Boundary Fitted Cs
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MISCELLANEOUS PAPER H-78-9
A DISCUSSION OF BOUNDARY-FITTED COORDINATE SYSTEMS AND THEIR APPLICABILITY TO THE NUMERICAL
MODELING OF HYDRAULIC PROBLEMS by
Billy J-4. Johnson, Joe F. Thompson
Hydraulics Laboratory U. S. Army Engineer Waterways Experiment Station
P. 0. Box 631, Vicksburg, Miss. 39180
September 1978 Final Report
Approved For Public Release; Distribution Unlimited
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Prepared for Assistant: Secretary of the Army (R&D) Department: of t:he Army, Washington D. C. 20310
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A DISCUSSION OF BOUNDARY- FITTED COORDINATE Final report SYSTEMS AND THEIR APPLICABILITY TO THE NUMERICAL MODELI NG OF HYDRAULIC PROBLEMS 6. PER FO RMIN G O RG. REPORT N UMBE R
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Bi l l y H. Johnson J oe F. Thompson
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Coordinates Mathematical models Numerical analysis
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A procedure for the numerical solution of nonorthogonal boundary-fitted coo rdinate systems, 1. e. , a coordinate line coincides with the boundary, l.S
presented. This method generates curvilinear coordinates as the solution of two elliptic partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other specified along the boundaries. No restrictions are placed on the irregularity of the boundaries, which are even allowed (Continued)
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20. ABSTRACT (Continued).
to be time dependent, such as might occur in problems involving the computation of the location of the free surface, flooding boundaries, and streambank erosion problems. In addition, fields containing multiple bodies or branches can be handled as easily as simple geometries. Regardless of the shape and number of bodies and regardless of the spacing of the curvilinear coordinate lines, all numerical computations, both to generate the coordinate system and to subsequently solve the system of partial differential equations of interest, e.g., the vertically integrated hydrodynamic equations, are done on a rectangular grid with square mesh.
Since the boundary-fitted coordinate system has coordinate lines coincident with all boundaries, all boundary conditions may be expressed at grid points, and normal derivatives on the bodies may be represented using only finite differences between grid points on coordinate lines. No interpolation is needed even though the coordinate system is not orthogonal at the boundary.
Several example sketches of coordinate systems for numerical modeling problems in the estuarine, riverine, and reservoir environments are presented. In addition, an actual computer generated plot of a boundary-fitted coordinate system for a region representative of the shape of Charleston Harbor is also presented.
Several features of boundary-fitted coordinate systems are especially suited to the numerical modeling of hydraulic problems. Some of these are:
a. The boundary geometry is completely arbitrary and is specified entirely by input.
b. Complicated configurations, such as channels with branches and islands, can be treated as easily as simple configurations.
c. Moving boundaries can be treated naturally without interpolation being required in the handling of boundary conditions or the expression of partial derivatives.
d. Grid points can be concentrated in regions of rapid flow changes.
e. General codes can be written that are applicable to different locations with different configurations since the code generated to approximate the solution of a given set of partial differential equations is independent of the physical geometry of the problem.
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PREFACE
The study reported herein was conducted during the period October 1977
to June 1978 by the Hydraulics Laboratory of the U. S. Army Engineer
Waterways Experiment Station (WES) under the general supervision of
Messrs. H. B. Simmons, Chief of the Hydraulics Laboratory, and M. B.
Boyd, Chief of the Mathematical Hydraulics Division (MHO). The study
was funded by Department of the Army Project 4A061101A91D, "In-House
Laboratory Independent Research," sponsored by the Assistant Secretary
of the Army.
Dr. B. H. Johnson, MHO, prepared this report along with Dr. J. F.
Thompson of the Aerophysics and Aerospace Department of the College of
Engineering at Mississippi State University. Special thanks are extended
to Messrs. N. R. Oswalt, C. R. Nickles, and H. A. Benson of the Hydraulics
Laboratory of WES for valuable discussions concerning hydraulic problems in
their respective areas of expertise. Review comments by Drs. G. H. Keulegan
and R. H. Multer are gratefully acknowledged.
Commander and Director of WES during the conduct of this study and the
preparation and publication of this report was COL John L. Cannon, CE.
Technical Director was Mr. Fred R. Brown.
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CONTENTS
Page
PREFACE . . . . . . . . . . . . . . . . . . . . . . · · · · · · · 1
CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT . . . . . . . • . . . . • . • . · · · · · 3
PART I: INTRODUCTION . . . . • • • • • • • • • • • • • • • • • • 4
Methods for Numerically Solving Partial Differential Equations . . . . . . . . . . . . . . . . . . • . . . . . 4
Need for Accurate Representation of Boundary Conditions on 7 . Irregular Boundaries . . . . . . . . . . . . . . . . . . .
Boundary-Fitted Coordinates . . . . . . . . . . . . . . • . 8
PART II: THEORETICAL ASPECTS OF GENERATING BOUNDARY-FITTED COORDINATE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . 10
The Basic Idea . . . . . . . . . . . . . . . . . . . . . . . 10 Mathematical Development . . . . . . . . . . . . . . . . . . 13 Types of Boundary-Fitted Coordinate Systems . . . . . . . . 19 Data Required for Generation of Boundary-Fitted
Coordinates . . . . . . . . . . . . . . . . . . . . . . . 21 Computer Time Required for Generation of Boundary-Fitted
Coordinates . . . . . . . . . . . . . . . . . . . . . . . 22
PART III: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS USING BOUNDARY-FITTED COORDINATE SYSTEMS . . . . . . . . . . . 24
Transformation of Equations . . . . . . . . . . . . . . . . 24 Complexities Posed by the Transformed Equations . . . . . . 27 Time-Dependent Problems with Moving Boundaries . . . . . . . 30
PART IV: APPLICABILITY OF BOUNDARY-FITTED COORDINATE SYSTEMS TO HYDRODYNAMIC PROBLEMS . . . . . . . . . . . . . . . . . . . 32
Estuarine Modeling . . . . . . . . . . . • . . . . . . . . . 32 Riverine Modeling . . . . . . . . . . . . . . . . . . . . . 34 Reservoir Modeling . . . . . . . . . . . . . . . . . . . . . 36 Pollution Dispersion Modeling . . . . . . . . . . . . . . . 37
PART V: CONCLUSIONS AND RECOMMENDATIONS • • • • • • • • • • • •
REFERENCES
FIGURES 1-18
APPENDIX A:
APPENDIX B:
• • • • • • • • • • • • • • • • • • • • • • • • • • •
DERIVATIVES AND VECTORS IN THE TRANSFORMED PLANE . .
NOTATION • • • • • • • • • • • • • • • • • • • • •
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41
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Al
Bl
CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT
U. S. customary units of measurement used in this report can be con-
verted to metric (SI) units as follows:
Multiply By To Obtain
degrees (angle) 0.01745329 radians
feet 0.3048 metres
miles (U. S. statute) 1.609344 ki l ometres
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A DISCUSSION OF BOUNDARY-FITTED COORDINATE SYSTEMS
AND THEIR APPLICABILITY TO THE NUMERICAL
MODELING OF HYDRAULIC PROBLEMS
PART I: INTRODUCTION
Methods for Numerically Solving Partial Differential Equations
1. In general, either finite differences or finite elements are
utilized in the numerical solution of partial differential equations.
There are, of course, both advantages and disadvantages to each of these
approaches and the situation changes as new techniques are developed.
Finite element method
2. In the finite element approach the field is divided into finite
elements, and the solution is approximated by a chosen function on each
element. This function contains several free parameters which are eval-
uated by requiring the function and perhaps certain of its derivatives
to equal the solution and its derivatives at certain points on the
element. If the partial differential equations can be expressed in
terms of integral variational principles, the variational integrals over
each element are evaluated analytically from the chosen approximation
functions on each element. The integrals over each individual element
are then summed over all the elements to produce the variational
integral over the entire field. This result contains the unknown
values of the solution and perhaps some of its derivatives at all the
points used above in the determination of the parameters in the approx-
imating functions. The variational integral is then minimized in terms
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of these point values of the solution and derivatives involved. If the
partial differential equations cannot be expressed in terms of varia
tional principles, then the method of weighted residuals (Galerkin) must
be used. Here the solution is again approximated on each element as
above. However, instead of evaluating variational integrals, integrals
of the products of weight functions and the partial differential equa
tions are evaluated on each element . This produces a set of simulta
neous algebraic equations to be solved for the values of the solution
and perhaps some of its derivatives at certain points on the elements.
3. The finite element approach is best suited to partial differ
ential systems that can be expressed in terms of a variational principle.
In this case, the boundary conditions can be incorporated naturally via
Lagrange multipliers. For more general systems, particularly nonlinear
systems that are not expressible in terms of variational principles,
the finite element approach must use the method of weighted residuals
(Galerkin) whereby a functional form of the solution in each element
is assumed and integral moments of the partial differential equations
are satisfied over the field as noted above. With this procedure, the
partial differential equations themselves are not actually satisfied.
Boundary conditions are incorporated in the assumed functional form of
the solution in the elements adjacent to the boundaries. A distinction
must therefore be made between finite element methods based on variational
principles and finite element methods based on weighted residuals as
applied to systems that are not expressible in terms of variational
principles. The former does satisfy the partial differential equation
and does incorporate the boundary conditions through the variational form.
