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FORMULATION OF NUMERICAL METHODS* r X.-

Go USED IN THE

XYZ THREE-DIMENSIONAL POTENTIAL FLOW PROGRAM

An Engineering Report

by

WILLIAM JAMES BEARY JR. DTICZLECTE 0

Submitted to the Faculty ofthe College of Engineering

Texas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF ENGINEERING

C,-

U-.' May 1986 -

Major Subject: Ocean Engineering

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FORMULATION OF NUMERICAL METHOD5

USEDIN THE

XYZ THREE-DIMENSIONAL POTENTIAL FLOW PROGRAM

An Engineering Report

by

WILLIAM JAMES BEARY JR.

Approved as to style and content by:

AIlen H. Magnuson (Cman)

David R. Basco(Member)

.eland A. Carlson (Member)

. . . .. . . . . . . .,,

:.'.-

FORMULATION OF NUMERICAL METHOD5

USED IN THEp..,,.

XYZ THREE-DIMENSIONAL POTENTIAL FLOW PROGRAM ""

An Engineering Report

by

WILLIAM JAMES BEARY JR.

Submitted to the Faculty ofthe College of Engineering

Texas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF ENGINEERING

May 1986

L Major Subject: Ocean Engineering

--' .*.. .. v.'..-. *.

The calculation of non-lirting potential flow about arbitrary three

dimensional bodies Is examined In detail with specific-interest In the XYZ

Potential Flow program developed by the David W. Taylor Naval Ship

Research and Development Center. The program uses a surface singularity

distribution to solve the Neumann boundary value problem by means of a

source panel method assuming a flat element with a constant source

density over the area of the element. Boundary conditions are applied at

control points on the elements producing a system of linear equations for

the source density. When the source density is known, velocities and

pressure coefficients may be calculated.

The main purpose of this paper is to present the details of the

approximation of an arbitrary three dimensional body using quadrilateral

elements, and to provide a detailed derivation of the exact source panel

integrations in order to gain insight for future research at Texas A&I1

University. A variation of the Hess method of surface discretization

using quadrilateral source panels Is described In detail as It Is used In the

-. XYZ Potential Flow program. The exact source panel integrations are

derived in detail.

A general discussion of other aspects of the program is included.

Velocities and pressure coefficients for flow about a triaxial ellipsoid

are calculated using the XYZ Potential Flow Program solution, and the 0:

results are compared with the analytical solution and the Hess Program

solution.Avail and or-

Dist Spe.cial

SA- 1'.

1-

AC KNOWLEDGEMIENTS

The author Is thankful to Janet S. Dean of the DaviJ W. Taylor Naval

Ship Research and Development Center, for her generous time on the

telephone explaining details of the XYZ Potential Flow program which she

. co-authored, and to John L. Hess of McDonnell-Douglas Corporation, who,

during a valuable phone conversation, suggested references which s,"

contained information necessary to complete the closed form source

panel integrations. Appreciation is also expressed to Dr. Allen H.

Magnuson, who provided direction as graduate committee chairman, and to

Dr. David R. Basco and Dr. Leland A. Carlson who served as committee

members.

%,""

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TABLE OF CONTENTS

Section Page

1.0 INTRODUCTION .. .. ...... ........1.1 Objectives 5

2.0 HISTORICAL DEVELOPMENT .. .... ...... 6

3.0 THEORETICAL DEVELOPMENT...........73.1 The Potential Flow Problem In 7

Three Dimensions3.2 Mathematical Model 103.3 Numerical Model 14

4.0 ORGANIZATION OF THE PROGRAM. .. .. ..... 17

5.0 DETAILS OF THE SURFACE APPROXIMATION. 19*5.1 Preparation of the Input 19 0

5.2 Source Panel Geometry 265.3 Locating the Centroid 30

*5.4 Coordinate Transformation 315.5 Moments of Inertia 32

6.0 THE MATRIX OF INFLUENCE COEFFICIENTS . . . 34

7.0 DERIVATION OF THE EXACT SOURCE PANEL . .39

INTEGRATION7.1 The Y Velocity Component 407.2 The X Velocity Component 457.3 The Z Velocity Component 46

*8.0 APPROXIMATIONS OF THE INDUCED VELOCITY .55

8.1 Quadrupole Method 558.2 Monopole Method 57

9.0 SOLVING THE MATRIX EQUATION FOR . . . . 58SOURCE DENSITY9.1 Jacobi's Iterative Method 589.2 Richardson Extrapolation 60

iv

Section P ..je

10.0 CALCULATION OF VELOCITIES AND ..... 64PRESSURE COEFFICIENTS

11.0 STREAMLINE CALCULATIONS .... ........ 67

12.0 DEVELOPMENT OF HIGHER ORDER ... ...... 69PANEL METHODS

13.0 VELOCITY CALCULATIONS FOR A ...... 72.-.....7TRIAXIAL ELLIPSOID

14.0 CONCLUSION AND REMARKS .... ........ 76

REFERENCES ........ ................. 79

APPENDICES• oh

Appendix I - XYZPF Section PF1 ...... ... 83Reads Input and Computes QuadrilateralParameters

Appendix II - XYZPF Section PF2 . . . . . . 95Computes Influence Matrix Coefficients

Appendix III - XYZPF Section PF3 .. ........ 101Solves Matrix Equation for Source Density

Appendix IV - XYZPF Section PF4 ... ...... 105Computes Velocities and Pressure Coefficientsfor On-Body Points

-. Appendix V - XYZPF Section PF5 ... ...... 109 .Computes Velocities and Pressure Coefficientsfor Off-Body Points

a-".% .

Appendix VI - XYZPF Section PF6 ......... 116Computes Velocities and Pressure Coefficientsfor Off-Body Streamlines

• 7 .- -.

• ' %,@V

7 ~

Page

Appendix VII - XYZPF Section PF7. .. ...... 123Computes Velocities and Pressure Coefficientsf or On-Body Streamlines

ApedxVl Tixa lisidIptFl 3

Appendix II - Triaxial Ellipsoid Intput File. . 131

Appendx lx Triaxal Elipsi uptFl 3

LIST OF FIGURES

Figure Page

I DISCRETIZATION METHODS ....... 4

2 POTENTIAL FLOW IN THREE DIMENSIONS .

.! 3 APPROXIMATION OF THE BODY BY. . . . 14SURFACE ELEMENTS

4 THE 3D QUADRILATERAL ELEMENT IN . . . 20GLOBAL COORDINATES

5 QUADRILATERAL INDEX NUMBERS . . . . 22

6 APPROXIMATING A THIN REGION WITH. 23ROUNDED PLANFORM

7 THE OUTER NORMAL TO THE QUADRILATERAL 26ELEMENT

8 THE SECOND LOCAL COORDINATE VECTOR 28

9 THE THIRD LOCAL COORDINATE VECTOR 28

10 LOCATING THE CENTROID OF THE ..... 30-: QUADRILATERAL

11 FORMING THE PLANE QUADRILATERAL ELEMENT 32

12 FUNDAMENTAL POTENTIAL FUNCTIONS FOR . 40SIDES OF A QUADRILATERAL

13 THE POTENTIAL DUE TO A FINITE LINE SOURCE 42 .

14 THE LAW OF COSINES. .. .-..-..... .43

15 CALCULATION OF ON-BODY STREAMLINES. 68

vii

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1.0 INTRODUCTION

This paper examines two aspects of the development of the XYZ

Potential Flow Program (hereafter referred to as the XYZPF Program), a

FORTRAN program which uses a source panel method to approximate

solutions to steady potential flow problems about arbitrary three

dimensional bodies. The aspects examined in detail are (1) the

description of the details of the approximation of an arbitrary

three-dimensional body using quadrilateral elements, and (2) a detailed-" derivation of the exact source panel integrations.

The XYZPF Program was developed specifically for applications in

- numerical ship modelling and hydrodynamics studies at the David W.

Taylor Naval Ship Research and Development Center (NSRDC) in Bethesda,

Maryland. The format of the program is based on the work of Hess and

Smith (1962) in the numerical calculation of non-lifting potential flow.

A similar program is maintained by the Aerodynamics Division of the

. McDonnell-Douglas Corporation, referred to in this paper as the "Hess

program." The XYZPF Program is a modification of what has come to be

known generally as the Hess Method. The most significant modifications

are improvements to the method of solving the matrix equation for the

source density, and greater flexibility in the input options (Dawson and

2% .Dean 1972).

Though potential flow is a product of mathematics, and is never

found in a real fluid, the results of potential flow calculations provide

usable information for flow regions external to a thin boundary lager,

hr

with little or no boundary layer separation. For such flow fields, the

region outside the boundary layer may be considered to be effectively

inviscid, and may be closely approximated by potential flow models.

Small viscous effects can be accounted for by "thickening" the body by the

- appropriate displacement thickness. Displacement thickness accounts for

the region of retarded fluid flow in the boundary layer inversely

proportional to the square of the free stream velocity. Downstream of

.: the point of boundary layer separation, the potential flow model no

longer applies.

Prior to the development of numerical methods, analytical

solutions were generally restricted to simple analytical shapes (Kellogg

Sg1929). The need to solve boundary value problems for arbitrary

* boundaries in continuum mechanics has fostered the development of

I numerical approximations to the integral equation expressions. While

. the integration methods have been well known for quite some time, only

o since the advent of high speed computers have many of the problems

* been practical to solve by numerical methods. Among the numerical

methods being used are finite differences, finite elements, and the

*: boundary element method.

"Finite differences" and "finite elements" are numerical methods

which satisfy the boundary conditions, and then approximate the

solution to the governing equation in the fluid domain. These methods

discretize the domain into a network of elements or cells. .

I

.Another approach is what is now known as the "Boundary Element

Method,O in which the governing equation is exactly satisfied in the

domain, and the boundar! conditions are applied through a boundary

discretization method. The boundary value problem is reformulated as a

boundary integral equation which is then discretized by subdividing the

qboundary into a finite number of surface elements. Each element is

represented by an analytical function, and the source density function is

>* integrated over the surface of each element. Two factors governing the

accuracy of the boundary element method are the boundary discretization

* method and the source panel integration. These two factors are examined

.. :. in detail in this report, as a detailed derivation of the exact source panel . . -

integration, including the development of the source panel geometry, has

not previously appeared in literature.

The difference between the domain methods and the boundary

methods is significant. The domain methods discretize the domain, while

the boundary methods discretize the boundary. Thus, the boundary method

reduces the dimension of the problem by one, as depicted in figure (1). In

the application of the XYZPF Program, the problem is reduced from a

three-dimensional problem in the domain to a two-dimensional problem

on the boundaries. This method is well suited to problems in which the

limits of the domain arre infinite or difficult to define, in that the

problem is applied to the boundary rather than the domain.

Sr: 3

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Just ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ , asteeae-ayvratosooomi ehds hr r

Uq FINITE ELEMENT DISCRETIATION

|

,~

.-

'" BOUNDAREY ELEMENT D ISCRET IZ AT ION

b Fkgre . D'iscretatIon M~e'dods

*-. Just as there are many variations of domain methods, there are-::

also a variety of boundary methods. In general, they can be classified as

' "indirect" or "direct" formulations. The "indirect" method assumes a

" . continuous source distribution over the surface of the body, and a

solution which satisfies both the governing equation in the domain, and

, - the boundary conditions on the body surface. The result is an integral

* equation on the boundary which has the surface source density function as

its unknown. By enforcing the boundary conditions at control points on

the surface, a system of equations is produced by which the source

density may be determined.

The "direct" method solves the velocity potential iunction through

V4

:!:V:'

: '

. . . .- . . -

an application of Green's Second Identity requiring the solution of a

source distribution and a dipole distribution on the boundary. The direct

method has more physical significance to the general boundary value

problem, and more versatility in its application as it can be applied to

Neumann problems, Dirichlet problems, or mixed boundary value problems p

(BrebDia 1984).

The simplicity and accuracy of the indirect method has made it

attractive for many applications The source panel method is an

application of the indirect formulation of the boundary element method

to the Neumann type of potential flow problem, for which the normal

derivative of thepotential function is prescribed on the boundary.

1.1 OBJECTIVESn

' The purpose of this paper is (1) to describe the details of the

approximation of an arbitrary three-dimensional body using quadrilateral

." elements, and (2) to provide a detailed derivation of the exact source

panel integrations for use in future investigations at Texas A&l1

University using panels of higher order geometries and source density

] functions. This paper is not intended to be a user's manual, though a

general discussion of other aspects of the program is also included.

NSRDC Report 3892 (Dawson and Dean 1972) is a summary of the XYZPF

Program for those strictly interested in its use.

5

r"I

2.0 HISTORICAL DEVELOPMEN

The foundations of the boundary element method were laid earl g in

this century beginning with Fredholm in 1903 when he established the

existence of solutions to the Neumann problem through a reconstruction

of the problem using a discretized boundary (Kellogg 1929). The solution

was determined to be the potential of a simple source distribution on a

* Lboundary with a continuous normal derivative for an infinite domain.

Later works by Kellogg (1929) in potential theory demonstrated the

application of the boundary integral equation method in electrostatics,

heat transfer, flow in porous media, and fluid flow problems, but

development was limited by the difficulty of obtaining analytical

solutions. No significant advances were made until interest in boundary

* integral equation methods was revitalized with the advent of high speed

electronic computers. Investigators were then able to discretize the

boundaries and solve the integral equations numerically. This method of

solution became known as the boundary element method. Early

I* development of such numerical methods was pioneered by Hess and Smith

(1962) and Jaswon and Symm (1963). Hess and Smith dealt primarily with

the indirect formulation eventually leading to a solution for the three

dimensional problem as described in this paper. In a parallel work,

Jaswon and Symm developed a direct formulation approach to the two

dimensional problem. The XYZPF Program is based primarily on the work

of Hess and Smith. Hess has since developed a higher order panel method

(Hess 1979) and Lefebvre modified the XYZPF Program for calculating

velocity potentials for five degrees of freedom (Lefebvre 1982).

6

rV

3.0 THEORETICAL DEVELOPMENT #.

3.1 THE POTENTIAL FLOW PROBLEM IN THREE DIMENSIONS

The governing equation for ideal (incompressible, inviscid,

irrotational) flow is Laplace's equation:i.5.

•2 =0 (i)V:.:-

where t is the velocity potential, and V2 is the Laplacian operator. The

XYZPF Program deals with steady, uniform flow of an ideal fluid about an

arbitrary three dimensional body. The velocity components at any point

within the flow field may be obtained from the negative gradient of the

velocity potential, i. e. .-.

hV = -V' = - j -- k (2)-" ax au az ;"'

The freestream flow V0 is defined as a uniform stream of unit

magnitude.

_ _ _ _ _ _ _ _ _ _ _(3) -

V V"Vv0A K +v + V2 -,.

The key to the boundary element method is the Divergence Theorem

i i (Green's Theorem) which relates a volume integral to an equivalent

surface integral reducing the three-dimensional problem to a

7- ...

-3-1 7 WW W W O - -- . .2 .. -. -, - . _X- -

two-dimensional one. The expression f or Green's second identity is

(Lamb 1924): Il

J(4D 2W WV 2 ,) dQ w cD-d - (4)

in which Q represents the integration over the three dimensional domain,

and r represents integration over the two dimensional boundary. The

* partial derivatives are taken with respect to the outward normal, n. The

* *. weighting function, w, is usually chosento be the fundamental solution

* for three dimensions, w =1/(4JTrr), where r is the distance from the

* source to an arbitrary point on the boundary.

*) I* rr

rr

Figur 2. Potential Flow inm irne wioqis

Consider an arbitrary three-dimensional body with surface r,having an equation of the form F(x, U, z) = 0 where x, y, z are Cartesian

coordinates of the global reference system as shown in Figure (2). The

unit outward normal, n, at any point on the surface is given by the

* * gradient of the function describing the surface divided by the magnitude

00..

of the gradient, i.e.oq ~+VF .G

n =(5)jVF" .

where the sign of the unit normal vector is chosen to ensure that the

vector is an outward normal. The potential function f describing the

flow field must meet the following boundary conditions:

a. V2= 0 (Laplace's Equation) (6)

b. For an impermeable boundarg, the velocity normal to the surface

must be zerorelative to the boundary (the Neumann boundary

condition):

c. The velocity potential approaches the freestream velocity

potential as the distance from the body goes to infinity: .

4* *4 as -*c'

* The total potential at any point In the domain Is composed of the

freestream potential and the disturbance potential due to the body,

T =oo Y (9)

9b

-h 9g.:

i . ° S. , o . *. ° . . o - . ..- .- . * *.. . .

t._..'m ' ._ ' . ... ''- -"- ," -" .

"' '"'" '"" " .- ' '.- . . . .. t ". . .. . - "*"_ .. " ' ". '-. -. . . ° . - • ,""

The disturbance potential, P, satisfies the following boundary

conditions: .0

a. V2( 0 10)

b. From equation (7), the velocity normal to the boundary due to

the disturbance and due to the onset flow must be of equal

magnitude, but opposite sign. Then from equation (9)

LT n(p). V0, 011)

,.oan4 r

Note that the normal vector is a function of position on the surface

of the bodyai

c. The disturbance potential approaches zeroas the distance fromthe body goes to infinity, i. e.

-a 0 as (12)

" 3.2 MATHEMATICAL MODEL

The disturbance potential of the body may be represented by a

.-. distribution of a source density function cf over the body surface. The

potential at an arbitrary point P(x, y, z) due to the surface potential is% o (Kellogg 1929): :'

10. . - . . -- 4-

(x, y, z) = 13 () -.ff r(P,q)

wher, q is the integration point onthe surface, and

r(P,q) I (x - xo )2(y Y)2 + (z- z) 2 ,

is the distance from the field point P to the integration point q.

The source density distribution function must satisfy the boundary

conditions for the disturbance potential. Boundary conditions (10) and

" (12) are automatically satisfied by the form of the integrand. However,

equation (11), the velocity normal to the boundary, combined with the

- Neumann boundary condition, equation (7), is the key to solving the

boundary integral problem.

. The integrand becomes singular as the surface of the body is

approached, i. e. r goes to zero. The singularity represents the local

fluid flux normal to the boundary due to the local source density. The

".* principal value of the singularity is -2vTc(p), determined through a

limiting process of the Gauss Flux Theorem (Kellogg 1929). The point p

represents a field point which lies on the boundary. The integral

expression is now composedof the contribution of the local source

density and the contribution of the source density function over the

remainder of the body surface. Solving for the velocity normal to the

surface yields the following expression:,

'11

.......................................... .* .

OT -2T1o'(p) + Fa(p)_] dr (14)On )r an Lr(p'q)

From equation (11). this expression becomes:

2 p) -J'f Lr(p,q)_0

This equation is a two dimensional Fredholm integral equation of the

second kind, which ensures a unique solution, and that the diagonalelements of the system matrix will be dominant, each having a value of

2Tr (Kellogg 1929). Once equation (15) has been solved for the source

b density 0, the velocity components at any point of the flow field may be

S'-.. obtained by differentiating the disturbance potential function (13) with

respect to the coordinate directions and adding the components of the

freestream flow, V0... .-

a pV y)-T k (16)

- The body shape does not have to be slender, axisymmetric, or

simply connected, allowing for analysis of Interior flow and a wide range

of applications of the method. The only restriction imposed on the form

of the body is that it must have a continuous normal vector.

