River2D - University of Alberta

River2D - University of Alberta
River2D
Two-Dimensional Depth Averaged Model of River
Hydrodynamics and Fish Habitat
Introduction to Depth Averaged Modeling and User's
Manual
by
Peter Steffler and Julia Blackburn
University of Alberta
September, 2002
1.0
INTRODUCTION TO DEPTH AVERAGED MODELLING
1.1
General Overview
Advances in personal computer capability and computational
software technology are making detailed analysis more routine in
almost all branches of engineering. In channel and river
engineering, two-dimensional (2D), depth averaged models are
beginning to join one-dimensional models in common practice.
These models are useful in studies where local details of velocity
and depth distributions are important. Examples include bridge
design, river training and diversion works, contaminant transport,
and fish habitat evaluation. This introduction is intended to give a
brief overview of 2D river modeling, highlighting considerations
for practical applications.
With possible high velocities and slopes, and relatively shallow
depths, river and stream models present a particularly difficult
computational challenge. This fact is likely a significant factor in
the lag of application of shallow water models in rivers compared
to coastal and estuarine problems. Recently, however, numerical
techniques originally developed for transonic fluid mechanics
problems have been successfully adapted. These techniques,
referred to as shock-capturing or high-resolution schemes, allow
freely intermixed subcritical and supercritical flow without the
imposition of artificial eddy viscosity. The bottom line for river
engineers is access to reliable and accurate solutions with a
minimum of fiddling.
2D river model applications normally focus on a relatively limited
extent of the channel, typically less than ten channel widths in
length. In most cases, discharge variation is relatively slow
compared to the water travel time through the reach, and steady
state conditions are sought. Most models are designed to give
transient solutions, however, and often the desired steady solution
is obtained as an asymptotic transient solution after a long elapsed
time.
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1.2
Depth Averaged Models
There are a number of commercial and public domain 2D models
available. They are based on a variety of numerical schemes and
offer a range of graphical pre and post processor modules. The
fundamental physics is more or less common, however. All 2D
models solve the basic mass conservation equation and two
(horizontal) components of momentum conservation. Outputs from
the model are two (horizontal) velocity components and a depth at
each point or node. Velocity distributions in the vertical are
assumed to be uniform and pressure distributions are assumed to
be hydrostatic. Important three-dimensional effects, such as
secondary flows in curved channels, are not included. As a rule of
thumb, caution should be exercised in any attempt to resolve flow
features less than about ten flow depths in horizontal extent.
In many problems, the lateral extent of land covered by the channel
is not known prior to the solution. In fact, for flood plain mapping
problems, this may be the primary desired result. Existing 2D
models take a number of approaches to handle the problem of
some areas being wet while others are dry and changing between
the two. Some models turn cells or elements on and off and insert
no-flow boundaries between them based on a minimum depth
criteria. Other models change the fluid properties at very small
depths so that a very thin layer of fluid is always present. Another
approach is to couple groundwater flow equations so that a
common free surface is calculated, above the ground in the
channel, and below the ground on dry land.
2D model schemes based on finite difference, finite volume, and
finite element methods are available. Each approach has
advantages and disadvantages. At the risk of grossly
oversimplifying sometimes religious arguments, I would say that
finite volume methods offer the best stability and efficiency while
finite element methods offer the best geometric flexibility.
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1.3
Data Requirements
It may be a cliche to say that a model is only as good as its input
data, but it is true. As input data, 2D hydrodynamic models require
channel bed topography, roughness and transverse eddy viscosity
distributions, boundary conditions, and initial flow conditions. In
addition, some kind of discrete mesh or grid must be designed to
capture the flow variations.
Obtaining an accurate representation of bed topography is likely
the most critical, difficult, and time consuming aspect of the 2D
modeling exercise. Simple cross-section surveys are generally
inadequate. Combined GPS and depth sounding systems for large
rivers and distributed total station surveys for smaller streams have
been found to be effective. In either event, you should expect to
spend a minimum of one week of field data collection per study
site. The field data should be processed and checked through a
quality digital terrain model before being used as input for the 2D
model.
Bed roughness, in the form of a roughness height or Manning's n
value, is a less critical input parameter. Compared to traditional
one-dimensional models, where many two-dimensional effects are
abstracted into the resistance factor, the two-dimensional resistance
term accounts only for the direct bed shear. Observations of bed
material and bedform size are usually sufficient to establish
reasonable initial roughness estimates. Calibration to observed
water surface elevations gives the final values. If unrealistic
roughness values are required, it is likely that there are problems
with the bed topography.
Transverse eddy viscosity distributions are important for stability
in some finite difference and finite element models and are often
assigned unrealistically large values. They can be also be used as
calibration factors for measured flow distributions. Stable shock
capturing and high resolution numerical schemes are not sensitive
to these values. In cases where accurate determination of eddy
viscosity is required, a coupled turbulence model should be
considered.
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Boundary conditions usually take the form of a specified total
discharge at inflow sections and fixed water surface elevations or
rating curves at outflow sections. Since 2D models make no
implicit assumptions about flow direction or magnitude, discharge
divisions in splitting channels and the discharge given inflow and
outflow elevations can be calculated directly. Locating flow
boundaries some distance from areas of interest is important to
minimize the effect of boundary condition uncertainties. Initial
conditions are important, even for steady flow, since they are
usually used as the initial guess in the iterative solution procedure.
A good guess will significantly reduce the total run time and may
make the difference between a stable run and an unstable one.
Mesh or grid design is the black art of 2D modeling. The total
number of degrees of freedom (number of computational nodes
times three unknowns per node) is limited by the computer
capacity and time available. For an overnight run on this week's
"modern" PC, something in the order of 100,000 nodes is feasible.
The challenge is to distribute these nodes in such a way that the
most accurate solution is obtained for a particular purpose. Closely
spaced nodes in regions of high interest or flow variation, gradual
changes in node spacing, and regularity of element or cell shape
are important considerations. A rough rule of thumb is that a
minimum of four and preferably ten or more cells or elements in
each direction are required to reliably resolve a flow field feature.
To remove concerns about mesh design effects on the final
solution, the solutions from more than one design should be
compared. Fortunately, graphical pre-processors are available to
aid the design (and later to interpret the results). Unfortunately,
these require far more programming effort, and therefore may be
more expensive, than the 2D hydrodynamic models themselves.
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2.0
PRINCIPLES OF 2D HYDRODYNAMIC MODELING
This section is intended to provide a brief background on the
physics and numerical procedures underlying 2D depth averaged
hydrodynamic models. The practical value of this background is
that it helps explain the significance of the input parameters and
also highlights the limitations and expected reliability of the model
results.
It is important to realize that there are two levels of approximation
inherent in most computer models and that, according to
“Murphy's Law” the errors at the two levels will tend to compound
each other. The first level of approximation is in the abstraction of
physical reality to a mathematical formulation. At this level, we
leave out a great deal to focus on what we consider to be most
important. Even for this limited subset of reality, we do not always
understand the behaviour completely. Turbulence is a classic
example.
The second level of approximation is the transformation of
mathematical statements to computer arithmetic. The essential
problem here is that while our mathematics are sophisticated
enough to recognize that real space is composed of an infinite
number of points, our computers must work with a finite number.
The approximation error decreases as we add more points, but we
are limited by the computer speed and memory resources available.
2.1
Physical Formulation
Depth averaged modeling is based on the basic physical principles
of conservation of mass and momentum and on a set of
constitutive laws which relate the driving and resisting forces to
fluid properties and motions. To illustrate the abstraction process,
the conservation of mass law is developed below in some detail.
The more complex conservation of momentum is more briefly
sketched out with the considerations for the necessary constitutive
laws.
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2.1.1 Conservation of Mass
Consider an imaginary column in a stream flow, rectangular in
plan (dimensions ∆x and ∆y) and extending through the depth of
water, H, as shown in Figure 1. Conservation mass for this box or
"control volume" states that:
The rate of change of water volume with time in the box is equal to
the net rate at which water flows into the box across the sides.
Figure 1: Definition sketch for the conservation of mass.
V4
y
U1
U2
V3
x
Since the volume of water in the box can only change if the depth
changes, we can calculate the rate of change of volume with time
as (A∆H)/(∆t), where A is the plan area = ∆x∆y.
For the net inflow, the component of velocity which is
perpendicular to the particular side is important. In the diagram
above, the general direction of flow is from lower left to upper
right, but only the perpendicular "crossing" components are shown.
The rate at which water crosses any side of the box is equal to the
perpendicular velocity times the side area. Since the depth can
change from side to side and some flows are into the box and some
are out, an expression for the net inflow rate is
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6
= H 1U 1∆y − H 2U 2 ∆y + H 3V3 ∆x − H 4V4 ∆x
The conservation of mass principle then can be expressed
quantitatively as
∆x∆y∆H
= H 1U 1∆y − H 2U 2 ∆y + H 3V3 ∆x − H 4V4 ∆x
∆t
which can be rearranged a bit into
∆H (H1U 1 − H 2U 2 ) (H 3V3 − H 4V4 )
=
+
∆t
∆x
∆y
which can also be written as
∆H ∆(HU ) ∆(HV )
+
+
=0
∆t
∆x
∆y
Calculus allows us to shrink the dimensions of the box and the
change in t to zero
∂H ∂ (HU ) ∂(HV )
+
+
=0
∂t
∂x
∂y
This is the differential equation of mass continuity. Defining
discharge per unit width components qx = HU and qy = HV gives
an alternate form of the continuity equation
∂H ∂q x ∂q y
+
+
=0
∂t
∂x
∂y
The continuity equation, in either form, gives one relationship for
the depth and two velocity (or discharge intensity) components at
every point in the flow. In general, two more relationships are
required at every point to have the same number of equations and
unknowns. The other two relationships are given by the two
components of momentum conservation.
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2.1.2 Conservation of Momentum
While the physics are quite different, the process for deriving the
differential momentum equations is very similar to that of the
continuity. Rather than dwell on that process (which does get long
and tedious), the physical relevance of the terms in the equation
will be emphasized.
If we reconsider the imaginary column or box in the stream,
conservation of water momentum in the x direction states that:
The rate of change of x momentum with time in the box is equal to
the net rate of inflow of x momentum across the sides of the box
plus the net force acting on the box in the x direction.
The momentum flows and forces are shown on the diagram below,
Figure 2.
Figure 2: Definition sketch for conservation of momentum.
F4
M4
P1
y
P2
Fb
M1
M2
Pb
F5
F6
M3
x
F3
On this diagram, P1 and P2 are pressure forces on the sides of the
box; Pb is the pressure force due to a sloping bed; M1 and M2 are
momentum fluxes in the component direction; M3 and M4 are x
momentum fluxes carried by the transverse velocity component; Fb
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is the friction force of the bed; F3 and F4 are the transverse shear
forces due to turbulence; and F5 and F6 are the normal forces due
to turbulence. Note that surface friction due to wind and Coriolis
force due to the earth's rotation have been left out.
The momentum conservation principle is then
∆M x
= (M 1 − M 2 ) + (M 3 − M 4 ) + (P1 − P2 ) + Pb − Fb + F4 − F3 + F6 − F5
∆t
All of these terms need to be evaluated in terms of the depth and
velocity components in order to set up a solution. This is where the
most serious approximations are made.
Since momentum is mass times velocity and mass is volume times
density, ρ, the rate of change of momentum can be evaluated as
∆M x
∆(HU )
= ρ∆x∆y
.
∆t
∆t
Assuming that the velocities are constant through the depth the
momentum fluxes are
M 1 = ρ∆yH1U 12 and M 2 = ρ∆yH 2U 22 .
and the cross momentum fluxes are similarly
M 3 = ρ∆xH 3U 3V3 and M 4 = ρ∆xH 4U 4V4 .
For the assumption of hydrostatic pressure distributions, the
pressure forces can be evaluated in terms of the depth as
P1 = ρg∆y
H 12
H2
, P2 = ρg∆y 2 , and Pb = ρg∆x∆yHS 0 x ,
2
2
where S0x is the bed slope in the x direction.
The shear on the bed and sides of the box can be expressed (for the
time being) as shear stresses multiplied by areas:
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Fb = τ bx ∆x∆y , F3 = ∆xH 3τ xy 3 , and F4 = ∆xH 4τ xy 4 .
The normal turbulent forces can also be evaluated in this manner
so that:
F5 = ∆yH 5τ xx 5 and F6 = ∆yH 6τ xx 6 .
Substitute these back into the conservation law and after some
mathematical steps, the conservation of x momentum equation is
obtained:
∂q x ∂
∂
g ∂ 2
+ (Uq x ) + (Vq x ) +
H
∂t
∂x
∂y
2 ∂x
1 ∂
 1 ∂
= gH S 0 x − S fx +  (Hτ xx ) + 
Hτ xy
ρ  ∂x
 ρ  ∂y
(
)
(
)

where Sfx = τbx/(ρgH) is the friction slope.
In order to complete the formulation, relations for the bed and side
shear stresses must be specified. Since these stresses arise
primarily from turbulent flow interactions, there is considerable
uncertainty in their evaluation. Typically, a two-dimensional form
of Manning's equation is used for the friction slope
S fx =
n 2U U 2 + V 2
H 4/3
.
and a Bousinessq type eddy viscosity is used for the transverse
shear
 ∂U ∂V 
 .
τ xy = ν t 
+
 ∂y ∂x 
The parameters n and νt are not constants or fluid properties, but
depend on the flow situation. As a result, they become the "tuning"
or calibration parameters that may be changed to bring a model
prediction into agreement with measured data. Compared to one-
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10
dimensional channel modeling, Manning's n is familiar, but the
eddy viscosity coefficient is new.
In the y direction, the process is the same and the resulting
equation is similar. With the continuity equation and two
momentum equations applying at every point in the flow, we have
enough information to solve for the depth and velocity components
at every point. Unfortunately, finding that solution is difficult and
can only be done approximately.
2.2
Numerical Modeling of Depth Averaged Flow Equations
Given a set of governing equations, there are two essential steps in
developing a computational model.
1. Discretization. The infinite number of equations for an infinite
number of unknowns is reduced to a finite number of equations
at a finite number of mesh or grid points in space and time. At
this stage, calculus operations are reduced to algebraic
operations.
2. Solution. A scheme or process is devised where the algebraic
equations developed in the first step can be solved for the
unknown nodal values. The algebra is reduced to arithmetic
which can be translated into computer code.
There are a number of alternatives for each step. Common
discretization methods include finite difference, finite volume, and
finite element methods. Solution methods include explicit and
implicit solvers, the latter of which depend on a variety of iterative
or direct non-linear and linear equation solution methods.
2.2.1 The Finite Element Method
The basis for the finite element method is a more general technique
known as the weighted residual method. The idea is that the
governing equations can be solved approximately by use of a "trial
function" which is specified but has a number of adjustable
degrees of freedom. In a way, the process is analogous to curve
fitting, say a straight line, to observed data. The straight line is a
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11
specified function which has two degrees of freedom. Values for
those two parameters are sought which gives the least error.
The continuity and momentum equations of depth averaged flow
can be written in abbreviated form as
C (H , U , V ) = 0 ,
M x (H , U , V ) = 0 , and
M y (H , U , V ) = 0 .
If we introduce trial functions for the variables H, U, and V;
denoted by Ĥ ,Û , and Vˆ , and substitute them into the equations,
they will not exactly satisfy the equations. A "residual" will result.
(
)
C Hˆ , Uˆ , Vˆ = RC ,
(
)
(
)
M x Hˆ , Uˆ , Vˆ = R x , and
M y Hˆ , Uˆ , Vˆ = R y .
The objective is now to make the residuals as small as possible.
The weighted residual approach is to multiply the residual by a
weighting function, integrate over the whole area and set the result
to zero. For every degree of freedom in the trial function, a
separate weighting function is used in order to generate the same
number of equations as unknowns. The continuity equation, for
example, becomes
∫ N i C (Hˆ ,Uˆ ,Vˆ )dA = 0 ,
where Ni is the ith weight or test function. Integration over the area
serves to reduce the spatial distribution to just a quantity.
The trick with the weighted residual method is to choose good trial
and test functions in a general enough way that will work for
different situations. The idea of interpolation becomes very useful
here. If the trial function for H is set up such that there are a fixed
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number of points with the depths at those points being parameters
to be solved for, then a function which interpolates between the
points can be used as a trial function. Interpolation functions can
be written in the form of "influence" or "basis" functions for each
point, so that the trial function is a sum of terms
n
Hˆ = ∑ H j N j
j =1
The Galerkin weighted residual method uses the same set of
functions for "basis" and for "test" functions, considerably
simplifying the choices that have to be made.
The finite element method takes the method of weighted residuals
a couple of steps further. First, the domain is divided into discrete
areas, called finite elements, which are defined by connections
between nodal points. The triangulation below, in Figure 3, is an
example.
Figure 3: A sample triangulation.
e
j
The nodes are at the vertices of the triangles. Simple interpolation
or basis functions are easy to construct, allowing linear variations
between nodes, for example. These basis functions are considered
to be "local" in that they only extend from a particular node to near
neighbouring nodes. Outside of that area, they are zero. The basis
function for node j can be visualized as a "tent" function with the
pole (of unit height) at node j, and pegs at all of the nodes which
are attached to node j by a triangle edge.
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The integration operation in the method of weighted residual is
carried out as a sum of element integrals in the finite element
method. Because of the local nature of the basis functions, only
those functions with nodes in the particular element need to be
integrated. For a linear triangle, with three nodes, a total of nine
integrals are required (combination of three basis functions with
three test functions), even if there are thousands of other nodes.
The biggest advantage of the finite element method is geometrical
flexibility. Elements can change size and shape readily, allowing
complex boundaries to be traced, as well as allowing refinement of
the mesh in particularly important or rapidly varying areas.
2.2.2 Discretization Error
Any discretization method, being approximate, engenders some
error. Having an idea of the source and magnitude of that error is
important in evaluating computational results and taking steps to
improving them. For the purposes of this discussion, error will be
defined as the difference between the numerical value at any point
(not just at nodes) and the analytical solution at that point. Note
that the difference between the analytical solution and the real
world is another matter!
For the finite element method, there are two important sources of
error related to the operations carried out. The first is interpolation
error. Considering the curves below in Figure 4 and two different
piecewise linear interpolations, it is obvious that the one with more
pieces is better.
Figure 4: Error related to piecewise interpolation.
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The second source of error is more subtle. The weighted residual
method is really a weighted average process. The nodal value at
any particular point reflects conditions surrounding that point. If
there is a large variation in conditions, then the point value may be
biased one way or another. A simple way to interpret this is to
consider the computational result at a point to be average of what
that value is in the surrounding elements.
There are two common examples of this problem in depth
averaged flow calculations. The first is near a sloping bank. The
node at the bottom of the bank reflects both channel (relatively
uniform velocity) and bank (slower, decreasing velocity)
conditions. The average tends to be lower than the velocity at the
specific point. Another case is an isolated bar in a stream. If a node
is placed on top of the bar and surrounding nodes are all in deep
water, then the velocity on top of the bar will be too fast, reflecting
surrounding conditions. This will cause trouble as the depth at that
point will be small, resulting in an excessively high, possibly
supercritical, Froude Number.
The solution to both types of discretization errors is smaller
elements. This means more elements and more nodes for the same
physical area. The art of computer modeling is to provide enough
refinement in important areas while maintaining reasonable
solution times.
2.2.3 Solution Methods
The result of the finite element method, or any other discretization
method, is a set of nonlinear algebraic equations for all the
unknown depths and velocities. The process of solving these
equations is what is demanding of computer resources.
Most computer models of depth averaged flow solve for transient
conditions, even if steady state results are desired. This is a
convenient way of providing a controlled and stable iteration
scheme from an arbitrary first guess or initial condition. Two
approaches are generally used, referred to explicit and implicit
methods.
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15
Explicit methods solve for the new variable values at any one node
based on the values at surrounding nodes at the previous time. This
approach has the tremendous advantage of calculating each value
independently. No matrix solutions are necessary so storage is
minimal and execution is fast. The disadvantage is that the time
step is restricted to keep the solution stable. The Courant-LewyFreidrichs (CFL) condition is a guide to the maximum time step
allowed.


