0.6
0.4
0.2
0
0.01
0.03
0.02
0.04
0.05
y
0.2
0.15
0.1
0.05
-0.05
-2
y
X 10 ®
Figure 6.11 Comparison of it-velocity profiles at several downstream locations; x =
1.214 (R5^ = 600) (o), .X = 2.157
= 800) (o), and x = 3.372
= 1000) (*).
Symbols indicate values computed using an immersed wall located at ^ = 0, and solid
lines results from the code nst2d of Meitz & Fasel (2000).
135
2.5
)OOQQOOO O O O O O O O
^OOOOOOO 0 0 0 0 0
>
0.5
-0.5
0
0.01
0.03
0.02
0.04
0.05
y
x 10'"
" V V V™ v
Figure 6.12 Comparison of velocity profiles at several downstream locations: x =
1.214
= 600) (o), X = 2.157 {Rs, = 800) (o), and x = 3.372
= 1000) (*).
Symbols indicate values computed using an immersed wall located at y — 0, and solid
lines results from the code nst2d of Meitz & Fasel (2000).
136
120
100
o o o o o o o o o o o o o o-
80
00000000 0 0 0 0 0 0
60
40
20
0
0
1
3
4
Figure 6.13 Comparison of a;-vorticity profiles at several downstream locations: x =
1.214 {Rs, = 600) (o), X = 2.157
= 800) (o), and x = 3.372 (% = 1000) (*).
Symbols indicate values computed using an immersed wall located at ^ = 0, and solid
lines results from the code nst2d of Meitz & Fasel (2000).
137
100
80
60
o*
40
20
0
-4
0
-2
4
2
6
Y
X10-'
120
100
ooooooo o o o
80
OOOOOO 0 0 0 0 io 0 0 0
60
40
20
0
0
1
y
2
3
.-3
Figure 6.14 Comparison of cj-vorticity profiles (zoom-in near the immersed wall) orig­
inally shown in figure 3.17 (the old IBM), top, and figure 6.13 (the new IBM), bottom.
Unlike the new IBM, the old IBM is seen to smear out the sharp jump near the im­
mersed wall.
138
X 10"
-1
-2
X 10
Figure 6.15 Comparison of disturbance it-velocity profiles (fundamental) at several
downstream locations: x = 1.214
= 600) (o), x = 2.157 (R^j = 800) (o), and
X = 3.372 (R^j = 1000) (*). Symbols indicate values computed using an immersed
wall located at y = 0, and solid lines results from the code nst2d of Meitz & Fasel
(2000).
139
2.5
>
0.5
-0.5
0.02
0.01
0.03
0.04
2
3
0.05
y
5
4
3
2
1
0
0
1
4
,-3
X 10
Figure 6.16 Comparison of disturbance velocity profiles (fundamental) at several
downstream locations: x = 1.214
= 600) (o), x = 2.157
= 800) (o), and
X = 3.372 (R^j = 1000) (*). Symbols indicate values computed using an immersed
wall located at ?/ = 0, and solid lines results from the code nst2d of Meitz & Fasel
(2000).
140
0.6
0.5
0.4
0.3
N
3
0.2
0.1
0
-0.1
0.6
0.5
-
-0.1
-
2
i
-
'
1
0
'
y
1
'
2
i
3
4
X10-'
Figure 6.17 Comparison of disturbance w-vorticity profiles (fundamental) at several
downstream locations: x = 1.214 (R^^ = 600) (o), x = 2.157 (R^j = 800) (o), and
X — 3.372 (R^^ = 1000) (*). Symbols indicate values computed using an immersed
wall located at j/ = 0, and solid lines results from the code nst2d of Meitz & Fasel
(2000).
141
000000 0 0 <> 0 0 0 0 0 0
***** * * * *
aoooooooooooooooooooo o o o o o o
0
0.01
0.02
0.03
0.04
0.05
y
Figure 6.18 Comparison of disturbance w-velocity phase profiles (fundamental) at
several downstream locations; x = 1.214 (R^j = 600) (o), x = 2.157 (R^j = 800)
(o), and X = 3.372 (R^^ = 1000) (*). Symbols indicate values computed using an
immersed wall located a,t y — 0, and solid lines results from the code nst2d of Meitz
& Fasel (2000).