Discontinuities in the right hand side of equation (15) will produce

unwanted singularities. Thus, in the process of approximating a body

• -which has distinct corners, where there is clearly a discontinuity in the

normal vector, the corner must be replaced by a surface with some finite

12F'I

............*...~ **.~.**..* . . • .

-: - 7- T V T7;C77717I

curvature. However, application of this method has shown that the flow

calculations give correct results for convex corners, while unrounded

concavecorners maw or mag not produce significant error, depending on

the angle produced by the corner (Hess and Smith 1962).

Because of the method of approximation, the calculation of flow

-" velocities on the body surface are restricted to the points at which the.", .-

boundary conditions were applied. Velocities at points other than those

must be obtained by interpolation. Direct calculation of velocities at the

edge of an element yields infinite velocities.

With the solution of the system of linear equations for the source

densities, the flow velocities at any point in the domain may be obtained

from equation (16), and pressure coefficients are then computed from the

I velocities using a form of the Bernoulli equation:

where P(t) is a constant independent of position. In the XYZPF Program,

the flow is steady. Therefore, equation (17) can be reduced to

p -- V = constant (18)2 ,'2

and the pressure field can be expressed in terms of a dimensionless

." pressure coefficient C as:

13

% V4

where p. is the static pressure at infinity.

3.3 NUMERICAL MODEL

In order to represent the surface of a body in the domain

, mathematically, the body may be described by analytical expressions

"" which may provide an exact representation of the surface. However, the

types of bodies which can be adequately described by such methods are

severely limited. Another way to represent the body is to use a large

number of analytical expressions, each describing only a small portion of

* the body. Hess and Smith (1962) suggested the use of an assembly of flat

quadrilateral elements to model the actual surface of the body, as shown

I in Figure (3). Each quadrilateral approximates a region of the surface

described by points which lie on the actual surface of the body. As planar

* elements, these quadrilaterals are clearly analytical, and when carefully

I constructed, the elements can approximate arbitrary three dimensional

. body surfaces without restriction.

Fiqwe 3. Approxtnaticn of the body byj virfae elements

F 14

o ,.. .

H-ssThe XYZPF Program uses the discretization procedure described by

Hess and Smith (1962) with some minor modifications. The threedimensional body surface may be described using a large number of plane

N-'.

'V quadrilateral elements, each assumed to have a constant source density

over the area of each element. Regions of the body requiring higher

resolution for sharp curvature or anticipated velocity gradients will

require a higher concentration of elements.

Because the plane quadrilateral elements cannot fit edge to edge on

a rounded surface, small gaps in the panel approximation contribute to the

error of the approximation. However, the error due to the gaps is

negligible when compared with the error of the basic model, that is, using

flat panels to approximate a curved surface. Triangular elements have

been suggested in an attempt to eliminate the gaps (Levy 1959), but the

" increased accuracy is so small that it may not justify the additional work

of organizing the triangular elements in lieu of the simpler quadrilaterals

(Hess and Smith 1966). The method presented is valid for an polygonal

:- element with any number of sides.

Equation (15) can now be decomposed into a summation of

integrals, each representing the contribution of one element of the body

surface. The unknown source density can be taken outside the integral,

since it is assumed to be a constant over each element. The integration is

performed over the area of the source element, and the boundary

condition equation (11) is then enforced at a single point p in each

*: remaining element. By performing this operation at each element of the

15

. . .. .

surface, a system of linear equations is generated which is equal in

number to the number of surface elements and the number of unknown

source densities. Equation (15) can now be approximated by the matrix

equation (Dawson and Dean 1972):

N he~re

C ii [

V ij 2,n n

*It is important to note that the influence coef ficients C and Ciare

* functions of geometry only, and once computed, need not be recomputed

for analysis of several different flows. From the solution of equation

(20) on the discretized surface, equatior (13) may be applied at any point

in the domain. Then, the velocity at an arbitrary field point P(x, y, z) in

the domain may be determined from equation (16). With the velocity

known, the pressure coefficient is determined from equation (19).

16.

I 7v

4.0 ORGANIZATION OF THE PROGRAM

The XYZPF Program is actually composedof seven independent

programs, referred to as sections PF1 through PF7, each of which builds

on data generated from a previous section. This type of organization

allows the user the flexibility of rerunning portions of the program using

different flow parameters without having to go through the time

consuming process of recalculating the influence coefficient matrix,

which is dependent only on the geometry of the body. While the NSRDC

program is very similar to the Hess program, there are also some

significant differences. The following list of differences is taken from

NSRDC Report 3892 (Dawson and Dean 1972):

(1) The input to XYZ PF is arranged to facilitate the preparation of

input for a series of problems in which only one part of the body is

changed. Also, a number of checks are made on the input to help detect - -

errors.

(2) An option was added for the recomputation of the source

density and velocities for only part of the body when only small changes

are made. This option also provides for the use of the solution of one

problem as an initial guess for the solution of another problem.

(3) The matrix of influence coefficients is computed column by

column instead of row by row. This column arrangement was used for the

original LARC computer version because it required much less high speed

memory. The computation is also about 10% faster this way than with

• . the row-by-row arrangement.

(4) A simultaneous d04;lacement iterative scheme is used to solve

17

................

the matrix coefficient for the sourcc density. The scheme is slower than JN

the successive displacement (Gauss-Seidel) scheme used in the Hessprogram, but it can be carried out using the matrix column by column

instead of row by row.

* (5) When possible, an extrapolation procedure is used to reduce the

number of iterations required for convergence. One such method is the

Richardson extrapolation.

The methods used in the XYZPF Program willI be discussed in detail

in the following sections.

b 18

S.., 5.0 DETAILS OF THE SURFACE APPROXIMATION

5.1 PREPARATION OF THE INPUT

- Section PFI is set up to read and store the input data, and to

examine the cornerpoints of the quadrilaterals to detect obvious errors in

the input. Because of the number of points which may have to be entered

*i for a complex geometry, the user input is a major source of program

error, and this first look for infut errors will save lot of run time in the

program as a whole. If Section PF1 detects major errors in the input, the

- program will not continue with the calculation of the coefficient matrix,

but will stop and identify the grid location of the error. Minor errors may

* not cause the program to stop, but will be noted in the output.

One of the major advantages to this program is in the organization

- of the input data. The surface is input in sections so that small portions

of the input geometry may be changed without having to recalculated the

points for the entire surface. The program also takes advantage of

:. symmetry to minimize the input effort. Only the portion of the body

which has no redundancy needs to be entered point by point. The

remainder of the body is reflected across the planes of symmetry by the

program to complete the surface representation.

The surface is represented by a set of points in three-dimensional

space which lie on the actual surface, and which will later be used to

define the plane quadrilateral source elements. These points are defined

in the global reference system. The points on the surface should be

19

2•........-...-'-...-.....-.-..,-..-,'. .- ..-......-..........-..-......-. ,...."--......---.....--.-....-.-.'-.-"--'

selected in such a way as to provide an accurate representation of the

surface with the fewest number of points possible. Portions of the

surface which are highly curved will require a larger number of points to

provide adequate resolution. Additionally, portions of the surface in

N ,which the flow field is expected to change rapidly will require a large

number of points to accurately determine the flow field in that region.

Some familiarity with fluid dynamics will provide a somewhat intuitive

approach to properly distributing the elements. Elements should change

gradually in size from areas of high concentration to those of just a few

large elements, changing no more than 50 percent in size between

adjacent elements (Hess and Smith 1966). The accuracy is only as good as

that provided by the largest element in a particular area. The use of

quadrilateral elements facilitates the use of known analytical equations

and body contours to determine the input points.

2' 3/

I 1 _-_- ---

ORIGIN

Filr 4. The 3D qailateral tlemnt i global coordinates

For the purposes of this program, the body surface is represented

by a large number of plane quadrilateral elements as shown in figure (3),

each of which is assumed to have a constant value of source density over

the area of the element. Each element is defined by four input points

*, 20

i . . ....... ..........

which lie on the actual surf ace as shown in figure (4). Since each input

point can be used as a corner for up to four elements, there is no need to

enter the same point four separate times. The input points are organized

in groups of four to form the quadrilateral element, and each point mag

also be associated with adjacent quadrilaterals. This is accomplished

through the use of a two dimensional coordinate system in which the user

assigns a pair of integers, m and n, to each point which identifies the

row" and the "column" in which it lies. A column of points will be given

a common value of n, and each point in that column will have a unique

value of m corresponding to the row in which the point lies. The

orientation of these "coordinate" integers determines the direction of the

outward normal for each element. Looking from the flow field toward

-. the section of elements, if the values of m are increasing upward, the

values of n must increase to the right. Increasing m and n can point in any

direction with respect to the global reference system. In fact, the

orientation can change from one section to another. However, any other

relationship between m and n will produce an incorrect normal vector.

IOnce assigned, the values of m and n also serve to identifg the element

for which the corresponding point is a corner. The four points which

form a quadrilateral element are two points of one column, or n line, with

consecutive m numbers, and two points of the next higher n line with the

same m numbers as the previous two points. Thus, the element m, n is

composed of the points identified by (m, n), (m+1, n), (m, n+1), and (m- 1,

nel).

L21

n -I-

4%

M= 5 n=5." m-,4 n=6 m=1"-

'"M: ' n=7 "--

m2m=2

"'r= 1 n=9 31

n=1 n"2 n 3 nw4 m=1 m=2 m3rn=4r5 n=1 n=2 n=3

Section 1 Section 2 Section 3

CORRECT CORRECT INCORRECT

FieWe 5. Quaillateral index i m6ers

Each section of the body surface is formed by specifying a set of

. corner points corresponding to the m, n pairs for all of the quadrilaterals

h of the section. The user will sequentially assign an m number to the

points for each n line, and also number the n lines for the section points

entered. The first point in each n line will always have m: 1. The n linesU* are also numbered sequentially, but the value of n is not reset for each u .

* new section. The sequence of n numbers runs through all the sections as

shown In figure (5). Points on a particular row or column do not have to

*. be strictly colinear. By forming nearly triangular elements, a rounded

planform can be approximated without conflicting with the numbering

convention, as shown in figure (6).

22

17 07 -1 W. .7 7 - - -

i.";

3 ,

.,72 2 2

2/" 2m. T= I __ _I -

n=1 2 $ 4 5 6 7 8

Figurt 6. Approxnatkg a thin region with rounded planform (Hess & Smtth 1962)

By entering data in sections, small changes in geometry can be

performed without having to reenter all the points associated with the

body surface. This feature is unique to the XYZPF Program, and offers a

great deal of flexibility in design work. However, with the added

flexibility comes more restrictions on both the input of the original

- geometry and on any modifications. There are four important

restrictions on the input which are required to provide quadrilateral

elements in groups of four to facilitate geometry calculations (Dawson

and Dean 1972):

(1) There must be an even number of elements in both the m and n

directions in each section of the body.