∆x

∆t < min
 V + gH 


This condition is related to the time taken by a shallow water wave
to travel from one node to the next. As mesh spacing is made
smaller in any area, the time step for the whole solution process
must be decreased.
The second approach is called implicit. Here values of all the
variables at the new time are considered to depend on each other as
well as the values at the previous time step. Since the unknowns
are interrelated, considerable effort using nonlinear and linear
algebra methods is required to solve the equations.
Typically, large matrices are formed. If there are N nodes in a
mesh, then there are 3N unknown variables to solve for at each
time step. The full matrix describing their interconnections would
be 3N X 3N. If N is 10,000, then the matrix would require 7.2
Gigabytes of memory to store. Fortunately, the full matrix is not
usually required. The vast majority of terms are zero. A more
realistic estimate of storage is 3N X 3B where B is the bandwidth
of the mesh. Roughly speaking, the bandwidth is the typical largest
difference between node numbers for connected points. For the
10,000 node problem, the bandwidth will be in the order of about
100. The matrix required is then less than 100 Megabytes - large,
but manageable.
Computation time for matrix solution is roughly proportional to
NB2. If B scales with the square root of N, then matrix size is
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16
proportional to N3/2 and solution time is proportional to N2. The
consequence is that adding a few more nodes to an already large
problem is expensive in storage and computer time. Again, the
problem of providing enough discretization to minimize errors and
get a solution in a reasonable time is the central trade-off in
computational modeling.
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3.0
River2D MODEL DESCRIPTION
3.1
General
The River2D model is a two-dimensional, depth averaged
hydrodynamic and fish habitat model developed specifically for
use in natural streams and rivers. It is a Finite Element model,
based on a conservative Petrov-Galerkin upwinding formulation. It
features subcritical-supercritical and wet-dry area solution
capabilities. Ice covers with variable thickness and discontinuous
ice covers can be modelled. It has been verified through a number
of comparisons with theoretical, experimental and field results
(Ghanem et al, 1995a; Waddle et al, 1996, Christison et al). A
complete description of the formulation and implementation of the
model is contained in Ghanem et al (1995).
3.2
Acknowledgments
The River2D model was developed at the University of Alberta
with funding provided by: the Natural Sciences and Engineering
Research Council of Canada, the Department of Fisheries and
Oceans, Government of Canada, Alberta Environmental
Protection, and the United States Geological Survey. The code was
written by F. Hicks, A. Ghanem, J. Sandelin, P. Steffler, and J.
Blackburn. T. Waddle has provided guidance and feedback on the
development of the model for real world applications. Ongoing
development of the model has been facilitated and supported by C.
Katopodis. Copyright is retained by these individuals and by the
University of Alberta.
3.3
Conditions of Use
The River2D model, in the form of a Windows
(95/98/2000/ME/NT/XP) executable program, is available in the
public domain. The program is supplied as is, with no warrant of
completeness or applicability to any particular problem. The
program and associated utilities, example data files, and
documentation may be freely copied and distributed as long as this
notice is included and use of the model is properly acknowledged.
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18
River2D and its associated utility programs are very much "works
in progress". Development of the programs and documentation is
continuous but sporadic. Whenever significant functionality is
added, updated programs are released. Constructive feedback, bug
reports, and discussion are appreciated.
Inquiries should be directed to [email protected] or
[email protected]
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19
4.0
FORMULATION
4.1
General
The River2D model is a two-dimensional, depth averaged finite
element model. It is intended for use on natural streams and rivers
and has special features for accommodating supercritical
/subcritical flow transitions, ice covers, and variable wetted area. It
is basically a transient model but provides for an accelerated
convergence to steady-state conditions. The fish habitat module is
based on the PHABSIM weighted usable area approach, adapted
for a triangular irregular network geometrical description. River2D
uses SI standard units (kg, m, s) for all input and output.
In cases with very large spatial domain, a more detailed solution of
a smaller internal area may be important for hydrodynamic or
habitat reasons. The River2D model has a component that gives
the user the ability to “cut out” sub-sites so that the user may treat
them separately. This component allows the user to define and
extract a boundary from within an existing coarsely defined mesh.
This boundary can then be used in R2D_Mesh to create a detailed
sub-mesh, which can be subsequently solved using the River2D
model. The boundary extraction process is based on the concept of
stream function, streamlines and stream tubes.
4.2
Hydrodynamic Model
The hydrodynamic component of the River2D model is based on
the two-dimensional, depth averaged St. Venant Equations
expressed in conservative form. These three equations represent
the conservation of water mass and of the two components of the
momentum vector. The dependent variables actually solved for are
the depth and discharge intensities in the two respective coordinate
directions.
Conservation of mass:
∂H ∂q x ∂q y
+
+
=0
∂t
∂x
∂y
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(1)
20
Conservation of x-direction momentum:
∂q x ∂
∂
g ∂ 2
+ (Uq x ) + (Vq x ) +
H
∂t ∂x
∂y
2 ∂x
= gH (S 0 x

1 ∂
 1 ∂
− S fx ) +  (Hτ xx ) +  (Hτ xy )
ρ  ∂x
 ρ  ∂y

(2)
Conservation of y-direction momentum:
∂q y
∂t
+
∂
(Uq y )+ ∂ (Vq y )+ g ∂ H 2
∂x
∂y
2 ∂y
= gH (S 0 y

1 ∂
 1 ∂
− S fy ) +  (Hτ yx ) +  (Hτ yy )
ρ  ∂x
 ρ  ∂y

(3)
where H is the depth of flow, U and V are the depth averaged
velocities in the x and y coordinate directions respectively. qx and
qy are the respective discharge intensities which are related to the
velocity components through
q x = HU
(4)
and
q y = HV
(5)
g is the acceleration due to gravity and ρ is the density of water.S0x
and S0y are the bed slopes in the x and y directions; Sfx and Sfy are
the corresponding friction slopes. τxx, τxy, τyx, and τyy are the
components of the horizontal turbulent stress tensor.
4.2.1 Basic Assumptions
1. The pressure distribution in the vertical is hydrostatic.
Generally, this limits accuracy in areas of steep slopes and
rapid changes of bed slopes. Roughly speaking, bed features of
horizontal size less than about 10 depths (typically dune
bedforms) will not be modeled accurately. Similarly, slopes in
in the direction of flow in excess of about 10% will not be
modeled correctly.
River2D - User Manual – September 30, 2002
21
2. The distributions of horizontal velocities over the depth are
essentially constant. An assumed velocity distribution may be
used in the interpretation of the provided depth average
velocity, but the distribution is treated as constant by the
internal calculations. Specifically, information on secondary
flows and circulations is not available.
3. Coriolis and wind forces are assumed negligible. For very large
water bodies, particularly for large lakes and estuaries, these
forces may be significant.
4.2.2 Bed Resistance Model
The friction slope terms depend on the bed shear stresses which are
assumed to be related to the magnitude and direction of the depthaveraged velocity. In the x direction for example,
S fx =
τ bx
U 2 +V 2
=
U,
ρgH
gHC s2
(6)
where τbx is the bed shear stress in the x direction and Cs is a nondimensional Chezy coefficient. This coefficient is related to the
effective roughness height, ks, of the boundary, and the depth of
flow through
 H
C s = 5.75 log12
 ks

 .

(7)
For a given flow depth, H, Manning's n and ks are related by
ks =
12 H
em
(8)
where m is
m=
H
1
6
2.5n g
River2D - User Manual – September 30, 2002
(9)
22
The effective roughness height was chosen as the resistance
parameter because it tends to remain constant over a wider range
of depth than does Manning's n.
For very small depth to roughness ratios (
H e2
<
), Eq. 7 is
k s 12
replaced by
C s = 2.5 +
30  H

e 2  k s



(10)
which gives a smooth, continuous, non-negative relation for any
depth of flow. There is no physical basis for this formula.
The effective roughness height (in m) is the resistance parameter to
be specified at every node in the mesh in the input files. For
resistance due primarily to bed material roughness, a starting
estimate of ks can be taken as 1-3 times the largest grain diameter.
The final values should be obtained by calibrating the model
results to measured water surface elevations and velocities.
4.2.3 Transverse Shear Model
Depth-averaged transverse turbulent shear stresses are modeled
with a Boussinesq type eddy viscosity formulation. For example:
 ∂U ∂V
τ xy = ν t 
+
 ∂y ∂x



(11)
where νt is the eddy viscosity coefficient. The eddy viscosity
coefficient is assumed to be composed of three components: a
constant, a bed shear generated term, and a transverse shear
generated term.
2
H U 2 +V 2
∂U  ∂U ∂V 
∂V
2
 + 2
ν t = ε1 + ε 2
+ ε3 H 2 2
+ 
+
(12)
Cs
∂x  ∂y ∂x 
∂y
where ε1, ε2, and ε3 are user definable coefficients.
River2D - User Manual – September 30, 2002
23
The default value for ε1 is 0. This coefficient can used to stabilize
the solution for very shallow flows when the second term in
equation 12 may not to adequately describe νt for the flow.
Reasonable values for ε1 can be calculated by evaluating the
second term in equation 12 using average flow conditions (average
flow depth and average velocities) for the modelled site.
The default value for ε2 is 0.5. By analogy with transverse
dispersion coefficients in rivers, values of 0.2 to 1.0 are reasonable.
Since most river turbulence is generated by bed shear, this term is
usually the most important.
In deeper lakesflows, or flows with high transverse velocity
outletsgradients, transverse shear may be the dominant turbulence
generation mechanism. Strong recirculation regions are important
examples. In these cases, the third term, ε3, becomes important. It
is essentially a 2D (horizontal) mixing length model. The mixing
length is assumed to be proportional to the depth of flow. A typical
value for ε3 is 0.1, but this may be adjusted by calibration.
4.2.4 Wet/Dry Area Treatment
In performing a two-dimensional model evaluation, the depth of
flow, as a dependent variable, is not known in advance. The
horizontal extent of the water coverage is therefore unknown.
Significant computational difficulties are encountered when the
depth is very shallow or there is no water at all over a part of the
modelled area. The River2D model handles these occurrences by
changing the surface flow equations to groundwater flow equations
in these areas. A continuous free surface with positive (above
ground) and negative (below ground) depths is calculated. This
procedure allows calculations to carry on without changing or
updating the boundary conditions. In addition, modelled area
selection and boundary condition specification are greatly
simplified. Specifically, the water mass conservation equation is
replaced by:
River2D - User Manual – September 30, 2002
24

∂H T  ∂ 2
∂2

(
H
z
)
H
z
(
)
=
+
+
+
b
b

S  ∂x 2
∂t
∂y 2

(13)
where T is the transmissivity, S is the storativity of the artificial
aquifer and zb is the ground surface elevation.
The transmissivity and storativity of the groundwater flow can be
set by the user. The transmissivity should be set to a low value
such that the actual groundwater discharge is negligible; the
default is 0.1. For a given area, the storativity is a measure of the
volume of water the ground will release per unit decline in the
water table. The default storativity is set to 1. For accurate
transient analysis or to speed up the groundwater response rate, the
storativity should be reduced.
4.3
Simple Ice Cover Model
River2D is equipped to model flow under a floating ice cover with
known geometry, that is, one with known ice thickness and
roughness. When an ice cover is present on a river, it affects the
flow hydraulics in a number of ways.
1. The ice cover increases the area on which shear stress
operates (up to a factor of 2 when a continuous ice cover is
present).
2. The roughness on the bottom of the ice cover increases the
total channel resistance by increasing the magnitude of the
shear stress on the flow.
3. Both 1 and 2 result in a reduction in the average flow
velocity, which causes the flow depth to increase to
accommodate the discharge.
4. The water surface elevation is increased to a value equal to
the new flow depth plus the submerged thickness of the ice
cover.
When an ice cover is input into River2D, the model changes the
surface water equations to account for these ice effects. The
River2D - User Manual – September 30, 2002
25
continuity equation remains the same but the momentum equations
are replaced by:
Conservation of x-direction momentum:
∂q x ∂ (Uq x ) ∂ (Vq x )
∂ H2
+
+
+ g 
∂t
∂x
∂y
∂x  2
= gD(S 0 x

∂H
 − gt s
∂x


1 ∂
 1 ∂
− S fx ) +  (Dτ xx ) +  (Dτ xy )
ρ  ∂x
 ρ  ∂y

(14)
Conservation of y-direction momentum:
∂q y
∂t
+
∂ (Uq y )
= gD(S 0 y
∂x
+
∂ (Vq y )
∂y
+g
∂ H2 
∂H