5
The latter, however, does neither of these things. Conclusions drawn
from the first typ.e of finite element method should not be applied to
the second.
4. A disadvantage of finite element methods is that they involve
dense matrices rather than the sparse matrices involved in finite dif
ference methods. This results in more time required for a finite element
solution having the same number of mesh points as a finite difference
solution. This disadvantage is particularly important with finite
element methods based on weighted residuals, since more points must
be used to compensate for the satisfaction only of integral moments
rather than the partial differential equations themselves. A related
disadvantage 1s that the finite element methods are more cumbersome
to code than the finite difference methods. Another disadvantage
1s that derivatives of some order are only piecewise continuous so that
the solution may be rough.
Finite difference method
5. In finite difference methods, the domain of the independent
variables is replaced by a finite set of points usually referred to as
mesh points, and one seeks to determine approximate values for the
desired solution at these points. The values at the mesh points are
required to satisfy difference equations usually obtained by replacing
partial derivatives by partial difference quotients. The resulting set of
simultaneous algebraic equations is then solved for the values of the
solution at the mesh points. If the boundaries do not coincide with
mesh points, then the finite difference approach requires interpolation
between mesh points to represent boundary conditions, while the finite
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el ements can always be constructed to use a boundary segment as an ele
ment side for elements adjacent to the boundary no matter what its shape.
6. A disadvantage of finite difference methods is the requirement
of a smooth mesh point distribution so that derivatives can be repre-
sented accurately by differences between mesh points.
Need for Accurate Representation of Boundary Conditions on Irregular Boundaries
7. The need for accurate numerical representation of boundary
conditions exists in all fields concerned with the numerical solution of
partial differential equations.
8. In irregular boundary domains, when utilizing finite differ-
ences for numerical solutions, the approach normally taken is to
construct a rectangular grid or net over the physical region in which
the solution is desired; such a grid is illustrated in Figure 1.
Partial derivatives are approximated utilizing algebraic finite differ-
ence expressions. In general, points on the boundary do not correspond
to the points at which computations are made. Computationally then,
interpolation must be used to determine net function values immediately
adjacent to the boundary. In addition, if derivative boundary condi-
tions are prescribed, interpolation is required to determine the boundary
values themselves. It is easy to see that such representation is best
accomplished when a coordinate line follows the boundary. In this case
numerical representation of the applied boundary conditions can be
achieved using only grid points on the intersections of coordinate lines
without the need for any interpolation between points of the grid. Such
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interpolation between grid points not coincident with the boundaries is
particularly inac~urate with differential systems that produce large
gradients in the vicinity of the boundaries, and the character of the
solution may be significantly altered in such cases. The Navier-Stokes
equations governing fluid motion, of course, are an example of such a
system.
9. In such systems, the boundary conditions are the dominant
influence on the character of the solution, and the use of grid points
not coincident with the boundaries places the most inaccurate numerical
representation in precisely the region of greatest sensitivity. The
use of a coordinate system generated such that a coordinate line always
follows the boundary (no matter how irregular) and such that coordinate
lines may be attracted near the boundaries removes such problems and
allows for much more accurate solutions.
Boundary-Fitted Coordinates
10. Examples of simple systems that possess a coordinate line
coincident with their boundaries are circular cylinders (cylindrical
coordinates) and rectangular bodies (Cartesian coordinates). However,
most practical problems involving solution of the Navier-Stokes equations
involve irregular boundaries, thus one is led to consider methods of
generating boundary-fitted coordinates for such arbitrary geometries.
11. Although orthogonal and conformal transformations immediately
come to mind, the requirement that the normal derivative of one of the
curvilinear coordinates be specified on the boundary, rather than the
value of the coordinate, removes the freedom to locate the mesh points
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as desired on the boundary. The ability to concentrate coordinate lines
in the field near'boundaries as desired is also lost. Conformal trans-
formations are difficult to generate for regions with complicated bound-
aries, especially when there are interior bodies present such as islands,
and are inherently limited to two dimensions.
12. A much more general method is to generate the coordinate
system as the solution of an elliptic partial differential system with
Dirichlet boundary conditions on all boundaries of the region. Such
an approach can be extended to three dimensions, allows arbitrary
specification of one coordinate on the boundary (with the other
coordinate being constant on the boundary), permits time-dependent
physical boundaries, and is applicable to multiconnected domains.
13. 1-5 Thompson et al. have developed a very general technique
for numerically generating such nonorthogonal transformations. This
subject of coordinate transformations is extremely important because it
provides generality to finite difference methods. 6 As noted by Roache,
with this method of treating irregular boundaries by finite difference
methods, statements to the effect that the finite element approach 1s
necessary for treating irregular boundaries are clearly in error.
14. The purpose of this report is to bring Thompson's work on
boundary-fitted coordinate systems, which were developed primarily for
flow around airfoils, to the attention of numerical hydrodynamicists
and to discuss their applicability to hydrodynamic problems.
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PART II: THEORETICAL ASPECTS OF GENERATING BOUNDARY-FITTED COORDINATE SYSTEMS
15. Thompson's work on the generation of boundary fitted
coordinates and their use Ln the solution of the Navier-Stokes equa-
tions can be found in References 1-5 and 7-12. The discussion below
is a summary of the more important theoretical aspects of the subject.
The Basic Idea
16. Suppose one is interested in solving a differential system
involving two concentric circles, such as shown in Figure 2, where
r = constant = n1
on the inner circle and r = constant = n2 on the
outer circle and e varies monotonically over the same range over both
the 1nner and outer boundries, i.e., 0° to 360°.
17. A cylindrical coordinate system is the obvious choice since
a coordinate line, 1.e., a line of constant radius, coincides with
each boundary. If one now pulls the interior region between the two
· 1 t t e -- 0° (or c1rc es apar a 8 = 360°) and folds outward, it is easy
to visualize the region D1
becoming the rectangular region D2 .
Likewise, it should be obvious that the right and left sides of
th t 1 t t b d · · 8 - 0° and e -- 360° e rec ang e are reen ran oun ar1es s1nce
are coincident in region D1
. If one computes a derivative in
the cylindrical system at 0 e - 0 , values at the points marked x
and o on both sides might be used. Thus, these same points, as shown
in the rectangular region, would be used for a similar derivative in
region D2 . This is the reason for calling these boundaries
10
reentrant boundaries. As shown, the boundary of the inner circle
becomes the bottom of the rectangular region while the boundary of the
outer circle becomes the top.
18. The general boundary-fitted system is completely analogous
to the system discussed above. In Figure 3 the curvilinear coordinate,
n , is defined to be constant on the inner boundary in the same way that
the curvilinear coordinate, r , is defined to be constant on the inner
circle in the cylindrical coordinate system. Similarly, n is defined
to be constant at a different value on the outer boundary. The other
curvilinear coordinate, ~ , is defined to vary monotonical ly over the
same range on both the inner and outer boundaries, as the curvilinear
coordinate, e , varies from 0 to 2TI around both the inner and outer
circles in cylindrical coordinates. It would be just as meaningless
to have a different range for on the inner and outer boundaries
as it would be to have e i ncrease by something other than 2TI around
one of the circles in cylindrical coordinates. It is this fact that
has the same range on both boundaries that causes the transformed field
to be rectangular. Note that the actual values of the coordinates, n
and ~ , are irrelevant, in the same way that r and e may be ex
pressed in different units in cyl indrical coordinates.
19. Now that the values of the coordinates, n and ~ , have
been completely specified on all the boundaries of a closed field, it
remains to define the values in the interior of the field in terms of
these boundary values. Such a task immediately calls to mind elliptic
partial differential equations, since the solution of such an equation
is completely defined in the interior of a region by its values on the
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boundary of the region. Thus if the coordinates, and n , are
taken as the solutions of any two elliptic partial differential equa
tions, say L(;) - 0 , D(n) - 0 , where L and D represent elliptic
operators, then and n will be determined at each point in the
interior of the field by the specified values on the boundary. One
condition must be put on the elliptic system chosen since the
same pair of values (;,n) must not occur at more than one point in the
field or the coordinate system will be ambiguous. This condition can
be met by choosing elliptic partial differential equations exhibiting
extremum principles that preclude the occurrence of extrema in the
interior of the field.
20. This may be illustrated with resort to the governing equation
for a stretched membrane. Consider a membrane attached to a flat plate
around a closed circuit of arbitrary shape as shown in Figure 4.
Now let a cylinder of arbitrary flat cross section be pushed up through
the plate, stretching the membrane upward. The vertical displacement,
h , of the membrane will be described by Laplace's equation, v2h = 0 ,
with h = h1
and h2
, respectively, on the circuits of contact with
the plate and cylinder. If equally spaced grid lines encircling the
cylinder had been drawn on the membrane before displacement, these lines
would appear to move closer to the cylinder when viewed from above after
displacement of the membrane. None of these line would cross, however.