142
3
2.5
8.93e-2
2
O
< 1.5
<
<2,23e-2
4.46e-2
_i
0.5
0
-0.5
700
600
900
800
1000
R.
1
0.8
<° 0.6
<
§• 0.4
_i
0.2
0
500
600
700
800
900
1000
1100
Figure 6.19 Lower figure: comparison of the growth rates of a 2-D T-S wave { A —
inner maximum of M(i,o),) for frequency F =1.4 x 10~^: LST (—), an immersed wall
(
), and results from the code nst2d of Meitz & Fasel (2000) (o). Upper figure:
figure 3.21, representing results from the old IBM (here, - • - represent results from
the old IBM,
from nst2d, and — from LST), is reproduced here for comparison
with results from the new IBM.
143
while in the outer layer
(6-10)
Ur
where 5 is the boundary layer thickness, and ^ is a pressure gradient parameter
^ ^ 5 dpe
Tw d x
6.11)
Equality of these two functions in the overlap layer requires that the [/-velocity vary
logarithmically here
[/+=-log?/+ + C,
K
(6.12)
where experimental data indicate that C ~ 5.0 and k, ~ 0.41.
That the RANS immersed boundary code is able to reproduce the above mentioned
characteristics of a turbulent boundary layer is confirmed in figures 6.20 through
6.23. In figure 6.20, the functional relationship of U versus y in the inner, outer,
and overlap regions appears as discussed above: the f/"*" versus
profiles collapse to
the logarithmic variation of equation (6.12) in the overlap layer (the upper figure),
while in the outer layer U plotted against y/5 collapses to one curve, as predicted by
equation (6.10) (the lower figure).
The wall skin friction C/ as a function of Re^ is
compared with a theoretical analysis (Hopkins &: Inouye, 1971) in figure 6.22 where
good agreement is found. The variables k'^ and uj'^ are presented in figure 6.23 for
reference.
For further validation, a similar RANS calculation was carried out for the turbu­
lent wall jet over a fiat plate (see Wernz, 2001, for details on the physics of turbulent
wall jets). The setup for the IBM is almost identical to the case described above for
the turbulent flat plate boundary layer. A typical result is shown in figure 6.24 in
the form of a [/"•" versus y'^ plot at one location along the fiat plate. Similar good
agreement was found over the entire computational domain.
Unlike the laminar calculations, variables and boundary conditions in the RANS
equations are dependent upon wall distance. In particular, the boundary condition
144
for oj at the wall was given in equation (2.32) as
UJ
where
60
ReA(Ay)2'
is the distance from the wall to the wall-next grid-point. The blending
function Fi, seen for example in equation (2.30), is determined as
Fx = tanh(arg^),
(6-14)
where
arg^ = mm max •
\/k
500
\
4 (7^2 k
(6.15)
0.09ujy^ Rey'^uj I '
and y is the distance from the grid-point at which the above expression is evaluated
to the wall. Because the immersed boundary is allowed to come arbitrarily close to
any given gridpoint, y and Ay in the above expressions must clearly be limited, e.g..
Ay = max(Aytrue) c)) where e is some prescribed limiting value.
An example of limiting wall-distance is shown in figure 6.25. The base-line case
is given by the sohd line (—), and is the same turbulent flat-plate boundary layer
calculated above: the immersed flat plate is located half-way between grid-points yiQ
and yii, i.e. y^ = (l/2)(?/io + ?/ii)) and the proper wall distance y and Ay are used, i.e.
— Vn ~ Vu) and y at yn is also equal to yn — y^. For the other two curves, y and
Ay are limited in magnitude to Ay = yn — yio. For the curve (
), the immersed
wall is placed at 1 x 10~® non-dimensional units below grid point j = 10. The curve
(- • -) is again placed half-way between gridpoints yio and yn. Limiting Ay and
y in this way appears to be a reasonable approximation, as the difference amongst
the three curves is well within acceptable error, given the approximate nature of the
RANS model.