 − gt s
∂y  2 
∂y

1 ∂
 1 ∂
− S fy ) +  (Dτ yx ) +  (Dτ yy )
ρ  ∂x
 ρ  ∂y

(15)
where H now represents the depth to the free surface while D is the
depth of flow from the bed to the bottom of the ice cover. These
two are related through
D = H − ts
(16)
where ts is the submerged portion of the ice cover . The submerged
thickness is related to the total ice thickness, t, using
ts =
ρi
t
ρ
(17)
ρi and ρ are the densities of the ice and the water respectively. U
and V are still the depth averaged velocities in the x and y
coordinate directions respectively but they are now related to the
discharge intensities, qx and qy, through
q x = DU
(18)
and
q y = DV
River2D - User Manual – September 30, 2002
(19)
26
All of the other terms in these equations are as defined previously.
In fact, this is the form of the momentum equations that are present
in the computational code but these equations simply reduce to the
open water momentum equations when no ice is present. These
equations have the same inherent assumptions as the ‘ice-free’
form including one additional assumption: the ice cover is fixed in
space. Therefore the ice will not react to any shear force applied by
the water on the ice.
4.3.1 Combined Ice and Bed Resistance Model
When ice is present, the friction slope terms depend on the ice
shear stresses in addition to the bed shear stresses. In the x
direction for example,
S fx =
τ bx + τ ix
ρgD
(20)
where τbx and τix are the bed and ice x direction shear stresses
respectively. If we define the average stress as
τ fx =
τ bx + τ ix
2
(21)
then
S fx = 2
τ fx
ρgD
=2
U 2 +V 2
U
gDC s2
(22)
The Chezy coefficient, Cs, is related to the depth of flow under the
ice cover, D, through
 D
C s = 5.75 log 6 
 kc 
(23)
where kc is a composite bed and ice effective roughness height and
is evaluated using the Sabaneev equation (Ashton, 1986), which
when modified for use with roughness height beomes:
River2D - User Manual – September 30, 2002
27
 k 14 + k 14
b
kc =  i

2





4
(24)
Users are encouraged to read Ashton (1986) in order to fully
appreciate the assumptions and limitations associated with the use
of Sabaneev’s equation.
When modelling ice-covered domains, some users may choose to
first determine the bed roughness by calibrating the model for an
open water event. Model calibration under ice conditions can then
be accomplished by adjusting only the ice roughness values.
However, it is important to note that bed roughness heights that
have been calibrated for open water conditions may not be
applicable for the ice covered case, even when depth or discharge
are the same. Consequently, the values of ice roughness height that
are necessary to achieve model calibration by this approach may
not seem physically reasonable. In these situations, it is important
to recognize that more than one combination of bed and ice
roughness height will yield the same composite roughness and the
user should remember that it is only the composite roughness that
is actually considered in the hydraulic calculations.
4.4
Fish Habitat Model
The fish habitat component of River2D is based on the Weighted
Usable Area (WUA) (Bovee, 1982) concept used in the PHABSIM
family of fish habitat models. The WUA is calculated as an
aggregate of the product of a composite suitability index (CSI,
range 0.0 - 1.0) evaluated at every point in the domain and the
"tributary area" associated with that point. In River2D, the points
are the computational nodes of the finite element mesh and the
tributary areas are the "Thiessen polygons" including the area
closer to a particular node than all other nodes.
The CSI at each node is calculated as a combination of the separate
suitability indices for depth, velocity, and channel index. Options
for triple product, harmonic mean, or minimum value, calculation
River2D - User Manual – September 30, 2002
28
of CSI are available. The suitability index for each parameter is
evaluated by linear interpolation from an appropriate fish
preference curve to be supplied separately. Velocities and depths
are taken directly from the hydrodynamic component of the model.
The channel index values may depend on channel substrate or
cover for different fish species and life stages. These values are
interpolated from a separate channel index file to the
computational nodes. The interpolation may be linear (continuous)
or nearest neighbour (discrete).
4.5
Boundary Extraction Module
The boundary extraction component is based on the concept of
streamlines and stream tubes. A streamline is defined as an
imaginary line in the field of flow such that at any point on the
line, the velocity vector is tangential to it. Therefore in theory,
there can be no flow across a streamline. For steady flow, a
streamline is fixed is space since there is no change in direction of
the velocity vector at any point in the field of flow. For twodimensional flow, the equation of a streamline is
q x dy − q y dx = 0
(25)
Let us now define a scalar function such that
qx =
∂ψ
∂y
and
qy = −
∂ψ
∂x
(26)
This scalar function, ψ(x, y), is called stream function. Consider
the total differential of ψ,
dψ =
∂ψ
∂ψ
dx +
dy
∂x
∂y
(27)
If we substitute equations 14, the above expression becomes
dψ = q x dy − q y dx
River2D - User Manual – September 30, 2002
(28)
29
which illustrates that the equation of a streamline (equation 25) is
just a special case of equation 28 where dψ = 0. This illustrates
that ψ is constant along any streamline.
In Figure 5(a), we have two streamlines, ψ1 and ψ2, some distance
apart in a two-dimensional flow field. Since there can be no flow
across either of the streamlines, the streamlines confine the flow
into what is called a stream tube. Now consider two streamlines
that are infinitesimally close, ψ and ψ + dψ, also shown in Figure
5(a). From Figure 5(b), we can compute the flow or discharge in
this elemental stream tube (across ds) as
dQ = q x dy − q y dx
(29)
which illustrates that,
dQ = dψ
(30)
Hence the change in ψ across this elemental stream tube is
numerically equal to the discharge through it. Now if the above
expression is integrated between ψ1 and ψ2, we find that the
difference in the stream function between any two streamlines is
equal to the flow between the streamlines:
2
Q1→2 = ∫ dψ = ψ 2 − ψ 1
(31)
1
In Figure 5(a), the flow is from left to right, sinceψ2 >ψ1. If ψ2
<ψ1, the flow would be from right to left.
Due to this relationship between stream function and discharge, if
one arbitrarily sets ψ1 = 0 at one side of a channel (assuming the
side of a channel is a no flow boundary, then by definition it is a
streamline), then the value of ψ2 at the other side of the channel
will be equal to the total discharge across the channel. In this case,
the value of the stream function at any point in the flow, ψp, will
be equal to the discharge in the stream tube defined by the
streamlines ψ1 and ψp. As ψp approaches the value of ψ2, the
River2D - User Manual – September 30, 2002
30
discharge in the stream tube approaches the total discharge in the
channel. For this reason, the value of the stream function across a
channel is also called the cumulative discharge.
River2D - User Manual – September 30, 2002
31
Figure 5: The relationship between stream function and
discharge: (a) a schematic of a stream tube and (b)
the calculation of discharge across an elemental
stream tube.
In order to solve the computational mesh of a sub-site that is to be
treated separately from the larger site from which it came, the
boundary conditions along the boundary defining this sub-site must
be known. If the boundary extends across the channel then the
River2D - User Manual – September 30, 2002
32
boundary conditions are known to be no flow at the sides of the
channel. However, if the entire boundary is within the flow, then
the problem is no longer trivial. Because streamlines by definition
are no flow boundaries, they are logical paths to trace when
defining a computational boundary within the flow. For this
reason, the boundary extraction module in River2D uses the
concept of streamlines and stream tubes when defining a
computational boundary for a new small scale mesh from within a
larger site.
To map out streamlines, the value of stream function must be
known at every node within the computational domain. If the
stream function is calculated assuming that one of the no flow
boundaries in the computational domain is a streamline where ψ =
0, then the stream function will also represent the cumulative
discharge in the channel. Only the location of the streamlines is
essential to River2D’s boundary extraction process. However,
when the values of stream function represent the cumulative
discharge in a channel, then the concept of stream function takes
on a physical meaning. In the River2D model, calculation of the
cumulative discharge is performed in two steps. The first step
involves calculating an approximate solution. In the second step,
the approximate solution is refined using the Finite Element
Method.
In the approximate solution, the equation of a streamline (28) is
used in its discrete form, that is
∆ψ = q x ∆y − q y ∆x ,
(32)
to obtain a value of ψ at all of the nodes in the domain. Equation
32 only gives the relative difference in ψ between two points
separated by a distance defined by ∆x and ∆y. Therefore, in order
for the values of ψ to represent the cumulative discharge across a
channel, the streamline along one side of the channel should be
given a value of ψ = 0. In the River2D model, equation 32 is used
to ‘sweep’ or ‘march’ across the domain, using nodes of known ψ
to calculate nodes of unknown ψ. The starting nodes of known ψ
River2D - User Manual – September 30, 2002
33
for the sweep are obtained by setting ψ = 0 at all the nodes along
one no flow boundary. The calculations then progress across
computational mesh until all of the nodes in the domain have a
value of ψ.
The refined solution is obtained by solving the vorticity–stream
function equation for the entire computational domain. In 2D
flow, the vorticity vector, η, has only a z-component:
η=
∂q y
∂x
−
∂q x
∂y
(33)
Substituting equation (26) into (33) results in the vorticity-stream
function equation:
∂ 2ψ ∂ 2ψ
+
= −η
∂x 2 ∂y 2
(34)
This equation is solved for all of the nodes in the domain by using
the Finite Element Method based on the Galerkin formulation.
This method produces a more refined solution than the sweep
method because it simultaneously solves for ψ at every node in the
domain. To solve any problem using the Finite Element Method,
the boundary conditions for the problem must be known. For
equation 34 there are two possible boundary conditions. No flow
boundaries will have a constant value of ψ while inflow and
outflow boundaries will be governed by
∂ψ
=0
∂η
(35)
which simply means that stream lines must be perpendicular to
flow boundaries. The second boundary condition is handled by the
Galerkin formulation; however the first boundary condition must
be known in order to obtain a solution. In the case of a simple
channel, with one inflow and one outflow boundary, the values of
ψ along the no flow boundaries can be set based on the total
discharge, Q, in the channel. Therefore the boundary conditions
become ψ = 0 along one no flow boundary and ψ = Q along the
River2D - User Manual – September 30, 2002
34
other no flow boundary. If the channel contains islands that are
enclosed by no flow boundaries, the boundary conditions for these
boundaries must also be set. Unfortunately, obtaining the value of
ψ for these boundaries is not so trivial. This is where the
approximate solution is beneficial. The sweep method will
calculate the values of ψ at all nodes regardless of whether or not
they are along boundaries. Therefore, the boundary conditions for
any nodes along internal no flow boundaries are set based on the
average of the values of ψ obtained from the approximate solution.
4.6
Hydrodynamic Model Implementation
4.6.1 Finite Element Method
The Finite Element Method used in River2D’s hydrodynamic
model is based on the Streamline Upwind Petrov-Galerkin
weighted residual formulation. In this technique, upstream biased
test functions are used to ensure solution stability under the full
range of flow conditions, including subcritical, supercritical, and
transcritical flow. As a result, there is no need for mixed (unequal
order) interpolations or artificially large transverse diffusivities.
A fully conservative discretization is implemented which insures
that no fluid mass is lost or gained over the modeled domain. This
also allows implementation of boundary conditions as natural flow
or forced conditions. A fixed elevation is imposed as an equivalent
hydrostatic force, for example, and known inflow discharges are
used directly.
The model discretizations supported include equal order linear,
quadratic, and cubic interpolation of all variables over triangular
and quadrilateral elements. The mesh generator supports only
linear triangles, however. These elements are the simplest possible
in two dimensions and result in the minimum execution time for a
given number of nodes.
Although the model is actually an unsteady one, it can be used
either to perform a transient analysis or to obtain a steady state
solution. For steady state results, an accelerated convergence
River2D - User Manual – September 30, 2002
35
procedure speeds the process to final completion by systematically
increasing the time increment.
4.6.2 The Newton Raphson Method
Using the Finite Element Method to solve the hydrodynamic
equations results in a system of non-symmetric, non-linear
equations which can be solved by explicit or implicit methods. In
River2D, an implicit approach is taken which requires a
simultaneous solution of the system of equations. Because the
system is non-linear, the Newton-Raphson iterative method is
employed. In this method, the iterative solution is achieved by
using the solution of the unknown values at the previous time step,
Φn, as the first guess of the solution at the new time step, Φn+1.
Unless the guesses are correct, then a residual will result, such that:
{f (Φ
n +1, m
)}= {R
n +1, m
}
(36)
where n and m denote the time step and the iteration respectively.
The corrections, ∆Φ, used to obtain the next guesses in the
iterative process are determined by solving the following system of
equations.
[J
n +1, m
]{∆Φ
n +1, m +1
}= {R
n +1, m
}
(37)
where [Jn+1,m] is called the ‘Jacobian’ matrix and is defined by:
[J
n +1, m
] =  ∂∂{{ΦR }}


n +1, m
n +1, m
(38)
After each mth iteration, the values of the solution variables are
updated using:
Φ n +1,m +1 = Φ n +1,m + ∆Φ n +1,m +1
The Jacobian can be evaluated analytically, using equation 38, or
numerically, using the following approximation:
River2D - User Manual – September 30, 2002
36
{
{
}
}
{ (
)
(
)}
n +1, m
− Ri Φ nj +1,m − δ j 
 ∂ R n +1,m   Ri Φ j
≈