21. Now let pressure be applied on the upper side of the membrane
as diagrammed 1n Figure Sa. This will cause the slope at the cylinder
to steepen, with the effect that the lines will appear to be drawn even
12
closer to the cylinder but still without crossing. This situation
2 corresponds to the Poisson equation, ~ h = - P , where P is the
applied pressure. If a variable pressure is applied on both sides of
the membrane to a sufficient degree, it is possible to make the membrane
assume an S shape as shown in Figure Sb. In this case the encircling
lines have crossed, and consequently, a point on the plate can no longer
be identified by specifying the encircling line that it lies below
(together with a radial ray). This latter case corresponds to a right-
hand side of the Poisson equation that is not of one sign over the entire
membrane, in which case the extremum principles of Poisson's equation are
lost.
22. Note, however, that if the differential pressure that is
applied across the membrane is not too large, the S shape will not be
reached. In this case the lines do not cross, but rather the lines
seem to concentrate near a line in the interior of the field. Thus the
existence of an extremum principle is a sufficient condition to prevent
double-valuedness in the coordinate system but is not a necessary con-
clition. Care must be exercised in its absence, however.
Mathematical Development
23. From the discussion above, a logical choice of the elliptic
generating system is Poisson's equation. Thus, based upon Figure 3, the
basic problem is to solve
~XX + ~yy - p
(1)
13
with boundary conditions,
n- constant- n1
on r1
(2)
n - constant - n2 on r 2
The arbitrary curve joining r1
and r 2 in the physical plane specifies
a branch cut for the multiple-valued function, ~(x,y) . Thus the values
of the coordinate functions x(~,n) and y(~,n) are equal along r 3
and r4
, and these functions and their derivatives are continuous from
r3
to r4 . Therefore boundary conditions are neither required nor allowed
on r3
and r4
. As previously noted, boundaries with these properties
are designated reentrant boundaries.
24. The functions P and Q may be chosen to cause the coordinate
lines to concentrate as desired, in analogy with the membrane discussed
above. As discussed 1n Reference 1, negative values of Q result in a
superharmonic solution and cause n lines to move toward the n-line
having the lowest value of n , while positive values have the opposite
effect. Considering the solution to be superharmonic results in the
interior of the ~ = constant lines being rotated in a clockwise direc-
tion in the physical plane; whereas if the equation is subharmonic,
i.e., P 1s positive, the lines are rotated in the counterclockwise
direction.
14
25. The form of these functions incorporated by Thompson, 4 based
upon much computer experimentation, is that of decaying exponentials.
For example, let , Q be taken as
Q- - a exp (- din - n. i) l
where a and d are constants, and n. l
. 1s some specified n-line.
This function reaches its maximum magnitude on the n. l
line and decays
away from that line on either side at a rate controlled by d .
26. This function would cause n-lines to concentrate on one side
of the n.-line and to move away from the other side. If, however, a l
sign-changing function is incorporated so that
Q - - a s gn ( n - n . ) exp (- d I n - n . I ) l l
where sgn(x) is simply the sign of x , the n-lines will concentrate
on both sides of the n.-line. In a similar fashion, it is possible to l
cause concentration of n-lines near a point (~.,n.) with the function l l
Q- - a sgn (n - n . ) exp l
r:-.)2 ( )2 .., + n - n. l l
•
Finally, concentration near more than one line and/or point is achieved
by writing Q as a sum of functions of the above form. In this case
the attraction amplitude a and the decay factor d may be different
for each line or point of attraction. The decay factor should be large
enough to cause the effects of each attraction line or point to be
confined essentially to its immediate vicinity. Thompson has found
that attraction amplitudes of 100 are moderate, 10 is weak and 1,000 1s
15
fairly strong. A decay factor of 1. 0 causes the effects to be confined
to a few lines near the attraction source, while 0.1 gives a fairly
widespread effect. Control of ~-lines is accomplished by an analogous
form of the function P .
27 . A maJor purpose of this coordinate system control is to con-
centrate lines in viscous boundary layers near solid surfaces, and some
automated procedures for this purpose have been developed (Reference 3).
Control is also useful to improve grid spacing and configuration when
complicated geometries are involved, e.g., estuarine hydrodynamic
modeling.
28. Since all numerical computations are to be performed in the
rectangular transformed plane, it is necessary to interchange the
dependent and independent variables in Equation 1.
29. Using the relations presented in Appendix A, which were ob-
tained from Reference 5, Equation 1 becomes
ax~~ - 28x~n - yxnn + 2
+ Qx ) 0 J (Px~ -n
(3a)
ay - 28y + YY + 2
+ Qy ) 0 J (Py~ -~~ ~n nn n
where
2 2 a - x + y
n n
8 - x~xn + y~yn
2 2 y - X~ + y~
(3b)
J - Jacobian of the transformation -
16
wit h the transformed boundary conditions
on r* 1
Again considering Figure 3, the functions f1 (~,n 1 ) , g1 (~,n 2) , and
g2 (~,n 2 ) are specified by the known shape of the contours r1
and
r 2 and the specified distribution of thereon . Boundary data are
neither required nor allowed along the reentrant boundaries r3 and
r4 . Although the new system of equations is more complex than the
original system, the boundary conditions are specified on straight bound-
aries and the coordinate spacing in the transformed plane is uniform.
Computationally, these advantages far outweigh any disadvantages resulting
from the extra complexity of the equations to be solved.
(4)
30. The boundary-fitted coordinate system so generated has a constant
n- line coincident with each boundary in the physical plane. The ~ -
lines may be spaced in any manner desired around the boundaries by
specification of x,y at the equispaced ~-points on the f* 1
and f* 2
lines of the transformed plane. As noted above, the entire side boundaries
are reentrant boundaries, and thus neither require nor allow specification
of x,y thereon.
31. Now the rectangular transformed grid is set up to be the size
desired for a particular problem. Since the values of and n are
17
meaningless in the transformed plane, the n lines are assumed to run
from 1 to the number of n lines desired 1n the physical plane. Like-
wise, the lines are numbered 1 to the number specified on the
boundaries of the physical plane. The grid spacing in both the and
n directions of the transformed plane is takPn as unity. Second order
central difference expressions are used in Thompson's coordinate
generation code, TOMCAT, 4 to approximate all derivatives in Equations 3a
and 3b.
32. Only one of a pa1r of reentrant boundaries 1s considered as a
computation line since the [x,y] are equal on both. As an example of
how a reentrant boundary is handled, consider the grid in Figure 6
where "o" indicates a computation point and "~" a boundary point. The
derivative of x with r espect t o along i = 1 would be written
as
ax a~ l · - (x2,j - xiMAX-l,j)/ 2 .
'J
33. Again, it should be stressed that all computations are per-
formed on the rectangular field with square mesh in the transformed
plane. The resulting set of nonlinear difference equations, two for
each point, are solved in TOMCAT by accelerated Gauss -Seidel (SOR)
iteration us1ng overrelaxation. Some discussion of this technique is
presented in Reference 4.
34. It might be noted that both orthogonal and conformal trans-
formations are special cases of the generation of boundary-fitted
coordinate systems as the solutions of elliptic partial differential
18
systems. In both of these cases the curvilinear coordinates satisfy
Laplace's equation with one coordinate constant on each boundary, and •
the normal derivative of the other coordinate equal to zero on each
boundary. A conformal system also requires a certain relation between
the range of the two curvilinear coordinates.
35. The same procedure may be extended to regions that are more
than doubly connected, i.e. have more than two closed boundaries, or
equivalently, more than one body within a single outer body. A river
reach containing more than one island would be an example. One such
transformation for such a problem is illustrated in Figure 7.
Types of Boundary-Fitted Coordinate Systems
36. Previous discussion of the generation of boundary fitted
coordinates has centered around the idea of using branch cuts to reduce
multiply connected regions to simply connected ones in the transformed
plane. Thompson's TOMCAT code employs such branch cuts. An example
using branch cuts is sketched in Figure 8. Here the body in the field
transforms to the entire bottom boundary of the transformed plane,
while the entire surrounding boundary, 1 - 2 - 3 - 4 - 5 - 6, transforms
to the top boundary of the transformed plane. The sides of the trans-
formed plane are reentrant boundaries, corresponding to the cut, 8 - 1
and 7 - 6, in the physical field. Thus, in the difference equations,
points lying just to the right of the right boundary are identical with
corresponding points just to the right of the left boundary. This 1s
the same type of circumstance that occurs with the familiar cylindrical
19
coordinate system, where e = 361° is the same point as e = 1°. Sim
ilarly, points just outside the left boundary are coincident with points
just inside the right boundary.
37. Many variations of this type of coordinate system can be
produced by the TOMCAT code. For instance, the transformed plane corre
sponding to the same physical field shown in Figure 8 can be rearranged
as shown 1n Figure 9. Now the reentrant boundary, corresponding to the
cut, is located on a portion of the bottom of the transformed plane.
The coordinate lines that result from these two types of arrangements
of the transformed plane are shown on each of the figures. As with
all the boundary-fitted coordinate systems, the grid is square in the
transformed plane regardless of the line configuration in the physical
plane.
38. Multiple-body fields also are transformed to simply connected
regions by the TOMCAT code, an example of which is shown in Figure 10.
Again, there are many different possible arrangements of the transformed
plane, all of which are created by sliding the boundary segments around
the rectangular boundary of the transformed plane. A number of examples
are g1ven in References 4 and 5.