(2) The common corner point of a group of four elements must not

coincide with any other corner point. The sides between the elements

serve to define the local coordinate system, and serve as the axis of

rotation when the surface is flattened for numerical differentiation of

2317

- . -'°

~~~~~~~~~~~~~~~~~~~~~~~.. .. '-.. .. ........... . ..... . -". "..-...... .. . ". °. -.. -...

°i

the velocity potential.

(3) Each set of four elements must have at least seven distinct

corner points to allow the elements to more closely conformto a curvedsurface. This also allows for convergenceof N-lines or M-lines as may

occur, for example, at the leading edge of an ellipsoid. Thus, only two ofthe four quadrilateral may degenerate into triangles by having two of

their corner points coincide. This does not necessarily eliminate the

possibility of more than two *triangular" elements since the adjacent

sides of a quadrilateral may be colinear as shown in figure (6).

(4) The normal vectors between two adjacent quadrilaterals in a

group of four must be less than 90 degrees and preferably less than 45

S.degrees. If a sharp edge is required, it should be a concavecorner with

I respect to the flow field, and the input should be arranged so that the

edge is along an outside boundary of the groups of four, and not through

I the center.

When making small changes to the original geometry, the number of

elements used in a new section must be the same as the number used in

the original section unless the part being changed is at the end of the

input data. Section configurations may be selected by natural divisions,

as a matter of convenience to more easily handle large numbers of points,

or as a tool to take advantage of symmetry.

ti- In setting up input data to use planes of symmetry, it is important

to note that the XYZPF Program has certain restrictions on the choice of

I. 24

...........

symmetry planes. The user only has the option to select the number of

symmetry planes. The planes which will be used as symmetry planes are

preselected by the program to optimize the calculation procedure.

Therefore, knowing this, the preparation of input data must consider the

following restrictions imposed by the program (Dean and Dawson 1972).

If only one plane of symmetry is used, the plane of symmetry is the

y 0 plane of the global coordinate system. As such, all the y

coordinates of the input points must be of the same sign, i. e., all

positive or all negative. If the body is closed and intersects the plane of

symmetry, the points touching the plane, i. e., corresponding to y 0,.b:. must also be entered with the input points.

If two planes of symmetry are used, the planes of symmetry are the

y 0 plane and the z 0 plane in the global coordinate system. Again, the

y coordinates of all input points must have the same sign, positive or

negative, and the z coordinates of all points must be of the same sign,

positive or negative without regard to the sign of y. If the body surface

intersects one or both of the planes of symmetry, the points which lie in

the plane, i. e., those corresponding to U 0 or z 0, must also be entered

with the input points.

If three planes of symmetry are used, clearly the planes are the

reference planes of the global coordinate system. As with the previous

cases, all the x coordinates of the input points must be of the same sign,

and similarly for the v and z coordinates. If any part of the body

intersects any of the planes of symmetry, the points which lie in that

25

"°. .° .' , . .. . . °. .. . . - a °, ° + ° . ° , ° .. a . ]

plane, i. e., x = 0, y 0 or z 0, must also be entered with the input L6

points.

5.2 SOURCE PANEL GEOMETRY -

NC N

-3244

/ .4

- /

Fito 7. The outer normal to the qladrlateral elermet

With the surface points identified by the location numbers, m and n,

and arranged in accordancewith program requirements, calculation of

various aspects of the source panel geometry and formation of the plane '"'-

quadrilateral element is the next step in the numerical integration ,._

process. Formation of all of the planar elements is identical, so the

p following discussion of source panel geometry will deal with only one

.. characteristic element. The four corner points forming the basic

quadrilateral are numbered in a clockwise direction from I to 4 as shown

in figure (7). It does not matter which corner point is identified with the

number 1 subscript, but the remaining points must be numbered

consecutively in a clockwise direction when observed from the flow field

in order to ensure an outward directed normal vector. These subscripts

will be used to identify the corner points for the remainder of this

discussion. For this example, the points will be identified as follows:

26

- - - - .. r '- -. ~ - W uq Uj w - 7Z - W- . -Z - . . ..

Position Numbers Global Coordinates

m, n X, YJ Z"

m+l, n X2 , Y2 , Z2

m+1, nl X3 Y3. Z3

m, n*l X4 Y4 , Z4

In forming the plane quadrilateral elements, the corner points,

which are generally not coplanar, are used to form the local coordinate

system, relative to the element. Recalling that the cross product of two

vectors yields a vector solution which is perpendicular to both of the

original vectors, the normal to the element may be obtained from the

cross product of the diagonals of the element,

N = T24 x T13 (21)

where T13 is the vector from point 1 to point 3, and T2 4 is the vector

'; from corner point 2 to point 4. The unit normal is then:

T24 x T1 3

T24 T= (-)

This unit normal now represents the first of the three local

• :coordinate directions, this one in the t direction. The side of the

quadrilateral from point 2 to point 3 is then used to obtain the second

coordinate vector.

27

r.

.:,.-. -,-°,-. : .-', .- ,.:. ., ... . . . .. *, . - . - --. . --. • . . . . . - . - . .

Tzs

Filwe S. The sooond oal ooordinat, vector

N2 =n T 23 (23)

and the unit vector

N2n2 (24)

IN21

Similarly, the third local coordinate vector is obtained from the

cross product of n2 and ni, the result of which is a unit vector.

r)~ X

Figare 9. The Uhid local oordae vector

n3 n2 n (25)

The unit vectors n3, n2, and n form an ortlionormal basis and define

the local coordinate system for the element in the TIr, and ~,directions

20

respectively. Other methods of obtaining an orthonormal basis could be

used just as well, and would make no difference to the remaining

computations. The origin of the local coordinate system would most

correctly be located at the "null point." the point at which the velocity

- potential has no contribution to the tangential velocity component on the

P source element. The null point is the point in each quadrilateral element

where the normal velocity boundary condition is applied. However, with

the exception of long, thin quadrilaterals, the physical difference between

the null point and the centroid of the quadrilateral is not significant. The

XYZPF Program will print a warning in the output when a quadrilateral is

: long and thin enough to jeopardize the accuracy of the approximation in

that region. By locating the origin of the local coordinate system at tme

centroid, rather than at the null point, the difficult process of locating

the null point for each element can be eliminated, later calculations of

the multipole expansion can be simplified, and the boundary conditions

can be applied at the centroid without contributing significant error to

the approximation (Hess and Smith 1966). Therefore, the origin for each

local coordinate system is located at the centroid for the respective

element.

29

t.i.

129 ..

... ... . . . . . _ .... . . ._ .. . a . ... a .A- .. .... ..-..a - J

5.3 LOCATING THE CENTROID

1P4

Fig.r 10. Locatig the oentroid of the quadnlateral

The centroid of the element may be calculated by first dividing the

Ilk , area of the quadrilateral into two triangular areas, the triangles being

separated by the line from point 2 to point 4. The area A, of the triangle

defined by corner points 2, 3, and 4 is

A, 1 4 T24x T37 (26)

Similarly, the area A2 of the triangle defined by corner points 1, 2, and 4

isa , = T x1 1 T 4 , - , ,

2 F

In the global coordinate system, the X component of the centroid is

given by' A I X1 + A2 > (2L)

- + A2

where X, and X2 are the averages of the X components of the corner

, -points of each triangle. Substituting the values for X1 and X2:

30

- ., . . . .~ -, -

LA (X42 + X3 +X4) + + X2 + W

A + A 2

T. [(A1 + A2) X2 + (Al + A2)X4 + Al X 3 + A2X 1-++4> A I + A2 "

A 1 A + A.X()

- X + [ + X 4 + ... . . -2Al + A 2

Similarly -

- -I'.(', , 3.

+ LY4 + AY +A 2 Y'-~ ~ A I' +4 .. .. 2 ,,..

3 Z [ + + 1 (31)L L2 AIl + A2 "

5.4 COORDINATE TRANSFORMATION

Now that the local coordinate system is formed and properly

located at the centroid of the element, the global coordinates of the

corner points (X, Y, Z) are transformed to local coordinates(,q,)

through the components of the reference vectors of the local coordinate

system as follows:

n3 n3 ~ n7

I -[.1"

K~ n. n (32)

nx~~ n.n 7-J,

31

r.r

The corner points are projected into the plane of the quadrilateral

element by setting the components to zero. The original diagonal

vectors, T13 and T24 , will be on opposite sides of the resulting plane. The

plane quadrilateral element is now completely defined. The program will

sweep through all of the input elements using the assigned location

numbers, and repeat this process for each element.

23

* -- 1 I. 4

'- ,I -

Figure 11. Forming the plane quadrlateral element

5.5 MOMENTS OF INERTIA

The calculation of the moments of inertia for each element are

performed for use in the computation of the velocity coefficients using

-. S the quadrupole method. Any calculus text will give the moments of

inertia of a planar section with a constant unit density about the origin

to be:

d, d

I Y T d d TAM

,-1 = $ '( d, dv r(3 )"- -

A

32

For the triangular region defined by the corner points 2, 3, and 4,

I xx- 12 W' ) + ) 4 t 2 (36)

IY - (1-2 + -t3 + 3 + "4 )2 + (1I4 + 1R-2 )2 ] (37)°12

X" t2 t3 'Q2 + Q3 ) + t3 + 4)( 13 + T4) (38)

+ ( 4+ 2 )( 14 + 1-2 T1

Similar equations can be generated for the triangular region defined by

the corner points 1, 2, and 4. The moment of inertia for the entire

quadrilateral is the sum of the corresponding expressions for each of the

triangles. The resulting equations are:

"-I Ix t 2 +- 312 +- (3 + +.14 ( 4 +- 2)2 (390) "A r"w."XX 12()

A+. )2 )2 ] .-Y -

+ 'q3 + (Q3 + -4 + '4 + 'Q2

+ A )2 )2 )2 (40)=~ ~~) 11 2 T, 2-3 4 ( 3 + 11"[4 ) + (1[4 + - T T 1212

+ (4 + 2 'Q4 + T22

A- [( 1 + 2)( 1 + -2 ) + ( 2 + 4)( ,R + - 4 )." ( 4 + 41 T( q'4 + 1-2 Q:':

12

- - .. . ' . '

7i

6.0 THE MATRIX OF INFLUENCE COEFFICIENTS -

With the quadrilaterals completely formed, the next step is to

calculate the velocities induced by the elements at the centroids of all

will be represented by N. Let the source element be the (j)th element, and

the element for which the velocity components are to be calculated at the

• " centroid is the (i)th element. It does not matter how the (i)th elements

are arranged in relation to each other as the sequence progresses.

However, the sequence must be consistent as the calculations proceed

from one source element to another. This program sweeps through the

(i)th elements in the order of their location numbers, m and n. For each

:. consecutive n line, the elements are swept in order of increasing m

numbers.

The result of the induced velocity calculations for the elements

i* with unit source densities is an N by N square matrix of the values of

-.induced velocities at each element, known also as the "matrix of

influence coerficients." The XYZ potential flow program calculates the

-" coefficients column by column, while the Hess program calculates them

row by row. The advantage of one over the other depends on the method

of later solving the matrix for the source densities. In calculating the

influence coefficients, twenty-five quantities which describe the

geometry of the source element are required to adequately calculate the

induced velocity at the centroid of the (i)th element. These quantities

- include the coordinates of the centroid in the global coordinate system,

the elements of the coordinate transformation matrix, the local

34-7. -

, -.1?

*o " -. .

coordinates of the corner points, the maximum diagonal, the area, and the

second moments of the quadrilateral element. Additionallg, the Hess

program uses the coordinates of the null point, making a total of

twentg-eight quantities for that method (Hess and Smith 1962).

When calculating row by row, the first (i)th element is selected,

containing the "null" point, and the influence coefficients are computed

- for all of the (j)th elements in sequence before proceeding to the (i~l)th

element. This procedure requires the twenty-five quantities for each

(j)th element to be available for calculation of the influence coefficient.

Because each of the N (j)th elements is used N times with this procedure,

calculating the geometric quantities or retrieving the values from low

speed memory would be very time consuming, since the calculations or

memory access would need to be performed N2 times. Therefore, it is

more practical to have the values available in high speed memory,

although large matrices may exceed the storage capacity of high speed

memory, imposing a limit on the number of elements which can be used.

The advantage to the row-by-row calculation is that solution of the

resulting matrix by the Gauss-Seidel reduction method does not require

transposing the matrix, which would be another time consuming process

- (Hess and Smith 1962).

Another alternative is calculation of the influence coefficients

column by column. This method calculates the influence coefficients by

sweeping all the (i)th elements for each (j)th element before proceeding

to the (j~l)th element. Therefore, the twenty-five geometric quantities

- are retrieved from low speed storage only once for each (j)th element,

35....--..-.

.....................................................................................

for a total of N times. This procedure is not limited by the capacity of

.9 high speed memory, and calculation of the coefficient matrix is

approximately 10% faster than the row-by-row method (Dawson and Dean

1972). This is the calculation method used by the XYZ Potential Flow

Program.

An influence coefficient represents the combined effects on one

element of the velocity potentials of all the other elements comprising

the body surface. For the quadrilateral element with a unit source density

* "in the xy-plane, from equation (13), the potential at point P (x, y, z) due

to the element is

r 7- dAd = d4"

A A (x - + (y -r)+ z2 (42

The integrand, l/r, can be expanded in a series about the origin in

* powers of _ and TI. Each term of the series will contain the product of

some powers of [ and il with a corresponding derivative of l/r0 , where r0

is the distance of the field point P from the quadrilateral origin.

2 .y 2

and let

Then the series expansion through the second order term is (Hess and

[" Smith 1962):

r 36

. ... ' - - -m b .~ W ~ 'X TT .X '~ '' z ~ q .~

I,i

Aw,- (MxwMy + MIw +120 xxWxx + 2 lxywxy + lyywyy) + (43)

The subscripts, x and y, used with w represent the respective partial

derivatives. This series represents the multipole expansion of the

*" velocity potential, since each term can be interpreted as a point

singularity of a particular order. The first term is the potential at point

P due to a point source of strength A located at the origin. The second

term is the sum of two dipoles of strengths Mx and MU located at the

• -: origin, oriented along the x and y axis respectively. The choice of the

centroid of the quadrilateral as the origin of the local coordinate system

causes the first moments, Mx and M to be zero. Therefore, the dipole

terms dissappear, and are not dealt with anywhere in the program. The

* third term is the sum of three quadrupoles of strengths Ixx, Ixy, and Iyy

located at the origin. Kellogg (1929) shows that this second order

approximation is absolutely and uniformly convergent, and Hess and Smith

..(1962) show that convergenceis rapid enough with an increase in r0 that

certain further approximations may be made without significant error at

: *. large distances r0 from the source quadrilateral. -Z

Hess and Smith (1962) presented a comparison of velocities

- •calculated using the exact formulas, a simple point source, and a second

order approximation. The comparisons were based on the ratio of the

*- . distance r0, between the centroid of the source quadrilateral and the field

. point P, to the length of the maximum dimension t, of the source

quadrilateral, typically the maximum diagonal. The non-dimensional ratio

is then r0 /t. The results show that sufficient accuracy is maintained

37

while using a simple point source at ratios of (ro/t) : 4, using the second

order source and quadrupole solution for the range 2.45 s (r0 /t) < 4, and

using the exact solution for ratios of (rolt) 1, 2.45. In anl case, the error

goes to infinity as the field point approaches the edge of the quadrilateral

where calculations indicate an infinite velocity. The XYZ Potential Flow

Program uses a monopole source for (r0/t) > 4, the source - quadrupole

formulae for 2 < (r 0 t) :- 4, and the exact formulae for (r0/t) - 2. Hess

and Smith (1962) reported a maximum error of 0.001 in approximating any

velocity component using the above criteria.

3.

............. ...

- . . r'r r r r, °

7.0 DERIVATION OF THE EXACT SOURCE PANEL INTEGRATION

From equations (2) and (42), the components of the velocity at the

field point P(x, y. z)due to the source quadrilateral are:

(X (- ) 0d-r! ax r3 (4 "

a , (y - G ) dt. d~r 45 :::A 3

d dZ _ Z d. (4F_, "I ::_.

a 0

A

Equations (44), (45) and (46) are evaluated by expressing each of

the integrals as the sum of four terms, each term representing the effect

of one side of the quadrilateral (Hess and Smith 1962). This method can

.. also be generalized for polygonal elements with any number of sides. The

potential function for each side of the quadrilateral is the combined

potentials of semi-infinite strips whose boundaries are the side of the

quadrilateral and two semi-infinite lines parallel to either the x or y

axis. When observed from the domain, and the sides are traversed in a

clockwise direction, the source strip on the right will have a source

- density of o +1/2 and the source strip on the left will have a source

density of o -1/2 as shown in figure (12). When the sides are

. recombined to form the quadrilateral, the source densities outside the

quadrilateral cancel each other, and the source densities within the

quadrilateral combine to form a source density of a *1. This will be

i39 . .. . ..

"2: " "" '" "' "" "' """" "" '." - -..... ' " "-"'"'"'".".. . .. . . . . . . . . . . ..'"'"" "' ""' "<4:

true for a planar element with any number of sides and in any relative

orientation within the plane.

2 2

1 , II" 2 3° ' +

* i F !

1.1 THE Y V

Fro eqain(5 , the veoiycopnn -i on b umn

qudiatrl Fo h sd.rm.on 11 opin..,11) h

I is i

d f (y-1~dl(y 12 2 2 .3

M (XI I + -

40

7.1 THE Y VELOCITY COMPONENT..-'

:::: ~From equation (45), the velocity component Vy is found by summing'-.,

m the four terms representing the contributions of the sides of the ,

' quadrilateral. For the side from point ( , mh) to point ( z 'ri), the

" contribution is expressed as the integral over the area of the .

semi-infinite strips with the source densities of 0 =+I1/2 and 05 = -I12.

rather than the unit source density of equation (45). -

[1 u +T] (y-(".-.(

F '(

Integrating with respect to I:

.. Vy,1 2 d;, [(X- 2 (y- I 2 ] no "

"p 1 1

I[(X - )2 + (y Tt )2 + Z2 - (x - - +(y - , +2 + .Z

[(X - + (y - 1 +Z2 1 .}-The terms evaluated at 'Ti +00 and 11 = -00 cancel, and the terms evaluated

at 'q T 12 add to obtain the following expression:

2 d4

1. 2 [ : ) + T[)2 +

" = (48)f r

Equation (48) is changed to a function of arclength s by the relation

___________- _______ - *. (4'Q)-:... 2 dl2jii,:ds + (()2 + (T2 "1

where d12 is the length of the side of the quadrilateral from ( q, ) to

: (-,4 T2) as shown in figure (13).

*" 4

r' 41

. . .- '

I ,

-S

Figme 13. The pottntial due to a finite Thie sowce (Hess & Smith 1962)

Substituting equation (49) into equation (48)

d12 r.,.

From figure (13), it can be seen that, in terms of arclength s, the distance

r from point P to any point on the line from point 1 to point 2 is given by

r F f 2 + (Sl-.

. Substituting into equation (50) yields

: $2- 1 fod1' ds41,-

-

-"d 12 o f 2 + (s-s,)2 (51)

42

* * * -:*:*

Evaluating the integral

- i r 2~ d1e

V 2d 12 log [(S SI)+ f2 (SS, j

1109~ {o[(dl2..S1 + fl + (d12 -S1 )2 ]-log [(SO+. f2 + S,~)2]

- 1 og r-. - (52)Io

Thequatites 1 2 i n 2 ue neuain(2 r ssoni

g figur (13). quation(52) issinglr2 whe r1's-wic2cushe h ~

Thcsi e (Hssandith i 2 12)2.sdineuton(2 aeaso ni

f ~ ~ ~ ~ ~ ~ ~ * igr 1)Jqain(2 ssnuarwe ,=swihocrwe h

field po~~itPif oae nweeaogte iedfndb h ieo h

F~w@ 14. The low of oostws

43

From equation (52)

- . (I__ ~ - COS 2 ) (7d12~ lo r, (1 - cois

where Pand fl2 are the interior angles shown in figure (13). Applying the

- law of cosines to f igure (13)

r r2di (5 4)

r, d, .2 r di d1 r

From equations (54) and (55):

2 r., 7

r1 ~J1 2 r"1

2 2 r,- d 2 r 2 + r -di

r 44 ,-

d r

Substituting equation (56) into equation (53) yields the final form of the

exact equation of the U component of the velocity induced by the side of

V the quadrilateral from point I to point 2:

V9= log

Equation (57) is applied to the remaining sides of the quadrilateral

*simply by substituting the appropriate point numbers for the corner

points of each side. The total contribution of the quadrilateral to the y

component of the velocity is the sum of the four terms representing the

contributions of each of the sides. The y component of the velocity at

the fijeld point P is now given by:

I7C) r; +r^,+ d2 C'3 r + + 2

d12 r + r., d+ -.

4 * r3 +r4 +d-.4 1-4 r4 +r, + d4l og + ~ 'og 4+r qd4r 3 +r4 -d74 r+ 1 -

7-2 THE X VELOCITY COMPONENT

A similar derivation process is used to produce the equation for the x

component of the velocity induced by the side of the quadrilateral from

t ipoint 1 to point 2. The semi-infinite source strips are constructed

parallel to the x axis, and the order of integration is reversed.

45

...................... .... ..... .... ..... .... ..... ....

7 -;LW W

The x component of the velocity at the field point P due to the

quadrilateral is given by:

1 2 -'Q1 r, + r2 + dil tI3-t2 r2 + r w 23Vx lo g + log

-d ! 2 r, + r2 - d12 dT23 r2 + r3 - d2 3

(59)1,4 "R.3 r3 + r. + d34 "Ql1 - -T14 r4 + r, + d41

+ 4 1oi logd" r 3 + r4 - d d41 r4 + r, - d4

7.3 THE Z VELOCITY COMPONENT

The z component of the velocity at the field point P due to the

quadrilateral is obtained in a similar fashion, using semi-infinite source

strips, this time parallel to the y axis. From equation (46), the

fundamental velocity potential of the semi infinite source strips is

integrated in a manner similar to that used to obtain equation (47), and

the z component of the velocity due to the side from ( , r) to (42. T2) is

" given by

-fT

L i 2 i ]., z;

", .

V.~

-- ' f °' -1 f L I (X - )2 + (y - n) + Z

-. Performing the integration with respect to T1, the integral

od .

• + (.-Q

46I-

.

fits the integral form

[C 2 . FI~ 2C 2 n-1) [

_f dF -l (2n-3)f dF[C2 + F2] n -2C2,(n-1) [C2 + F21' - 1 C(2n -3 F2]nI .

where.j ~c 2 (x - )2 z2 -

F T(-I)

dF -dT"

n= 3/2

Then, from equation (60)

L. f [' ' - + )+

X - iL)

(X - +. Z[(X --

Z2 - (X) ( :.DO

:::: - ~~[(x - &:) .z 2 ][(- ) (y ) "::

(y - 11)

t ) + z I-f: -- ''(U + Z ' '

Again, the terms evaluated at +oo and -oo cancel and the terms evaluated at

T112 add to obtain the following expression:

r - (Y - ' (12

Y -[(':- z+ z1I] (x -()X (2 - T 2) + (y 'ti (62)

47

Without a convenient substitution with which to integrate equation (62),

the integration is performed directly. Recognizing that along the line

defined by the side of the quadrilateral from (.1 rh) to ( 1. 2 ri), 1112 may

be expressed as a function of :

* T = mrn,~ r + b12 (63)

where the s!ope of the side, M12 is given by

M12 - 12(64:. ml12 - _ ::'

and b12 may be determined knowing that Tri12 TI, when 1

L2 T1 k5

Substituting equation (63) into equation (62) yields

Z 2 f- Z +,z[x-+z + ][(x )2 ( -m, -b,) 2 Z 2 (66)

Define the quantities

q12 Y - b12 - m12x (67)

u =x - (68)

Then du = -d, (69)

. - b12 - m 12 mu +q2 (70)

48

By a change of variable, equation (66) is now expressed as a function of u:

XZ2 (m, 2u + q12) du

VZ2 Z U (71)2 2+

XS2 (m,, u + ql2) du

2 [U l~2 1 2+-, [(M,2 12 C. 2,~ 2 Z (72)

which fits the form of

F (L t + M) d Uf (Au2+ 2eu + C)4 (au2+ 2bu c)'

where -

L M12 N q12

A= I a= (M12 2 +

B =O b= M12q12

= z C c=q 12 2

*From Hardy (1944), this integral form may be integrated by the

substitution

t + I (

where p and u satisfy

apiu b(ja u) c 0 (75)

Apu + B(P +u) +C 0 (76)

and are the roots of the equation

r(aB -bA) Q - (cA- aC) + (bC -cB) 0 (77)

49

Substituting the appropriate values into equation (75), the roots of the,"

quadratic equation are

% q1212 (78)

.- \-

ic - (7 9 )912

It can be verified that these values satisfg equations (75) and (76).

. Substituting equations (78) and (79) into equation (74)

m12z' q.t

St+ I .

2. ~q 2 t m 12 Z

du = mn1 1) , dt ,: 1)I. L (t + I)

By substitution and a change of variable, equation (72) becomes a function

of the parameter t. After simplification, the integral now fits the form

of

f dt t(2)

where

K = -(m12 q12z + 2m12 4q12 z q12 /m"22)

Ox = q124 M12 2q122Z 2

50

7

M12 m122q12

2z2 ,--

= q124 + m12

2q122 Z2

S m12 z4 + m124 z4 + 2mI 24q122z2 + m122q124 + m122q122z2

Equation (82) can be rationalized by the substitution

-. tt,,67+

from which it can be shown that

t2 (6 4)

[ '., ]~~1/ 2- .(,)""

i:!: 0t = 1 - vZ ',' -

2.6 d (65)

* Substituting equations (84) and (85) into equation (82) and simplifying

yields the integral in terms of the parameter v:

(f,.t t K f +cj,)v (66)

which fits the form

a + yv ab t a

• where a2 =

b2 = (CxKS- e)

51

" 5 1 * .;<

Performing the integration

f dv K- ta6t v (88)

From equations (80), (83), and the expressions f or o, , 8, and 2 from

equation (82), and after a considerable amount of algebraic manipulation

and simplification, equation (88) becomes

K____ _ ,tarn 1 vcr:- j4- taT)

1 [ ' M 12 Z 2 - q12 u( ")

SFrom equations (63) and (67), equation (89) becomes

_Irn 1 ,,Z 2 U j:

Z, tJri L 2 + U 2+ (.rrl 12 U+ q12)'.. F.-.a °

.*, ." [ 12 -M 12 LuS + - 72 - 'U

1 tan- g )-.. - z z u + y - -rt,2)' + z ':::

52¢"ir

Finally, applying these results to equation (71)

(m, 2u + q 2) du I"

Vz12: z [u I+ z2 U (f,2u q,2)' z 2

Z2)( -.(y

" m12 (u + zT)1- (yU- 1 2 ) u

= tan- ,- tar L U2 + (y - Tht2 )2

+ 2 ' Jx= aC [+2) (y-T ( .. .. ... ]

"'; tan'l~J z ,l(x - )2+ ( - T : ° :.:.'" (91)

M12 ( -. + z) (2) -)(x 1-tan.'

Z - )2+ ( - Tt.-)2 + 22 J

Recall that when x =, y = T and when x = Y2' y = Ti2. Then, for the sake

of a more compact equation, define the following quantities:ei e= (X - 1)2 + Z2 e2 (X - 2 )2 + Z2 -.,"

h, (y - -Ti)(x - I)h2 = (Y - q2)( x - 2) ;;::

I The quantities ri and r2 are as shown in figure (12), where

ri=,i(:; -)2 +(y TI) 2 Z2 r2 (x -)2 + (Y !,)' + z&• r , (Y 112l ,

Substituting these quantities into equation (91) yields the exact z

component of velocity due to the side from point ([ 1 ) to (4qr2) in the

form used by the XYZ Potential Flow Program:

Vz2 = tan[3 -tan -' f(92) ".,. (,12 z r, Z r2

I.5

,,; . : ., -- .- .. ,, ,. -. ... - , , .. ... ,. , .. . . . . ... . .. . ... . . .. .. .. - . . -. . . .. , -'-

The total z component of the velocity at the field point P(x, y, z) due to

the quadrilateral element is the sum of the four sides:

nM12ei - h1 -tan" [r 2 e2 -h2

,z L Z r2

j M m2 Ik2 - h2 [2 m 3 e 3-h1II + tan- tan-'z r, z r3" (93) "

+ tar-' [tn-' [-" 'l' Z r3 Z r,4,

+ tan"- -tan-' i-4 e h

z r4 z r,

II

54

. . . . . . . . . . . ...

8.0 APPROXIATIONS OF THE INDUCED VELOCITY

8.1 QUADRUPOLE METHOD

'' As previously mentioned, as the ratio of (r0 /t) exceeds the value of"' '4

2, then certain approximations may be made which greatly reduce the

calculation effort otherwise required by the exact method. In the range

of 2 < (r0/t) < 4, the XYZ Potential Flow Program uses the second order

approximation of the potential described by equation (43). With the

origin at the centroid of the quadrilateral, the first moments are zero,

-- and the second order approximation is

Aw (1/2)(I xxWxx + 2 1xyWxy I yyWyy) (94)

where the first term is a point source of strength A, the second term is

I composedof three quadrupoles of strengths Ixx, Ixy, and Iyy located at

the local origin, and the subscripts on w indicate the partial derivatives

• ..: of w with respect to those variables as before. The quantity A is the area

of the element, and the terms Ixx, Ixy, and l are the respective moments

of inertia of the source element given by equations (39), (40), and (41).

To obtain the velocity components at the field point, equation (94) is

-. - differentiated with respect to the coordinate directions giving:

55

K[. .. :*-'-**.,'.*

'-" . " . . . . - . . . , .. . . . . . ' ,. . . . . . . t. . ,. . P . , , . , , . . . . . t ! q , . ; , . £, ..

Lb V

- s- [4wX + 21 1 XX WXXX+ IxyWX~~ + -L lyyWXiy] (9) COO

-y V [A + iXXw~ x w + Iyy] (96)

-- AW, + 'xx WXXz+ LXyTWXLI 2LI~Ywz

Recalling that wX2 + q2 + r.

the derivatives of w, as expressed bg Hess and Smith (1962) and as used in

the XYZPF program, are

UW..= - X r o -I"

rH r- 3 ~9

': s=x(3p + lOx 2 ) r 7

I Wxxy zy p ro-7

W 3x q r&7

'. -" ;"?Vuy yy 13 q;Y -' + IOLI '2 to7( O ,,: :. 7, 1/ ,, "r7

(100

, .w > .Y Z 3 z p r o 7

NA W =y - 15xyz r - 7Y11 -7.yyz q r 7

_ where

p = Y2 + 2 - 4X2

q = x2 + Z2 - 4y2

56

-" . .r ""L: :.;: .< '. -_ " ' "" ' " "' " " " . . . . . . . . .. "

8.2 MONO~POLE METHOD

When the ratio of (r0/t) is greater than 4, then the quadrilateral

may be approximated by a simple source corresponding to the first term

* .. of equation (43). Then the velocity components at the field point due to

the quadrilateral are given by

h~~ - AYW 1O

*where the partial derivatives of w are those given in equation (99).

57

- . . -- -. . - - o- o.

9.0 SOLVING THE MATRIX EQUATION FOR SOURCE DENSITY

~ 9.1 JACOBI'S ITERATIVE METHOD

From equation (20), the matrix equation may be solved for the

constant source density 1i for each element which satisfies the boundary

condition equation (11). Equation (20) suggests the use of Jacobi's

iterative method of matrix solution in the form

(m"I)N (i.-n)

"iV + ' Cij i ' i: 1,2,..., N (104)

j

where N is the number of elements composing the body surface, and m is

* the number of iterations completed. A partial sum of equation (104) is

computed for each of the ith elements before proceeding to the next jth

element. The iteration is complete when the summation of equation

In (104) includes all of the jth elements. Because the values of the source

densities at all of the elements are recomputed before any of them are

N. used in the iteration, this method is also called the simultaneous

displacement method (Ralston 1965). This is contrasted with the

Gauss-Seidel iterative method used in the Douglas program. In the ,..

Gauss-5eidel method, as each new oi is computed, it is used immediately

. in the iteration process for calculation of 0 (i+l) . This is also known as

the successive displacement method and is expressed as

L.:58• " "5"

.' " -: I V X i+ '-w- -j= 1 j=i+l .4

i 1, 2, N""

Though the Gauss-Seidel iterative method is faster, the Jacobi iteration

method was selected for use in the XYZPF program in order to be able to

perform the iterations column by column, since the coefficient matrix is

also computed column by column, and the matrix does not have to be

transposed for solution.

When the (m+l)th iteration is complete, the values of the source

densities are compared with those of the (m)th iteration and the

differences summed for all of the elements. The total difference between

successive iterations is then compared to a convergencecriteria input by

the user. If the difference is less than the convergencecriteria, then the

matrix solution is complete and the values of the source densities are

stored for later use in computing velocities and pressure coefficients. If

the convergencecriteria is not met, then the iteration process is

repeated. After every five iterations, if the convergencecriteria is still

not met, then an extrapolation is attempted in order to accelerate the

-- convergence. The XYZ Potential Flow Program uses a Richardson

extrapolation method, a numerical procedure which uses two approximate

results to obtain a third approximation which is closer to the exact

solution (Ralston 1965). ',.

59.

59'-

-. ,i7

9.2 RICHARDSON EXTRAPOLATION

The Richardson extrapolation assumes that the iterative process is

convergent. For the iterative solutions So. S1, and S2, where So is the

most recent approximation and 52 the oldest, the solution is convergent

ifso- S1 x < (1O0) ""

51- 52

While a Richardson-type extrapolation can take may forms, the XYZPF

program uses a procedure developed from the following approximations

(Dawson and Dean 1972). If there is only one dominant eigenvalue and a

sufficient number of iterations have been completed, the iterative

solutions may be approximated by

S0 5f E Xm

51 5f + E Xm-1 (107)

52 Sf

*.I.: S2 Sf +E Xm- 2 ':""

Si Sf + E Xm-I

where Sf is the true solution

X is the eigenvalue

E is the eigenfunction

m is the number of completed iterations

Define the linear combination which, from equation (107), may be

60

" %:.'1

approximated as

AS 0 + (1- A) S1 Sf+ E Xn-l (AX+ 1 A) (108)

The value of A may be chosen such that

(AX + I -A)= 0 (109)

Then, from equations (108) and (109)

AS 0 + (1 - A)S: - Sf (110)

where the expression on the left converges to the exact solution.

From equations (106) and (109)

X - 5 1( "

Solving for A,$1- S 2 S 1

2 1 " "

C ' I"1 + C2 [

• Since the value of A generally changes from element to element, a

weighted average of A is used in the extrapolation, where

N.i: (52(0) S W) (sign of D(i))

*- . - ,, .(.-.-

N:.: Z O(i)D i= 1

Equation (113) is recomputed after every fifth iteration. If the difference

between the new value and the old value is less than 0.02, then the

solution is extrapolated. From equation (110), the extrapolated solution

isA = A (1 - A) i (114)

61 ° r-

When there are two dominant eigenvalues, then the iterative solutions ..

may be approximated by

Si - Sf + E1 X m-i + E2 X~m-i (115)1 2

where Sr is the true solution

X and X2 are the eigenvalues

E1 and E2 are the eigenfunctions

:. m is the number of completed iterations

Define the linear combination which, from equation (115), may be

approximated as

B2 SO + BI 1 + 0 B -B1 2) S2

'5f EX, 2 B2 l2+B 1X1 +(1 -B1 -B2 (116)

+ E2X 2 [132 X22+ 1 X2 +(1 -B1 2)]

The values of B1 and B2 may be determined for which the eigenvalues :X

- -:" and X2 are roots of the quadratic equation

B2 X2 +B1 X+(i -BI-B 2)= 0 (117)

- Then, from equation (116)

B2 50 +B 1 1 +(1 -B 1 -B 2) 52 5f (118)

62

* . .- * .*% - -- ..-. *. ' ... .*.''"*. ,.

. . .. .I

where the left side of the equation (118) convergesto the exact solution. ..

"- Using equation (115) and eliminating terms containing E2:

(50 - 51) - X2 (St - 52) = 1 xr-2 (X, -X2 )(X1 - 1)

(sj- 52) - X2 (S2 - 53) E, X 1 3(X, -X2)(X- 1) (119)

(S2 -S3) X2 (S3 - S4) = E X1r(X -X2 )(X - 1).:.: '-..

oi-.-.o

Solving for X-

S(S S!) X2 (S1 - 5) (Si S2 -X ( 2- ) (12):,, = • =(120) ' "

(31 -52)- X2(52- S.: (2-b 3 X A2S3 - 54)

bi From equations (117) and (120)

(5- 5) (5o-25 4+ )-(54- 5)(31 - 5 2)-(5- (S21)," B : (12 1)"" "D

(34- 52) (4-2S3+S2)-(54- 5-[(51 -5)-( 3- 54)]B2= (22)D

where D= (54 -25 3 -52)(50 -25 2 54) -(S 1 - 52 -53 +5 4 )2

The weighted averages of B, and B2 are used for the extrapolation as done

with A in equation (113). If the sum of the absolute values of the

weighted averages of B, and B2 changes by less than 2%, then the

extrapolation is performed. Then from equation (118), the extrapolated

- solution is

825 + + (1 - )2 (123))5

63

U-. . l I * -- ' i ". -. I U I U UU%, i i -... - . .. I ]

• -- ' - .. . . , -. - .. % . . ... . .- .. *•.. . .. . -. . .-..- . - -,* .U,. - ... - . * ., , .. . .U .

10.0 CALCULATION OF VELOCITIES AND PRESSURE COEFFICIENTS

With the influence coefficients and the source densities

determined, the calculation of velocities is a relatively simple matter.

From equation (9), the total velocity is the sum of the freestream

velocity and the disturbance velocity due to the body. The product of the

source densities and the influence coefficients are summed for all of the

* elements, and then added to the freestream velocity to determine the

total velocity at any point in the domain. Velocities on the surface of the

body are calculated at the null points only, as the boundary conditions are

enforcedonly at the null point of each element, and velocities at other

points in the element would produce significant error due to the method

of approximation. The components of the velocity at the centroid of the

ith element are

j=1 .' [N

• = v oj (124)

V = ,:z + CIjj=1

From equation (15), the velocity induced by an element at its own null

point has a magnitude of 2"r directed along the outward normal vector of

the element.

64

.- '..-..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .- .- --... . .