n +1, m 
δj
∂ Φ
 

(39)
where δ is a small perturbation and the indices i and j represent the
equation number and the variable number respectively. In River2D
the user has the option of evaluating the Jacobian matrix
analytically or numerically. The analytical Jacobian is faster to
evaluate than the numerical one, but due to assumptions in the
mathematical derivation, it may be less accurate than the numerical
Jacobian. In particular, the analytical Jacobian does not consider
changes in the upwinding matrix or the effect of changing water
edge location.
When performing steady state analyses, it is best to start the
solution process using the analytical Jacobian. If the solution
cannot attain the desired level of convergence, it is advisable to
switch to the numerical Jacobian for the rest of the simulation.
The analytical Jacobian should be adequate for transient
simulations, as the water edge and upwinding matrix are assumed
to be fixed over the time interval.
4.7
Model Equation Solvers
Within each Newton-Raphson iteration, the system of linear
equations described by equation 37 requires solving. In River2D,
the user has the choice of two linear equation solvers: a direct
solver and an iterative solver.
4.7.1 Direct Solver
The first solver, which is called the active zone equation solver
(Stasa 1985), employs a direct solution to the non-symmetric linear
system. That is, a solver based on Gaussian elimination. In the
active zone equation solver, the coefficient matrix [A] in the linear
system implied by
[ A]{x} = {b}
River2D - User Manual – September 30, 2002
(40)
37
is decomposed into lower and upper triangular matrices [L] and
[U] such that
[ A] = [ L][U ]
(41)
Therefore, using equation 41, equation 40 can be rewritten as
[ L][U ]{x} = {b}
(42)
which is, in effect, two systems of equations:
[ L]{c} = {b}
(43)
and
[U ]{x} = {c}
(44)
The system defined by equation 40 is then solved by forward
substitution of equation 43 to compute {c} and then by back
substitution of equation 44 to obtain the solution for vector {x}. In
River2D, the coefficient matrix is a very large Jacobian Matrix
with the dimension 3N by 3N, where N is the number of
computational nodes. The Jacobian Matrix is typically sparse
(many of its entries are 0) and banded (most non zero entries are
gathered along the main diagonal). The active zone equation
solver takes advantage of these properties to reduce the storage
requirements by using a skyline storage method (Stasa 1985). This
reduces the memory requirements significantly to 3N by 3B, where
B is bandwidth of the matrix. However, unless the bandwidth is
small, the storage requirements can be quite significant. The
bandwidth of the matrix is a function of the geometry and node
numbering of the mesh. Narrow channels will produce small
bandwidths while wide channels will produce large ones.
Unfortunately, the speed of the active zone equation solver is a
function of the memory requirements for the Jacobian Matrix. As
a result, the size limit of a problem that can be solved will be a
function of the available RAM.
River2D - User Manual – September 30, 2002
38
4.7.2 Iterative Solver
The second solver uses an iterative method for solving the nonsymmetric linear system called the Generalized Minimal Residual
method or GMRES method (Saad and Schultz 1986). This solver
was incorporated into River2D because it does not have the large
storage requirements of the direct solver. This therefore allows for
larger meshes to be solved than can be solved using the direct
solver.
In the GMRES method, an initial guess of the solution to vector
{x0} in equation 40 is chosen. This results in:
{r0 } = [ A]{x0 } − {b}
(45)
where {r0} is called the residual vector. The formation of the
residual illustrates why this method, like all iterative methods,
requires very little storage. There is no need to store the
coefficient matrix, [A] explicitly as is necessary in direct methods;
it need only be stored implicitly in the product vector [A]{x}. In
River2D, this product vector is obtained by summing the matrixvector products at the element level.
The residual vector then is used to form an orthogonal basis
(orthogonal set of vectors):
Vk = {v1 , v 2 , L, v k }
(46)
of, what is called, a Krylov subspace Kk. Hence, Vk is called a set
of Krylov vectors. The GMRES solution to the non-symmetric
linear system, described in equation 40, is formed using a linear
combination of these Krylov vectors. That is:
{x} = {x0 } + y1{v1 } + y 2 {v 2 }L y k {v k }
(47)
where yi are the linear combination coefficients. For details on
how the vectors and their coefficients are derived, please see the
original paper on the GMRES method by Saad and Schultz (1986).
River2D - User Manual – September 30, 2002
39
The iterative nature of the GMRES method arises by evaluating the
residual using {x} after each new Krylov vector (and its
coefficient) is calculated. Once the residual is minimized to an
acceptable tolerance, then the solution is said to have converged
and no more vectors are required. The number of vectors, or the
number of iterations, k, that are required depends on how strict the
tolerance is. However, the number of iterations cannot exceed the
number of unknowns, N. This is due to the fact that the Krylov
subspace is defined by N vectors. Unfortunately, the execution
time of this algorithm increases geometrically with every
additional iteration. This is because the calculation of vector vi is
based on the vectors v1 to vi-1 inclusively. To get around issue, we
can restart the algorithm after the development of m vectors, that is
after m steps, using the solution vector at m steps, {xm}, to develop
a new residual vector, {rm} and a new set of Krylov vectors, Vm.
The algorithm is restarted as many times as required to achieve a
desired level of convergence. The restarted version of GMRES,
denoted by GMRES(m) is the iterative method used in River2D.
In River2D, GMRES(m) is implemented so that it continues to
restart until the residual is minimized to within a goal tolerance or
until a maximum number of restarts, or iterations, has occurred.
Currently, the goal tolerance is compared to the magnitude of the
residual at the end of every iteration divided by the magnitude of
the {b} vector. That is:
N
{rm }
{b}
∑r
2
i
=
i =1
(48)
N
∑b
2
i
i =1
The number of steps until restart, m, and the maximum number of
iterations, k, are also user definable parameters. The default values
for these parameters are 10 restarts and 10 iterations. The default
goal tolerance is 0.01. These values were chosen based on a small
number of tests. The optimum values for these parameters are
problem specific. In general, the greater the value of m, the fewer
the number of iterations that are required to converge and vice
River2D - User Manual – September 30, 2002
40
versa. Although the time per iteration increases exponentially with
every additional step before restarting, every additional iteration
only causes a linear increase in the total time.
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41
5.0
INSTRUCTIONS FOR USE
5.1
Installation
The River2D executable code is entirely contained in the file
River2D.EXE. This file is about 650 KBytes in size and may
simply be copied to any convenient directory.
5.1.1 Model Requirements and Performance
The
River2D
model
is
written
for
Windows
95/98/NT/2000/ME/XP. In general, fast floating point performance
and large memory (at least 256, preferably >512MB) spaces are
required. Typically, a model using about 10,000 nodes and 20,000
elements takes about 100 MB of memory and will execute in less
than an hour on an intermediate Pentium III class computer.
5.2
Preparation of Input Files
5.2.1 Computational Mesh - .cdg files
A separate application, R2D_Mesh, is used to generate the input
mesh files for the River2D model. This is an interactive, graphical,
Windows (95/98/2000/ME/NT) based mesh generation program.
This program operates by first reading in a bed topography file
which contains digitized position, elevation, and bed roughness
information for scattered points and feature lines. Another utility
program, R2D_Bed, is available to aid in the development of the
.bed file from field data.
In R2D_Mesh, the finite element mesh boundary and layout are
defined graphically by the user on an overlay to the topography
map read in. A variety of mesh design tools are available which
allow mesh refinement while ensuring reliability. When the user is
satisfied, a complete .cdg input file is generated. This file contains
default runtime parameters for the River2D model.
Note that once a .cdg file is generated by R2D_Mesh, it can be
read back in to R2D_Mesh for further editing. However any
hydrodynamic information in the .cdg file is lost. Detailed
River2D - User Manual – September 30, 2002
42
instructions for R2D_Mesh are contained in a separate user
manual.
5.2.2 Ice Topography - .ice files
An ice file is required by the simple ice cover module to evaluate
nodal ice properties. These properties include ice thickness and ice
roughness height. The structure of the .ice file is identical to the
.bed file except that bed elevation and bed roughness height are
replaced with ice thickness and ice roughness height respectively.
The R2D_Ice utility can be used to develop an ice file from field
data.
5.2.3 Channel Index - .chi files
A channel index file (extension .chi) is required by the habitat
module to evaluate channel index suitability indices for each
computational node. The structure of the .chi file is identical to a
.bed file except that the channel roughness parameter for each
point is replaced by the channel index value. The R2D_Bed utility
can be used to aid the generation of a channel index file from field
data. As with .bed files, the points in the .chi file are usually at
different locations than the computational nodes in the .cdg file so
an interpolation process is required in the habitat analysis.
Note that a channel index file is not required for use of the
hydrodynamic component of River2D.
5.2.4 Fish Preference - .prf files
Separate fish preference files (extension .prf) are required for each
distinct fish species/life stage to be considered in the habitat
analysis. Each .prf file contains tables representing velocity, depth,
and channel index preference curves for that particular species/life
stage. The tables may be interpreted as either a piecewise linear
approximation between points or as a direct lookup table for
discreet channel index categories.
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43
5.3
Running River2D
The River2D program executes as a standard Windows program
and can be started by double-clicking its icon. The main River2D
window will open (initially blank) and the About River2D notice
window appear as well. The About River_2D window contains a
brief introduction, including the conditions of use notice. The
appearance of River2D once the about box is closed is shown in
Figure 6.
River2D - User Manual – September 30, 2002
44
Figure 6: The elements of the River_2D screen.
Title Bar
Menu Bar
Tool Bar
Viewing Window
Status Bar
The normal sequence of events is to first read in a .cdg input file
defining the finite element mesh and initial flow conditions. Then
display options are set up to allow visualization of the solution
process. Optionally, boundary condition values can be changed.
The hydrodynamic solution is then run. Further display and
visualization is used to determine if the solution is acceptable. If it
is, then a habitat analysis may be performed, if desired. Finally, the
results can be output in different formats to files. The default
output format is a .cdg file which can be used as an input to a
subsequent run with perhaps new boundary conditions. The only
difference between the input and output .cdg files is that the time
and flow variables (depth and discharge components) are updated
to the latest solution values in the output.
If the result of the hydrodynamic solution is not acceptable, there
are options available for improving the solution. If minor
adjustments to the mesh are all that is needed, the mesh editing
features in River2D can be used to refine the discretization or
River2D - User Manual – September 30, 2002
45
modify the bed elevation or bed roughness. In some situations, it
may be necessary to build an entirely new mesh, in which case, a
new .cdg file must be created using the mesh editor (R2D_Mesh).
In either event, the mesh is rerun. This editing process is typically
continued until an acceptable solution is achieved.
In line with the typical process outlined above, there are basically
six groups of operations involved in using River2D.
1. Standard Windows operations. These are grouped under the
standard Windows File, Edit, View, and Help menus.
2. Display operations. These operations affect what information is
displayed on the screen and how it is to be displayed.
Commands for extracting the displayed variable to comma
delimted (*.csv) files for further analysis are also available.
These operations are grouped under the Display menu. Some
are also available as toolbar shortcuts.
3. Hydrodynamic model operations. These operations are
available under the Flow menu. The command for calculating
the cumulative discharge is also grouped under the Flow menu.
4. Mesh editing operations. These operations are for refining the
mesh discretization and can be found under the Mesh Edit
menu. Some of these operations are also available from the
toolbar.
5. Habitat model operations. These operations are available under
the habitat menu. WUA calculation is also available from the
toolbar.
6. Option setting operations. These operations are for setting
various options in River2D and are found in the Options Menu.
Detailed descriptions of the operations available are presented in
the following sections.
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46
5.4
Standard Windows Operations
5.4.1 File -> New
Not currently supported
5.4.2 File -> Open
The standard Windows file open dialog is presented with the filter
set to files with a .cdg extension. When an appropriate file is
selected and the OK button is pressed, the .cdg file will be read in.
Information from any previous .cdg file open will be deleted (from
the program's memory, the original file will be unaffected).
There is a significant amount of processing required before the
.cdg file is ready to use. It may take several minutes to complete,
depending on the number of nodes and breakline segments. When
done, an outline of the modeled area will appear in the main part of
the River2D window.
5.4.3 File -> Save
The current mesh information is saved back into the original .cdg
file. Normally, the newly solved depths and discharge intensities
are saved. Other changes, such as boundary conditions, roughness
values and bed elevation are also saved. No habitat information is
saved. The newly saved .cdg file may be reopened at a later time
and used to initiate further analysis.
5.4.4 File -> Save As
The Save As command allows the specification of a new .cdg file
for the output using the standard Windows File Save As dialog.
The original file is untouched and subsequent Save commands use
the new .cdg file.
5.4.5 File -> Print
The Print command sends whatever is displayed in the main
River2D window to the selected printer. All current display options
are used for printing. The display is scaled to a single page, such
that no part of the current window display is lost. Because the
River2D - User Manual – September 30, 2002
47
shape and/ or orientation of the paper may be different from the
screen, the output may appear smaller or may include areas not
visible on the screen. Some trial and error (using Print Preview) in
adjustment of the window shape may be necessary to obtain the
best result on paper. Generally, setting the widow shape to match
the paper shape (portrait or landscape) works the best.
Output to the printer uses Windows drawing commands which will
take advantage of whatever resolution the printer is capable of.
Note that very high resolution printing, particularly of colour fill
drawings, may be very slow.
5.4.6 File -> Print Preview
Print Preview displays an image of the printer output, considering
the current Display and Printer Setup options in effect.
5.4.7 File -> Print Setup
Print Setup is the standard Windows dialog for setting printing
options for the selected printer.
5.4.8 File -> recent documents
The four most recent .cdg files opened will be listed here for
convenience. Selecting one of them will open it as if it had been
selected in File -> Open.
5.4.9 File -> Exit
The program will cease running and the window will close.
Clicking the "X" box in the top right hand corner will have the
same effect.
5.4.10 Edit
The standard Windows operations of Undo, Cut, Copy, and Paste
are not supported in the present version of River2D.
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48
5.4.11 View
The two items under the view menu toggle the display of the
Toolbar and the Status Bar respectively. The Toolbar provides
some useful shortcuts for common operations and the Status Bar
provides details of the point at the current mouse position. Neither
is essential, however, and a larger display area is available if they
are toggled off.
5.4.12 Help
The only function currently available under the help menu is a
command to display the About River2D dialog box. This box is the
same as initially displayed at start up.
5.5
Additional File Operations
5.5.1 File-> Save As EMF…
This command allows the user to save the current view to an
Enhanced Windows Metafile. This file format saves the view in a
vector based format which is excellent for presentation purposes.
Currently, the EMF’s generated by this command can be imported
into any Microsoft Office application that accepts image files.
5.5.2 File -> Compress Video…
This command allows the user to compress any AVI video files
that may have been generated as output from a transient analysis.
Selecting this command will open a series of dialog boxes. The
first will require the user to select a video file for compression and
the second is a standard “save as” dialog box which allows the
compressed file to be saved under a new name. (Note: the
compressed file cannot overwrite the uncompressed file. If this is
attempted, the compression process will end without compressing
the file.) Once a name is chosen, the “Video Compression” dialog
box opens, as shown in Figure 7 below.
River2D - User Manual – September 30, 2002
49
Figure 7:The Video Compression Dialog Box.
This allows for the selection of a compressor, the compression
quality, data rate and the frequency of key frames. In addition,
there is an “About…” button which provides information on some
of the different compressors. It is also possible to preview the
results by selecting the “Preview” button.
Care should be taken in selecting a compressor as each
compression method (codec) has its advantages and disadvantages.
For example, Microsoft RLE was designed for animations and
cartoons. It works well for 8 bit graphic images (256 colour) even
though it gives relatively poor performance for real-life images. In
contrast, compressors such as Cinepak are excellent for action
sequences but less appropriate for low-motion sequences. Other
options include Indeo, which is appropriate for low-motion, and
Video 1, which is the standard compressor shipped with video for
Windows.
Since the video output produced by River2D is a low motion
sequence with 8 bit graphic images, Microsoft RLE is probably the
most suitable compressor, while Indeo and Video 1 are other
appropriate alternatives. Cinepak and other “high action”
compressors would likely not be suited to River2D videos. Note
that the list of compressors in the Video Compression dialog is
machine dependant. Therefore you may or may not have all of the
described choices. You may also have compressors that are not
described here which you might want to research before using.
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50
5.6
Display Operations
The River2D program is built around a single display window
which shows a map of the area being modeled. When this window
is active and in the foreground, the Status Bar at the bottom of the
window is continuously updated with information from the current
mouse pointer position. The x and y coordinates and one other
parameter are given on this line. The specific parameter is the same
as that selected for contour/colour fill. If none is specified, then the
bed elevation is given.
The Display commands, found in the Display Menu, control what
and how the model information is displayed in the window. The
appearance of the River_2D Display Menu is shown in Figure 8.
The Display settings are also used for printing.
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51
Figure 8: The River2D Display Menu.
5.6.1 Display -> Redraw
The Redraw command simply redraws everything shown in the
display window, updating any changes that may not yet be visible.
The Toolbar button
provides a shortcut to this function.
5.6.2 Display -> Scale to Fit
The Scale to Fit command chooses a center point and scale factor
such that the entire modeled area is displayed in the window with a
small margin. The Toolbar button
function.
provides a shortcut to this
5.6.3 Display -> Zoom In On Rectangle
The Zoom In On Rectangle command allows the user to zoom
directly to a rectangle selected by clicking on one corner and
dragging the cursor (holding the mouse button down) to the
opposite corner (then releasing the button). The display is centered
and rescaled to contain the selected rectangle. If the shape of the
display window and the selected rectangle are not the same, some
River2D - User Manual – September 30, 2002
52
additional area may also be displayed. The Toolbar button with the
two rectangles provides a shortcut to this function. This command
can also be accessed using the F2 function key.
5.6.4 Display -> Zoom In On Point
The Zoom in by Point command allows the user to click on any
point in the display and enlarge the view around that point. The