39. The other type of coordinate system transformation available
leaves the multiplicity of the region unchanged. In this case, bodies
in the interior of the physical field are transformed to rectangular
slabs or even slits in the transformed plane. Three different possi
bilities are shown in Figure 11 for the physical plane shown in Fig
ure 8. In the case of slits, the physical coordinates and solution
variables generally have different values at points on the two sides
of the slit, even though such points are coincident in the transformed
20
plane. This does not introduce any approximations, but simply adds a
little more bookkeeping to the code. Fields with more than one body
in the interior simply result in a like number of slabs and/or slits
in the transformed plane.
40. Comparison of all of the above figures shows that different
types of transformation may be more appropriate for different physical
configurations. A further example of this is the configuration in
Figure 12 shown with the original TOMCAT form and with two variations
of the slit/slab form. Generally, the slit/slab form is more appropriate
for channel-like physical configurations having bodies in the interior,
while the TOMCAT form works particularly well for "unbounded" reg1ons
involving external flow about bodies and for regions having an outer
boundary that forms a continuous circuit without pronounced corners
around the field. The slab is generally superior to the slit unless
the boundary has a sharp point. The case of a single channel without
any interior bodies is the same in either form. An example of a river
reach containing two islands, using horizontal slits rather than the
branch cuts previously presented in Figure 7 is given in Figure 13.
Data Required for Generation of Boundary-Fitted Coordinates
41. The basic input or data required to generate a boundary-fitted
coordinate system are the physical coordinates of points on the bound
aries. For example, with reference to Figure 8, the coordinates of
points on the body from 8 around to 7 would be required, with thes e
21
points being spaced in any manner desired as long as there is a contin-
uous progression from 8 to 7. Similarly, the (x,y) values for points '
on the outer boundary from 1 to 2, etc., on around to 6 would be re-
quired. Again these points may be spaced around the boundary as
desired, with no restriction as to how many points lie on each boundary
segment, e.g., between 1 and 2 or between 4 and 5, provided that only
the total number of points from 1 around to 6 is the same as from 8 to
7. The coordinates of points on reentrant segments of the boundary in
the transformed plane, e.g. 1 to 8 and 6 to 7, are not specified but are
free to be determined by the solution.
42. Similarly, with reference to Figure lla, the coordinates of
outer boundary points are required in the slab/slit transformations.
In addition, body points from 6 to 1 on the lower half of the body and
from 1 to 6 on the top half are required. No calculations would be made
on the slab sides of Figure llc or slits of Figures lla and llb since
values at such points are fixed. Points in the interior of a slab are
irrelevant. As with branch cuts, points may be spaced as desired around
the bodies and outer boundary segments.
43.
Computer Time Required for Generation of Boundary-Fitted Coordinates
4 Thompson indicates that the typical time required to generate
a one-body coordinate system without coordinate system control (the
functions P and Q are set to zero) is about 2 min on a UNIVAC 1106 computer
for a 70 x 30 field (70 points on the body). If P and Q are not
zero so that the spacing of coordinate lines is controlled, the computation
22
time increases. Multiple-body coordinate systems typically requ1re about
6 min for a 70 x 4U field. If these same computations were to be made on
a CDC-7600 computer, the times quoted above would be reduced by perhaps
an order of magnitude or more. Therefore, the cost of generating
boundary-fitted coordinate systems for use in numerical hydrodynamic
modeJs will be insignificant if the coordinate system is generated only
once. Time-varying coordinate systems as related to problems in which
the free surface is computed, flooding boundaries, or streambank erosion
will be discussed later.
23
PART III: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS USING BOUNDARY-FITTED COORDINATE SYSTEMS
44. As previously noted, either finite elements or finite differences
are generally used to obtain numerical solutions of partial differential
equations. In the past, many advocates of the finite element method
have implied that finite elements were necessary to accurately handle
irregular boundaries. With Thompson's method for generating boundary-
fitted coordinate systems, this is no longer true.
Transformation of Equations
45. It is desired to make all computations on the rectangular,
square-mesh grid, transformed plane. Thus solutions of partial differ-
entia! equations are performed on the boundary-fitted coordinate system
by transforming each partial derivative and boundary condition according
to the relations taken from Reference 5 and presented in Appendix A, so
that the independent variables become the curvilinear coordinates,
and n , rather than x and y . As an example, consider the ver-
tically averaged equations governing flow momentum and continuity in
shallow, well-mixed estuaries, with the assumption that flooding does
not occur.
Continuity: ar.: a(hu) +
a (hv) 0 -+ -at ax ay
2 2 1/2
x-Momentum: au au au ar_: u(u + v ) 0 -+ u- + v - + g -+ g -at ax ay ax
hC 2
24
y-Momentum: av av av -+u-+v-+ at ax ay
2 2 1/2 g a~ + g v(u + v ) a Y ---!...--.;...2 .....L__ - o
hC
Using the relations below from Appendix A:
f -X
the transformed vertically averaged equations become
Continuity:
x-Momentum:
y-Momentum: av -+ at
where J is the Jacobian of the transformation.
2 2 1/2 gu (u + v )
hC2
2 2 1/2 gv (u + v )
hC2
46. The coordinate program (e.g. TOMCAT or equivalent) produces
- 0
- 0
values of x and y at each (~,n) point and stores these as arrays on
a file. Another program would then read this file and calculate the
25
J needed above at each (~,n) point and store
these values in arrays on a file for subsequent use. Since the deriva-
tives of x and y needed in the above equations always appear divided
by J , this division would properly be performed by a front - end program
that reads the file containing x~ , y~ , etc. The program that solves
the partial differential equations, e.g. the vertically averaged
hydrodynamic equations above, then reads this file and the arrays stored
by the coordinate program.
4 7. Using centered differences, the derivatives,
would be written in difference form as
(ft") .. - (f. 1.- f. 1 .)/2 <:. 1,] 1+ ,] 1- ,]
(f ) .. - (f .. 1 - f .. 1)/2 n 1,J 1,J+ 1,J-
f~ and f ' n
or the equivalent in one-sided representation, etc. As in this example,
first derivatives remain in tridiagonal form in the transformed plane.
Second derivatives, however, involve a cross derivative in the trans-
formed plane, e.g.,
f 2 2y~ynf~n + 2f ) I J2 - (ynf~~ XX Y~ nn
2 2 + [Cy ny ~~ - 2Y~YnY~n + v y )(x f - X f )
· ~ nn n ~ ~ n
2 - 2y~ynx~n + y~xnn)(y~fn - ynf~)] /J3 + (ynx~~
and hence lose the tridiagonal form. This cross-derivative 1s
26
generally not too large, being zero where the coordinates are orthogonal,
and can be lagg~d at the previous outer iteration in a nonlinear solution.
48. Since the curvilinear coordinate system has coordinate lines
coincident with the surface contours of all boundaries, boundary con
ditions may be expressed at grid points. Normal derivatives at bound
aries may be represented using only finite differences between grid
points on coordinate lines without need of any interpolation even though
the coordinate system is not orthogonal at the boundary. These rela
tions are presented in Appendix A.
49. Since the coordinate system program TOMCAT uses iteration
to solve the difference equations for the values of x and y at each
~,n , point some convergence tolerance must be specified. This tolerance
is properly taken as some fraction of the distance scale of the problem.
Thus a proportionally larger tolerance would be used for a problem
spread over 100 miles than for a spread of 1 mile. The x and y
values produced by TOMCAT have whatever units that the boundary values
were specified in, and may be taken as nondimensional if nondimensional
boundary values were used.
Complexities Posed by the Transformed Equations
SO. Once the boundary-fitted coordinates are generated for a given
physical domain, as noted above, the set of partial differential equa
tions of interest and their associated boundary conditions are trans
formed utilizing the relations given in Appendix A. It is of primary
27
importance that the equations do not change type, e.g., if an equation
in the physical plane is hyperbolic it must remain hyperbolic in the
transformed plane. This invariance is demonstrated by Thames. 8 In
addition, Thames also demonstrates that the divergence property is not
lost in the transformed plane. Thus an integral conservation relation
in the physical plane over the nonsquare area formed by intersection
of the curvilinear and n coordinate lines is shown to be equiv-
alent to the conservation relation over the square area formed by the
intersection of the x and y coordinate lines .
51. The use of the boundary-fitted coordinate systems does add a
number of extra terms to a partial differential equation being solved
thereon, and thus increases the number of operations that must be per-
formed at each point. For instance, a first derivative, f , becomes X
1 J (ynf~ - y~fn) when transformed. Here the factors yn/J and
y~/J , (J = x~yn - xny~ , Jacobian) are evaluated from the results of
the coordinate code and are stored on a file by that code for subsequent
use in the solution of any partial differential system of interest. In
difference form, the evaluation of f X
in Cartesian coordinates would
involve a subtraction followed by a multiplication. In the transformed
plane, three subtractions (one each for f~ and f , and one to comn
bine the two) and two multiplications are required. The number of oper-
ations per point for the evaluation of the single derivative f X
lS
thus two in the physical plane and five in the transformed plane. How-
ever, the evaluation in the physical plane assumes that there IS a co-
ordinate line of constant y through the point in question. If this
is not the case then interpolation is required before the x-difference
can be calculated. Now interpolation in one dimension involves five
28
operations (three multiplications, one subtraction, and one addition).