~~~~~~W5

W & A2~~~ ~ ~ Y2V~IV rrr rip I?. -F -W* w'J7'- I~ V - iW w.- -- .W wu wg r - -- --

At a point of f the surface of the body, the components of the --

velocity are determined just as if the point of interest was a null point of

a single element. The total velocity at the field point is the sum of the

freestream velocity and the contributions of each of the elements of the

body surface. The contribution of each of the elements is determined by

calculating the influence coefficient based on the element geometry, and

multiplying the result by the source density for the element. The total

velocity at the field point may be expressed as

" N

Vp Vc, . + Z Cpq Oq (125)

where p and q represent the field point and the source element

respectively and the influence coefficient,

Lpq J ~71 rp,&' dl'(126),.5.

.: Cpq 1 .r' rpr dr :.-

As discussed in section 6.0, the influence coefficient may be

calculated by the exact method, or it may be approximated by the

quadrupole or monopole method depending on the ratio of the distance, r0 ,

between the field point and the centroid of the source element to the

maximum dimension, t, of the source element. .

65

r. ' ,, , . "" 3 ." , ". -'.' : -. "' -,-.-.° ' ' .. -.- -. ." ".•., .. . . . .

The magnitude of the velocity at either the on-body or of f-body Vpoints is given by

VI IV + V, +v (^127)

The pressure coef ficient is calculated by using the result of

- *. equation (127) in equation (19), renumbered here for clarity.

r

66

-. -7Z W. -. W -7-

11.0 STREAMLINE CALCULATIONS

For the calculation of streamlines off the surface of the body, a

timestep procedure is performed by calculating the velocity at the

-' starting point of the streamline from equation (125), and advancing the

streamline one time increment by a fourth order Runge-Kutta integration

to a new point (Ralston 1965). The timestep procedure is repeated, thus

*-i creating a streamline composedof finite segments.

For the calculation of streamlines on the surface of the body, the

streamline is started at a specified point and quadrilateral number. Therd local velocity is calculated from equation (124). and the values of a

. stream function are computed for each corner point. The stream function

is chosen so that it has a value of zero at the last point on the streamline

in the quadrilateral. The side of the quadrilateral through which the

.. streamline exits is determined, and coordinates of the point on the side

which has a stream function value of zero are computed. The direction of

the streamline is verified by comparing it with the known direction of

positive velocity. The next quadrilateral through which the streamline

passes is determined by calculating the proximity of the new

quadrilateral to the most recent point on the streamline. A circular area

is computed which encloses the new quadrilateral with an additional 10%

margin. If the last point of the streamline falls outside the circle, then

- the quadrilateral is discarded and a new one selected until the streamline

is adjacent to the new quadrilateral.

67

7 ,..... . . .

L2 3

":I =0

T=OO

IllF

1, /4

Ftgme 15. Calouation of on-body stveamles

o .-

This procedure is repeated along the surface of the bodg until all of

the surface elements have been tested. The result is a streamline

composed of segments from one side of an element to another.

il i*j'"68

° .

.............................................

12.0 DEVELOPMENT OF HIGHER ORDER PANEL METHODS

,,",€_i.

3I The XYZ Potential Flow program assumes a constant element

source panel as described in Section 3.3. Extensive use of the constant

element source panel method has shown that the primary diasadvantage of

- the method is that, in order to obtain a highly accurate solution, a large

number of surface elements must be used to discretize the body surface.

* The method has been applied to problems of increasingly complex

configurations (Hess 1977). By doing so, the size of the coefficient

matrix is increased resulting in increased computer time and cost.

i* Additional cost is accrued due to the manhours required to prepare the

input. Therefore, while the constant element methods have proven to be

.- very successful, the cost has motivated the development of higher order

methods.

The higher order surface singularity methods discretize the body

surface with curved elements having a variable source density, as

. compared to the flat elements of constant source strength used in the

. basic method. Hess (1973) showed that the effect of a curved surface and

the effect of a variable source density are of the same order of

magnitude. Therefore, the two effects must be used together to provide a

"consistent" solution. The consistent higher order panel method provides

*" the increased accuracy and speed desired for three dimensional Neumann

• " - problems (Hess 1979).

U Accordingto Hess (1979), the evolution of the higher order panel

69

method from the constant element method involved the derivation of new

influence coefficients based on the integration of a variable source

Lo density over a curved element. Other portions of the method were

unchanged. However, the development of the higher order velocity

equations also required different programming logic.

In examining the potential for development of the higher order

methods, Hess (1979) noted that "a consistent approach always uses a

source polynomial one degree less than the panel polynomial." Through an

independent ef fort, Brebbia (1984) presented a higher order approach

using the direct method to solve for a surface potential polynomial

stating that the potential function must be of a degree at least equal to

the degree of the polynomial describing the element. Knowing that the

velocity function is the derivative of the potential function, these two

* observations agree. As a result of his derivations, Hess showed that thesolution of a f lat element with a constant source requires one integral, a

paraboloidal panel with a linearly varying source density requires six

* - integrals, arnd a cubic element with a quadratic source density requires

- twenty-three integrals. Development of higher order methods has

* focused on the paraboloidal element with the linearly varying surface, as

solutions of higher order than that offered little benefit for the amount

* of ef fort required to produce a working program (Hess 1979). Hess

(1979) and Eriksson (1983) have independently developed programs f or

* - three dimensional higher order panel methods. The higher order Hess

program evolved from the constant element program which he developed

* in the early 1960s, while Eriksson developed a new program based on the

70

- ----

.1:~W.7P va~' #-~ ~ NFN~rwyuw. -~ - L - - - LW VV FW Kr.V .

work of Johr.Jon and Rubbert (1975). Continued work in the near future is

expected to deal primarily with refinement of the paraboloidal element

with a linearly varying source (Eriksson 1983).

In order to alleviate the burden of preparing the input, a geometry

package for input data generation has been developed which is

incorporated into the Hess higher order panel program. This allows the

user to enter relatively few points to describe the body. The geometry

package enhances the surface representation by distributing additional

points on the surface based on one of many algorithms or recurring

geometries (Halsey 1978).

As the state of the art in fluid dynamics has progressed, the XYZ

Potential Flow program has seen increasingly complex applications

*requiring a great deaf of effort in preparing the input, and requiring long

* computer run times. Hess (1979) reported the use of the Hess constant

* element program for a configuration utilizing 7000 effective elements.

* Realizing that the computation time increases as the square of the number

* of elements, it is easy to see the motivation for developing the higher

order panel methods. Though modern computers off er storage capac it ies

which can handle most applications of the constant element method, the

higher order panel methods can provide equal accuracy f or much less user

* ~..effort. While the constant element method is still a versatile tool,

future generations of the surface singularity methods will be able to

* ~handle the more complex applications being demanded in fluid dynamics. *-

71

13.0 VELOCITY CALCULATIONS FOR A TRIAXIAL ELLIPSOID

As the only true body for which an analytical solution exists, a

triaxial ellipsoid was selected for the sample calculations in order to

compare calculated results with the analytical solution. Hess has made

use of the triaxial ellipsoid throughout his works in developing both the

constant element method and the higher order panel method. Therefore,

the XYZPF program will be compared with existing results of the Hess

method (Hess 1979).

The triaxial ellipsoid utilized for the calculations has semiaxes

A dimensions of Li 2. and 0.5 in the x, U, and z directions respectively. The -"-*

surface was discretized by selecting fixed intervals of 0.1 in the y

direction, and fourteen equal divisions of the 900 sector in the x-z plane.