display is centered on the selected point and the scale factor is
doubled. Repeated Zooms can be used to further refine the focus.
The Toolbar button with the large triangle provides a shortcut to
this function.
5.6.5 Display -> Zoom Out
The Zoom Out command leaves the display centered on the
previously selected zoom in point and the scale factor is halved to
show twice the extent. Repeated Zoom outs can be used to show
ever greater extents. The Toolbar button with the small triangle
provides a shortcut to this function.
5.6.6 Display -> Move Center to Point
The Move Center to Point command allows the user to click on
any point in the display and shift the display to center on that point
with the scale unaffected. This command can be used to "pan"
across the model at small scales, looking for detailed features. The
Toolbar button with the dot in the rectangle provides a shortcut to
this function.
The “panning” command is also available using the keyboard. The
arrow keys have been mapped as follows in the River2D
environment:
⇐ key: shift the display to left
⇒ key: shift the display to the right
⇑ key: shift the display up
⇓ key: shift the display down
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53
Letter keys on the other side of the keyboard have also been
mapped with “panning”:
A key: shift the display to left
D key: shift the display to the right
W key: shift the display up
S key: shift the display down
5.6.7 Display -> Mesh
The Mesh command toggles the display of the finite element mesh.
The triangles shown are the finite elements and the vertices of the
triangles are the computational nodes. Figure 9 shows a sample
problem with the mesh displayed.
Figure 9: A sample .cdg file with mesh displayed.
5.6.8 Display -> Node Numbers
The Node Numbers command toggles the display of numbers of all
of the computational nodes. The display may need to be zoomed in
to distinguish the individual numbers, as shown in Figure 10. This
option is sometimes useful to identify problem points and to get an
River2D - User Manual – September 30, 2002
54
indication of the node numbering efficiency. The node numbers are
also useful when editing boundary conditions.
River2D - User Manual – September 30, 2002
55
Figure 10: Using node numbers to identify problem points.
Zoom in on
area to identify
problem points
River2D - User Manual – September 30, 2002
56
5.6.9 Display -> Water's Edge
The Water's Edge command toggles the display of a blue line
which represents the boundary between wet and dry areas in the
model. As the water depth changes, this line is updated to reflect
drying or inundation. As most other display options are only valid
for wet areas, this line often appears as a boundary of the
interesting area. Figure 11 shows a sample problem with the
water’s edge displayed. This feature is displayed by default. It
should be noted that the computational boundary is always
displayed and cannot be “turned off”. The colours of the boundary
represent the various boundary conditions. Red denotes a “no
cross-flow” condition, while green and blue and orange represent
inflow and two kinds of outflow conditions respectively, as shown
in Figure 11.
Figure 11: A sample .cdg file with the water’s edge displayed.
No cross-flow
boundary
Inflow
boundary
Outflow
boundary
Water’s
edge
5.6.10 Display -> Contour/Colour
The Contour/Colour command brings up the Contour dialog box,
shown in Figure 12, which controls parameter contouring and
colour fill displays. The top of the dialog allows the user to select a
River2D - User Manual – September 30, 2002
57
single parameter or variable a drop down list. The selected
parameter is also the one that will be tracked with the mouse
pointer on the Status Bar. If no habitat calculations have been
done, the various habitat indices will be zero.
The check boxes on the lower half of the dialog allow the user to
select any, all or none of the display type options.
Figure 12: The Contour dialog box.
If Display Contour Lines is selected, then the Contour Interval
value is used to draw a contour map of the selected variable. Since
linear triangles are used for interpolation, the contour lines will be
straight across each individual element. Contour values are not
shown on the map, but may be found by pointing the cursor at any
line and reading the parameter value from the Status Bar. Figure
13 shows a sample problem with the bed elevation contours with a
0.5 contour interval. The cursor is pointing to the contour with a
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58
value of 100 as indicated by the Bed Elevation value in the Status
Bar.
Figure 13: A sample .cdg file with bed elevation contours
displayed.
If the Display Colour Shading option is selected, then a colour
spectrum (red is highest, blue is lowest) will be scaled to the
parameter values. Currently, there are 20 distinct colours in the
spectrum. Figure 14 shows a sample problem with the velocity
magnitude displayed using colour shading. The colour spectrum
and parameter mapping can be shown as an option with the
Display -> Annotation dialog. The scaling can be set to automatic,
with the maximum and minimum values set by the maximum and
minimum present in the model. Alternatively, the maximium and
minimum can be set manually. Values above and below the set
limits are assigned no colour (white). If Clip Colour Shading to
Water’s Edge is chosen then the colour spectrum is displayed only
inside the water’s edge. When the clipping feature is selected, the
maximum and minimum values are set based on the maximum and
minimum of the nodes that are inside the water’s edge. In Figure
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59
14, the clipping feature is selected. If the clipping feature is turned
off, the colour spectrum fills the entire computation domain.
Figure 14: A sample .cdg file with colour contours used to
display velocity magnitude.
5.6.11 Display -> Vector
The Vector command brings up the Vector Plot dialog box shown
in Figure 15. This dialog controls the display of velocity or
discharge intensity vector arrows. The Vector display is
independent from other display options and may overlay a contour
or colour fill display. The combination of depth shown as a colour
fill and velocity shown as vectors is a particularly effective way to
visualize the overall flow field, as shown in Figure 16.
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Figure 15: The Vector Plot dialog box.
Figure 16: A sample .cdg file with bed elevation displayed
using colour shading and velocity magnitude
displayed using a vector plot.
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61
The Vector Location option indicates where the vectors are to be
plotted. The At Nodes option is most useful for evaluating the
solution quality. The On Grid option usually gives a better picture
of the flow field.
The Scale value determines the length of the vectors drawn. For
example, a scale value of 10 means that a velocity of 1 m/s will be
drawn as a vector of length 10 m on the map. The grid spacing
value is the distance on the map between grid points. The
Minimum Depth sets the minimum flow depth for displaying the
velocity vectors. In Figure 16, the vectors are displayed at the
nodes with a scale value of 5 for a minimum flow depth of 0.02.
5.6.12 Display-> Bed Contours
The Bed Contour command toggles the display of a contour map
of the bed topography. This command is only active if a .bed file
has been loaded using the Load Bed File command found under
the Mesh Edit menu. This feature is displayed by default when a
bed file is loaded to indicate to the user that the bed data has
loaded properly. The setting for changing the contour interval of
the bed topography map is found in the Mesh Edit Options dialog
box which is accessible from the Options menu.
5.6.13 Display -> Annotation
The Annotation command brings up the Annotation Options dialog
box, shown in Figure 17. This dialog controls the display of legend
and axis information on the map. By default, all annotation is
turned off to allow the maximum screen area to the map itself. The
annotation information may be useful, particularly when printing.
Any, all, or none of the annotation options may be selected
simultaneously. Figure 18 shows the sample problem of Figure 16
with all annotation options displayed.
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62
Figure 17: The Annotation Options dialog box.
Figure 18: A sample .cdg file with annotation displayed.
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63
The Axes option simply causes the coordinates of the bounding
box to be shown on the display.
The Colour Legend shows the mapping of the colour spectrum to
the parameter values (and gives the name of the parameter being
displayed).
The Distance Scale causes a specified reference horizontal distance
to be printed with subdivided ticks.
The Vector Scale causes a reference vector of specified value to be
drawn.
The Title will be printed across the top of the display if this option
is selected.
Selecting Inflow and/or Outflow Discharge will cause the
respective value(s) to be printed at the top of the display.
Finally, the Time option displays the current model time.
5.6.14 Display -> Dump nodal csv file
The Dump nodal csv file command is an optional output from the
River2D model. There is one line in the csv file for every node in
the mesh. Each line contains the node number, the x coordinate,
the y coordinate and one parameter for that particular node,
delimited by commas. The parameter is the one selected in the
contour/colour dialog and indicated on the status line at the bottom
of the window. The .csv form of output is convenient for further
processing of the results in a spreadsheet or GIS program.
5.6.15 Display -> Dump grid csv file
The Dump grid csv file command is similar to the dump nodal csv
file command except that the points are set on a grid. The file
format is identical with one line in the csv file for every point in
the grid. A dialog box prompts the user for the coordinates of the
area to be gridded in terms of the lower left (x1, y1) and top right
(x2, y2) values. The default values are set to be just outside the
domain of the actual mesh points. Any point on the grid which is
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64
not "inside" the mesh will have negative node number and a zero
parameter value assigned. The point spacing value is the grid
increment in both directions. The grid will start with lines at x1
and y1 and extend to the last grid lines which are less than x2 or
y2.
5.6.16 Display -> Extract points to csv file
The Extract points to csv file command is similar to the dump
nodal csv file command except that the points are already defined
in an existing .csv file. The input file format is identical to the
output format of any of the .csv file options. Having defined a grid
or section once, this option allows output of different variables at
the same points, or the same variable at different discharges.
Another possibility is to compare different meshes of the same
area. A dialog box prompts the user for the input .csv file and then
for the output .csv file.
5.6.17 Display -> Extract section to csv file
The Extract section to csv file command is similar to the dump grid
csv file command except that the points are set on a line defined by
two endpoints. The file format is identical with one line in the csv
file for point along the line. A dialog box prompts the user for the
coordinates of the start (x1, y1) and end (x2, y2) points. Any point
on the line which is not "inside" the mesh will have negative node
number and a zero parameter value assigned. The point spacing
value is the increment spacing along the line. The set of points will
start with a point at x1 and y1 and extend to the last incremented
point before the end point is reached. The line may be defined in
any direction. The endpoints need not be within the mesh.
A profile line can be developed by concatenating individual line
segment files in a spreadsheet. Note that once defined, the
concatenated file can be used with the Extract points to csv file
command for other variables and discharges.
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5.7
Hydrodynamic Model Operations
The hydrodynamic component of River2D is controlled through
the commands under the Flow menu, as shown in Figure 19.
Figure 19: The River2D Flow Menu.
Although the hydrodynamic model is in fact a transient one, at any
given time, it can be run in two different modes: steady and
transient. Before describing the details of the specific commands in
the Flow Menu, a general overview of the solution process for each
of the two modes is useful.
5.7.1 Steady Mode Solution Process
In steady mode, the process used to converge the solution to steady
state is actually a pseudo-transient one, and therefore the iterations
are moderated by time increments and a final time is used to end
the simulation. The objective is to reach steady state with as few
calculations as possible while remaining stable under any flow
circumstances.
The basic approach is to start with a very short time step. This is
because the initial condition may be very different from the final
equilibrium and the rate of change of the flow response is likely to
be very high. As the solution progresses toward steady state,
usually the rate of change decreases and longer time steps are
possible. Due to the wavelike nature of open channel flows,
including moving bores and hydraulic jumps, the progress toward a
final steady state may not be uniform.
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The program logic proceeds as follows. At the end of each time
step, the overall relative change is calculated and compared to the
goal change. The overall relative change is defined as the square
root of sum of the squares of the changes in all variables divided
by the square root of sum of the squares of all of the variable
values. If the actual change is less than 1.25 times the goal change,
then the current iteration is accepted and the solution proceeds.
The time increment for the next step is then calculated as the
previous time step multiplied by the ratio of goal to actual changes.
If this ratio is greater than 1.5, it is reduced to 1.5. As long as the
time is less than the defined end time, another time step iteration is
calculated.
If the actual change is greater than 1.25 times the goal change, then
the current iteration is rejected. A new time increment is calculated
as half of the unsuccessful time step multiplied by the ratio of goal
to actual changes and the iteration is retried. This process may
repeat a few times until a small enough time step is achieved to
allow the solution to progress.
In most cases, the time step will eventually become large and the
solution will converge quickly to steady state. The total inflow and
outflow discharges are indicators of how close the solution is to the
final solution. As the solution converges, these numbers will
become constant and approximately equal (relative to the total
discharge). In principle, these values should be exactly equal, but
their calculation is approximate and depends on the degree of
discretization in the vicinity of the boundary.
The model time required to reach steady state varies from problem
to problem. A useful scale is to estimate the time for a fluid
particle to flow the length of the reach. Doubling or tripling this
time will usually provide a reasonable estimate of the model time
to steady state.
5.7.2 Transient Mode Solution Process
In a transient simulation, the objective is to obtain accurate spatial
results throughout the duration of a specific temporal event. This
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requires an accurate solution of the governing non-linear system of
equations at every time step in a simulation. In the model, the
system of non-linear equations is solved by approximating it as a
linear system and then iterating to a solution, with a specified level
of accuracy, using the Newton-Raphson technique. The transient
mode allows the user to control the amount of solution
convergence at every time step, unlike the steady mode where the
model is set to perform only a single Newton-Raphson at every
time step.
Prior to any transient simulation, boundary conditions and initial
conditions must be specified. Assuming subcritical flow, the
boundary conditions must be specified as discharge at inflow
sections and water surface elevation at outflow sections. In a
transient analysis, these conditions will typically take the form of a
discharge hydrograph (graph of discharge versus time) at an inflow
section and a stage hydrograph at an outflow section. If an
elevation hydrograph is unavailable at any of the outflow sections,
then a depth-discharge relationship is a viable alternative.
Initial conditions must be specified at every point in the
computation domain and should reflect the flow conditions prior to
the simulation event. Initial conditions can be obtained by running
the model in steady mode using constant boundary conditions that
reflect that the boundary conditions at the beginning of the planned
transient simulation until a steady solution is achieved.
Alternatively, the model can be run in transient mode from an
arbitrary initial condition with appropriate boundary conditions to
the point in time when the desired simulation is to start.
The choice of time step increment will depend on the speed of the
phenomenon and the desired level of resolution of the results. If
we desire detailed spatial resolution, then we use small elements.
Similarly, if we desire detailed temporal resolution, then we should
use small time steps. For a small-scale flow transients to be
modelled accurately, the Courant number should be no greater than
one where Courant number is defined as
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Cr =
Vw ∆t
∆x
(49)
where Vw is the speed of the phenomenon, ∆t is the time step
increment, and ∆x is the mesh spacing. Since mesh spacing will
vary across the domain, due to varying element sizes, the Courant
will not be the same throughout the domain at a given point in
time. In addition, the Courant will also vary at a point in space
through time because the speed of the phenomenon will vary. For
this reason, estimates of velocity and mesh spacing in the above
equation should provide a suitable value for the time step
increment.
If a large scale phenomenon is being modelled, such as a flood
wave, then the model can be run using a larger Courant number
and hence a larger time step. However, the larger the specified
time step increment, the more likely nonlinear instabilities are to
arise which will result in a reduction in the time step increment
until the instabilities are resolved.
Once the model is running, the solution proceeds as follows. At the
end of each Newton-Raphson iteration, the value of each solution
variable (3 for each node) is compared to their respective values
from the previous iteration. If the change in all of the variables is
less than the allowable change then the time step is accepted. The
time increment for the next step is then calculated as twice the
previous time step (given this value does not exceed the user
specified time step increment). Provided that the current time is
less that the defined end time for the simulation, the solution at the
next time will be evaluated.
If the change in any of the variables is greater than the allowable
change at the end of one Newton-Raphson iteration, then
subsequent iterations will be performed until the change is less
than the allowable change, and the time step will be accepted, or
once the maximum allowable number of iterations is reached. If
the maximum number of iterations is reached before the allowable
change criterion is met, then the time step is rejected. A new time
step increment is then calculated as half the unsuccessful time step.
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In some instances, a number of time steps may be rejected until a
small enough time step is achieved that will allow simulation to
progress.
The model time required for a transient event will really depend on
the time it takes for the phenomenon to propagate through the
domain (which should be determined prior to the simulation so that
transient boundary conditions are defined for the length of the
event) and the size of the time step increment.
5.7.3 Flow -> Run Steady
The Run Steady command brings up the Run Steady dialog box
shown in Figure 20. This dialog monitors and controls the
execution of the hydrodynamic model in steady mode. This dialog
box will remain open until it is explicitly closed by clicking the
box. The hydrodynamic model runs as a background process,
which means that display settings and other operations can be
changed while the model is executing. The values in this dialog
box are explained in general terms in the section above entitled
“Steady Mode Solution Process” and their specific uses are
detailed below.
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Figure 20: The Run Steady dialog box.
The "Present time" value is the point in model time at which the
solution is currently running or has stopped at. This value can be
reset before the start of a subsequent run.
"Final time" is the time at which execution of the hydrodynamic
model will be stopped. Actually, execution is halted once the
present time exceeds the final time.
"Time increment" is the size of the current time step. It may be set
at the start of a run. If it is too long the program will automatically
adjust it downward. Over the course of the iterations this value
normally increases steadily. Occasionally, it will decrease
dramatically and then gradually grow again. If this happens
frequently, there may be a problem with the boundary conditions
and/or the mesh.
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"Max time increment" is a user specified setting for the largest
time step allowed. This provides an extra measure of control on the
convergence process.
"Solution change" is the relative overall change in the solution