Two-dimensional ,interpolation would involve even more. Thus the use
of the transformed coordinates is faster than any Cartesian scheme re-
quiring interpolation. If the Cartesian scheme is constructed entirely
of points on lines of constant x or y , interpolation will still be
required at general boundaries. In this case the Cartesian grid lines
would have to be closely spaced in both directions to represent an
irregular boundary with any accuracy. The boundary-fitted system,
however, has a coordinate line coincident with the curved boundary by
construction and therefore requires fewer points than does the Cartesian
system for accuracy in representation of boundary conditions. Thus,
even though more operations per point are required in the transformed
system, fewer points are required, so that the transformed system will
be faster when general boundaries are involved.
52. Now consider the case of a more general combination of deriv-
atives, such as af X
The transformed expression may be written
as ( a Y n - b xn ) f + J J s Here the Cartesian form
requires five operations and the transformed requires nine. Thus the
relative increase in operations per point 1s less when combinations of
derivatives occur. The equation + bf y
2 + c\1 f = 0 would re-
quire 17 operations in the Cartesian system, while the transformed
version would require 28 operations. The time per grid point for the
transformed equations is thus somewhat less than twice that for the
Cartesian equations. The boundary-fitted coordinate system would
require significantly less than half the points required by a Cartesian
system for general boundaries.
29
Time-Dependent Problems with Moving Boundaries
53. The boundary-fitted coordinate systems can be used to solve
time-dependent problems with moving boundaries by performing all
numerical solutions on a fixed rectangular field with a uniform square
grid in the transformed plane. The only interpolation required is to
determine values of input quantities such as bottom elevations and
roughness coefficients at the new locations of net points in the physical
plane. The physical plane grid system is generated by solving a set of
elliptical partial differential equations with one of the coordinates
specified to be constant on the boundaries of the physical plane, and the
other coordinate distributed along the boundaries as desired. If the
boundary values of x and y are changed in the physical plane by the
movement of perhaps the free surface contours, a new solution of the
elliptic system with the changed boundary values is obtained over the
same range of values of the curvilinear coordinates in the field. Thus
the transformed plane remains unchanged as the coordinate grid system
moves in the physical plane. Only the values of the physical coordinates
(x,y) change with time at the fixed grid points in the transformed plane.
taf) t-at
54. The transformed time derivative is
x,y
a (x, y' f) a(E:;,n,t)
a(x,y,t) a(E:;,n,t)
All derivatives are expressed in the transformed variables (E,;,n), thus
eliminating the need for interpolation between points in the physical
plane. The movement of the physical plane grid points is accounted for
30
by the time rate of change of X and Y , ( ~~)~.n and (~~)~.n , in
the above express1on. For the case of a fixed grid, time derivatives
transform directly to the rectangular grid.
55. An example of the use of such time-dependent coordinate
systems involving the computation of the free surface wave generated by
a moving hydrofoil 1s provided by References 9-11. In the past, numerical
solutions for free surface flow problems have generally tracked the moving
free surface through a fixed grid, using interpolation among the fixed
regularly spaced grid points to represent the surface boundary conditions.
Similarly, solid body shapes in the flow have either been simple, so as to
coincide with rectangular or cylindrical grids, or have been represented
also by interpolation between grid points (Reference 13).
56. The numerical solution for such problems with a free surface
is complicated by the fact that part of the boundary of the calculation
region, i.e. the free surface, is deforming. This makes the accurate
representation of boundary conditions on the free surface difficult; yet
this solution, as other partial differential equation solutions, is most
strongly influenced by the boundary conditions. The most critical need
for accuracy thus lies in the region of the most difficulty of attain-
ment. The use of a time-varying boundary-fitted coordinate system is
particularly attractive for such problems, since, of course, a co
ordinate line remains coincident with the free surface as it deforms.
31
PART IV: APPLICABILITY OF BOUNDARY-FITTED COORDINATE SYSTEMS TO HYDRODYNAMIC PROBLEMS
57. The boundary-fitted coordinate systems are especially suited
to problems such as fluid flow problems involving the solution of
partial differential equations on fields having arbitrarily shaped
boundaries. With these coordinate systems, solution codes can be
written that allow the specification of the boundary shape entirely from
the input of the set of points defining the boundary. A single flow
program can thus be written that is applicable to arbitrary harbor,
island, and river configurations. Flow obstructions can be inserted
easily and can be moved around and changed in shape via input without
rewriting the code. Time spent in writing a single code can thus pro-
duce a design tool applicable to many different locations with which
the effect of different configurations on the flow can be analyzed
simply by changing the input to the program. Bodies, such as islands,
1n the field can be handled easily by such a code. It is also possible
to treat cases with moving boundaries, such as free surface waves,
silting beds, flooding boundaries, and eroding shore lines, with coordinate
systems that automatically follow the moving boundaries without requiring
any interpolation or approximation of the boundary location. Specific
problems are discussed below.
Estuarine Modeling
58. Various estuarine hydrodynamic finite difference models
exist,14
with perhaps the earliest operational~two-dimensional model
32
being Leendertse's15
vertically averaged model. Since Leendertse's
original work, others have developed similar models. Later models,
for example, have included the capability of handling flooded cells. 16
Leendertse's most recent work has centered around the development of a
quasi-three-dimensional model for estuarine hydrodynamics as well as
17 the modeling of water quality parameters. In addition to such finite
difference models, two-dimensional finite element models of estuarine
hydrodynamics, water quality, and sediment transport in both the hori-
18 19 zontal and vertical planes have been developed. ' A discussion of
the relative merits of finite differences and finite elements has pre-
viously been presented.
59. In the finite difference models, the boundary geometry is
represented in a staircase fashion on a rectangular grid. Invariably,
the grid spacing must, for economical operation of the models, be of
such magnitude that the irregular boundaries encountered in estuarine
modeling cannot be represented with great accuracy. Such problems are
ideally suited to boundary-fitted coordinate systems since a coordinate
line coincides with the boundary. Figure 14 demonstrates an actual
computer plot of a boundary-fitted coordinate system generated for a
region representative of the shape of Charleston Harbor.
60. With the current approximate method of handling boundaries
with relatively large grid spacing, the effect of small structures such
as jetties to deflect the flow away from beaches can only be handled
in a very empirical fashion. The common approach is to Increase the
roughness coefficient sufficiently to stop flow through the cell that
33
the st ruct ure is assumed to be within. With boundary-fitted coordi-
nat es, a coordinate line would actually coincide with the out l ine
of the structure. Such an approach would allow for a more accurat e
description of the effect of such flow control structures. An example
sketch of such a coordinate system is presented in Figure 15.
61. For the case of a flooding boundary, one would simultaneously
compute the movement of the boundary, based upon comparing computed
water-surface elevations with land elevations, with the horizontal flow
field. Then, even though the physical locations of the grid points
change, all computations would still be done on the initial fixed-
rectangular grid with square mesh. The complexities of working with the
physical coordinate system have been, in effect, eliminated from the
problem at the expense of adding two elliptic equations to the original
system. It should be noted that the coordinate system would not necessarily
be recomputed after each hydrodynamic flow field computation. The need for
such computations would depend upon the frequency and magnitude of flooding.
In addition, the computer time required for recomputing the coordinate
system would be significantly less than the times presented in paragraph 43.
This is because the "initial guess" for each recomputation would be much
closer to the true solution than that for the very first coordinate
generation, and thus less time would be required for convergence.
Riverine Modeling
62. Most unsteady riverflow models are one-dimensional, i.e., the
govern1ng equations are in essence averaged over the river cross sec-
tion. 20 Such models are adequate for predicting flood stages on open
34
rivers or perhaps surges generated by power operations. However, for
detailed studies, such as determining velocities near banks for stream
bank erosion studies or perhaps the effect of rock dikes on the flow,
one-dimensional models are not sufficient. At least a two-dimensional,
vertically averaged model that accurately handles the bank and/or dike
geometry is needed. Again, the boundary-fitted coordinate system is
ideally suited to such problems. Essentially all statements previously
made concerning two-dimensional, vertically averaged models are appli
cable here also.
63. Figure 16 illustrates one possible coordinate system for the
case of dikes placed in a river for the purpose of channelizing the flow
to perhaps reduce maintenance dredging. Figure 17 demonstrates a pos-
sible coordinate system, with arbitrary spacing of lines and strong
coordinate control of n lines near the banks, for use in determining
velocities for streambank erosion studies. Assuming one can estimate
the rate of erosion, given velocities near the bank, time-dependent
coordinate systems could be generated with a coordinate line always
coincident to the eroding bank line. As previously indicated, a river
with one or more islands or perhaps structures such as bridge piers
could also be easily handled in such problems.
64. Once again, it should be noted that Thompson's method of gen
erating boundary-fitted coordinates as solutions of elliptic partial
differential equations is not limited to two dimensions. Therefore, 1n
th~ory, a three-dimensional model could be developed to analyze in de
tail flows through gated structures, e . g . the Old River Control Struc
ture, for various combinations of gates and gate openings.