6.4
Turbulent Flow over a Step
The step geometry is specified as
y ( x ) = y 2 + ( l / 2 ) ( y i - y 2 ) ( c o s ( 7 r - ^ — — ) + 1 ) , x i < x < X2,
X2 — X i
(6.16)
145
b 15
1/.41 Log(y"') + 5.0
-5
3
8
0.5
Figure 6.20 RANS results for the zero pressure
pressure gradient
gradient, turbulent flat-plate
flat- plate boundary
rr-L
I
' layer using the model of Menter (1994). U'^
versus y+, top, and
~ [/)/•Ur versus
y/6, bottom, for Re^* = 5020 (o) and Re^ = 7575 (
146
30
25
20
3 15
10
5
10'
10
,2
10'
,3
10'
,4
10
Figure 6.21 RANS results for the zero pressure gradient, turbulent fiat-plate boundarylayer:
versus
for Re^i = 5020. Present IBM results using the model of Menter
(1994) (—) compared with a body fitted code using the k-u model (o). Note: (1)
Every 4th point of the body-fitted data is plotted. (2) * denotes first point next to
wall from IBM/RANS results.
with xi = 1.0, X2 = 1.04, yi = 0.02, y2 = 1.25 x 10"^. At the inflow, a zero-pressure
gradient, turbulent flat-plate boundary layer profile with Re^i = 2400 is specified; the
global Reynolds number is set to Re^ = 10^.
6.5
Additional Examples
In this section, several different geometries are presented to give an indication of the
flexibility of the immersed boundary code. The first example, shown in figure 6.29, is
that of flow over a curved waU or ramp. The curved section of the wall is described
by half of a cosine wave, and connects two flat-plate
walls, one at the inflow, and
one at the outflow. At the inflow of the domain, a zero-pressure gradient, flat-plate
boundary layer is prescribed, and the free-stream boundary condition is a slip-wall.
The incoming boundary layer separates near the top of the ramp, forming a free
shear-layer and a recirculation zone. The free shear-layer is highly unstable with
147
4.5
3.5
3
2.5
1000
2000
3000
4000
5000
6000
7000
8000
9000
Figure 6.22 RANS results for the zero pressure gradient, turbulent fiat-plate boundary
layer. Skin friction c/ versus Re^: RANS results using the IBM and model of Menter
(1994) (o), k-u RANS results using a body-fitted code (x), and theory (—) from
Hopkins & Inouye (1971).
respect to disturbances in the flow, resulting in the vortical structures seen in the
figure. Further downstream, the flow can be seen reattaching to the wall.
The second example is that of uniform flow past a 6:1 ellipse. In figure 6.30, the
ellipse is shown at zero angle of attack. A vortex street, similar to that seen for the
circular cylinder in section 6.1, is seen behind the eUipse. The start-up flow around
the ellipse at a 30° angle of attack is shown in figure 6.31. The ellipse is moved
impulsively, so the initial condition is taken as the inviscid solution. A separation
bubble forms on the upper side of the ellipse, and grows as time progresses. The
starting vortex which is shed is shown in figure 6.32. Eventually, a vortex street
forms behind the ellipse, as can be seen in figure 6.33.
Finally, a higher Reynolds number case (Re//=5000) is shown in figure 6.34 to
avoid leaving the impression that the IBM code is only suitable for low Reynolds
numbers. The case presented is for flow over a bluff body that sticks out of the inflow
148
3.5
2.5
+
0.5
,+
„ x
2|
10®
1.5h
"3 n
Figure 6.23 RANS results for the zero pressure gradient, turbulent fiat-plate boundary
layer using the model of Menter (1994):
versus y~*~ (top) and
versus y'^ (bottom)
for Reg = 5020 (o) and Re^ = 7575 (x).
149
20
1/.41 Log (y'^) + 5.0
=D 10
y,+
Figure 6.24 RANS results for the turbulent wall jet ; U~^ versus
. Present results
(—) compared with results of Wernz (2001) (o). Note: (1) Every 4th point of data
from Wernz (2001) is plotted. (2) * denotes first point next to wall from IBM/RANS
results.
30
25
3 15
y,+
Figure 6.25 RANS results for the turbulent boundary layer : U~^ versus y~^. For (—),
the immersed wall is located half-way between grid-points, y^ — (1/2)(yn -|- yw)The actual Ay = yn — y^ was used to determine u) at the wall, as well as wall
distance for Fi and Ut in the RANS model. For the other two curves. Ay is hmited
to Ay = yu — yiQ, and the immersed wall is located at y^, = yw — 1 x 10~® (
),
and yw = (1/2) (j/u + yw) (~ • ~)- Curves are shown at the same location in x.
