J

• •

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

•

Prepared for Assistant: Secretary of the Army (R&D) Department: of t:he Army, Washington D. C. 20310

Under Project: 4A0611 0 I A 91 D

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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

20.. AIIIS~ACT ~ .. re..,._ NO U ~ aztl ldenlifT by block nwnber)

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.

1

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 • • • • • • • • • • • • • • • • • • • • •

2

41

45

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

3

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

4

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

6

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

7

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

8

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.

9

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

11

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