35
Reservoir Modeling
65. The U. S. Army Corps of Engineers 1s very much interested in
the effects of density flows, resulting from thermal stratification, on
withdrawal water quality from Corps reservoirs. As a result, the Ohio
River Division contracted with John Edinger Associates, Inc., for the
development of a laterally averaged, two-dimensional hydrodynamic model
which includes thermal effects. Edinger's mode1 21 uses finite differ
ences to solve the unsteady two-dimensional, laterally averaged hydro
dynamic equations with the effect of density variations due to temper
ature changes included. The effect of a varying density as a result of
the varying temperature field enters the equations of motion through
its influence on the horizontal pressure gradient. Similarly, the vary
Ing flow field influences the temperature computations through the con
vective terms 1n the temperature equation .
66 . In the numerical computations, the grid spacing along the axis
of the reservoir must be constant. Therefore, in order to economically
model a reservoir several miles long, using Edinger's model, a fairly
large longitudinal grid step must be employed. However, initial testing
of Edinger's model has revealed strange flow patterns near the outlet in
the downstream solid boundary for the case of large ~x's . It appears
that a relatively small spatial step may be needed to properly resolve
the gradients imposed by the boundaries. However, since the ~x must
be constant, a large reservoir might require an unreasonable number of
grid points. In addition, the model requires a staircase representation
of the bottom geometry similar to the manner 1n which a coastline must be
represented in the estuarine models.
36
67. As with any problem involving the need for accuratic represen
tation of irregular boundaries and the concentration of net points near
solid boundaries, the boundary-fitted coordinate system is ideally
suited for such a model. The bottom geometry can be accurately repre
sented since a coordinate line is always coincident with it, and the
arbitrary spacing of lines, such that net points are concentrated
near the downstream boundary, removes the boundary problem discussed
above. Figure 18 demonstrates a coordinate system that might be gen
erated for such a problem.
68. It should be noted that the ILIR study reported herein has
directly led to the Water Quality Branch of the Hydraulic Structures
Division of the U. S . Army Engineer Waterways Experiment Station (WES)
Hydraul ics Laboratory contracting with Mississippi State University (with
Thompson as the principal invest i gator) for the development of a two
dimensional, laterally averaged model for use in selective withdrawal
studies in reservoirs. This model will consider vertical accelerations
of fluid particles. The boundary-fitted coordinate technique , of course,
will allow for an accurate description of the bottom geometry as well as
the outlet structure itself.
Pollution Dispersion Modeling
69. Current general dispersion models 22 for computing concentra
tions of some substance or water quality parameter usually assume a flow
field 1s known, which i s then used as input to solutions of the
convective-diffusion equation below:
37
ac + a(uc) at ax
+ a(vc) _ a ay ax ( 0 ac)
X ax +-
ay ( D ~) Y ay
These solutions are normally obtained either through a direct finite
d1fference solution on a fixed grid or perhaps through a Lagrangian
23 approach such as Fischer's backward convection scheme. With the
Lagrangian approach, particles occupying each grid point are traced
backward in time, through use of the input velocity field, to determine
their location one time-step before. New concentrations are then taken
as some suitable average of the old grid concentrations surrounding the
particle's previous position. In either approach, the computations are
looped at each time-step over the complete fixed grid even though a
majority of the net points may possess a zero concentration.
7(). 24 Reddy and Thompson have combined an integrodifferential formu-
lat1on with the technique of numerically generated boundary-fitted curvi-
l1near coordinate systems to develop a numerical solution of the time-
dependent, two-dimensional, incompressible Navier-Stokes equations for the
laminar flow about arbitrary bodies. With the integrodifferential formu-
lation, the solution is obtained in the entire unbounded flow field, but
with actual computation required only in regions of significant vorticity.
(The velocity is calculated from an integral over the vorticity distri-
bution, and the vorticity development is governed by the vorticity
transport equation.) The computational field thus expands in time.
The finite numerical calculation field in the integrodifferential
formulation is, in effect, infinite; and the necessity of locating
"Infinity" at a finite distance is avoided. Although it is not
necessary in this numerical method to calculate the velocity at points
outside the region of nonzero vorticity, the velocity at these points
38
and, in fact to infinity, is determined by the solution via the integral
over the vorticity distribution.
71. In previous applications of the integrodifferential formula
tion, cells of fixed size have had to be added to the field as the
region of vorticity spreads. This necessitates either a complicated
cataloging procedure to keep track of neighboring cells, or e lse the
storage advantages of the formulation are lost as many useless cell s
with no vorticity are included in the field array. With the boundary
fitted coordinate systen1, however, the number of cel l s can be kept fixed
and the size of the cells allowed to vary in time rather than the
reverse as in previous applications. Although the computational region
changes in shape and extent as time passes, the coordinate system con
tinually deforms in such a way that a coordinate line follows the outer
boundary of this reg1on; and the transformed field on which the numer1-
cal computation is actually done remains fixed and rectangular.
72. This formulation could have appl ication to po llution dis
persal, since the pollutant concentration can change at each time- step
only at or ad jacent to points having a nonzero concentration at a given
time. The concentration calculation region thus could then be re
stricted to the current region of nonzero concentration. This concen
tration region could be covered by a time-dependent coordinate system
fitted t o the deforming outer boundary of the region in the same way
that such a coordinate system covers the region of nonzero vort icity
in Reference 24. (The concentration equation is of the same form as the
vorticity transport equation, of course . ) The coordinate system then
would expand with the developing concentration region as it spreads due
39
to convection and diffusion of pollutant. As noted previously, no in
terpolation to handle boundary conditions or to express partial deriv
atives in difference form is required with the time-dependent coordinate
system, even though the coordinate lines are moving over the physical
field, and all computation is still done on a fixed, square grid in the
transformed plane. Since the coordinate system expands with the spreading
concentration region, it is not necessary to add points to this region as
it grows. This formulation is not limited to two dimensions, and some
earlier work in three dimensions is reported in Reference 11. Efforts
in three dimensions to date, however, must be classed as only prelim
lnary. As noted in Reference 24, the calculation of velocity from the
integral over the vorticity distribution is the time-consuming part of
the flow solution in integrodifferential formulation. If the velocity
field is predetermined, however, irrespective of the pollutant, this
part of the calculation is avoided and the velocity at points on the
moving coordinate system in the spreading concentration region could be
determined by interpolation between the points where velocity has been
specified.
40
PART V: CONCLUSIONS AND RECOMMENDATIONS
73. A very general technique, developed by Thompson et a1, 1 for
generating a nonorthogonal boundary-fitted coordinate system has been
discussed. Unlike orthogonal transformations (defined either analyt
ically or numerically), this nonorthogonal transformation allows an
arbitrary spacing of the body-fitted coordinate lines on the boundaries.
No restrictions are placed on the shape of the boundaries, which may
even be time-dependent, and the procedure is not restricted to two
dimensions or fields containing only one body such as an island. Coor
dinate lines may be concentrated as desired along the boundaries.
74. The boundary-fitted coordinates are generated as the so lution
of two elliptic partial differential equations with Dirichlet boundary
conditions, one coordinate being specified to be constant on each of the
boundaries, and a distribution of the other being specified along the
boundaries. Spacing of the coordinate lines in the field may be con
trolled by adjusting parameters in the generating partial differential
equations.
75. Regardless of the shape of the physical domain or whether or
not the boundaries are time-dependent, all computations are performed in
a fixed, transformed, rectangular grid with square mesh. The complexities
of working with the physical coordinate system have been, in effect,
eliminated from the problem at the expense of adding two el liptic
equations to the original system.
76. Several f eatures of boundary-fitted coordinate systems are
especially suited to problems, such as hydrodynamic probl ems, involving
41
the solution of partial differential equations on fields having
arbitrarily shaped boundaries. These are summarized below.
a. The boundary geometry is completely arbitrary and 1s
specified entirely by input.
b. The boundary geometry can be changed radically without
altering the code.
c. Complicated configurations, such as channels with branches
and islands, can be treated with the same basic procedures
as simple configurations.
d. Moving boundaries can be treated naturally without
interpolation. The only interpolation needed is to provide
values for quantities such as bottom elevations and roughness
coefficients at the new locations of the net points in the
physical plane.
e. Moving boundaries can be coupled with the flow equations,
the boundary moving in response to the developing flow.
f. Accuracy can be obtained with fewer points than required
by rectangular grids.
£· Grid points can be concentrated in regions of rapid flow
changes similar to the way in which finite elements can
be concentrated, but without the complicated coding re
quired of finite element models.
h. General codes can be written that are applicable to dif
ferent locations with different configurations since the
code generated to approximate the solution of a given set
of partial differential equations is independent of the
42
physical geometry of the problem. All physical regions
have the same appearance in the transformed plane.
77 . Recommendations for additional research (some of which will be
submitted through the WES ILIR program) which will greatly enhance the
Corps numerical modeling capability are given below.
a. Thompson ' s basic coordinate generation work should be
obtained and made compatible with computing facilities
at WES .
b . A two-dimensional, vertically averaged hydrodynamic-water
quality model util izing the boundary-fitted coordinate
technique should be developed to compliment the two
dimensional, laterally averaged model Thompson is currently
developing for the Hydraulic Structures Division's Water
Quality Branch.
c . Both of the two-dimensional models should be fully tested
utilizing both laboratory and field data.
d . As the two, two-dimensional models are being developed,
ideas for the development of a fully three-djmensional
hydrodynamic water quality model should be finalized.