150
0.08
0.06
>• 0.04
0.02
0.08
0.06
>-0.04
0,02
0.96
0
0.08
0.06
>. 0.04
0.02
0.96
0.98
1
1.16
Figure 6.26 Streamlines for turbulent flow over a step. Shown are, from top to bot­
tom, streamlines, contours of vorticity, and [/-velocity profiles. Note: contours are
exponentially spaced, and [/-velocity magnitudes have been reduced by a factor of
10.
151
0,014
0.012-
0.008
0.006
0.004
0.002
0
0.6
Figure 6.27 Turbulent kinetic energy k for turbulent flow over a step.
x=1.0
,-2
>-10'
x=1.04
x=1.08
10 "L-0.2
0.2
0.4
0.6
0.8
u
Figure 6.28 Comparison of [/-velocity profiles at several x-locations for turbulent flow
over a step: IBM method with SST model (—), and k-uj model (
), compared
with compressible code results
152
of the computational domain. Zero pressure-gradient, fiat-plate flow is prescribed as
an inflow on the top and bottom surfaces of the body. The flow separates at the blunt
base, and forms a vortex street downstream of the body. Part of the computational
grid used in this calculation is shown in figure 6.35, where it can be seen that, because
an AMR approach has not been implemented, the high resolution near the wall is
carried downstream and into the free-stream, where it is not required. For such higher
Reynolds number cases, then, the problem becomes to provide enough resolution for
the computation to capture thin shear layers near the wall, but not to the point that
the wasted resolution elsewhere causes the calculation to become too big in terms of
the number of grid-points involved.
153
spanwise vorticity to
passive scalar
\
1
—P
\
-
-111 1
12
16
20
\
(i(
24
X
\ <
28
J
32
y
36
streamwise velocity profiles u
Figure 6.29 Flow over a curved wall, Ren = 2500. Shown are instantaneous values
of spanwise vorticity u, a passive scalar introduced at the wall between x = 12 and
X — lb (diffusivity a = 0.001), and streamwise velocity profiles u.
154
4
2
>< 0
-2
-4
5
10
15
20
25
30
35
40
30
35
40
x
spanwise vorticity u
4
2
0
-2
-4
5
10
15
20
25
x
passive scalar
3
2
1
>>
0
_
-1
-2
-3
4
6
8
10
12
14
16
18
20
X
streamwise velocity profiles u
Figure 6.30 Flow over a 6:1 ellipse at zero angle of attack, Rec=2400. Shown are
instantaneous contours of spanwise vorticity u, a passive scalar (with diffusivity a =
0.01) introduced at the wall, and streamwise velocity profiles u.
155
3
2
1
0
1
•2
-3
6
X
3
2
1
0
1
-2
-3
6
10
X
3
2
1
0
1
-2
-3
6
8
X
Figure 6.31 Startup flow around a 6:1 ellipse (Rec = 300) at a = 30° angle of attack
that has been impulsively set into motion. The time interval between each figure is
At = 3.0.
156
4
1
1
1
1
1
r
3-
-3 _4
6
I
I
1
1
1
1
8
10
12
14
16
18
20
x
Figure 6.32 Startup flow around a 6:1 ellipse (Rec = 300) at a = 30° angle of attack
that has been impulsively set into motion. Shown are instantaneous contours of
vorticity.
>•
0
f
Figure 6.33 Vortex street formed by flow around a 6:1 ellipse (Re^ = 300) at a = 30°
angle of attack. Shown are instantaneous contours of vorticity.
157
3
2
1
1
-2
-3
40
42
44
46
48
50
x
Figure 6.34 Flow over a bluff body, Rei/=5000. Shown are instantaneous contours
of vorticity, top, stream function, middle (a zoom-in plot is provided to the right
of the main stream function plot to show detail), and stream wise w-velocity profiles,
bottom.
158
Figure 6.35 Computational grid for results shown in figure 6.34. For clarity, every
fourth grid point is shown.
159
7.