The values of x and z were then solved in terms of y and an angle e. This

method yielded 280 elements in the first octant for a total of 2240

" effective elements after employing symmetry. A FORTRAN program was

* Pused to generate the corner points and to prepare the input file for later

use by the XYZPF program.

.7 Figures (16) and (17) show excellent correlation with the analytical

solution and little difference from the Hess solution using 4320 effective

* .. elements. The use of the centroid as the control point is an

approximation used to simplify the multipole expansion of the potential

7 about the origin of the local coordinate system. This approximation is

valid for most elements. However, for elements which are long and thin,

72F~.,

.. . . . . . .. . . . . . . . . .

. . . . . . . .. . . .... . . . . . . . . . . .

the physical difference between the centroid location and the null point

location is significant, and use of the centroid can produce significant

error as may be observed in figure (16) when the value of y approaches

2.0.

Recent calculations on the same body (Hess 1979) showed that

-- results of at least equal accuracy could be obtained using only 480

effective elements using the higher order panel method. These results are

a significant demonstration of the value of the higher order panel method.

Using the higher order panel method rather than the constant element

method, the user has the option of obtaining equal accuracy with cruder

discretization or higher accuracy for the same discretization effort.

While the results of the triaxial ellipsoid show relatively little

improvement in accuracy, the most significant advantages of the higher

, order panel method are evident for a body with concave regions (Hess

,, 1977).

I

73

F

]

2 8

ANALYTIC SOLUTIONIa HESS PROGRAM (4320 ELEMENTS)0 XYZPF PROGRAM (2240 ELEMENTS) I__________

2A

2.0

* 08

- 04 1

0 04 085 1.2 1.6 20

* Figure 16. Comparison of analy~tic and calculated velocity distributions on an ellipsoid with

ax-ts ritios: 1 :2:0 5. Velocities in the xz-plane. (from Hess and Smith 1962)

74

2.8

ANALYTIC SOLUTIONo HESS PROGRAM (4320 ELEMENTS)

* El XYZPF PROGRAM (2240OELEMENTS)

2.41

2.0

.0.2

V/a

*C 08VV6'

-A N

0.4

C) 0.2 04 06 Og .C

* rigure 1 7. Comparison of analytic and calculated velocity distributions on an ellipsoid with a

axes ratios 1:2:0.5. Velotities in the yz-plane. (from Hess arid Smith 1962)

75

[7

14.0 CONCLUSION AND REMARKS.

The objectives of this paper were (1) to describe the details of the

approximation of an arbitrary three-dimensional body using quadrilateral

elements, and (2) to provide a detailed derivation of the exact source

panel integrations. Both of these objectives were met.

The method of surface discretization and source panel geometry is

easily described using basic principles of geometry and vector algebra.

By using quadrilateral surface elements, many surfaces can be discretized

in a very straight forward logical fashion. The user can frequently

visualize the contour lines of the surface which may be used to form the

quadrilaterals, with some help from an intuitive approach to the fluid

dynamics problem. The method of forming the planar quadrilateral

element in the XYZPF Program differs slightly from the method presented

by Hess and Smith (1962). The differences lie in the formation of the

local coordinate system and the use of the centroid rather than the null

point as the control point for applying the boundary conditions. The

method of forming the local coordinate system has no effect on the

potential flow calculations as long as one of the coordinate vectors is

the outer normal to the planar element. The use of the centroid as the

control point is an approximation used to simplify the multipole

expansion of the potential about the origin of the local coordinate

.system. This approximation is valid for most elements. However, for

elements which are long and thin, the physical difference between the

centroid location and the null point location is significant, and use of the oi

76

....................................................

centroid can produce significant error as may be observed in figure (16)

when the value of y approaches 2.0.

A detailed derivation of the exact source panel integration has not

previously appeared in literature, though the results are summarized by

Hess and Smith (1962). The derivations presented in this paper verify the

equations presented by Hess and Smith (1962), and the equations used in

S- the XYZPF Program. The integral expressions for the velocity

components were evaluated exactly with no assumptions or

approximations used in the course of the integrations. Since the method

. of integration reduces the surface integral to a line integral around each

of the sides of the element, the integration method can be generalized for

•* a planar element with any number of sides, though the surface

discretization used in the XYZPF Program uses only quadrilateral

elements.

The calculation of potential flow about arbitrary three dimensional

bodies is an engineering tool which is basic to design involving fluid

dynamics. The XYZPF Program is a useful tool which has proven its value

over the past 14 years. However. the increasing demands placed on this

method are exposing the errors of the approximation as evident in the

sample calculations presented in this paper. The requirement for

increased accuracy has motivated the development of the higher order ."-.

.panel methods. Some of the limitations imposed on the XYZPF Program

were due to computer memory and speed limitations. Advancesin

computer performance may allow future investigators to eliminate some

77

. ..-. .

- . -. . . . .. . . . .. .. .. .1. . .. . . ... .. .A

of the simplifying approximations used in the XYZPF Program, allowing

increased accuracy without violating computer limitations. Some

modifications might include the use of the null point as the control point

rather than the panel centroid (as is used in the Hess program), or

extending the range in which the exact velocity calculations are

performed. The gains in accuracy by modifging the "constant element

*" method" are limited by the basic approximations of the planar element

and the constant source density for each element. Significant gains are

most evident in the higher-order panel methods. This author concurs with

Eriksson (1983) in expecting advances in the surface singularity methods

* focus on the "development and refinement" of the higher order panel

methods.

76

*. . . . . . . . . .. . . . .~,. 5. - . -

S * S S * 55 * . 5 . . .. . . . . .

.; -,..,

1. Brebbia, C. A., The Boundaru Element Method for Engineers, Pentech ".'-Press, London1984.

; -2. Brebbia, C. A., Telles, J. C. F., Wrobel, L. C., Boundary ElementTechniques Springer-Verlag Publishing Company, Berlin, 1984.

3. Dawson, C. W. and Dean, J. S., "The XYZ Potential Flow Program,"

Naval Ship Research and Development Center, Bethesda, MD., NSRDCReport 3892, June 1972.

4. Eriksson, L. E., "Development of a computer code for athree-dimensional higher-order panel method for subsonic potential

flow," Aeronautical Research Institute of Sweden, Stockholm, Report

No. FFA-138. Aug 1983.

5. Halsey, N. D.,"A Three-Dimensional Potential Flow Program with a

Geometry Package for Input Data Generation," NASACR 145311,March 1978.

6. Hardy, G.H., A Course of Pure Mathematics, MacMillan Company, NewYork, 1944.

7. Hess, J. L., "Higher-Order Numerical Solution of the Integral Equationfor the Two-Dimensional Neumann Problem," Computer Methods in

ADlied Mechanics and Enineering. Vol. 2, No. 1, February 1973.

8. Hess, J.L., "Review of integral equation techniques for solvingpotential flow problems with emphasis on the surface-sourcemethod," Computer Methods in ADDlied Mechanics and Engineering.Vol. 5, No. 2, March 1975, pp. 145-196.

9. Hess, J. L., "Status of a higher order panel method for non-liftingthree dimensional potential flow," McDonnell Douglas Corp., Long

Beach, CA., Report No. MDC J7714-01, Aug 1977.

10. Hess, J. L., "A higher order panel method for three dimensionalpotential flow," McDonnell Douglas Corp., Long Beach, CA., Report No.

MDC J8519, June 1979.

79

" " " % "" "% *" "'" ' ..... """ 1' " "n ' * '*""" - 5 ".. .." ' '. ."5 . ." " """ ""•*• S ' ' S-"' '. ".". ". " ; "' " ,

. ' :., . . " . ",'

x;-~~~. 7 W_-W7V-

11. Hess, J.L. and Smith, A.M.O., "Calculation of nonlifting potential flowabout arbitrary three dimensional bodies," Douglas Aircraft CompanyReport No. ES 40622, March 1962.

12. Hess, J.L. and Smith, A.M.O., "Calculation of nonlifting potential flow. about arbitrary three dimensional bodies," Journal of ShiD Research.

Vol. 8, No.2, p.22, September 1964.

* i13. Hess, J.L. and Smith, A.M.O., "Calculation of Potential Flow aboutArbitrary Bodies," Progress in Aeronautical Science. Vol. 8, 1966.

14. Hunt, B., "The panel method for subsonic aerodynamic flows: A surveyof mathematical formulations and numerical models and an outlineof the new British AerospaceScheme," Computational Fluid ** -

Dunamics. March 1978, von Karman Institute for Fluid Dynamics,Lecture Series 1978-4.

15. Jaswon, M. A.,"Integral Equation Methods in Potential Theory. I,"Proceedings of the Roual 5ocietu of London Series A, Vol. 275, No.

.. 1360, 20 August 1963.

16. Jaswon, M. A., Integral Equation Methods in Potential Theory andElastostatics, Academic Press, London, 1977.

I 17. Johnson, F.T. and Rubbert, P.E., "Advanced panel-type influence. :coefficient methods applied to subsonic flows," A]AAPaper No.

75-50, 1975.

18. Kellogg, O.D., Foundations of Potential Theory. Springer-Verlag -

Publishing Company, Berlin,1929.

19. Lamb, H., Hydrodynamics Cambridge University Press, London,1924.

20. Lefebvre, P. J., "Theoretical considerations and user's manual for a' modified XYZ Potential Flow program for calculating five

degrees-of-freedom velocity potentials," Naval Underwater Systems* .- Center, Newport, Rhode Island, Technical Memo No. 82-2076A, Nov.

1982.

s0 -. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21. Levy, R.H., "Preliminary report ona proposed method of calculatingpressure distributions over arbitrary bodies," Canadair LimitedReport No. RAA-00-113, October 1959.

22. MacMillan, W.D., The Theory of the Potential, McGraw Hill BookCompany, Inc., New York,1930.

23. Newman, J.N., Marine Hydrodynamics, MIT Press, Cambridge, 1977.

24. Noblesse, F. and Triantafyllou, G, "On the calculation of potentialflow about a body in an unbounded fluid," Massachusetts Institute ofTechnology, MIT Report No. 80-8, Dept. of OceanEngineering,Cambridge, MA, Sept 1980.

25. Ralston, A., A First Course in Numerical Analusis. McGraw-Hill BookCompany, New York, 1965.

26. Smith, A.M.O. and Pierce, J., "Exact Solution of the Neumann problem.Calculation of Plane and Axially Symmetric Flows about or within .-arbitrary boundaries," -Douglas Aircraft Company Report No. 26988,April 1958. (Summarized in the Proceedings of the Third U.S.National Congress of Applied Mechanics. 1958).

27. Symm, G. T.,"Integral Equation Methods in Potential Theory. I1,"Proceedings of the Royal Society of London Series A, Vol. 275, No.1360, 20 August 1963.

h.

. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

PageI.,,-

Appendix I - XYZPF Section PF1 ...... 83.......8Reads Input and Computes QuadrilateralParameters

Appendix 11 - XYZPF Section PF2 ...... 95Computes Influence Matrix Coefficients

Appendix III - XYZPF Section PF3 ......... 101 -.

- Solves Matrix Equation for Source Density

Appendix IV - XYZPF Section PF4 ... ...... 105Computes Velocities and Pressure Coefficientsfor On-Body Points

Appendix V - XYZPF Section PF5 ...... 10 9, Computes Velocities and Pressure Coefficients

for Off-Body Points

Appendix VI - XYZPF Section PF6 ......... 116Computes Velocities and Pressure Coefficientsfor Off-Body Streamlines

Appendix VII - XYZPF Section PF7 .......... 123Computes Velocities and Pressure Coefficientsfor On-Body Streamlines

Appendix Vill - Triaxial Ellipsoid Input File . 131

Appendix IX - Triaxial Ellipsoid Output File . 138

82

-..................................................... .

RPPEHDIX I - XYZPF SECTION PIFI

PROGRAM PFPI ( INPLIT= 128, OUTPUT= 128, TAPE5= INPUT, TAPE6=OUTPUT, TRFEO3,I TRPE3=TRPEO3, TAPE04, TAPE4=TAPE04, TRPE5O= 128)

C XYZ POTENTIAL FLOW PROGRAM VERSION 4 SECTION 1 A-C READS INPUT AND COMPUTES QUADRILATERAL PARAMETERScC FOR INFORMATION CONTACT

.,-C BILL CHENG OR JANET DEAN

C NUMERICAL FLUID DYNAMICS BRANCH CODE 1843 ,tC NAVAL SHIP RESEARCH AND DEVELOPMENT CENTERC BETHESDA, MARYLAND 20084

DIMENSION INDEX(,SG(%,tFgCZ~g) IF(9>,XP'9),YP(9),ZP(g)1,MSK(lOO), WS(240),PROB(15),Dr(5O)COMMON X(SOO ),Y(800 ),Z(800 ),ID(41,71),&(250),T(460)KP(lo0)

*i EQUIVALENCE (CZ(I>,F(1) >EQUIIVALENCE(WS(1),KP(1)),(WS(1O1),MSK( )),(WS(201),NF ),(WS(202' -.2 NSP),(WS(2O ),NEP),(WS(204),NSE),(WS(205),MIX),(WS(206),MIY),3 (WS(207),MIZ7w),(W-2),I-,:M) (WS(210) ISM", (S(211).K), (S(213),4 EPS),(WS(208), IPS ),(WS(212), IPF ),(WS(217),Xl ),WS(218) YJ), ..,5(WS(219), ZI )EQU I VALENCE (Y 12, Y23), (Y34, Y4 1)INTEGER FP1,P2,PS,P4,PC,PS,P6Cp7,P8 FgWRITE (6,5)

5 FORMRT(49HIXYZ POTENTIAL FLOW PROGRAM SECTION 1, VERSION 4 )10 FORMAT (11, 15A4)20 FORMAT (1X, 1514)50 FORMAT (1X,217,6E12.5)30 FORMAT (1X,5F12.9)

C A. READ IN CONTROL PARAMETERSWS(220 )=4.K1=O

I D1=O .-102=0

103=004=0 IN-IDS=O

* 1015=0

MAXN=70MRXM=40 .".MAXNQE=650 "C'MAxPC=800 ".CHTRL= 1EOF50=O.READ (5 ,IO)J, (PROB(I), I=1, 15)IF (EOF(5) EQ. 0.) GO TO §WRITE(5, 8)

8 FORMAT(3§HONO TITLE CARD FOUND - PROGRAM ABORTED )STOP

9 CONT I HUE* J=O

.'. =

, -" WRITE (6,10) J.,(PROB(I),I=1,15)SA=.O86=0.

. 21 FORMAT (17HONO. OF QUAS.A =,14 /17H NO. OF SECTIONS=, A'214/31H MAX. NO. OF ITERATIONS X FLOW 13,9H Y FLOW ,313,9H Z FLOW ,13)READ (5 11)NQE,NSE,MIXMIYMIZ IS1M,EPSIUC:T IPS, IPF, ISP

63rr

, *;. .- *v,- .... - --. I..- .4

. < '- * . . *. * . -*

- - <~-Z

1 ,IEDIT1,IEDITS,IEDIT4,ITRPE,XCENTER,YCENTER,ZCENTER11 FOAMAT(614,FB.5,8I4,1X,3F5.3>

IF CEOF(5) EQ. 0.) GO TO 19WRITE(6,18) '

18 FO~RlRT43HOIO PARAMETER CARD FOUND -PROGRAM ABORTED )STOP

19 CONTINUE '1IF CIEDITIEQ.I) ICNTRL=OWRITE (6,21) NQE,NSE,MIX ,MIY,MIZ

31 FORMAT ("CONVERGENCE CRITERIA",FS.5)41 FORMAT (4H-0 M,7X,2HX1,12X,2HX2, 12X,2HX3, 12X,2HX4,12X,2HXP,12X,I 2HXN,12X,1HA,13X,3HC24/4H N,7X,2HYI,12x,,2HY2, 12X,2HYS,12X,2HY4,2 12X,2HYP, 12X,2HYN, 12X,2HFL, 12X,3HC25/4H P,7X,2H1, 12Y, 2H22,12X,

42 2H23. 12X,2H24, 12X.2HZP, 12X,2HZN, 12X,4HCZ1 10X,3HCZ6/)42FORMAT (4H1 I1,7X,2HXI, 12X,2HX2, 12X,2HX3, 12X,2HX4, 12X,2HXP, 12X,1 2HX9, 12X, IHA, 13X,3HCZ4/4H N,7X,2HY1, 12X,2HY2, 12X,2HY3, 12X,2HY4,2 12X, 2HY', 124X,2HYN, 12X,2HFL, 12X,3HCZS/4H P,7X,2H21, 12X,2HZ2-, 12,3 2H23, 12X,2HZ4, 12X,2HZP, 12X,2HZN, 12X,4HCZ I,lo,3HCZ6/)

24 FORMAT (IM 1 I1, 19H PLANES OF SYMMETRY)240 WRITE (6,24) ISM2?0 WRITE (5,31) EPS2801- IF (IPS.LE.O) GO TO 290285 WRITE (6,36> IPS,IPF

36 FORMAT (45HONEW SOURCE DENS ITY TO BE COMPUTED FOR QUADS., 14, 3H-114)

290 K=OWRITE(6,39)ISP

39 FORMAT (9HO ISP =,13)

WRITE(e6,37) IEDITi. IEDITS, IEDIT4, ITAPES'? FORMAT (QHOIEDIT1 =, I3/9H IEDITS1 =, I3/91- IEDIT4 =,13/9H- ITAPE

1 3)W I TE(6, 38) XCENTER,YCENTER,ZCENTER

3 FORFiT (1 OHOXCENTER =,F5 .2/1iOH YCENTER = ,F5 .2/1 OH ZCENTER =, F5 .2)MM=OMN=OP=1

00 291 1=1,41DO 291 J=1, 71

p291 ID( I.JJ=O

C 8. READ FIRST PT.IERR=OIF (ITAPEEQI1) GO TO 292

2000 READ (5,40) XI ,YI .1 ,NI ,MI ,NS.,NE.,UNIF (,EOP(5).NE.O. OR. NS.LE.O) GO TO 2050WRITE(6, 45) IC:NTRL -,NS

45 FORMATU11,9H- SECTION ,14)LINE=OGO TO '29 3

2050 IF(IEF;R.EO.O) GO TO 2200WRITE(6,2 100)

2100 FORMAT(39HONO POINT CARDS FOUND - PROGRAM ABORTED )STOP

-2200 IERR =1 :

ITAPE=1WRITEk6, 2&OO)

2300 FORtIAT(47HOERROR I N I NPUT - PO INT CARDS NOT ON INPUT FILE ,IOX,

1 SS3HPROGRAM WI LL CHANGE I TAPE TO 1 AND TRY TO READ TAPE50 cUIF ( EOF(5).EQ.O ) WRITE(6,2400) XI ,Y I,.2ZI

* 2400 FOPMAT$ 1 HOEXTRA FLOW, 3F12.5, 5X, 2OHW ILL NOT BE COMPUTED)

.4-4

292 READ(50,40') XI,YI,ZI,NI,MI,NiS,NEIN

40 FORMAT (3F12.9,414,F12.9) NEOF5O=EOF(50) "S IF (EOF5O.NE.O. OR. 115LE.0) GO TO 2450

WRITE(6,45) ICNTRL,t1SL INE=O

GO TO 293 .2450 IF CIERREQO0) GO TO 2500

250LRITE(6,.2100) 2250IERR=l

ITAPE=O260WR ITE (6, 2600)

260 FORMAT("ERROR IN INPLIT - POINT CARDS NOT ON TAPE5O', lox,I"PROGRA1 WILL CHANGE ITAPE TO 0 AND TRY TO READ' INPLIT FILE")GO TO 2000

293 UNR=UNNSS=NSPC= IIF (NE .EQ.0) GO TO 2700

MMIA=NI

NM I N=Ml I

GO TO 300km 2700 MIN~=MI 1

NM IN=N I

00 TO 300295 IF (ITAPE.EO..l) GO TO 297

READ (5,40) xI,YI,ZI,Ni t,NS,ME,VN3 IF (EOF(5).EQ.O.) GO TO 299

YI=O.