With the development of a fitl l y three-dimensional opera
tional model, utilizing the boundary-fitted coordinate
technique, hydraulic engineers will have J tool that can
qu1ckly prov1de answers to many important hydraulic problems.
~s. ~dditional areas for basic research to enhance the utility of
the boundary-fitted coordinate technique are presented belO\~ .
a. The intriguing possib~lity exists of taking the coordinate
control functions (P ·1nd Q) to be dc.pendent on the vorticity
43
magnitude, or other gradients, and thus causing the
coordinate lines to concentrate automatically in regions
of high gradients in the flow field, allowing the co
ordinate system to be time-dependent.
b. The complete coupling of the partial differential equa
tions for the coordinate system with those of the physical
problem of interest, so that the coordinate system as such
is effectively eliminated, is also an area worthy of fur
ther pursuit.
. '
44
REFERENCES
1. Thompson, Joe F., Thames, R. C., and Mastin, W. c., "Automatic Numerical Generation of Body-Fitted Curvilinear Coordinate System for Field Containing Any Number of Arbitrary Two-Dimensional Bodies," Journal of Computational Physics, Vol 15, No. 3, July 1974.
2. Thompson, Joe F., et al., "Use of Numerically Generated Body-Fitted Coordinate Systems for Solution of the Navier-Stokes Equations," AIAA Second Computational Fluid Dynamics Conference, Hartford, Conn., June 1975.
3. Thompson, Joe F., et al., "Solutions of the Navier-Stokes Equations in Various Flow Regimes on Fields Containing any Number of Arbitrary Bodies Using Boundary-Fitted Coordinate Systems," Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics, July 1976.
4. Thompson, Joe F., et al., "TOMCAT- A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Containing Any Number of Arbitrary Two-Dimensional Bodies," Journal of Computational Physics, Vol 24, No. 3, July 1977.
5 . Thompson, Joe F., et al., Boundary-Fitted Curvilinear Coordinate Systems for Solution of Partial Differential Equations on Fields Containing Any Number of Arbitrary Two-Dimensional Bodies, NASA CR-2729, National Aeronautics and Space Administration, Washington, D. C., July 1977.
6. Roache, Patrick J ., "A Review of Numerical Techniques," First International Conference on Numerical Shi p Hydrodynamics, David W. Taylor Naval Ship R&D Center, Bethesda , Maryland, October 1975.
7 . Thames, Frank C., et al ., "Numerical Solutions for Viscous and Potential Flow About Arbitrary Two-Dimensional Bodies Using BodyFitted Coordinate Systems," Journal of Computational Physics, Vol 24, No. 3, July 1977.
8. Thames, Frank C., Numerical Solution of the Incompressible NavierStokes Equations About Arbitrary Two-Dimensional Bodies, Ph.D. Dissertation, Mississippi State University, May 1975.
9. Thompson, Joe F. , et al., Numerical Solution of the Navier-Stokes Equations for 2D Hydrofoils, AASE-77 - 160, Aerophysics and Aerospace Engineering, Mississippi State University, Feb 1977 .
10. Thompson, Joe F. and Shanks, S . P., Numerical Simulation of Viscous Flow About a Submerged 2D Hydrofoil over a Flat Bottom, AASE-77-164, Aerophysics and Aerospace Engineering , Mississippi State University, March 1977.
45
11. Thompson, Joe F., and Shanks, S. P., Numerical Solution of the Navier-Stokes Equations for 2D Surface Hydrofoils, AASE-77-165, Aerophysics and Aerospace Engineering, Mississippi State Univers ity, February 1977.
12. Thompson, Joe F., et al., "Numerical Solutions of the Unsteady Navier-Stokes Equations for Arbitrary Bodies Using Boundary-Fitted Curvilinear Coordinates," Proceedings of Arizona/ AFOSR Symposium on Unsteady Aerodynamics, University of Arizona, 1975.
13. Harlow, Francis H. and Welch, E. J., "Numerical Calculations of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface," The Physics of Fluids, Vol 8, No. 12, Dec 1965.
14. Estuarine Modeling: An Assessment by TRACOR, Inc., Austin, Texas, February 1971.
15. Leendertse, Jan J., Aspects of a Computational Model for LongPeriod Water-Wave Propagation, RM-5294-PR, Rand Corporation, Santa Monica, California, May 1967.
16. Butler, Lee H. and Raney, D. C., "Finite Difference Scheme for Simulating Flow in an Inlet-Wetlands System," ARO Report 76-3, Proceedings of the 1976 Army Numerical Analysis and Computers Conference.
17. Leendertse, Jan J., et al., A Three-Dimensional Model for Estuaries and Coastal Seas: Vol 1, Principles of Computation, R-1417-0WRR, Rand Corporation, Santa Monica, California, Dec 1973.
18. Norton, William R., Ian P. King, and Gerald T. Orlob, A Finite Element Model for Lower Granite Reservoir, Walla Walla District, U. S. Army Corps of Engineers, Walla Walla, Washington, March 1973.
19. Ariathurai, Ranjan, R. C. MacArthur, and R. B. Krone, A Mathematical Model of Estuarial Sediment Transport, Technical Report D-77-12, U. S. Army Engineer Waterways Experiment Station, Vicksburg , Miss., October 1977.
20. Johnson, Billy H., A Mathematical Model for Unsteady-Flow Computations Through the Complete Spectrum of Flows on the Lower Ohio River, Technical Report H-77-18, U. S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., October 1977.
21. Edinger, John E. and Edward M. Buchak, A Hydrodynamic TwoDimensional Reservoir Model; Development and Test Application to Sutton Reservoir, U. S. Army Engineer Division, Ohio River, Cincinnati, Ohio, August 1977.
46
22. Brandsma, Maynard G. and David J. Divoky, Development of Models for Prediction of Short-Term Fate of Dredged Material Discharged in the Estuarine Environment, Report D-76-5, U. S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., May 1976.
2~. Fischer, H. B., "A Method for Predicting Pollutant Transport in Tidal Waters," Water Resources Center Contr ibution No. 132, March 1970, University of California, Berkeley.
24. R. N. Reddy and Joe F. Thompson, "Numerical Solution of Incompressible Navier-Stokes Equations in the Integra-Differential Formulation Using Boundary-Fitted Coordinate Systems," Proceedings of the AIAA 3rd Computational Fluid Dynamics Conference, Albuquerque, N.M., 1977.
47
I'"'
t ~
- - -LEGEND
• - COMPUTATIONAL POl NTS 6- BOUNDARY POINTS
... .. '
A
• ""'
• ... ...
-~
~
- - - -
Figure 1. Discretization of a continuum region
r
0
r •CONSTANT•llz
o,
I I 0 t__..o~~lll--___ _
360
-----REENTRANT BOUNDARIEs---
?igure 2. Transformation of domain between concentric cylinders
y
..____X
PHYSICAL PLANE
or,* 2
't)-112 I
I I
r.* REGION OM I r.* 4 • 3 _.
I I
l'l-ll, I I
-r.* I
"'------ ; R REENTRANT BOUNOA IES
TRANSFORMED PLANE
Figure 3. Transformation of an irregular domain
h2-------
I I I I
I I I I
SIDE VIEW
TOP VIEW
Figure 4. Illustration of extremum principle for Laplace's equation
I I I I I
J.-----4---1
Cll.
b.
Figure 5. Illustration of extremum principle for Poisson's equation
-------REENTRANT BOUNDARIES-------
Figure 6. Computational grid in transformed plane
4
PHYSICAL PLANE
II 10 9
REENTRANT BOUNDARIES
------REENTRANT BOUNDARIES------
~· 7 .. 1gure ..
TRANSFORMED PLANE
Boundary-fitted coordinates for a river containing two islands
5
2
3
PHYSICAL PLANE
2. 3 4 5 6
.. ,.
8 7
REENTRANT BOUNDARIES
TRANSFORMED PLANE
Figure 8. Example of coordinates generated using a branch cut. Placement of body is such that sides are reentrant boundaries.
~----------~----~----~-5
3 ----- - -1 ____ ..___-L._..L___J2
PHYSICAL PLANE
2 3 4
~ 8 /////////////////////l '//////////////•7 _/
REENT RA NT BOU NOARIES
TRANSFORMED PLANE
Fi gure 9 . Examp l e of coordinates generated using a branch cut . ? l acement of body is such that reentrant boundaries lie on bot tom line of the trans=ormed plane .
5
6
5
PHYSICAL PLANE
2 3 4 5 6
-.. .
ti/// // / /, "/7711 I 10//////////////// / / ' ///~ 8'' // ,, /// /' ////7
REENTRANT BO U NDARIES
REENTRAN T BOUNDARIE S
TRANSFORMED PLANE
Figure 10. Coordi nat es generated f or a mu ltiple-body f i e l d
~~~~----~------~r-_,~s I
t5 I
3 2
a.
6 ' ~&\:.\~~ t@ :} ·:::·,;:::_::: ~::_::_: ~::
&lW ~?.;:22 ;::~~:·.:::
7 8
3 2. 3 2
c.
4 5
8
I
3 2
b.