Conclusions
In this dissertation, a high-order immersed boundary method for the computation
of unsteady, incompressible fluid flows on two-dimensional, complex domains was
proposed, analyzed, developed and validated. The new jump-corrected IBM was
proposed to remedy problems in an older existing method that had originally been
selected for use in numerical flow control investigations.
In particular, the older
method suffered from considerably reduced accuracy near the immersed boundary
surface where sharp jumps in the solution, i.e., jump discontinuities in the function
and/or its derivatives, were smeared-out over several grid points. Results obtained
using the old IBM tended to show a considerable amount of "noise" at the immersed
boundary surface. The large error is a result of the nature of the forcing terms used
to form the surface inside the computational domain. In the absence of AMR, the
resolution available in 2-D was usually only just sufficient to obtain qualitatively good
results; in 3-D, the situation was considerably worse. In hindsight, the use of compact
finite-differences in the old method did not help to improve the situation. Compact
differences, with their spectral-like numerical performance, should not, in general, be
directly applied to functions with discontinuities.
To avoid smearing-out jump discontinuities, a sharp interface method, originally
developed by LeVeque & Li (1994) and Wiegmann & Bube (2000) in the context of
elliptic PDEs, was introduced where the numerical scheme takes such discontinuities
into consideration in its design. By comparing computed solutions to jump-singular
PDEs with their known analytical solutions, the new IBM was shown to maintain the
formal accuracy, in both time and space, of the underlying finite-difference scheme
used. In addition, the method was shown to be numerically stable.
Further validation of the new IBM code was accomphshed through its application
to several two-dimensional flows for which results are available in the literature for
160
comparison. The first test case was that of flow past a circular cylinder. This case was
selected because of the abundance of results available in the literature which could
be used to compare to present results. In addition, a circular geometry contrasts well
with the underlying rectangular Cartesian grid, bringing out the point that the body
is indeed an immersed one. All of the parameters selected for study, e.g. Co, St,
etc., were found to agree well, within the scatter of the data, with values pubUshed in
the literature. The second physical test case selected was that of T-S waves in a flatplate boundary layer. That the new IBM is able to pass this test is important because
it means the method can be used to accurately carry out flow control simulations,
particulary where the flow control strategy exploits flow instabilities such as the TS instability. Flow control applications were identified earlier as one of the main
motivations for the development of the new IBM.
With the feasibility of a fourth-order accurate IBM code established, there remain
several areas that still need to be investigated before the code will be capable of being
used for CFD production runs. Each topic listed below may require a considerable
amount of effort for its implementation. For this reason, the list is presented for
future consideration.
i. Adiaptive Mesh Refinement (AMR). The idea of adaptive mesh refinement
has already been introduced in section 3.4.4. AMR will be, without a doubt,
an important feature of any future IBM research code. Indeed, an IBM code
suitable for use in production runs will require AMR for all but the simplest
geometries and/or lowest Reynolds number investigations. The reason for this
is that, to resolve thin boundary layers, grid clustering usually is required close
to the surface. A canonical example is shown in figure 7.2 for a body-fitted grid
that might be used to compute the flow past a circular cylinder. For the same
geometry, a stretched Cartesian grid is considerably less efficient at improving
near-wall resolution, as seen in figure 7.3. AMR can be used to locally refine
161
the grid, without clustering points elsewhere in the domain where they are
not needed. This unnecessary clustering of grid points considerably increases
the size, and therefore, the run-time of CFD simulations. To further increase
computational efficiency, AMR with explicit time integration schemes will most
likely need to be combined with local time stepping in domains with high spatial
resolution.
ii. Arbitrary Geometry. The geometries investigated in this dissertation were
intentionally hmited to those which could be described by simple analytical
expressions: wall-bounded flows had the wall height y relative to some datum
as a function of x, i.e. y = y{x), while bodies immersed in a uniform free-stream
were described by polar functions r = r{9). Future implementations would
have to be more flexible in terms of allowed geometries, and, eventually, accept
arbitrary geometries. One additional difficulty that arises when the geometry
is not restricted is that points such as the one labelled o in figure 7.1 will cause
difficulties when there are not enough points on either side to compute jumpcorrections ™ in this particular example, jump-correcting x-derivatives will be
difficult. The simplest remedy would be to further refine the grid in the x
direction. However, the same problem would now arise closer to the immersed
boundary at larger y. A second possibility would be to use a 2-D finite-difference
stencil to compute the jump correction. In this case, the logic that determines
which points to include in the finite-difference stencil will be considerably more
complicated than the one currently implemented.