GO TO 299297 READ(5O,40) Xi,YI,2I,NIMI,NS,ME,.N

I EOF5O=EOF(5O)IF (EOF5O .NE.O) 115=0

299 PC=PC+ 1- .IF (SNE.NSS-.) GO TO 330*300 IF (NE.EQO0) GO TO 304

301 1 WNlI

MI=IW4*C C. STORE PT. IN PT. ARRAY

304 IF (MAYPO. i-PC) 295, 305,310305 WRITE(6,306) NS,MI,NI306 FORI1AT(bOH ERROR I N INPLUT - THERE ARE TOO MAINY DATA POI NTS I N SEC'

*ITION ,14,30H - POINTS BEGINNING WITH Mi =,14,5H N = 14,2 17H WILL BE IGNORED)LlNE=LINE+1104=104+1

' GO TO 295

Y(PC>=YIZ(PC)=2IIF (MI .LE.MAtM AND. NI.LE.MAXN) GO TO 315WRITE(6.311) MI,NI

S 65

;7'C 77vr"rr"Ir"Tr7

C~ LlNE=LlINE+ I -311 FORMAT(38H ERROR IN INPUT -INVALID tl,N INDICES I1OX,

1 14HPOINT WITH M 14,5H N = 14,17K WILL BE IGNORED)105=105+1PC='C-1GO TO 295

:315 10(MI,NI>=PCrIMAX=rIAXO(rMARX,rlI)PMIN=MINOa(rIIN,MI)NMAX-MAXO(IIAX, NI>NIIN=1INO(NIIIN,NI )GO TO 295

330 IF (IEDIT1.EQ.1) GO TO 294IF (LINE.LT.40) GO TO 333

p WRITE(6,42)LINE=OGO TO 294 '

*333 WRITE(6,41)294 CONTINUE

C E. 00 LOOPS TO SWEEP PT. ARRAYN1=NJ1IN

* rlri2=r 1AX-lMr1* NMN2=NIIRX-NMIN

IF ( IIOL(M,2).ECI.O AND. rlOD(NN2,2).EO.O) GO TO32LJRITE(EJ,321) NSS, MI'lIN, MItFIX, NrlI N, NrIXLINE-LINEti

331 FORNAT( J6HOERROR -SECT ION 1, ,45H DOES NOT HAVE QLUADS ARRaiNGED- IN1 BLOCKS OF 4 ,9H MM I N= ,12, 6k MIIAX= ,12, 6k NN IN= ,1 225H NMIRX= .12)

106=I06+1332 MM2=Iir112 /2

r NN2/200 404 NN=1.NN2

IM I 1=tlp I NDO 402 M'l= 1, MM2NQ= 1

C F. HAVE 9 CORNER PTS. BEEN G IVkENI T= IED(K i1, N1 )* IEI011+ 1 ,Ni1 )* I DQI1+2,Nl 1)1I0(M1 .. N1+1 1) I(t 1+ 1, Ni1+i

1 I D(M1+1, N1+2)ID(M 1, ,N 1+2)* 1IM 1+ 1 , NI1+2)-+I1D(M1+2, N1+2)IF( fT.EO.O ) GO TO 402'IERR=OM2=rl1+1ti2=NI+1I =M t, M2DO 400 N=N 1,N2GO TO (3,333,3)NQI

334 PI-ID(M ,N)P2=ID(Ml+1,N)Pl3=I0(M4-l,N+1)P4=10M1 ,N+l)P5=iDk'1P2, N >P5=IEI<M+2,N+1)

* . P7=ID(M.1,N+'2)Po'=10h ,N+2)P9=P1

* - IF((XrP).NE.X(P2).ORY(P1).NE.Y(P2).OR.Z(PI).NE.Z(P2.> AND.I (X(PI).IE.X(P4).WYPI).NE.Y(P4YOR PI. -P).NE.Z (F'4))) GO TO 340

P9=) D1+2 N+2)GO TO 340

335 P1=10(11 N+1)

P3,=I0(M+1,N)

P4=10D(Mt 1.1+ 1)P5=10(11 H-i>

* P6=10(M14-1-)P7=10-1+2,19 )P8=IO(1+2,I-i-)pg=pjIFX(X(PI)A-1E.X(P2).OP.V(P'I>.IE.Y(P2).OR.Z(P1 ).tE.Z(P2)> AND.1 (X(P1).ME.X(P4).OR.Y(P1).HE.V(P4).OR.Z(PI).NE.Z7(P4))) 030 TO 340

0O TO 340336 P1=ID(r+1N)

P2=10(l-'-,19--)P3=fD(M N1+1)P4=10(t1 AINPS=I0(M-i1,19-i2)Pb=10(11 Nf+2)P7= 10(M--1,19-i-)PB=IDU1-1,Hpg=plIF(X(Pl.E.X(P2).OP,.YPl.NE.Y'.uP2 .OP,.z(F1xHtE.z 7P2') RHO.I (.X(FI).19E.X(F-P4).OF.Y(P1).IIE.Y(F-4).OP,.Z(F-).NE3(F-4))) 0O TO 34C0

*Pg=10(11 J- 1, N1+2>GO TO 340

'33 P 1= I DQi+1, + 1P',= 0I ,N+I)

b6 P3=1001 ,r-N )P4=I00-14IA )P5= I DM- I 19N+1)

P6= I D "-1 N19 )F-7= I0 D<M 19N-1)

F*(X(Fl ),tEX('P2).ORP.Y(%PI )JiE.Y'(F2) . R.2(PI) .. E7P4)), AND.I- 4

34 0 I P(I1)=PI1

I P(3 )=P.S -

I P'(4' )=PF4p1 IP(5)=P55

I P (7 )=P7IP(3)=PS

C 02 COMPUTE MOW1AL VECTOR (XII. '91, 219)1=X (P3 S-X( P I)

Xz-Y (P4)-X)(P2)YJ-=Y(P3 )-Y(P1I

-P4)-Z('P)

X''"I + , M-22X4Z

219 2*YI 1-X 14-Y2R=ES02(-XH, YN,ZH)I IF (PO T. .00000000001) 00 TO 3?45LIPITE(6,343)

343'- FORMAT (33H ERROR 119 IN1PUT -ZERO AREA OUROLINE=LINE+1101=101+1

* 67

7 D-AI68 L67 FORMULTION OF NUMERICL METHODS USED IN THE

XYZ 2 48THREE-DINENSIONAL. POTENT..(U) TEXAS A AND M UNIY

.7 COLLEGE STATION COLL OF ENGINEERING W J BERRY NAY 86UNCLASSIFIED N@0229-05-0-3393 F/0 20/4 NL.

mhmhhhhh

4.4

IIII/4 .

-,.4 1 ~4~~

4.-.:....*

_ . j. +L-,_-. , .. ." - . .- .. - , .,.] + ,,+ + - -. .. . ", .+ ., . ,. ., -. .- ,. .,.. .--° , , ... .. . , . . ., -. . , ,- , ,

,, -,- , -. , - ,.. .. -,,, . .-. .,.;. .,.,. .> . . '....-,-. . .'< .,--.--.-. -.-u..u.-.-i . °1-..,3. ,6.

- C C'C' ' '' C ~''1N~.~ t~VW ,W ~ r C' tX~'V ~ LV VW ir LU . 7w L t WL-W 'lVW L .- UN, Ir I. I

xC=O.YC=O. -

pE FL=O.CZ )OCZ (4)=O.

I ERR= 100 TO 351

345 CONT INUEXN=XN /PYN=VN /F;ZN=,ZN /R

all 80O=. 5*RP*C COMPUTE CENTROID

XI=X(P3)-X(P2)* Y1=Y(P3)-Y(P2)

* . 2=Z(P-Z(P2)X5=Y 1*Z2-Y2*Z 1

Y5=2l*X2-22*X 1

A1=S02(XS,YS,Z5>82=Ri-81IT=1XC=(X(P2 ")+X(P4 >+(A 1*x(P3 >+A2*X(P I ))/R)/3,

.(C= (Z(P2 )+Z.(F4 )+ (A I*Z (P3 )+A'2*-Z(P 1) )/PR )/3.C COMPUTE SECOND AND TNHIRD VECTORS

945 X4-YN*Z 1-Y 1*ZNY4=ZN*X1J-Z 1*XN4=XN 1-X 1 *"1NS A=1./SQ2(X4, Y4 ,Z4)

Y4=Y4*fl

Y4=Y4*A24=Z4*RX3=ZM*Y4-Z4*YHY3=XN*24-X4*ZNZ3-,=YIelN*4 4T4*XN

* C COMPUTE POI NTS !N QUAD SYSTEMDO 947 1=1.9gL=IP(l )XP I )X3*(X(L )-XC:)+Y34 (YL)YC )CSZ3* 'Z (L )-ZC>YeP( I)= t('L)X)+4(()-C)4(Z1)-)

947 ZP (I )=XM* X L)-XC:)+YN* (V( L)-YC )tZN* 2 (L )-Z C)COMPUTE M1ATR IX COEF. TO F IND SURFACE EQ.

DO 949 1 =2, 9

O(1,2)=XP(l )V l, 3 >=YP(l )GD1,4)=XP<l)**2

G r I , E,'=YP'( I )*'YP( I949 Fr 1 ZP( I)

DO 95E3 1=1,6 -4

r-(1>I)=G(5, I )+Q(5,I)9& 5.t(E, 1 )=0(7, I >tOi(8, I)

F( 1)=F(9) -

F(6>=F(7)+F(8)

66-A

SC SOLVE MATRIX EQ. G*CZ=F FOR C2CALL MATINS(G,9,6,F,6, 1,OETERM, 1DM, INDEX)IF (IDM.EQ. 1) GO TO (955,960) ITIERR=1WRITE(5, 954)

954 FORMAT (33H ERROR IN INPUT -SINGULAR MATRIX )LINE=LINE+1102=102+1GO TO 060A

A955 IT=2-%C FIND NEW NORMAL VJECTOR

VN=VN-CZ(2 )*Y3-CZ(3 )*X4XN=XN-CZ(2 )*XS-CZ(3 )*X4ZN=ZN-CZ(2 )*Z3-CZ(3 )*Z4Ai1./S.2(XN,VN,ZN)

YN=YN*A2Ii=ZN*AGO TO 945

C STORE DATA960 8(J+1)=XP(1)

8(J+3)=XP(2)8(3+4 )YP(2)B(J+5 )=XP(3)

8(3+? )=YP(4)

8(3.S )=XS a8(3+9 )=Y38(J+ 10 )=238(3+11 )=X4E(3+ 12 )=148(3J+13)=Z48(J+ 14 )=CZ( 1)8(3+15 )=CZ(4)9(3+ 16)=CZ(5)8(J+ 17 >=CZ(6)IF (K .LT. ?*MRXrifE) GO TO 965ID7=ID7+1K1=K+K1K=O

§65 CONTI NU ET (K+ 1 )=CT(K+2) a..T(K+3)= cCT(K+4) ='tT(K+5)=YNT(K+ti)=ZT(K+?)=AC,

C COMPLUTE QUADRUPOLE MOMENTSXl 11=XP ( 1)+APk2)X12=XP( 1)+XP(4)X I 3=XP(3)tXF(2)X14=XP(3)+X.P(4)XIS=XP(2)+XP(<4)---VI ThYP( 1)+'TP(2)Y12=YP( I)+YP(4)Y13=YP(3)+YP(2)Y14=YP (3)+YP(4)VI 5=YP(2 )+%'P(4)R1=A1/24.

r 89

R2=112/24.R3-AQ / 12. IL

AXX=(X I **2+X I2**)*R 1+(X f3**2+X J4**2 )*R2+XI 5**2*R3 ,AXY=(Xl 1*YI 1+X12*YI2)*R1+(X13*Y134X14*Y14)*R2+XI5*YI5*R3AYY=(YI **2+Y12**2 )*R 1+(YI 3**2+YI 4*.2 )*R2+YI5**2*R3

C COMPUTE SOLID ANGLEXX=XC-XCENTERYY=YC-YCENTE.ZZ=20-ZCENTERX 1-XX*X3+YY .Y3tZZ4.Z3'/=XX*X4+YY*Y4+ZZ*Z4

*Z 1=XX*XNYY*YN+ZZ*ZNRD=I. /SQ2(X1,Y1,Z1)

q4 RCU=RD*4.3RSV=RCLI**2*RDSA=SA+Z 1*(AQ*P.CU-( (AXX*(Y 1**2+Z 1**2-4.*X 1**2)1 +AYY*(X1**2+Z1**2-4.*Y1**2))*1.5-15.*XY*PAY)*RS')B(Jt 18 >AXX8J+ 19 )=AXY8(J.20 )=AYY

C ERROR TESTSE'1=52((XP(3)-XP()),(YP(3)-YP(1)),O. )D2=8Q2((XP(4)-XP(2)>, (YP(4)-YP(2)),O. )FL=.5*AIAX1(D1,D2)CZ23=ABS( 02(2) )+ RBS( 02Z(3))IF( ABS(C(2))+ABS(C(3)) .GT. FL*.O01 ) GO TO 970IF ( ABS( 02(1)) .LT. FL*.3 ) GO TO 977

970 WRITE(6,975> C223975 FORMART (29H QUEST IONABLE POINT -POOR F IT 6E 14.3)

I ERR=1ILINE=LINE+1

977 IF (YP(1)*XP(2)-YP(2)*XP(1) GE. 0. -AND.1 YP(2)*XP(3)-YVP(3)*XP(2) -GE. 0. .RND.2 YP(3>*XP(4>-YP(4)*XP(3) GE. 0.. AND.

3 i(4 )*XP( I)-YP(1I )XP(4). GE. 0. ) GO TO 984980 WRITE(6, 1000) CXP(I ),YP(I ), 1=1,4)1000 FORtIAT(4 1H ERROR IN INPUT - CROSSED OR CONCAVE QUAD

1 4(2F10.S ,3X))IERR=1LINE=LINE+I103=103+1

964 CP0CF=S02.((XP "(2)-XP(1)),(YP(2)-YP(1)>,,O)+X-i*c3;.-xP(2) 6Q2((X,.P(1)-I XPi 4 ).(YF( 1 )-YF(*4') ), 0. )+S Q2 (X, ( ')- ( ,(P.4)Y( 0.)

IF ( 36.*RQ .GT. 0R0F**Z2. > GO TO 986LINE=LINEtIWR I TE 6, 95 1)

961 FORMAT (24H WARN I NG LONG THIN QUAD.)9$6 IF ( 21I GE. 0. ) GO TO 351 -

350 WRITE (6,25).85 FQP.MAT f~- O45 IFS ONABLE PI NT -I NUARD NORMAL)

IJEPP,=1.9. LINE=LINE+1 l

C J. ED! TE QUAD I NrORMAT I ON351 IF OIEDIT1.EQ.2.AND. IERF:.EQ 0) GO TO 354

IF ( IEDITI EQ. 1 ) GO TO 354GO TO (356,357,358,359) NO

356 WRIE(6,51) M,Y(P1),X(P24),X(F'3),X(P4),C.N,AIQ ,CZ(4),I N.Y(P1>,Y(P2),Y(P3),Y(P4),YC,YN,FL 0Z(5),

2 P',2(PI),Z(P2),2(P3),Z(P4),ZO,ZN,CZ(1),02(6)GO TO 360

357 WRITE(6,51) M,X(F2),X(PS),X(P4>,X(PI),XC,XN,AIQ ,CZ(4),

r 90

* :.'~..----...t-Q-.xt~--JK'.-.-vCk--jK-xv~... J.'~-> *y*:-.-

go.~~~~ 1r F I;I

1N,V(P2),Y(PS>,Y(P4), Y(PI1), YC, YN, FL ,CZ(5)2 P,Z(P2),Z(P3),Z(.P4),Z(P1 ),ZC,ZN,CZ(1I),CZ(6) popGO TO 360

358 WRITE(6,51) M, X(P4 ), X(P1I), X(P2 ), X(P3 ), XC, XN, AQ ,CZ(4)1 N,Y(F4),Y(P1),(P2)Y(P3).YC.YNFL ,CZ(5I)2 P,Z(P4),Z(P),Z(P2),Z(P3),ZC,ZN,CZdI),CY-.6)GO TO 360

359 WRITE(5,51) i,X(P3),X(P4),X(P1),X(P2),XC,XN,AQ ,CZ(4)11,Y(P3),Y(P4),Y(PI).,Y(P2),YC.YN,FL ,CZ(5)

2 P,Z(P3), Z(P4 ),Z(PI),Z(P2),ZC,ZN,CZ(I),CZ(6)360 CONTINUE51 FORMAT (11- 13,8E14.5/lX,13,8E14.5/1X,13,8E14.5/)

LINE=LINE+4p ~IF (LINE.LT.50) GO TO 354 --

352 WRITE (6,42)LINE=O

.354 CONT INUEJ=J+20I=P

DM( I)=UNPPF+ 1NO=NQ+ 1V.IEPR=O 4--

349 K=K+7C K. WRITE OUT BLOCK OF 6 RRAY IF FULL

IF (J.LT.240) 00 TO 40055 WR IT E (04 ) Q, ( B( I), I= 1,240)

cI=PJ=0

C L. END OF DO LOOP O('ER PT. ARRAY400 CONT INLUE4 02 !MI11i+2404 N1I=N 1+2

C M. SET FOR NEXT SECT I ONNSS=NS0Do 40'5 t=MMIN,tMAl~XDO' 405 N=til! N,Nt1Xf

*IF MHE EC-O) GO TO 410MMRIX=NlI

tilI N~iINM IN=t1 I

V3 7CO 420410 M~IR,"=M I

- MrIN=tlIK:N NIIN=N IL N!IRX=N I

420 NE=fIE12,L'RIF=(N

IF (NS.LE.0) SC' TO 500C WRITE(6,45) ICrITP.LNS

L INE=OGO TO 300

500' UPI TE ('4 ) QE),(I I= ,24C)550 NP='.. K+1)fl

ISH slp Ism + 1GO TO (5g5,5gC',580,510),stliP

570 SA=SA+SA580 SA=SA+SA

91

5g0 SR=SR+SR" T,

C 01 WRITE PARAMETERS AND T ARRAY ON TAPE%', ,595 J=1

IF (ITAPE.EQ.1 AND. EOF50 .NE.O ) GO TO 601597 WS(J) = XI

WS(J+20) = YlWS(J+40) - Zl

_ ~~J = J+l ,-IF(XI**2 + YI**2 + ZI**2) 599,599,59k'