variables over the latest time step. The size of this value relative to
the goal solution change will govern how quickly the time
increment increases. Once the maximum time increment is
reached, the solution change should decrease with each subsequent
iteration. When this becomes sufficiently small (approx 0.00001),
the solution can be considered converged.
In some cases, the solution will reach a relatively small value of
solution change (of the order of 0.001) and refuse to diminish
further, regardless of the number of subsequent time steps.
Usually, this indicates a small, persistent, oscillation at one (or
sometimes more) points in the flow field. Often, the oscillation is
associated with a shallow node that alternates between wet and
dry. Another possibility is periodic shedding of eddies from
obstructions. Eddies forming at outflow boundaries are also
unlikely to stabilize. Steps that can be taken to stabilize the
solution include: change to numerical Jacobian (see below),
change eddy diffusivity (see below), refine the mesh in the
problem area, and relocate or revise the outflow boundary.
Finally, the oscillating solution may be considered acceptable, as
the size of the variation may be within the desired accuracy of the
simulation.
"Goal Solution change" is the user specified target relative overall
solution change. The bigger this value, the bigger the time
increments that will be used and hopefully the fewer the number of
iterations required to reach steady state. However, nonlinear
instabilities are more likely to arise, leading to more frequent
episodes of time step reduction followed by gradual increase
phenomenon.
"Log file name" is the name of the text file to which a record of the
program execution is written. This record includes the time, time
increment and solution change for each iteration. If the iterative
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solver is used, then additional information about the performance
of the solver is included in the log. If an iteration is rejected,
further information which may help localize the problem is also
written. The log file is always appended to and never overwritten.
Its default name is “steady.log” and it resides in the same directory
as the current .cdg file.
"Total Inflow" and “Total Outflow” represent the total discharge
flowing through inflow and outflow boundaries, respectively. One
characteristic of a steady state flow condition is that these two
values should be equal. In practice, due to approximation errors,
they will not be precisely equal. Instead, there will be a slight
difference (relative to the total discharge) between the two values.
The progress of the solution toward steady state can be observed
by tracking these numbers.
"Update display every ____ time steps" is used to set how often the
display updates. As drawing the display takes some processor time
away from the computations, it may be advisable to limit the
number of times that the screen is redrawn.
"Current step #" keeps track of the number of time step iterations
since the start of the latest run.
After the desired parameters are entered, the "Run" button will
initiate the solution process. While the model is running, the button
caption will change to "Stop". Pressing the "Stop" button while the
model is running will cause it to stop after the end of the next
iteration. While waiting for the iteration to finish, the caption will
change to "Stopping". The solution may be resumed after a halt by
pressing the “Run” button again. Stopping the solution in mid-run
allows parameters to be changed.
5.7.4 Flow -> Run Transient
The Run Transient command brings up the Run Transient dialog
box shown in Figure 21. This dialog monitors and controls the
execution of the hydrodynamic model in transient mode. This
dialog box will remain open until it is explicitly closed by clicking
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in the
box. As with a steady run, a transient analysis runs as a
background process. This means that display settings and other
operations can be changed while the model is executing. The
values in this dialog box are explained in general terms in the
section above entitled “Transient Mode Solution Process” and their
specific uses are detailed below.
Figure 21: The Run Transient Dialog Box.
The "Present time" value is the point in time at which the model is
currently running or has stopped at. In contrast to steady mode,
time cannot be loosely specified when using the model in transient
mode. The present time is used to set the model boundary
conditions based on input hydrographs (discharge and/or elevation)
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and to generate output at appropriate times. It is recommended
that the time only be reset at the beginning of a simulation.
"Final time" is the time at which execution of the hydrodynamic
model will be stopped. As in steady mode, execution is halted once
the present time exceeds the final time. It is recommended that the
any input hydrographs are defined to exceed the specified final
time, otherwise any transient boundaries will be taken as the value
at the maximum time in the hydrograph file (file format described
below in the section on boundary conditions).
"Time step increment, ∆t " is the size of the current time step. It
may be set at the start of a run, provided that it does not exceed the
Goal time step increment. The program will automatically adjust it
downward or upward as necessary in order keep the solution
within the specified tolerance after every time step. If the model is
running smoothly it should remain at the value of the Goal time
step increment.
"Goal ∆t” is the user specified time step increment for the
simulation. It is also the maximum time step which the model will
allow. This value is also used in defining when file output from
the analyses will be generated. For example, if the model is run
with a goal ∆t = 2, then the model will ensure that a solution is
produced at t = 2, 4, 6, 8…etc, even if the actual time step
increment must be less than 2 to maintain stability.
“# iterations per ∆t” is simply an indicator of how many NewtonRaphson iterations were required to converge at the last time step.
“Max # of iterations per ∆t” is a user specified setting that limits
the number of Newton-Raphson iterations for each time step. If the
actual number of iterations reaches this value before the solution
change is within the tolerance criterion, then the time step will be
rejected and the time step increment will be reduced.
"Solution tolerance" is a user specified value that controls the
amount of solution convergence required at every time step. This
tolerance is defined as the maximum change in any variable at
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node in the mesh. A larger value of tolerance will result in fewer
iterations per time step. However, accuracy may be compromised
and numerical instabilities may occur. Reducing the tolerance will
lead to more iterations per time step. As the Newton-Raphson
procedure converges quadratically, each additional iteration may
be associated with a tolerance reduction of several orders of
magnitude.
“Implicitness, θ” is a user specified value that controls the way in
which the model solves the system of governing equations. A
value of 0 indicates fully explicit while a value of 1 specifies fully
implicit. The solution to the governing equations is most accurate
when the model is run semi-implicit, that is θ = 0.5. However, the
solution is more stable with θ = 1.0. The model should not be run
with θ less than 0.5 as it will become unstable. The rule of thumb
for setting θ is as follows. If you are modeling small flow features
then you will probably want to set the goal time step increment
such that the Courant number is one and set θ = 0.5. If, on the
other hand, you are only interested in the general trend of the
solution, that is large scale phenomenon such as a flood wave, then
you can get away with using a larger goal time step increment and
should set θ = 1.0. This is because the use of a larger time step
increment will obscure any accuracy improvements provided by
using θ = 0.5. Fully implicit solutions tend to be smoother and
converge more quickly.
"Log file name" is the name of the text file to which a record of the
program execution is written. The record that is produced is
essentially the same as the one produced in steady mode except
that it includes the number of Newton-Raphson iterations
performed at every time step. Its default name is “transient.log”
and it resides in the same directory as the current .cdg file.
"Total Inflow" and “Total Outflow” represent the total discharge
flowing into and out of the model, respectively. In a transient
analysis these will change according to the transient boundary
conditions.
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"Update display every ____ time steps" is used to set how often the
display updates. As drawing the display takes some processor time
away from the computations, it may be advisable to limit the
number of times that the screen is redrawn.
The “Output Options” button simply opens the Transient Output
Options Dialog box. This dialog allows the user to select and
customize output from the transient analysis. Details of this dialog
are explained in section entitled ‘Option Setting Operations’. The
desired output must be chosen prior to starting the simulation.
Once all of the parameters are set appropriately, the "Run" button
will initiate the solution process. While the model is running, the
button caption will change to "Stop". Pressing the "Stop" button
while the model is running will cause it to stop after the end of the
current time step. While waiting for the time step to finish, the
caption will change to "Stopping". The solution may be resumed
after a halt by pressing the “Run” button again. However, if the
model is stopped and then started again when the model is running
at a time step that is less that the goal time step increment,
subsequent transient output will not be produced at even intervals.
5.7.5 Flow Boundaries
In the model, boundary conditions must be set for each boundary
element in the model. A flow boundary is a convenient format for
applying boundary conditions. For example, an inflow section
may be comprised of multiple boundary elements. Having to set
the boundary conditions at each boundary element would be very
cumbersome. Not only would you have to specify the element as
being an inflow element but you would also have to assign the
element a portion of the total inflow discharge so that the discharge
in all of the inflow boundary elements would add up to the total
inflow discharge. A flow boundary allows the user to set the
boundary condition along a section of the boundary rather than at
each boundary element individually. They are called flow
boundaries to distinguish them from no flow boundaries. The
necessary information to define a flow boundary includes: a
starting node (along the boundary), an ending node (also along the
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boundary), the type of flow boundary (inflow or outflow), and
information relevant to type of boundary (e.g. total discharge for
an inflow boundary). Flow boundaries are always defined walking
around the boundary with the computational mesh to the left.
Therefore flow boundaries on the external boundary are defined in
the counter clockwise direction while flow boundaries on an
internal boundary are defined in the clockwise direction. There are
two flow boundary commands available to the user. They are Edit
Flow Boundary and Define New Flow Boundary. They are found
under the Flow menu and are described below.
5.7.6 Flow -> Edit Flow Boundary
This command allows the user to edit existing flow boundaries.
Once selected, the user must use the mouse (single click) to select
either an inflow or outflow boundary. This opens up the Edit Flow
Boundary dialog box, as shown in Figure 22.
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Figure 22: The Edit Flow Boundary Dialog.
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The extent of the boundary can be changed in the top of the dialog
by changing the flow boundary starting and/or ending nodes (in
terms of node numbers. The type of boundary condition can also
be modified simply by selecting the appropriate radio button. The
choices are an inflow condition, an outflow condition or a no flow
condition.
For an inflow condition, the user must also specify either a fixed or
time varying discharge. For a fixed discharge, only a single value
is required while if it the discharge is varying with time, then the
location of a file containing an inflow hydrograph (table of times
and the corresponding discharges) must be specified.
For an outflow condition, the user has four possible options: a
fixed outflow water surface elevation, a time varying water surface
elevation, an elevation(stage)-discharge relationship (rating curve),
or a depth-unit discharge relationship.
The first two options are similar to the inflow options. The fixed
elevation requires a single value while the time varying elevation
requires a file, containing a stage hydrograph, to be specified.
Examples of discharge and stage hydrograph files are shown below
to illustrate the file format in Table 1. Both are text files,
differentiated by their file extension: .bcq for a discharge file and
.bch for a stage file.
The third option is a stage-discharge relationship also known as a
rating curve. This is also in the form of a table (discharges and
corresponding stages) and therefore the location of a file,
containing the table, must be specified. A rating curve filename
has the same format as the hydrograph files and must have a .bcr
file extension. An example of a rating curve file is included in
Table 1.
All three files types can have any amount of descriptive text at the
beginning of the file provided that there is no open bracket ‘(’ in
the text. There is no length limit to the size of the table. The only
requirement is that the table is enclosed with brackets as shown in
Table 1.
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Table 1: Format for Boundary Condition files
discharge hydrograph for River X
first column is time is seconds
second column is discharge in cms
(
0
200
400
600
800
1000
1200
1400
1600
1800
)
175
209
239
262
273
273
262
239
209
175
stage hydrograph for River X
first column is time is seconds
second column is water surface elevation in metres
(
0
175.0
200
209.2
400
239.3
600
261.6
800
273.5
1000
273.5
1200
261.6
1400
239.3
1600
209.2
1800
175.0
)
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Table 1 (continued)…
rating curve for River X
first column is discharge in cms
second column is water surface elevation in metres
(
203.24
99.50
207.29
99.55
212.96
99.60
220.25
99.65
229.16
99.70
239.69
99.74
251.84
99.76
270
99.77
290
99.78
)
At a given point in time, all three of these outflow conditions will
force the water surface elevation to be constant across the outflow
boundary. In some cases, the model will have difficulties
converging when the same elevation is forced across an outflow
boundary. When this occurs, the forth and last option, a depth-unit
discharge relationship, may help with convergence. This option
forces a weir type flow through each boundary element in the
outflow boundary. Because the equation is applied at the element
level rather than across the whole outflow section (which would
require a depth-total discharge relationship), the elements can
respond individually to the flow in the domain resulting in a water
surface elevation that varies across the outflow section. This type
of boundary condition is particularly handy when the downstream
boundary is chosen at a bend in the river where the water surface
elevation across the section is unlikely to be constant.
If a no flow condition is selected, then any existing flow conditions
and the location of the flow boundary will be removed, as is stated
in the dialog. The flow boundary can only be reinstated by
redefining it using the “Define New Flow Boundary” command
which is described in the following section.
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5.7.7 Flow -> Define New Flow Boundary
Selecting this command will open the Define New Flow Boundary
dialog box. This dialog allows the user to define new flow
boundaries in the same way as the Edit Flow Boundary dialog
allows the user to edit existing flow boundaries. The starting and
ending nodes must be defined in terms of node number. To
determine the node numbering for a new flow boundary simply use
the Display->Node Numbers command.
Figure 23: The Define New Flow Boundary Dialog
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5.7.8 Flow -> Reset Initial Conditions
This command allows the user to reset the initial conditions. Initial
conditions are required at every node in the domain as an initial
guess in the iterative solution procedure used to obtain the steady
state hydrodynamic solution. These conditions include water
surface elevation and discharge intensity components (in the x and
y directions). When this command is selected, a dialog box
appears requesting an estimate of the inflow water surface
elevation. The estimate supplied by the user is used to set up an
initial water surface throughout the computational domain.
Depending on the value of the inflow estimate (typically chosen to
be higher than the outflow because water flows downhill), an
initial water surface slope is obtained. This water surface is
estimated based on a numerical solution of the Laplace equation
using the same finite element mesh. Note that this initial condition
is somewhat different than that provided by the mesh editor
R2D_Mesh. Once the water surface slope is obtained, the
discharge intensities are reset to zero.
5.7.9 Flow -> Check Memory
When the direct solver is selected, the solution process for each
iteration involves the formation of a very large Jacobian Matrix.
This matrix represents the interaction of all of the solution
variables (three per node) on all of the other solution variables.
While a sparse matrix storage scheme is used (the skyline storage
method), this matrix remains very large and the RAM capacity to
store the matrix is the critical limitation to the number of nodes in
a mesh. The size of the Jacobian Matrix depends on the complexity
of the mesh as well as on the number of nodes.
The memory for the Jacobian Matrix is determined and
allocatedduring the first iteration. At any time after this, the Check
Memory command will display an information box indicating the
amount of memory allocated as well as the number of nodes and
elements in the mesh, as shown in Figure 24. The information box
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also indicates to the user which solver is currently being used in
the solution process. If the memory allocated is larger than the
physical RAM installed, then hard disk space will be used via the
standard W95/98/2000/ME/NT/XP dynamic memory mechanism.
Hard disk access is relatively slow however, and the overall
solution time will be much longer.
Figure 24: Check Memory information box for the direct
solver.
The iterative solver does not form the Jacobian Matrix. This is the
main advantage of this solver, since it requires far less memory
allocation. As is the case for the direct solver, the memory that is
necessary for the iterative solver is determined and allocated
during the first time step iteration. After the first iteration using
the iterative solver, the Check Memory command will display an
information box, as shown in Figure 25.
Figure 25:Check Memory information box for the iterative
solver.
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5.7.10 Flow -> Cumulative Discharge
This command is used for calculating the cumulative discharge,
also called stream function, throughout the computational domain.
When this command is selected, the cumulative discharge will
automatically be displayed as a colour spectrum once the
calculations are complete, as shown in Figure 26. Cumulative
discharge is one of the possible display parameter in the Contour
dialog box, as shown in Figure 12. Therefore, the display of this
parameter can be turned off once the calculations are complete.
Unlike the other flow parameters that are available for display,
such as depth, velocity and discharge intensity, the stream function
value at every node is not updated every time the hydrodynamic
solution is updated. Therefore, if the flow solution changes, this
command needs to be recalled if the cumulative discharge is to
reflect the current flow solution.
Although the cumulative discharge can be used as an illustrative
measure of the flow distribution in a channel, its main purpose in
the River2D model is in the boundary extraction process. The
streamlines that are defined by the stream function are one of the
possible choices that can be selected by the user to map out the no
flow boundaries of a boundary being extracted. If the model has
not been run to convergence, then the streamlines that result may
not reflect those of the steady state solution. Therefore, if the
cumulative discharge is being calculated in order to use the
boundary extraction command, the user is advised to ensure that
the model has been run to steady state before calculating the
cumulative discharge.
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Figure 26:A sample .cdg file with cumulative discharge
displayed.
Since this command is in its beta stage, in some cases, it requires
user intervention to ensure that it is executed properly. These
interventions are described as follows.
If the problem at hand has only one inflow boundary, River2D can
easily calculate the cumulative discharge for the domain. Along
the external computational boundary, the required boundary
conditions for calculating cumulative discharge are set by
‘walking’ around the boundary. For the boundary conditions to be
set correctly, the model must start at the boundary node that marks