iii. Extension to 3-D. The extension of the IBM to 3-D should present no par­
ticular difficulties beyond the increased complexity required to determine in­
tersections of the immersed boundary with the underlying Cartesian grid, and
the corresponding jump corrected finite-difference stencils. In addition, a suit­
able formulation of the Navier-Stokes equations will have to be selected. As
162
X
Figure 7.1 At the point o, an x-derivative requiring jump-correction would cause
problems. As before,
denotes the domain outside of the immersed boundary dQ^,
and
the inside.
mentioned earher, the stream function-vorticity formulation used in the present
investigation is only valid in two dimensions.
iv. Moving Immersed Boundaries. The application of the IBM to moving
boundaries was only briefly touched upon in section 5.4.
It is in the area
of moving boundary simulations where the IBM really becomes a very com­
petitive numerical method. The main difficulty, as described in section 5.4,
appears to be determining values for freshly uncovered grid points. The logic
required to update the underlying finite-difference stencils and corresponding
jump-corrections may also prove to be challenging.
163
Figure 7.2 Body-fitted grids for the computation of the flow around a circular cylinder:
unstretched, left, and stretched, right.
Figure 7.3 Cartesian grids for the computation of the flow around a circular cylinder:
unstretched, left, and stretched, right.
164
Appendix A:
Computational Parameters
Value
Parameter
Physical parameters:
Re
Domain size:
grid size
Xi
X-i
V\
V2
Geometry:
IB shape
radius
center
65
10.0
X
65, 129 X 129, 257 x 257
0.0
1.0
0.0
1.0
circle
0.1
(0.5,0.5)
Miscellaneous;
h
t2
At
time integration
a
P
1.45
X
0.0
0.4
10-4 (2,750 steps h
second order R-K
-1.0
2.07r
^2)
Table A.l Parameters for the spatial convergence study of section 5.1.4.
165
Value
Parameter
Physical parameters:
Re
1.0
Domain size:
grid size
Xx
X2
Vi
y2
Geometry:
IB shape
radius
center
61 X 61
0.0
1.0
0.0
1.0
circle
0.1
(0.5,0.5)
Miscellaneous;
h
At
time integration
a
P
4
X
0.0
0.2
10-^ 2 X 10-^ 1 X 10-^ 5
second order R-K
-1.0
2.07r
X
10"®
Table A.2 Parameters for the temporal convergence study of section 5.1.4.
Parameter
Value
Domain size:
grid size
41
x
xi
X2
Vi
y2
Geometry:
IB shape
radius
center
Miscellaneous:
UJ
41, 81
81, 161
0.0
1.0
0.0
1.0
x
x
161
circle
0.1
(0.5,0.5)
2.0 tt
Table A.3 Parameters for the spatial convergence study of section 5.2.3.
Parameter
Value
^0
0.2
125.6637
1/3
0.2
8.0
LOQ
u
k
Domain size (space & time):
0.0
Xi
1.0
0.0
0.1
t2
Table A.4 Parameters for 1-D moving IB computations of section 5.4.
167
Value
Parameter
Physical parameters:
20, 40, 50, 100, 200
Reo
9.9956
XQ
0.0
Vc
Domain size (A = 0.056, A = 0.023);
0.0, 0.0
Xi
46.5795, 46.5795
-8.9401,
-21.3278
Vi
8.9401, 21.3278
y2
Grid size (A = 0.056, A = 0.023);
641, 641
281, 321
Uy
1.386 X 10-^
1.148 X 10"^
^ymin
Miscellaneous:
At
time integration
XB
5.0
10-^ to 1.0 X 10-^
second order R-K
35.2033
X
Table A.5 Parameters for the circular cylinder computations of section 6.1.