598 WRITE(6,600) XI,YI.,ZI.'""1',"600 FORMAT( 11HOEXTRR FLOW, 10X,3F12.5) ,,

601 READ(5,40) XIYIZIIF (EOF(5) .EQ. 0) G TO 597@ xi=o." I=O.

ZI=O. . *%

GO TO 597599 IF (ISP.LT.O) GO TO 605

WRITE (03) (PROB(I),1=1,15)WRITE (03) (WS(I),1=1,220),IEDITS IEDIT4WRITE (03) (T(I),I=1,IK)WRITE (03) (DM(I),I=1,NP)

605 CONTINUEC Ni CHECK SOLID ANGLE

5-10 WRFITE (6,9 ) SR80 FORMAT(14HOSOLID ANGLE = ,F8.3)

ml 520 REWIND 04REWIND 03

622 IDS=ID1+ID2+ID3+ID4+ID5tlID+ID7IF (IDS.EQ.O AND. NP.EQ.NQE) GO TO 638WRITE(6,625) C;IF (ID! .GT.O) WPITE(, 628) 101IF (102 GT.O) WRITE(6,629) 102IF (I3 .GT.O) WRITE(6,630) ID3IF (ID4 GT. 0) WRITE(6,631) ID4IF (ID5 GT. 0) WRITE(6,632) ID5IF (ID .GT. 0) WRITE(t'.533) ID"IF (ID? GT. 0) WRITE(6,634) NP,MAXNQEIF (NP.NE.NQE) WRITE(6,637) NP,NQE

* STOP625 FOR1AT(38HOFATRL ERROR IN DATA - PROGRAM ABORTED)528 FORMAT(IHO,15,31H QUADRILATERALS WITH ZERO AREA629 FORMAT(1HO, 15,44H QUADRILATERALS GENERATE A SINGULAR MATRIX )630 FORMAT( 1HO, 15,26H CROSSED QUADRILATERALS )1631 FORMAT(1HO,15,32H SECTIONS HAVE TOO MANY POINTS )632 FORRT(1HO, 15,34H POINTS HAVE INVALID M.N INDICES )633 FORMAT( 1H0, 15,52H SECT IONS 00 NOT HAVE QLIADS ARRANGED IN GROUPS 0

IF4 )634 FORMAT,'(HO,IS,48H QUADRILATERALS GIUEN, EXCEEDING THE LIMIT OF

1 14)- 637 FORMAT( 1KO, 15,28H QUADR ILATERALS GIVE0EN, NOT 14)

638 IF (ISP.LE.O) GO TO 640WRITE(6,639) ISP

639 FORMAT(7HO ISP= ,14, 19H - PROGRAM ABORTED )STOP

C640 CONTINUE02 READ PEPS2 AND TRANSFER TO IT

STOP 1END

FUNCTION SQ2(X,Y,Z)

92

.:cTtt&.**---*- *...* *k . ; I -;.s- ,. y-d..h

lilC COMPUTE SQUAR ROOT OF R**2

R- ABS(X)+ABS(Y) +ABS(Z) +.0000000000001700 RS=X**2+Y**2 +Z**2

R=R+RS/RR= 25*R+RS/RR=R+RS/RSQ2=. 25*R+RS/RRETURNEND

CSUBROUTINE MATINS(ANR,N1,B,NCMI1,DETE;r, I D, INDEX)

C MATRIX INVERSION WITH ACCOMPANYING SOLUTION OF SIMUL. EQ.C PIVOT METHODC FORTRAN I V S I NGLE PREC I S I ON W I TH ADJUSTABLE D I MENS i ONC NOVEMBER 1971 S GOOD NAVAL SHIP R & D CENTERC WHERE CALLING PROGRAM MUST INCLUDEC DIMENSION A(NR,NR)., B(NRNC), INDEX(NR,3)C WHERE NR,NC ARE DIMENSIONS OF A,B, INDEX

- C NI IS THE ORDER OF AC Ml IS THE NUMBER OF COLUMN VECTORS IN B (MAY BE 0)

.*.. C DETERM WILL CONTA I N DETEF;M INANT ON EX I TC ID WILL BE SET BY ROUTINE TO 2 IF MATRIX A ISC S I NGULRRA, I I F I NV.ERS I ON WAS SUCCESSFULC MATRIX A (INPUT MATRIX) WILL BE REPLACED BY A INV

SC MATRIX B: THE COLUMN VECTORS WILL BE REPLACEDC BY CORRESPONDING SOLUTION VECTORSC INDEX' WORKING STORAGE ARRAYC IF IT IS DESIRED TO SCALE., THE DETERMINANT CARD 29 MAY BEC DELETED AND DETERM PRESET BEFORE ENTERING THE ROUTINEC

DIMENSION A(NR,NR;), B(NR,NC), INDE(NR,3)EQUIVALENCE (IROL-,JROU), (ICOLUM,JCOLUM,, (RrlX, T, SWNP.

CC INITIALIZATIONC

N=N-M=M1DETERM=1 .000 20 J=I,N

U C20 INDEX"J,3)=O

DO 550 I-1, NCC SEARCH FOR PIVOT ELEMENT

Atloy. = 0. 0SLIDO 105 J=1.N

IF(.INDEX(J,3)-l) 60, 105, 5060 DO 100 K=1,N

IF(INDEX(K,3)-l) 80, 100. 71580 IF ( AriA)"-ABS (A(,.I,K))) 85, 100, 10085 IROW = J 46

ICOLUM = KAMAX = ASS (A(J, K))

100 CONTINUE105 CONTI NUE

INDEXICOLUI1,3) = INDEX(ICOLUM,3) + IINDEY(t, 1) = IROWINDE.(1,2) = ICOLUM

C INTERCHANOE ROWS TO PUT P I'OT ELEMENT ON DIAGONAL

9 317

IF~ (IROW-ICOLUI) 140, 310, 14010DETERM= -DETERI00 200 L=1,NSWJAP= A(IROW,L)A( !R0W,L)=R( ICOLUr,L)

200 AUICOLUI'1L)=SWAPIF(M) 310, 310, 210

210 DO 250 L=1,MSWAP-B(fAOW,L)B(IROLJ,L)=B(ICOLII,L)

250 8IC0LUr1,L=SIAPCC DIVIDE PlUOT POW BY PIVOT ELEM1ENTC310 P IVOT = A(rICOLUiT, I COLUI)

DETERM=DETERI*P I UT330 A(I COLUrI, I COLLIF) = 1.0

00 350 L=1,N350 R( IC0LUL),=R( IC0LUM,L)/FI(V0T

IF (M1) 380, 380, 3503'60 DO 370 L= 1, M370 P'ICCLLWIL)=8(ICOLLWL)/PIV0T q--

C REDUCE NON-F I OT R.OWS -

cKO: DO 550 L1=1,N

IF(LI-ICLLIM1) 400, 550, 400400 7=8 eL 1. iCCLUM1

-L 1, COLLUM)=0. 0D0 450 L=1.N

4 50 Fl(LI, L)=A (L1, L )- AKI COL UM, L)Tu 460IF (N." 550, 55C, 4604000 500 L=1,Ii

500 B(L1,L)=B(L1,L)-(ICOLUII,L)*T550 CO0NT INUE

CI NTERCHANGE COLUMINSC

00 710 I=1,tiL=N+l1IF 1 ?iEyL, 1)-I NOEY(fL,2 ) 6 30, 710.,3

620 JRC'14 INDEX' ( L, 1)JCOLU, 1= I NDEX (L, 2DO' -705 K=1, NSWAP = A (K, JP OW.-A(,K, IFIOW )=R(K( JCC'LLWIA K, JCOLU1 )=0WAF

705 CONT INUE710 CLINT f NIJE

LC "0 K=1 , N-I FI(c fNODY:f 3)-I 7 15, 720, 7 15

72C1 CC'N!T INLE730 CONT I NUE

810 R.ETLUFRN715 10=2

00 TO 810END

94

l~r K- 1T k rWmv-rnsr w- - r qw .r -v -

APPENDIX 11 - XYZPF SIECTION1 PJF2

PROGRAM PF2(OUTPLJTh128, TAPE&=OU- TPUT,TRPE03, ThPE02, TRPEo',

2 TAPE3=TAPEO3, TAPE2=TAPEO2, TAPE8=TAPEOS, TAPE9=TAPEO9)

C XYZ POTENTIAL FLOW PROGRAM VERSION 4 SECTION 2pE R S CE100,C 0t2

1 ,(WS(212),IPF) (145(209), I P (MKINTEGER SYMBLK=1 .0I 1W=0READ(O?) (PRC'B(l ), 1=1, 15)UP ITE(6 ,5)

*-5 FO-RMiAT(l49H0X'*YZ POTENT IAL FLOW PROGRAM SEC:T ION 2, VERSION 4)90 FOFRIRTk IHO, 15A4)

WRITE (6 A0) PFRODB("I )., I=1, 15)C A. READ PARAMETERS, T ARRAY, FIRST BLOCK OF B ARRAY

READ(OS) W I (), 1,220)h READ(03) (T(I ) I-I ,M1-K> c

READ(04) (B(I), 1=1,241)C B. START LOOP OVER OLIADRALATERALS

P~ 1[C0= 1

20IF (BER )-P >5951,295 ,5959FORMAT('POINTS OUT OF ORDER B(1)=" 1F4

295=2(J2

Y3=T(IJ+4I XZN=T(V-'-)

P A=T(K-'-f A

YY=B'E-1C.Q+

ZY=P(,1+ 12)c. C I CO;W'UTE LENGTH OF S I DES OF QUAD

023=S02FG,XSX.Y2.YS, 0..0)

034=SQ2F,'X),X4,Y3SY4, .0,.0)D41=502F(X4,X1,Y4,YI,.0, .0)

C C2 COMPUTE SLOPE OF SIDESIF(X2-X3)305.,300,305

300 Ct 2=1I.GO TO 310

305 CM2S=(Y2-Y3 )/(X2-X3)C123=0.

310 IF(X3-X4)315,311,315311 C134=1.

GO TO 320315 Ct34=(Y4-YS3)/(X4-XS-)

C134=0.p320 IF(X4-X1)r25,32 1,325

32 1 C141=1.GO TO 330

325 CM4 1=(Y 1-Y4)/'(X 1-X4)C 141=0.

330 I F("). I-2 )33 5,331 ,335331 C1 12=1.

0O TO 340335 CM 12=(V2-Y 1)/0(X2-X1)

C1 12=0.C CS"E COMPUTE QLIAOFAPOLE MOM"ENTS

340 CIXX=E'(J+17)CI XY=B(J+ IS)C IIYY=B (J+ 19)CY 12=0.0CX 12=0.00Y23=C'. 0CX23=0.OCY34=O.OCX34=O.IJCY4 1= .0

C C4 COM -PUTE S IN AND COS OF SLOPE ANGLE FOR EACH SI DEIF (1)94 ,3294

g..4 CY12=(Y2 Yl'D[3 -

CX123=(0Y12 iD 1

IF',04 4 3, 492- Y2 19349

C234=(X3-X4? '39344 1IF ([14 1 "6 "9'34'_

9S47 CY4 1=' Y' 1-Y4 ) *41CX41 (X4-X 1)/E1

*C C5 COMPIJTE Mciy LENGTH OF OURD

3T2=SC!2F ( X2, X4, Y2',Y4, C0, CtST=AMAX 1 (ST*, ST2, ,E12,023.3,0D41)

C 0. STARpT LOCIF' C" 'Er NULL FTSc342 KO=1

L= 1343 1I=K0

I F( I PS gO9260,g9350- * ~9350 I F(.L- IPS) % 2~i*.9352 1IF,(L- [PF ) 36C5, 936.0, 9$55-

9355 CI 1! 0.

CS(,Jc)= 0)

96

GO rO 5419360 9S=1

4 XCQ=T( I6 .YCO=T(1+ 1) .

ZCC!=T( i+2) VXNO=T( 1+3) SS YNQ=T( 1+4)

34ZI9Q=T(I+5)C E. COMPUTE DISTANCE BETWEEN QUAD AND NULL PT.C. DETERMIN METHOD

345~' RQ$F(XC, XCQ, YC, YCO,., 0,201 F(RPO-ST*4 )350, 3150,460'

350 X= (XCQ-XC )*XX+(YCQ-YC )*YX (ZC0-ZC )*j'.YIF Y=CXCQ-XC )*XY4.(YCQ-YC )*YY+(2CQ-ZC )*Z7Y

Z= ( XCO-XC >*XN+ (YCQ-YC )*Yri+ (ZCO-2 Z)*Zfl

IF(RPQ-ST*2.O)35!5 355,400C F. COMPUTE UELOCITY COEF. BY EXACT METHOD

355 R1=802F(X,X1,Y,Y1,2,0.)R2=SQ2F(X, X2, Y,Y2,Z,0. )RS=SQ2F(X,X3;,Y,YS,2,0O.)R.4=SC!Q2R (Y.,Y4.,Y. y4, Z,0F ((F:1+R2) LE. 012) 00 TO 1000IF ((R2+R3) .LE. 023) 00 TO 1000IF ((P2j+P4) LE. 034) G0 TO 10000IF ((P,4+R.1) .LE. 041) GO TO 1000

I di CLFI1=ALOG (AR1 +R2-D 12 )/(R I±P2+0i12)S ~~~~CLR2*ALO3,P+JD3)/R+.t2)

CLRS=FILOG( 'P3+R4-D341)/(RF4+D3[4l))

TUX=CY I ThCLA 1 +CY23,*CLA2+CY34*CLA3l:+.Y4 1I:CLFA4TUlY=CX 1 2*CLA 1 +CX.23*:CLA2.,CX34*,CLA3-"+*X4 14CLAi4

36lF(ABS(;,Z/lST)-.O1O) 375,361,361362SQ=2**2

E I .Zcr:n+ (X-X 1IE'S'O+(X X22

E4=ZSu+(X-X4 )*:+:2H I =(v-Y X1)X(1X)

*H2 (YY2dl)*(X-X2 )HS"VY3)147X-x3)

H:3=,:Y 94*(x>)

;F(c: 12 25 5* 363 ~lJ=(CMt12*E-H)/(2-*F;1)

V1$2-(Cl 12-tE-H-2 )(Zt ;2)AT 1=A~44 1)AT2=ATArI(32)TUZ=RT 1-RT2

364 IF(0123)355.266,357366 ATS=ATAI( (Cr123E2-2)/V(2*R2))

AT4=ATF~VC13 4 SH~: 'p)TVZ=TJJZ+AT3-AT4

316 7 IF(0C134 )se35 2)FO, t3b8 A T=A T AN rs W-r14:+:E3 H3 )/Z:+R

AT6=ATAN( (CrI34A:E4-H4 )/(Zt+rR4))T Tk.Z+2ATS-ATb

3659 IF(C141)370.,3?O.3?5370) RT7?ATAti( (Cr14 1*E4-H4 )/(2*R4)

AT8=RT~RN((CM4i*E 1-Hi )/(Z*F; I))

37 5 00 TO 450*C G. COMIPUTE UELOI TY COEF. BY QUADRAF":'LE tME-THC;L

97

40RPQ7=(RPQ3**2 )*RPO1431= X/RPQSXSQ=X**2V$OY**2ZSC!=Z-**2PS=YSQ±ZSQi-4.*c;QS=XSQ+ZSQ-4.*YS;WJS-X* ( 9 *PS:+30 *XSQ)/PPQ7LJS3=3. *Y*PS/RPQ71434=3. *X*-O$/RPQ7-4TVX=ARLJS -Cl XY*14S3-C IXX*t4S2-C i YflS4 h

W4$ =Y/RPQ3*W32-=V* (9 tO$Q3', 0 *YOf) ) /PPQ?

Tk.Y=AflJS: I1-C I' ^,Cf-r I XVY*WS4-C I YY?*'Wr,2Tt)ZZ*(A /RFO3- *(Cl XXYFS-5 .*CXY-%tX-* +C IYQ$ )/PPFC7)

450 'JX( iS)=TUX*XX+TLIY*XY+TQZ *XHQY ( IS )=TUX*YXtTVV*YYl+TY2*YH

$Z~ )=TXC+TUVY*Y+TlJ2*2NGO TO 470

c H COMPUTE UELC I TV CC'EF BY MONOPOLE METHO0D450 APO3A/PP+3.

UV( IS )= (YCC!-'C )*APO3UZ( ls)= 2ZCO-Zc),APO.

C 1 . REFLECT HULL PT. IN PLANE OF SYMlETRY4 70 GO TO ",4'80 45, 490, 495, 5OO;, 5O5, 510, 515), I S

O: 00 LOOPS SET LW TO FORCE USE OF I NDEX PEG ISTEPS

J2=JC

k')DX='YX 1

U(J( I )=LJz(l 1)

0.2( j1 +1 )=UVI 1)

QJ34 I 2)=U'yj 1)

12,' 1 +2 )Z(1)3'j 1+2 )=lZ( 1)F(S~rD 530,530,481

C XZ7 SYMETRYIC=-C

GO TO 34543P_5 1lF(SY-1 i).517 517, 435"

C Xl CYrIETFRY4865 15=3

GO TO34490 =

Vr ri=-~yrjGO TO 345

495 F ($SR'l- 2 F5 t, 1 t,.495tC Y7 SWr1ETRY

*496 113=5( XcO=-xCc!

GO TO 3:45500 IS=6

Yc0=-yCO

98.

1. -,7 .0

55GO TO 345

* - 260=-ZOOGO TO 345

510 IS=8YCQ=-YCQGO TO 345

51 J. ROD CONTRIBUTIONS OF RLL REFLECTIONS515l(J1)=U(J)+UX(S)+X(7)+X()+JX(5)

V JI)=V 2WJI >-9X (8 )+VX (7 )>tV9.(6 )-UX (5)

3(31 )=3(1 )9X()-kJX8+(7)+(6UX(5)

V23(J31+1 '=VU(J 1+1I )+VY (S )-UY'(7 )+VY (e )-UY(5 )

li1(31+2 )=V 1(W1+2 )4-(6 )-UZ( )+9(.j ()+Yk'75'V2(3 1 +2)=92(W 1+2 )+Z (8)-VZ(7)+.(5 )-9Z(5))03(3J1+ 2 )=(3 1 +2 )+UZ7(S )I(7 );(6 )Z(5 ).