the transition from the inflow boundary to the no flow boundary
while walking around the boundary in the counter-clockwise
direction. If there is more than one inflow boundary, then there is
more than one node that the model could choose as a starting point.
Unfortunately, not all of the inflow to no flow transition nodes will
result in the correct boundary conditions if they are used as starting
nodes. A transition node can only be used as a starting node if the
next transition node, traversing the boundary in the counterclockwise direction, is a no flow to outflow transition node.
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Therefore, if a domain in question has more than one inflow
boundary segment, the user will be prompted to select the
appropriate starting node so that the solution may progress
correctly.
For sites with more than one computational boundary, that is, one
external plus one or more internal boundaries, the order of these
boundaries in the .cdg file is crucial to the correct progress of the
cumulative discharge solution. The computational boundaries are
described in the list of boundary elements. For the cumulative
discharge to be calculated correctly, the external boundary must
appear first in the list of boundary elements. The original order of
the boundaries in the boundary element list is determined by the
order in which they were defined when the mesh was created. If
the external boundary does not appear first in the list, a text editor
can be used to relocate it. If the location of the external boundary
in the list is not known, it can be ascertained by examining the list
in a text editor in conjunction with displaying the node numbers in
the River2D environment.
5.8
Mesh Edit Operations
While most mesh editing takes place in the mesh generator,
R2D_Mesh, there are some cases where it is very useful to be able
to make minor changes, particularly refinements, to the
computational mesh without having to back to the R2D_Mesh
environment. In other cases, large sites may require sub-sites to be
‘cut out’ and treated separately. For these reasons, mesh editing
capabilities have been incorporated into River2D. The mesh
editing operations that are available to the user in the Mesh Edit
menu, as shown in Figure 27, are described below.
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88
Figure 27:The River2D Mesh Edit Menu.
5.8.1 Mesh Edit->Load Bed File…
To make any changes to the computational mesh, the original bed
topography that was used to create the mesh must be loaded into
the River2D environment. This is achieved by selecting the Load
Bed File command. A standard Open File dialog is presented with
the filter set to files with a .bed extension. Loading a new bed file
replaces the one currently loaded.
The Load Bed File command can also be used to change bed
elevations and roughnesses in the computational mesh. First, the
.bed file that was used to generate the mesh is revised either
manually or by use of the R2D_Bed utility. Then the Load Bed
File command is used to read in the edited .bed file. The bed
elevation and roughness at all computational points in the mesh are
then interpolated from the revised bed points. Making changes to
the bed elevations and/or roughnesses can be useful in two
contexts. The first is during model calibration, where changing
roughnesses locally may improve the match to measured data. The
second is to investigate anticipated changes to the channel
geometry.
5.8.2 Mesh Edit->Extract Boundary…
Once a mesh has been run to steady state, it may become apparent
that a more detailed solution in some regions would be beneficial
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89
for hydrodynamic or habitat reasons. The Extract Boundary
command allows the user to define and extract a boundary that can
be used to develop a more detailed computational mesh of a region
inside the boundary.
This command uses the concept of streamlines (which are
theoretical ‘no flow’ boundaries within the flow) to define the ‘no
flow’ boundaries in the boundary extraction process. In order to
map out streamlines in the flow, the stream function, or cumulative
discharge, must be known throughout the computational domain.
Therefore, before a boundary can be extracted, the cumulative
discharge must be calculated. This is achieved by selecting the
Cumulative Discharge command, located under the Flow menu.
Once the cumulative discharge has been calculated, the Extract
Boundary command will become active. Selecting this command
opens the Extract Boundary Dialog box as shown in Figure 28.
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Figure 28: The Extract Boundary Dialog Box.
At this time, only simple boundaries can be extracted from the
flow, that is boundaries with only one inflow segment, one outflow
segment, and two no flow boundary segments (left and right
boundaries).
The no flow boundaries can be defined using a streamline, a depth
contour, or a computational boundary. If a streamline is used to
define a no flow boundary, then the value of cumulative discharge
that represents the streamline must be specified as shown for the
left boundary in Figure 28. Note: left and right boundaries are
defined looking downstream. In cases where the discharge in the
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domain is very low, the streamline in the flow may not be very
smooth and therefore they may not make very good no flow
boundaries. In this case a depth contour, such as the water’s edge
(0.02m contour line), can be used as an alternate no flow boundary.
If a depth contour is used to define a no flow boundary, then the
value of the depth contour must be specified as shown for the right
boundary in Figure 28. Since the computational boundaries are no
flow boundaries, they can also be used to define the no flow
boundaries in the extraction process. If a computational boundary
is selected, no value is required.
The inflow and outflow boundaries are determined based on points
specified by the user. The location of these flow sections should
be such that the features that are being used to define the no flow
boundaries are continuous lines (not branching) between the inflow
and outflow sections. The inflow section should be selected at a
location where the majority of flow across the section is parallel to
the no flow boundaries. This is to ensure that the inflow boundary
condition that will be set along this boundary (direction of flow is
assumed to be perpendicular to the inflow section) is representative
of the actual flow. The outflow section should be selected at a
location where the water surface elevation is relatively constant
between the left and right boundaries so that the constant elevation
boundary condition at the downstream boundary is valid
The defining points must be selected along the desired location of
the flow boundary such that they are spatially between the left and
right no flow boundaries. In addition these points should be
selected at a locations along the section where the velocity is well
defined and representative of the direction of flow of the section
they are defining. This is because the velocity vector of a defining
point is used to determine the strike of the section being defined.
The last box is used for selecting the node spacing for the new
boundary. The node spacing must be set to a value greater than
zero.
Because the boundary extraction process is accomplished through
a dialog box, the values for the no flow boundaries and the points
used to determine the location of the inflow and outflow
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92
boundaries must be determined before the dialog box is opened.
To choose the values for no flow boundaries being defined by a
streamline or depth contour, the user should set display the
cumulative discharge or water depth using either a contour map or
a colour spectrum. This allows the user to see where the
streamlines or depth contours are located in the flow. Once the
location of the left and right boundaries has been determined
visually, the values of the cumulative discharge or depth contours
that define these boundaries can be determined using the mouse.
When the mouse is placed over one of the streamlines or depth
contours, the corresponding cumulative discharge value or contour
value will appear in the status bar. The mouse and status bar can
also be used to determine the x and y coordinates of the points
required to define the inflow and outflow boundaries.
Once all of the parameters are entered into the dialog box and the
OK button is selected, a standard save dialog box will open so that
the boundary can be saved to file. The boundary is saved in the
same format as a mesh file with the .msh extension. In addition to
the nodes and boundary elements that are used to define the
boundary, the mesh file contains the inflow and outflow boundary
conditions. The value of the discharge at the inflow boundary
segment is determined by taking the difference between the values
of cumulative discharge of the end points that are used to define
the inflow segment. At the outflow segment, a fixed water surface
elevation boundary condition is set. The value of this boundary
condition is based on the average water surface elevation between
the two end points that are used to define the outflow boundary.
Therefore, if the outflow segment is selected at a location where
the water surface elevation is not relatively constant across the
section, then the boundary condition at the downstream end may
not reflect the flow conditions. (It should be noted that, if the
boundary was defined to include any internal no flow boundaries
(i.e. an island), these boundaries will not be included in the mesh
file. If it is imperative that the internal no flow boundaries be
maintained, they will have to be added manually to the mesh file.)
To develop a mesh based on this boundary, the .bed file that was
used to make the original mesh (from which the boundary was
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93
extracted) must be opened in the mesh editor, R2D_mesh. Once
the bed file opened, the new boundary can be opened and a new
mesh can be developed. Once the development of the mesh is
complete, a cdg file can be created and solved in River2D.
5.8.3 Mesh Edit->Mesh Merge
The Mesh Merge command is used for combining information
from two meshes. This command takes information from a mesh
that is on file (secondary mesh), and places it in the mesh that is
currently displayed (primary mesh). Meshes can be merged in two
ways:
1. Region Replace: This option removes all of the nodes and
breaklines (except boundary nodes) from the primary mesh
that are within the computational boundary of the
secondary mesh and replaces then with nodes and
breaklines from the secondary mesh. All nodes from the
secondary mesh are included into the primary mesh, even
the boundary nodes, although the boundary conditions are
not included. The type of the node (fixed, sliding, or
floating) is retained when it is transferred from the
secondary mesh to the primary mesh.
2. Information Transfer: This option simply interpolates all
flow and bed information from the secondary mesh and
transfers it to the nodes in the primary mesh that are within
the computational boundary of the secondary mesh.
Typically the computational boundary of the secondary mesh falls
within that of the primary mesh, although this is not a necessary
criterion. The secondary mesh could overlap the primary mesh or
it could include the primary mesh.
Selecting this command with a primary mesh in the display opens
the Mesh Merge dialog box as shown in Figure 29.
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Figure 29: The Mesh Merge Dialog box.
Once the merging option is selected and the OK button is clicked,
a standard open dialog box will appear so the .cdg file that defines
the secondary mesh can be loaded. Once the file is selected and
the OK button is clicked, the secondary mesh will be merged with
the first mesh. If the Region Replace option is selected, then the
changes to the primary mesh can be easily observed by displaying
the element boundaries in the mesh. This is accomplished by
selecting the Mesh command under the Display menu. If the
meshes are merged using the Information Transfer option, the
changes to the primary mesh will be more difficult to detect, and
be dependent on how different the flow and bed information is
between the two meshes.
The Mesh Merge command is useful in conjunction with the
Boundary Extraction command. Once a boundary is extracted and
the resulting mesh has been run to steady state, the user can use the
Mesh Merge command to combine the results of the secondary
submesh with the results from the primary mesh from which the
submesh was derived.
5.8.4 Mesh Edit->Add Floating Node
The Add Floating Node command sets the mouse to add floating
nodes mode. This is indicated by a change in the cursor to a cross
hair. At every point (within the computational boundary) that is
subsequently selected by clicking the left mouse button, a new
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floating node will be generated. To exit add floating nodes mode,
reselect the Add Floating Node command. The Add Floating Node
command is also available by selecting the
toolbar button.
5.8.5 Mesh Edit->Region Refine
The Region Refine command places a single new node in each
existing element within the region polygon defined by the user. To
define the region, first select the Region Refine command. This
sets the cursor to a cross hair. Then click the left mouse button on
any point within the mesh boundary to start defining the region to
be refined. Then move to the next point keeping the area to be
refined to the left of the red line being generated (counterclockwise
direction). Click at this location and at subsequent locations until
the region is almost completely defined. Close the polygon by
clicking the mouse button very close to the starting point. When
the region has been closed, the red line defining the border of the
region will disappear and a node will appear at the centre of every
triangular element that is contained within the region. After
triangulation the area will have a node density of roughly twice
(actually 3 ) that previous to refinement. Figure 30 and Figure 31
illustrate the process of region refinement.
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Figure 30: Defining a region to be refined.
counterclockwise
Figure 31: Nodes that are inserted once the region is defined.
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5.8.6 Mesh Edit->Auto Refine
The Auto Refine command works in a similar manner to the
Region Refine command. It also refines the mesh by placing nodes
at the centre of existing triangular elements but it uses variable
criteria rather than spatial (in or out of the polygon) criteria.
Selecting the Auto Refine command will open the dialog box
shown below in Figure 32.
Figure 32: The Auto Refine Dialog Box
This dialog box allows the user to provide the variable criteria for
refinement in the form of an inequality statement. The edit boxes
are for setting the limits of the inequality statement while the two
drop down list boxes at the centre of the dialog box are for
selecting the variable that must fall within these limits. The left
edit box is used for setting the lower bound of the inequality
whereas the right edit box is used for setting the upper bound. The
second list is for choosing which nodal parameter will be used for
refinement. The parameter choices are the same as those available
to the user is the Colour/Contour dialog box. The first list is for
specifying whether an element will be targeted for refinement
based on the average value of this parameter within the element or
the absolute change in this parameter over the element.
5.8.7 Mesh Edit->Delete Floating Node
The Delete Floating Node command sets the mouse to delete
floating nodes mode. This is indicated by a change in the mouse
cursor to a cross hair. Every floating node that is subsequently
selected by clicking the left mouse button, will be marked with a
red X. Once the mesh is re-triangulated, these nodes will be
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removed from the mesh. The Delete Floating Node will mark
floating or sliding nodes for deletion. Fixed nodes or nodes that
were used to define the computational boundary cannot be deleted.
The Delete Floating Node command is also available by selecting
the
toolbar button.
5.8.8 Mesh Edit->Clear Untriangulated Nodes
The Clear Untriangulated Nodes can be used to clear the mesh of
all new nodes that have not yet been triangulated. This includes the
nodes that have been added by either the Add Floating Node
command or the Region Refine Command.
5.8.9 Mesh Edit->Triangulate
The Triangulate command retriangulates the mesh incorporating
any node changes into the mesh. The command invokes a constrained Delauney triangulation routine which gives the “best”
possible triangles. That is, it generates triangles which are as close
as possible to equilateral. The constrained nature of the
triangulation is necessary to insure that no triangle oversteps the
defined domain boundary. The Triangulate command is also
available by selecting the
toolbar button. Note: this command
is only available for use once nodes have been inserted or marked
for deletion.
5.8.10 Mesh Edit->Smooth
The Smooth command is used to make the triangles more regular
in shape, to give a more gradual transition between triangles of
different size, and to give a better representation of the underlying
topography. The smoothing process moves each point to a more
central position with respect to neighbouring points, as defined by
the triangles and towards breaks in topographic slope. After the
points have been moved, the entire mesh is re-triangulated to
insure the best possible triangulation. The smoothing process may
be repeated as often as desired. The mesh will become smoother
and more regular, but the discretization contrast will gradually
diminish.
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The smooth command actually causes the floating and sliding
nodes in the mesh to move to new positions using a combination of
element shape and topographic representation as criteria. The
direction of the node movement will tend to be toward the
adjoining triangle with the largest elevation difference between the
bed and the mesh. This difference is measured at the centre of the
element. A parameter controlling the relative importance of the
topographic criteria, called the Bias Smoothing Parameter, can be
set in the Mesh Edit Options dialog found under the Options Menu.
A value of zero indicates no topographic effect and a value of one
indicates no consideration for element shape. With higher values of
this parameter, larger differences in adjacent element size are
allowed.
The Smooth command is also available by selecting the
button.
toolbar
5.8.11 Mesh Edit->Mesh Information…
The Mesh Information command opens an attention box, shown in
Figure 33, that displays the total number of nodes, the total number
of triangular elements and the Quality Index (QI) of the mesh. The
QI is the minimum “triangle quality” value for all triangles
generated. The triangle quality is defined as the ratio of triangle
area to circumcircle area (the circle which passes through the three
points defining the triangle) normalized to the corresponding ratio
for an equilateral triangle. Thus, an ideal mesh (all equilateral
triangles) would have a QI of 1.0. Real meshes will have a QI of
less than one. Typical acceptable values may be in the order of 0.1
to 0.5. Smoothing will usually increase the QI dramatically, if it is
very low. Meshes with many breaklines will have lower QI values,
especially if the breaklines are closely spaced.
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Figure 33: The Mesh Information box.
5.9
Habitat Analysis Operations
Once an acceptable hydrodynamic solution is obtained, a habitat
analysis may be performed, if desired, using the Habitat Analysis
Operations under the Habitat Menu shown in Figure 34. The
operations available are described below.
Note that a .cdg file must be loaded before using any habitat
functions.
Figure 34: The Habitat Menu.
5.9.1 Habitat -> Preference
The Preference command is used to load a preference curve file. A
standard Open File dialog is presented with the filter set to files
with an extension of .prf. Once a file has been selected and loaded,
a check mark will appear before the Preference command in the
Habitat Menu.
Each .prf file contains the depth velocity and channel index
preference curves for a single fish species/life stage. Each
preference curve is delimited by brackets and the points on the
curve are on separate lines with the point number, the parameter
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value, and the corresponding suitability index separated by spaces.
Table 1 presents a sample .prf file for Juvenile Brown Trout.
If more than one species/life stage is to be analyzed, then the
preference files may be loaded sequentially and a WUA calculated
for each of them.
Table 2: A sample .prf file.
Kaninaskis River 97/01/07
Fish Preference Curves for
Brown Trout - Juvenile
Velocity
(
)
Depth
(
1
2
3
4
5
6
7
8
9
0
0.08
0.23
0.37
0.52
0.67
0.83
0.98
100
1
1
0.5
0.1
0.1
0.1
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
0
0.08
0.22
0.37
0.52
0.67
0.82
0.97
1.12
3.05
100
0
0
0.5
0.5
1
1
1
1
1
0.5
0.5
1
2
3
4
5
6
7
0
7
21
35
49
63
100
0.1
0.1
0.1
0.5
0.5
1
1
)
Channel Index
(
)
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5.9.2 Habitat -> Channel Index
The Preference command is used to load a channel index file. A
standard Open File dialog is presented with the filter set to files
with a .chi extension. Once a file has been selected and loaded, a
check mark will appear before the Channel Index command in the
Habitat Menu.
Channel index files have the same format as .bed files except that
the channel index value replaces the bed roughness height.