PHYSICAL PLANE TRANSFORMED PLANE
Figure 11. ExamJles of coordinates generated using slabs/slits
t--+--+---+--+--+---t 5
~+-+--+---+--+~ 6
t--+--+--+--+--+---t 7
a.
4
3
6 5 7
7
2.
b.
3 4
5
6
~itr:~::: [::::::::~~ t_::::::~:.::
,~~~~~\: ~:::::::
5
:·:·s·~·.
7 8
z
e.
PHYSICAL PLANE TRANSFORMED PLANE
Figure 12. Comparison of TOMCAT and slit/slab generation of coordinates
3
3
2
PHYSICAL PLANE
4
8 7 6 5
TRANSFORMED PLANE
Figure 13. Coordinates generated with sl its for a river with two islands
i I
Figure 14. Computer-generated plot of boundary-fitted coordinates for a region representative of Charleston Harbor
PHYSICAL PLANE
6 7 ?
5
4
8 lg
II 110
3
TRANSFORMED PLANE
Figure 15. Example of coordinates generated in a fiel d containing a jetty and an is land
2
z 5
I
~,~''''''''' ,,,~~,' 6 ,,,,,,,~'~'' ''''''''x ' ' ' ... '.) "(( ~ ~ ' '
, .. , .. ,,,,-;.,,9,,,,,,,,,, 1
·>->" \ v )\ \ ~./"--' \ \ ~--""' \ \~ "v;" ' ~ \ , "v1s'\ \ \
\""3 ' '
'\/\ ..... ~ 1
\\ ' \ I-"
\
10
, ....
) 7 ~, .... J ~ 'l -
-~ z-1- ~rtJ .. L~:fi3 -, \ \)1' ~ 7/-f~
7f. " , J ./-/7· '/''''7 ~~,,, rr _, , .,., ~ /// / / / ' / // / '/I',,,T)
~ /,, · // 19 I " '', /,,,,., 7 //, ,. , ~ / /~ /7
zo 1.5 12
I I
PHYSICAL PLANE
~+=+=~~1s~~,~~~~~~.:~::tt'7~=1~==t:'4l"w~~=·l~::m.tr'I3~-T-T~~ ~:
TRANSfORMED PLANE
Figure 16. Boundary-fitted coordinates for a river containing dikes
2
PHYSICAL PLANE
8 7
TRANSFORMED PLANE
Figure 17. Boundary-fitted coordinates for a Strearnbank erosion study
6
6 4 5 '\) -~
.
~ / -/ / i""" -/ ,_.,. i""" -- ..... -
3~ - ~ ~~ i;i77 ~ - -~//:;
2 / / /
- ---=~ / /
~ 1.- 1.----
~ .,..,. ;;,7 I /I/~ (l . J ~ '
PHYSICAL PLANE
4
3
2
TRANSFORMED PLANE
. ~igure 18 . Boundary- fit t ed coordi nates f or a r eservo1r
APPENDIX A
(Taken f rom Reference 5 )
DERIVATIVES AND VECTORS IN THE TRANSFORMED PLANE
This appendix contains a comprehensive set of relations in the
transformed (~,n) plane. A few relations involving x andy derivatives
of the coordinate functions ~(x,y) and n(x,y) are also included.
Since the intent here is to provide a quick reference only, most of
the algebraic development is omitted. The following function defi-
nitions are applicable throughout this appendix:
f (x,y, t) - a twice continuously differentia~le scalar functi.>n
of x, y, and t.
F(x,y)-! F1
(x,y) + ~ F2
(x,y) ~a continuously differentiable
vector valued function of x and y. i and j are the - -conventional cartesian coordinate unit vectors.
Derivative Transformations
f ( af) _ X - ax y,t (y f - Yc-f )/J n ~ ., n
(A . l)
f <af) _ y - ay x,t
(A.2)
(A. 3)
Al
f - f(x~y +X y~)f~ - X~yL.f -X y f~~)/J2 xy ., n r, ., ., n ., ., nn n n .,.,
•
+ [(x~y - x y~ )/J2 + (x y J~- x~y J )/J3]f~ ., nn n .,, n n ., ., n n .,
(A .6)
Derivatives of E;(x,y) and n(x,y)
; = y /J x n (A.8)
; = -x /J Y n (A. 9)
n = -y /J X £; (A.lO)
(A.ll)
(A.12)
E; = - (n x + E; x )/J - (E; n J + ; 2J~)/J YY Y nn Y E;n Y Y n y ., (A.l3)
A2
(A.l4)
(A.l5)
(A.l6)
(A.l7)
Vector Derivative Transformations
Laplacian:
V2f = (af~~- 2Sf~ + yf )/J2 + [(ax~~- 2Sx~ + yx )(y~f - y f )
~~ ~n nn ~~ ~n nn ~ n n ~
(A.18)
or,
V2f = '..xf - zaf + -.•f -:- of + ·:f )/J2 ' ~~ ~ ~n · nn n ~ (A.l9)
Gradient:
'iJf = [{y f~ - y~f )i + (x~f - x f~)j ]/J - n ~ ~ n - ~ n n ~ -
(A. 20}
Divergence:
(A. 21)
Curl:
A3
Unit Tangent and Unit Normal Vectors in the szn Plane
In many applications components of vector valued functions
either normal or tangent to a line of constant ~ or n are required.
Similarly, directional derivatives in these directions are often
needed to evaluate boundary conditions. These quantities may be
obtained by trivial calculations if unit vectors tangent and normal
to the ~ and n-lines are available. These vectors are developed
below.
It is well known [9] that the unit normal to the graph f(x,y)
- cons~ant is given by
Associating the coordinate function n(x,y) with f(x,y), we have
Vn n(n) - ---...
Utilizing equation (A.20) this reduces to
- (A. 23)
which is the unit vector normal to a line of constant n. In a
similar manner the unit vector normal to a line of constant ~ is
given by
v~ n ( ~) - --- - (y i - x j) Ira.
n... n-- lv~l -(A.24)
These vectors are illustrated as they appear in the physical plane
A4
in Figure A.l below.
y
j -i ...
X
~=K 2
(n) n -
Figure A.l. Unit Tangent and Nonnal Vectors
The unit tangent vectors are then given by
t(~) = n(~) x k = - (x i + y j)/~ - - - n- n-
n=-C 1
(A. 25)
(A. 26)
Vector Components Tangent and Normal to Lines of Constant ~ and ~
F (n) n -
-
- n(n) • F-- -(A.27)
(A. 28 )
AS
-
-Directional Derivatives
1f = n(n) • Vf = (yf11
- 8fr)/J/Y an(n) - - ~
~ = _t{n) • V_f = fr//Y at(n) ., -a£ -an(E.:) - --
a£ -at (E.:) = t(E.:) • Vf - - f ,ro_
T) -Integral Transform
Scalar Function:
J f(x,y)dxdy -D
Vee tor Function:
- -
J f(x(E.:,n) , y(E.:,n))jJidE.:dn D*
(A. 29)
(A. 30)
(A. 31)
(A.32)
{A.33)
(A. 34)
(A. 15)
Consider a vector integral in the physical plane of the form
I - J f(x,y) n(x,y) dS (A. 36) - s
where S is the closed cylindrical surface of unit depth whose
perimeter is specified by the contour rl in the physical plane
(see Figure 2) and whose outward unit normal at any point is
given by n(x,y). -
A6
y
dr -
------·------·----- X
Figure A.2. Integration Around Contour r 1
If r - r(x,y) is the position vector describing r1
, then
dB = (1.0) jdrl -
which implies that (A.36) becomes
I = ~ f(x~y) n(x,y) ldrj (A. 37) - -
We now wish to transform (A.37) to the (~,n) plane. Consider ldrj : -
-= IY d~ (A. 38)
since r1
transforms to r1*, a constant n-line (n = n1). Noting that
n(x~y) -becomes
-- (-y i + x~j) //Y (Eq~tion (A.23))~ Equation (A.37)
~- .,_
A7
~max
I - J f(x(~,n1 ) , y(~,n1)) (x~~ - y~:)d~ - ~min
~max
= f f(~,n1 ) (x~~ - y~~)d~ ~min
(A.39)
where ~min and ~max are the minimum and maximum values respectively of
~ on r 1 *· Note that all quantities in (A.39) are evaluated along
If the vector n(x,y) is incorporated into the function f(x,y), n =- n • 1 -
we can define the vector function f(x,y) as -f(x,y) : f(x,y) n(x,y) - -
Equation (A.37) now becomes
I - f ~max
(A.40) - ~min
which is merely an alternate form of (A.39).
A8
APPENDIX B: NOTATON
a Amplification factor in coordinate control functions
a,b Arbitrary constants
C Chezy coefficient
c Concentration
D,L Elliptic operators
D Diffusion coefficient 1n x-direction X
D Diffusion coefficient in y-direction y
d Decay factor in coordinate control functions
g Acceleration due to gravity
h Vertical displacement of a membrane; water depth
J Jacobian of the transformation
P Applied pressure
P,Q Coordinate control functions
r,e Cylindrical coordinates
t Time
u Ambient velocity component in x-direction
v Ambient velocity component 1n y-direction
x,y Cartesian coordinates
Coefficient equal to
B Coefficient equal to
Coefficient equal to
Water surface elevation
c Curvilinear body-fitted coordinates \.., ,n
Bl