168
Value
Parameter
Physical parameters:
Uoo
U
Rcl
L
Domain size:
Xi
X2
y2
Grid size:
rix
Uy
Ax
^dp2
/
A
Miscellaneous:
At
time integration
XB
30.0 m/s
x 10"^ m^/s
10^
0.05 m
0.4 (Re^i = 344)
5.195 (Re^, = 1240)
0.4 (w 11(5)
601
112
7.992 x
^ymin
Forcing dipole slot:
^dpl
1.5
10-^ (At-s ~ 2lAx)
1.5
x
10-4
0.8375 (Re<5i = 498)
0.9225 {Res, = 523)
1336.8 Hz (F =1,4 x 10"^)
0.01%
1.122 x 10"^
fourth order R-K
4.755 {Res, = 1187)
Table A.6 Parameters for T-S wave computation of section 6.2 using the code nst2d
of Meitz & Fasel (2000).
169
Parameter
Physical parameters:
ROL
L
Domain size:
Xi
X2
y2
Grid size;
IT'x
Uy
^^min
Value
10^
0.05 m
0.4 (Rej, = 344)
5.326 (Res, = 1256)
0.4
11(5)
541
117
7.992 X 10"=^ (AT-S « 2lAxmin)
1.566 X 10-^
Volume-forcing dipole:
0.8375 (Re^, = 498)
^dpl
0.9225
(Re^, = 523)
^dp2
1336.8 Hz (V =1.4 X 10"^)
/
1.0
A
Miscellaneous:
1.122 X 10-^
At
time integration
fourth order R-K
4.554 (Re^i = 1161)
XB
Table A.7 Parameters for T-S wave computation of section 6.2 using the immersed
boundary code nsib.
Parameter
Value
Physical parameters:
10^
RBL
Domain size:
0.0
Xi
43.315
X2
1.745
2/2
Grid size:
49
153
Uy
Ax
0.9024
3.449 X 10-4
Table A.8 Parameters for flat plate RANS calculations of section 6.3.
170
Appendix B:
Finite-difference Coefficients
Instead of using the standard practice of hard-coding finite-differences stencils into
the IBM code, a more flexible alternative was developed. Specifically, the form of
the finite-difference stencil to be used is specified, and the finite-difference
stencil
coefficients are determined numerically as functions of the coefficients of a linear
operator L[(j){x)] and the underlying computational grid. The details of this procedure
are the subject of this appendix.
A general linear finite-difference stencil can be written as
^ciik{cijk(t>k\xi)) = 0,
(B.l)
ijk
where i denotes the grid point, j denotes the derivative order, that is,
—
(f)/dx^\x=xi) k denotes a separate, individual term at point z, Cijk is the coefficient
of (f)^^\xi) which belongs to term k, and aik is the finite-difference coefficient. The
maximum values of j, and k are denoted by J, and K, and the number of points used
in the stencil by I. As an example, take L[4){x)\ to be
L[4>{x)\ = a{x)(l)xx + h{x)(l)x + c(a;)0.
(B.2)
One would like to write a three-point compact finite-difference stencil for this operator
0'i-i,iL[(j)i-i\ -l- aj_iL[(^j] +
+ <^1,2^1 + ^ii+i,20i+i-
Two separate terms k at point i are declared; L[(l)i] is term k = 1, and
(B.3)
is term
k = 2. One has Cj2i = a(xj), Cm = fe(xj), and Qoi = c{xi)^ and Cij2 = — 1 if j = 0,
and zero otherwise. In this example / = 3, J = 2, and K = 2. If the finite-difference
coefficients an and aj2 can be determined, then a high-order, compact solution to the
problem
L[(j){x)] = p{x)
(B.4)
171
is easily obtained by solving the tridiagonal system of equations
201-1 + <2i,20j + 0.i+l,24'i+l —
(^-5)
The finite-difference coefficients Uik are obtained by performing a Taylor expansion
of equation (B.l) about an arbitrary point x = x*, resulting in a linear system of
equations for the ajfc. The coefficient
of
in the m}^ equation is
JIL
(xqmik = y^c,^^—,
where m =
—
x*)"^~^
V-'
(B.6)
... ,IK — 2. The last equation, m = IK — 1, must be used to set a
normalization condition on the coefficients ajfc. Here aik = 1 is taken, for some index
{i, k) = {ni, nk). The resulting system
^ ] QmikO'ik
ik
^ni,nk
0,
771 =
0, 1, ... , IK
— IK
1
2
(B.7)
(B.8)
can be numerically solved using a direct matrix solver, yielding the desired finitedifference stencil coefficients aik-
172
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