Loading a new channel index file replaces the one currently
loaded. Table 2 shows a few lines from a sample .chi file.
Table 3: A section from a sample .chi file.
Node
Number x coordinate y coordinate
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
5023.629
5027.854
5028.133
5029.339
5029.342
5030.814
5031.653
5030.541
5026.543
5020.711
5009.677
5001.159
4992.345
4979.924
4965.825
4956.073
5055.417
5048.185
5037.883
5029.443
5019.357
5008.098
4997.251
4983.261
4969.127
4954.302
4937.6
4933.663
4927.675
4913.866
4897.613
4876.482
Bed
Elevation
Channel
Index
199.223
199.102
199.024
199.137
198.945
198.817
198.94
199.231
198.824
199.301
199.472
199.496
199.599
199.592
199.65
199.722
6.2
6.2
4.9
4.9
4.9
4.9
4.9
4.9
5.25
5.25
5.25
4.95
4.95
4.95
4.95
4.95
5.9.3 Habitat -> Weighted Usable Area
The Weighted Usable Area command computes the weighted
usable area for the current hydrodynamic solution, preference file
and channel index file. The WUA result is presented in an
information box, as shown in Figure 35, along with the total mesh
area for reference. Once the WUA calculation is performed, then
display of the various habitat indices is possible.
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Figure 35: The Weighted Usable Area information box.
5.9.4 Habitat -> Save Node Attribute File
The Save Node Attribute File is an output option which saves
habitat as well as hydrodynamic information for each node. Note
that the WUA calculation must be performed first so that all the
habitat indices are defined at each node.
The format of the Node Attribute File is two information lines
followed by one line for each node in the mesh. The first
information line holds the names of the mesh file(.cdg), the
channel index file (.chi), and the preference curve file (.prf) that
was used to generate the most recent WUA. The second
information line contains column headings for the node attributes.
Each line contains the node number, the x coordinate, the y
coordinate, the depth, the velocity, the channel index, the depth
suitability index, the velocity suitability index, the channel index
suitability index, the composite suitability index, and WUA
associated with that node. Commas are used for delimiters.
5.9.5 Habitat -> Save Physical Attributes
The Save Node Attribute File is an output option which saves
hydrodynamic information for each node. The WUA calculation
need not be performed. The channel index file may or may not be
set.
The format of the Node Attribute File is three information lines
followed by one line for each node in the mesh. The first
information line holds the names of the mesh file(.cdg) and the
channel index file (.chi), if defined. The second line contains the
current total inflow discharge and the number of nodes in the file.
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104
The third information line contains column headings for the node
attributes. Each line contains the node number, the x coordinate,
the y coordinate, the depth, the velocity, the channel index, and
tributary area associated with that node. Commas are used for
delimiters.
5.10
Ice Cover Operations
River2D has the ability to simulate flow under an ice cover. The
Ice Cover Menu provides the user with two commands: one for
incorporating an ice cover into the computational domain, and one
for removing it once it is no longer required. These commands are
described in detail below.
5.10.1 Ice Cover-> Load Ice File
Before simulating flow under an ice cover, the geometry of the ice
cover must be defined throughout the computational domain. That
is, ice thickness and ice roughness must be specified at every
computational node. These values must be interpolated from a
separate ice topography file. An ice topography file is similar in
format to a bed topography file except that bed elevation and bed
roughness are replaced with ice thickness and ice roughness. For a
description of the construction of ice topography files please refer
to the R2D_Ice User Manual.
Selecting the Load Ice File command launches a standard Open
File dialog box with the filter set to files with an .ice extension.
Once a file is selected and the OK button is clicked the ice
topography is interpolated to the computational nodes as follows.
1. A TIN is generated from the nodes in the ice file.
2. The location of each node in the computational mesh is
checked. If it falls spatially within an ice covered element in
the ice TIN then nodal ice parameters (thickness and
roughness) are linearly interpolated from the element. If it falls
within an ice-free element, then the nodal ice parameters are
set to 0. Ice covered and ice-free elements are distinguished by
their nodal values of ice thickness. If any of the nodes in an
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105
element have an ice thickness of 0, then the entire element is
ice free, otherwise it is ice covered.
After the ice parameters are interpolated, the ice thickness value
for the domain will automatically be displayed using a colour map
as shown in Figure 36. Ice covered elements (all nodes must have
and ice thickness greater than 0) are coloured varying shades of
blue dependent on the nodal ice thicknesses while ice-free
elements are coloured dark blue to represent the open water areas
in the domain.
Figure 36: A sample cdg file with ice thickness displayed.
If a new ice file is loaded when one is present, then the existing ice
TIN will be replace with a new one based on the nodes in the new
ice file. Also nodal ice parameters will be updated to reflect the
new ice TIN.
5.10.2 Ice Cover -> Remove Ice
As the name suggests, this command removes any ice from the
model. The ice TIN is removed and all of the nodal ice thickness
values and ice roughness values are reset to 0.
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5.11
Option Setting Operations
There are a number of parameters that the user can set in the
River2D environment. These include options related to the flow
calculations, the mesh edit commands, and the habitat analysis
calculations. These option-setting operations are described as
follows.
5.11.1 Options -> Habitat Options
The Habitat Options dialog box, shown in Figure 37, allows the
setting of Channel Index Interpolation and WUA calculation
Method options for the habitat analysis.
Figure 37: The Habitat Option dialog box.
Channel Index values need to be evaluated at all computational
nodes in the mesh. They are specified at another set of points,
however, in the .chi file. The Channel Index Interpolation from
specified to computational points may be either continuous or
discrete. Continuous means that the channel index variable can
vary continuously over a range. Percentage cover might be an
example. In the model the channel index for any point will be
evaluated by linear interpolation from the values at surrounding
points. A discrete channel index, on the other hand, represents a
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107
specific class or category. If the discrete option is selected, then the
channel index at any point is taken to be equal to the channel index
at the nearest specified point.
The WUA Calculation Method refers to the procedure for
calculating the composite suitability index from the three separate
suitability indices (for depth, velocity, and channel index). Product
multiplies the three indices together. Geometric mean is the cube
root of the product. Minimum is the smallest of the three.
5.11.2 Options->Mesh Edit Options
The Mesh Edit Options command opens a dialog box that allows
the user to set two parameters. The first is the Bias Smoothing
Parameter which must be set to a value between 0 (no topographic
effects) and 1 (no consideration for element shape) inclusively. By
default, this value is set to 0.5. The second is the contour interval
for the bed topography contour map. This value is set to 0.5 by
default. The Mesh Edit Options dialog box is shown in Figure 38.
Figure 38:Mesh Edit Options dialog box.
5.11.3 Options->Flow Options
The Flow Options command opens a dialog, shown in Figure 39,
which allows the user to set various parameters that affect the
computational solution. These parameters appear at the top of the
.cdg input file. As a result, they can be changed by manually
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108
editing the .cdg file as well as being changed in the River2D
environment. The default values for these parameters are set when
a .cdg file is first created in the mesh editor, R2D_Mesh.
The first parameter is the upwinding coefficient of the PetrovGalerkin finite element scheme used to solve the hydrodynamic
equations. The upwinding coefficient, ω, should be between 0 and
1 inclusively. Hicks and Steffler (1992) recommended a value of ω
= 0.25 for transient problems and a value of of ω = 0.5 when using
the model to converge to steady state. This is also the default
value.
To avoid the computational difficulties associated with very
shallow flow or dry elements, the equations change from the
surface flow equations to groundwater flow equations in these
problem areas. The second, third, and fourth user parameters are
associated with the way in which the groundwater flow is handled
in the model.
The second parameter is the minimum depth for groundwater flow.
This is the depth at which the transition is made in the model from
the surface water flow equations to the groundwater flow
equations. By default this has been set to 0.01.
The third parameter is the groundwater transmissivity, which is
found in equation 13. Although the user is free to enter any
positive value for the transmissivity, using a low value is
recommended to ensure that the groundwater discharge is
negligible compared to the surface flow. By default, this is set 0.1.
A larger value may be used temporarily to speed up the water
surface elevation adjustment of large dry areas.
The fourth parameter is the groundwater storativity, also found in
equation 13. The storativity is defined as the volume of water the
ground will release from storage per unit surface area of the
ground per unit decline in the water table. For unconfined
groundwater (non-compressed) as assumed in the model, the
storativity can range from 0 (ground with 0% porosity) to 1.0
(value for surface water). The default value has been arbitrarily set
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to 1.0.
Lowering the value of storativity will allow the
groundwater surface elevation to react faster to changes in net
groundwater flow.
The last three parameters are used to define the eddy viscosity
coefficient, νt, defined in equations 11 and 12. In the .cdg file ε1,
ε2, and ε3 are named eddy viscosity constant (may be labelled
Mimimum Depth for Diffusive Wave Calculation in older .cdg
files), eddy viscosity bed shear parameter (may be labelled as
DIF), and eddy viscosity horizontal shear parameter (may be
labelled as Groundwater Flow Artificial Diffusion).
The default value for ε1 is 0 (but it will be 0.01 in files made with
older versions of the River2D programs). This coefficient can
used to stabilize the solution for very shallow flows where the
second term in equation 12 may not be adequate to describe νt for
the flow. Reasonable values for this coefficient can be calculated
by evaluating the second term in equation 12 using average flow
conditions (average flow depth and average velocities) for the
modeled site.
The default value for ε2 is 0.5. By analogy with transverse
dispersion coefficients in rivers, values of 0.2 to 1.0 are reasonable.
Since most river turbulence is generated by bed shear, this term is
usually the most important.
In deeper lakesflows, or flows with high transverse velocity
outletsgradients, transverse shear may be the dominant turbulence
generation mechanism. Strong recirculation regions are important
examples. In these cases, the third term in equation 12, ε3, becomes
important. It is essentially a 2D (horizontal) mixing length model.
The mixing length is assumed to be proportional to the depth of
flow. A typical value for ε3 is 0.1, but this may be adjusted by
calibration. The default value for is ε3 0 (but it will be 0.1 in files
made with older versions of the River2D programs).
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Figure 39: The Flow Options dialog box.
5.11.4 Options->Solver Options
The Solver Options command opens the dialog box shown in
Figure 40. This dialog allows the user to choose the linear solver
that is used in the solution process. The user has the choice of
either a direct solver that is called the active zone equation solver
(Stasa 1985) or an iterative solver based on the GMRES(m)
method (Saad and Schultz 1986). The direct solver is faster for
small (<10K nodes) but requires large memory allocations. The
GMRES solver is faster for large meshes and requires much less
memory. The GMRES solver is limited to relatively small time
step lengths.
The iterative solver has three user definable parameters. The first
is the number of steps before the algorithm is restarted, the second
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is the maximum allowable number of restarts or maximum number
of iterations, and the last is the goal convergence tolerance. These
values are set to 10, 10 and 0.01 respectively. These values were
chosen based on a small number of tests such that the time per time
step using the iterative solver is comparable to that for the direct
solver. The optimum values for these parameters are problem
specific and should be treated as such. In the future, after more
extensive tests are performed, more guidelines will be available as
to how to choose appropriate values for these parameters.
This dialog also allows the user to select the method of evaluating
the Jacobian matrix: analytical and numerical. Both methods have
tradeoffs. The analytical method is faster while the numerical
method is more accurate. The model is set to use the analytical
Jacobian by default. The advice to the user upon selecting a
method of evaluating the Jacobian is as follows. For steady state
analyses, start by using the analytical Jacobian. If the solution
reaches a relatively small value of solution change (of the order of
0.001) and refuses to diminish further regardless of the number of
subsequent time steps, switch to the numerical Jacobian. The
analytical Jacobian is adequate for transient simulations.
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Figure 40:The Solver Options dialog box.
5.11.5 Options -> Transient Output Options
This command opens the Transient Output Options Dialog, as
shown in Figure 41 below. This dialog enables the user to specify
and to format of any desired transient output when the model is
running in transient mode. This dialog box can also be accessed
by clicking the Output Options button on the Run Transient dialog.
There are three possible types of transient output available to the
user: video output, point output, and cdg output.
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Figure 41: The Transient Output Options Dialog
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The upper section of this dialog box allows the user to output video
to file in the AVI (Audio Visual Interface) file format. The
resulting video animation will be a compilation of sequential
frames captured directly from the display window. Therefore, the
appearance of the display window (% zoom, display parameter,
annotations, etc) during the transient simulation will be identical to
the appearance of the frames in the video file. The option “Output
frame every ___ goal time steps” makes it possible to specify the
frequency of this output. For example, if a goal time step of 5 and
a present time of 0 are specified in the Run Transient Dialog prior
to a simulation, then frames will be output will be output to the
AVI file at t = 0, 5, 10, 15…until either the simulation is stopped
or the final time is reached. The user can also stipulate a playback
rate and resolution for the video.
The middle section of the dialog box allows the user to output
transient data at specified points in the domain. This point output
is in csv (Comma Separated Values) file format. These points,
which can be in an arrangement (e.g. scattered, cross-section(s),
and/or grid), must be specified in a separate csv file. The csv file
commands in the Display Menu offer an effective way to obtain
the desired output points. To illustrate the input csv file format,
sample file is provided in Figure 42.
Figure 42: A sample input point csv file.
1058,101.9089
1058,101.8977
1058,101.8864
1058,101.8752
1068,101.909
1068,101.8979
1068,101.8868
1068,101.8756
1078,101.9093
1078,101.898
1078,101.8867
1078,101.8757
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1088,101.9089
1088,101.8978
1088,101.8866
1088,101.8756
The user has a choice of four output variables. They are depth,
water surface elevation, x discharge intensity, and y discharge
intensity. If selected, each one of the variable choices will produce
a separate csv file. These files are named according to an
abbreviation for variable name and a user defined output file
prefix. For example, if “Fortress” was chosen as the prefix, then
the generated files would be Fortress_h.csv, Fortress_wse.csv,
Fortress_qx.csv and Fortress_qy.csv corresponding to depth, water
surface elevation, x discharge intensity, and y discharge intensity,
respectively. In addition to the file prefix, the user must also
specify the location of the folder where the point output files will
be written and the frequency at which to output to the files. The
format for the output csv files is shown in Figure 43.
Figure 43: A sample output point csv file shown in MS Excel.
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The first four rows of the file show the path of the cdg file that was
used to generate this transient data, the parameter (here it is water
surface elevation) that was output at the specified points, the date
when this csv file was created, and the time when this csv file was
intialized. The fifth and sixth rows are the x and y coordinates of
the specified points respectively. Each subsequent row contains the
model time in the first cell followed by the values of the output
parameter at that time for the specified points. For example, at t =
30s the water surface elevation at the point x = 4998 and y = 5066
is 199.219.
The bottom section of the dialog box allows the user output
transient data in the .cdg file format. This means that, if desired,
River2D input files can be generated throughout the transient
analysis. As is the case for the video and point output, the
frequency at which this output is generated can be specified in
terms of the goal time step. Generated cdg files are named
according to a user defined file prefix and the current model time
(e.g. Fortress500.cdg) and are written to a user specified folder.
After specifying all desired output, pressing the Initialize Output
button will accept the output choices and initialize the output files
that are appended to throughout the transient simulation such as
point output files and video output files. Once the Initialize Output
button has been pressed, the dialog can be closed using the Close
button. If the close button is pressed without first pressing the
Initialize Output button, then any selected output will be
disregarded.
5.11.6 Options->Ice Options
When ice is present, this command will open the Ice Options
dialog box as shown in the Figure 44.
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Figure 44: The Ice Options Dialog Box.
Currently, this dialog has only one ice parameter that the user can
set: the specific gravity of the ice. The value that appears in this
box will correspond to the value of specific gravity at the bottom
of any ice file that has been saved using the ice file editor,
R2D_Ice. However, River2D will accept ice files that have been
created with a text editor that do not have a value assigned to this
parameter. If the current ice file does not contain a value for the
specific gravity of ice, then the value that appears in this box will
be the default value of 0.92. It should be noted any changes to this
value will be applied to subsequent computational runs within the
same River2D session but they will not be saved to either the
current .ice file nor the current .cdg file.
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6.0
REFERENCES
Ashton, G.D., Editor (1986). River and Lake Ice Engineering. Water
Resources Publications, Littleton, Colorado, 485pp.
Bovee, K. D., 1982. “A Guide to Stream Habitat Analysis Using the Instream
Flow Incremental Methodology.” Instream Flow Information Paper
No. 12. U.S. Fish and Wildlife Service. FWS/OBS-82/26.
Ghanem, A. H., P.M. Steffler, F.E. Hicks and C. Katopodis. 1996. "Two
dimensional finite element flow modeling of physical fish habitat",
Regulated Rivers: Research and Management, Vol. 12, pp. 185-200.
Ghanem, A., P.M. Steffler, F.E. Hicks and C. Katopodis, 1995a, "Dry area
treatment for two-dimensional finite element shallow flow modeling",
proc. of the 12th Canadian Hydrotechnical Conference, Ottawa,
Ontario, June, 10 pp.
Ghanem, A., P.M. Steffler, F.E. Hicks and C. Katopodis, 1995b, "Two
dimensional finite element model for aquatic habitats", Water
Resources Engineering Report 95-S1, Dept. of Civil Engineering,
University of Alberta, 189 pp.
Hicks, F.E., and P.M. Steffler, 1992, “Characteristic Dissipative Galerkin
scheme for open-channel flow”, ASCE Journal of Hydraulic
Engineering, Vol 118, No.2, pp. 337-352.
Saad, Y., and M.H. Schultz, 1986, “GMRES: A generalized minimal residual
algorithm for solving nonsymmetric linear systems”, SIAM J. Sci.
Statist. Comput., Vol 7, No. 3 (July), pp. 856-869.
Stasa, F.L., 1985. “Applied Finite Element Analysis for Engineers”, Holt,
Rinehart and Winston, New York, N.Y. 657 pp.
Waddle, T., Steffler, P.M., Ghanem, A., C. Katopodis, and A. Locke, 1996,
"Comparison of one and two dimensional hydrodynamic models for a
small habitat stream" Ecohydraulics 2000 Conference, Quebec City,
June, 10 pp.
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