5j1ff. V I W I)=0 I(J1I)*VX(4)>+V'(3)02(31 )=k92(3J1 )+UX (4)-()9J3(31)=U3(J1 )-UX(4)-9X(3)V 1 W3 1 + 1 )=!U 1( 1 + 1 )+QY (4 )-IY (2)

92- "( J I + 1 )-V2 W.I + 1 )-'VY' (4 )+Uiy (2:

V 1(W 1+2 )=V 1(W 1+2 )-92 (4 )-9Z (3 ')V2(31 +2)=V2(31 +2)-92(4)+JZ(3)V:3(J 1+2 )=03'(3 1+2 )+UZ(4 ),+V7 ft

51 17 913 J=' I fI(J 1 )+Y J)(292'(31 #924(31)-UX(2)93(W 1 )=V3 (JLI ,+Vy (291(311+1 )=U 1(31+1-UVY(2)92L(J1+ )=92('J1+ 1)+9Y(2)

I.3( 1 .#)(J 1 + 1)-Y2)

112(,l 1+2 )=U32'"J 1+2)-UZ(l2)

* .- ' C:I(2(3 XQ* I-r+ W~ (1 )+YNC'V I (3 1 + 1 ')+ZNM0* I2(J 1+2)

-~ .. - -. . .3(J1 )+YNtAO11kJ3(3l 1+ 1'.'+7t41:+95:(,J 1+2

5403=33

r 0 LiY TE £0EF I C! ETSc K -WR I TE COEF. 0ON TAPE OR DRUM I F STORFlGiE AREA IS FULL

545 F,-!:,-1IDO51C c-i CC

1 (1 )=E:LK

U(1 -)=PLK

IF BLK5Th.0)550 , 55'3,555

GO TO c-FZ:

56 L.RiTE(1) E:LK.,U1, f2,U315 5 Et &K=B:LK+ 1.570 I F <jo-go 1 )56-. 57 1,57 1

- ~ 571 IE'W=I1DW'-IWRITE to::' IDW1,C1WRITE(0)I,2

C WA I TE (0?)I IW, 013

L=L+1

*c L END OF LOOP OVER NULL PTh.

1~ 99

IF(KQ-KM )343,343,581581 CI(JC)=O

C2(JC)=OC3(JC)=otF(KQ-KMM)541585,585

585 P=P+1K=V+7J=,.+20

IF(K-KM )5;6,586 600 .A.IFEDC N. END OF LOOP OVER QUADS.

M t. PRD FiEYT BLOCK OF F RRPY IF NEEDEDn I• ~586 IF(J-241 296,510., 590 :.l

590 READ(04)(B(I )., 1=1,241>IF(B( I )-F'"s'22',59

595 WRITE (6,98) B( 1 ), PSTOP

600 IF(BLK-636.0) 610,620,630 ".--: 0. WRITE REMAINING COEF. ON TAPE OP DRUM

610 WRITE(OI)ELK, U1 , 2, U3R EWIN D 0 1GA TO 640

620 RE41NO 01630 WRITE(11) BLK,U1 , V2, VU--"

REWINO 11640 WRITE (02) IDW, Cl

WRITE (09) IDWCOREW fIND 02RE "IND O-SREWIND 04REW IND OS.,I. N.. &D -.

c F. TRtfSFER TO PFPSi3GO TO 500r,

1000 lWF;ITE(6,2000) L,P20u'F,, AT(,H L= 15,20X.,.3H P= .,F5. 1)5 C-ik-Il'l,.7-NT I NUE

STOP 2* END

FUNCTION SQF(XI., X2, Y1., Y2, Z 12) t=X I-X2

Z'=Z IZ . .2=21-220F:=P:E:S(X ;,+AE:S (Y ",+AES(2 )+ 1 .OE-20 '"

R=FR+ R S,3/RSR=.25*R+RS/R-

R= R+FS/RSQ2F= .25*R+/R"RETURNEND

100'.*''**- ,

APPENDIX III - XYZPF SECTION PIF3

PROGRAM PFP3(OLITPUTh 128, TRPEO2, TAPEOS, TAPEOQ,1 ~TAPE 12,TAPEO3, TAPE6=OUTPUT,TAPE2=TAPEO2, .

2 TAPES=TAPEO8, TAPE9=TAPE'9, TAPE3=TAPEOS)

C XYZ POTENTIAL FLOW PROGRAM VERSION 1 SECTION 3*C SOLVES MATRIX EQUATION FOR SOURCE DENSITY

CCOMMON SN(554>,VIP(65O),S(5,550),PROB(15),WS(220),DM(650),I B(220),COEF(900),XN(650),YN(650),ZN(650)EQUIVALENCE (WS(213),EPS) ,(KK,8(201))

q EQUIVALENCE (MIX,WS(205)),(MIY,WS(2068/I1MIZ,WS(207))EQUIVALENCE (W3(2OIX.NP)tWS(2OS ,IPS),U4S(212 ,IPF) (WS(2l1),K-),1('KMMK'

5 FORMAT(49HIXYZ POTENTIAL FLOW PROGRAM SECTION 3, VERSION 4)WRITE (6,5)READ (03)(PROB(I>),l= , 15)W4RI TE ( 6, I11)(PROB( I),fI= 1, 15)

1001 FORMAT(1HO, 15A4)READ (03) (143(I ), 1=1,220), IEDITS, IEDIT4R EADc (0:3 ) (SK I FP,SK! F, SK IP, XN( I ), I ),ZN< I ),SKI P, I =li. NP)

=-/.14159265READ (osOi(Itl)I=1,NP)

fie K2=NP240 FORMAT (ISHOCHANGES IN PAGE -, 1584)

IlF (IP S)12 20, 1220, 123112 ?PRAD (12)( B (K), K= 1,15)

WRITE (6,240X86(K) K=1 ,15)READ (12)( 8(K), K=1 ,220)READ (12)SKIPIREAD (12)SKIP

K2=JPFC A. SET CONDITIONS FOR FLOW OF -1 IN X DIRECTION1220 FA=-1

FY=OFZ=O

124 NF. COMPUTE INITIAL APPROXIMATION TO THE SOURCE *-

120DO 1250 K=1,NI Pt K)=XNt(K )*FX+YN(K )*FY.ZN(K )*FZ-DM(K ) V

S (5, K)=- I Pt K 1* 19;S5C 0. SET FART IAL SUM VECTOR TO ZERO

1250 SIN( K 0= . 4

a- SN(NP+1)=O.

SN(NP+3 )=O.

SH( NP+4 )=0'.WRITE(6,997) FiFYJFZWR ITE (6. 996)

998 FORMAT (A'27 HO TERF.T! ONlf SUMl OF CHANGES 9X., 1 HA., 1 OX.,2H61, 1OX*.,2HE; 2)

IF (IPS) 12tC, 125O, 12551255 REAO(e12) (S3(5K), K=l ,KK)

DO 1255 K=1 ,KK C( 00 1255.F 1=1,41256 S( I.,K)=S(5,K)

C D. START ITERATION

101

IF (NF)1261, 1262, 12631261 READ (02)IDW,COEF

00 TO 1264h*.1262 READ (08 )IDW,COEF

GO TO 12641263 READ (09)IDW,COEF164 J=O

f

12 0. READ FIRST BLOCK OF COEF2 E. 12 ART LOOP OVER QURDS.DO 120 K1, NP

C F. PICK LUP SOURCE DENSITYSP=S( ICK)

C 0. START LOOP OVER NULL PTS.DO 1290Z- KP=1,NP,5IF'J-900)SO,6-5,65

55, IF (NF)I57,6S8,696? READ (02)1014,COEF

60 TO 7068 READ (O8)IDW,COEF

GO TO 70ti9 READ (09)DWC2ZZ-70 J=O

C H. COMPUTE PART IAL SLIM FOR NEXT 5 PTS.SO Sfi(KP )=SriCKP )+COEF(J+ 1)*3P

SN(KP+ 1)=SN(KP+ 1)+COEF(,J+2 7)*S-PLS SN(KP+2)>=SN(KP+2 )+COEF(J+3 )*SP

SN (KP+3 )=SNi( KP+3 )+C:OEF (J+4)*5PSNl(KP+4 )=SN(KP+4 )+COEF(J+5 )*SFJ=Jt5

C J. END OF LOOP OVJER NULL PTS.C K. END OF LOOP OVER QUADS.

1290 CONTINUEC L. COMPFUTE NEW SOURCE

IF (NF) 91,92,9391 PEW; tD 02

GO TO 9492 REWIND 08

0O TO 9493REWiND 09p94 PRSSI.O

DO 100 K=K1, K2c5,NkK)=( SN(K)+,IP(K) ):

SUIl,'t!1tTESTIF (TEST .0T. EPS):', PAiSS=-1.Ol

100O CONT I NUEIF (PASS EQ. 1.0) GO TO 180IF (ITGE.MIX) GO TO 1SOIF (IEDIT3 .EO. 0) L.RITE(6, 99) IT, SUM -

IT=IT+1

IF (!C: EQ. 0) GO TO 120D0 110 KzK1.K2S( IC,K)=SN(K)

*110 SIND=0KGO'TO 1260

* 120 A=0.r* B61=0.

62=0.DAI=O.D1=0.

* 102

DO=140 K- KI,K2DS9=2*S( l,K)-SN(K)-S(2,K)

I- IF(DS§ .GT. 0.) G0 TO 122A=RtS(2,K)-S( 1,1<OA=DA-DS9GO TO 125

122 A =A +S(1, K)-S(2,K)DA=05+DSQ

125 051-9(4, K>-S(3, K) >DS2=S(3,K)-S(2,K)

0SS=32,K)-S( 1,K) t4

084=062-6(1 ,K)+SN(K)OS7=063*0S4-0S5*0S6DSB=DSS*D5-OS4*063IF(DST .GT. 0.) 60 TO 128B 1=S1-DS 1*0S4+D62*OS601=01-OS7'30 TO 130

128 B I13 1 +03 1 *03 4 -S2*0:S6

130 IF (09 GT. 0.) GO TO0 1332B2=B2-D6 l*0S5+D32*DS3 t

GO TO 140132 2~r+D -D02*Dc:

02=02+038140 CONT INUE

B2=B2/D2IF(IT EQ. 6) GO TOI 155

IF (A GT. .02) '30 TO 148'0O 145 K=KI, K2

145 Sti(K)0O.1,1.I1TE (6, 6000)

6000 FORMAT (29X, 17HA EXTRAP OLAT IONMGO TO 160

148 BB1I=EI1-ES1BB 1=AE:C;(E$1)

a,' B115O.*B1

1212=59(22)B82=50 .*6BE$=AE:$kE!i) + A:'E2IF ( (66.1 GT. 13881) OP. (22 GT. BBC. ) GO TO 155DO' 150 K=KI,K2&S , ., K =6(2, K )+E, I1*(S( 1, K) (2K))6*(tK '-S (, K)

150 SN(K)=O.14P!TE (5,70MOO

7000 FORrAT(2QX, 1?HB EXTRPOLATIOIN )00 TO 16

155 00 152 K=KI K2S6(5, K )=SNQ(K

158 Sti'(K)=O.160 IC=5

r 103

11WRITE(6, 16:) A, 81, 82

BS1=B1BS2=82GO TO 1260

180 WRITE(6,99) IT,.SIFp DO 182 IK1,K2182 S( 1,K)=SN(K)

WAITE(03) (8(1, K>, K,NP)99 FRT(4X, 13,EIS6.5)

007 FORMAT (13140 X UELOCITY=,F4.1, 15$ VUELOCITY=,F4.1,115H Z 'ELOCITY=,F4 1)IF(FZ)1400, 1Q90, 1400O

c P1 IF THIS WAS NOT LAST FLOW, SET FOR NEXT FLOW1390 FZ=FV

FY=FXFXrOMIX=flIYN8; -=i2z*NF=NF+ IGO TO 1240

1400 REWIND 03c P2 READ INl PFPS4 AND TRANSFER TO IT

STOP 3END

.10

APPENDIX lU XYZPF SECTION PF4 4

4 ~PROGRAM PFP4(OLITPUT, TAPE6=OUTPIT, TAPEOS, TAPED1, TAPE11,I ~TAPE3=TAPEO3,TAPEI=TAPEOI) I.

C S

C XYZ POTENTIAL FLOW PROGRAM VERSION 1 SECTION 4C COMPUTES VELOCITIES AND PRESSURE COEFFICIENTS FOR

C POINTS ON THE BODY

CCOMMON VXI(50>,VYI(650),VZI(650>, VX2(650>,VY2(.65O>,VZ2(650>1 VX3(650),VY3(650),VZ3(650), SI(650), S2(650), S3(650)

2 ,X(650), Y(650), Z(650), T4(650), T5(65O), T6<650)3 *DM(5O)

DIMENSION PROSB(15), WS (220 ),CV 1 (1000 ),CO2(lO t31000;'I0c)EQUIVALENCE (WS(201),NP),(WS(2lltKN),(WS(21?tX ),(S21tY)1(WS(219),VZI),(WS(208),IPS),(WS(212),IPF)EQUIVALENCE (MIX,WS(205)),(MI(Y,WS(2'05)),(MIZ,WS(20?))

5 FORMAT(4§HOXYZ POTENTIAL FLOW PROGRAM SECTION 4, VERSION 4 >WRITE (6,5)PRA (03 )(PROS (I ), I= 1,15

100 FORI1AT(1H , 15A4)WRITE (6,100X(PROB(I),I=1,15)

C A. READ PARAMETERS AND SOURCEREAD (03) (WS(I),I=1,22DtIEDIT3,IED.IT4READ (03)(X(I>,Y(I),Z(I),T4(itT5(ItTE(I),SKIP, I=l,NP)READ (0J3>(DM(I)l, 1=1,NP)D=-.51..1415925READ (03)(S 1 (I),I=1, NP)READ (03)8(i(I),I= 1,NP)REAID (03) (S3(l ),I =1, NP)

K 2=NPIF(IPS)1O8, 108, 102

102 KI=IFSI-K2=IPF v

108 BBR=1.C B. READ F IRST BLOCK O-F 'COEF.

ITRF*E=Ol >PRA (01) BB,CV1,CV2,CVS-IF (BE-B66R) 300,120,300

120 DO 12511l, K2VX1(I)=-1.O -S1I)*T4(I) /DUYl(I ) -81(1 )*T5(I ) /D

VX2(I)= -52(1)*T4(I) /DVY2(I)=-1.O -S2(i)*T5(I) /DV Z(I) -S20I)*-Tt( I) /DVx3( I)= -S3(I)*T4(i) I'DVY3(1I)= -83(1 )*T5( I> /D

C125 V23(I)=-1.O -S'3(I )*T6() IDC C. SET UP LOOP OVER QUADS.

LIC=2

C D. PICK UP SOURCE4:130 S II=51I(3)

S2J=Sr2(J)53J=S3CJ)

C E. SET UP LOOP OVER NULL PTS.DO 180 JP=K1,K2 A

C F. COMPUTE PARTIAL SUM FOR 3 COMPONENTS OF SVELOCITIESVX1VJP)VX1(JP)+S1J*CVI(JC)V)Y1(JP)=VY1(JP)+S1J*CV1(JC+1)

105

I V .~. . . . . . .. ..& / .--.

w1 Iv-in weP=V Il MY)+ flJ*f I, M(W f f

V2(JP)=VY2(JP)+S2J*CV2(JC+1)LZ2(JP)=V22(JP)+S2J* CV2QJC+2)VX3(JP )VX3(JP )+S3,J*CV3(JC)VY3 (J.P )VY3 (JP )+S3j*CV3 (JC+ 1)U23(JP )=VZS- (JP >+SSJ*CV3QJC+2)JC=JC+3

C G. READ MORE COEF. IF NEEDED.IF (JC-1000)180, 135, 135

135 JC=2IF(BBR-635.O) 140, 150,155

140 READ (01) BB,CJ1,C'2,C.3GO TO 160

150 REWIND 01155 READ ( 11 ) BB,CV1,CU2-,CV3160 BBR=BBR+.

IF (BBR-BB) 300, 180,300C H. END OF LOOP O'.ER NLILL PTS.

1SO CONTINUEJ=J+ 1-

*c C 1. END OF LOOP OVER QUJADS.IF(J-NP>130, 130,200

200 IF(BBR-535.0) 231,231,2:3223 1 REW IND 0 1

GO2 0TO233-A''REWIND 11

C I. EDIT THE VELOCITIES ETC. AND WRITE THEM ON TAPE233 WRITE (03)(VX1( I ,VY1I( I ), VZ1 I) =,~

WRI TE (03 )(UX2( I), 'VY2(I),VZ2(I) ,I=1,NF) CWRITE (O3XuX3(I),VY3(1 ,VZ3(I) ,I=1,NP,)

23'5 FORIRTT1H, 1594,S8H PACE =, 115REWIND 03aIP=K 1+49I S=K 1IPRGE=1IF (IEDlT4 EQ. 1) 00 TO 293IF (MIX .LE.O ) G0 TO 25

242 FORMAT(SHO X FLOW)*240 FORMART(4H PT. , IlOX, I HX,Q9X, I1HY,Q9X, 1 HZ, 3X, 2HUX, SX, 2HYY, SX, 2HU2, lOX.,

1 5HRES . , 7X, 2HCF'. X, SHSOUF;CE,4X , HV NORMAL)245 FORM1AT (IX, 13,4X,3F10.5,4X,3F10.5, 1F13.5.2F11.5,E12.2>250 IF(IP K2)255,255,260

C J. COMPUTE PRESSURE AND ABS. VALLUE OF OELOC ITY25WRITE (6, 23 PROI), 11, 15, 1IPAGE

WRiTE (5, 242)WRITE (5,240)DO 25? IS', I P

Q=UX 1( I )*c*2+UY 1( I )**2+UZ 1 1

'IUt+USQ M.1 'M= 25*: 1k.,i

M= . 5*(Ijflq.IJQ/Ifrj)CP =1.-USC

257 WRITE (6, 245) I , X( I ), Y( I )Z( I )VX1(1 1 , 1 ,Z 1( 1I -1I ,CF' ,S1(I ),UNR

15=15+50IP=IP+5OI PAOE=I1PAGE+ 1IF(K2-IS) 255,260,250

*260 IP=K2

106

GO TO 255265 IP-K1+49

IF MYS .LE.O0 GO TO 2802V7 IF( IP-K2)275,275,27O270 IP=K2275 WRITE (5,235)(PROIc(I ), 1=1)15), IPAGE

WRITE (6,277)277 FORMAT(8H0 Y FLOW)

WRITE (15,240)

U)SQ=VX2(i )**2+UY2( I )** +U2(

VM=(ABS(UX2( I ) )+A:S'Y2( I) )*ABS('22( I)) )*.79Vr1=UM+USO /ur*1=. 25* VMI.VSO./Vrvrr =.5*(Vl*USO/Vh)UNR =UX2(I )*T4(I) +uY2(I )*T5(I) +U22(I )*TS(I)CIP 1. -USO

278 WRITE (5, 245) 1l, XI), Y(I) Z(I) VX2 (I ),Y2( I) U2( I) VtI1 ,CP S2(I),UNRI8=18+50IP=IP+5OIPRGE=IPRGE+1IF( iS-K2)2b7, 257,280IJ 1FK+49IS=K1IF (1hZ LEO0 ) GO TO 293:

2 82 IF(IF-K12)2490,290, 285285 I P=K229: WRITE (6,235)(PROB(l ), 1=1,15), PAGE

WRITE (E.,292).292 FORR(SHO Z FLOW)

WRI TE (6.240)00 291 fS,fPLJSO=JXS( I )**24JYI.3(I )**2+U-Z3(I I*Utl=(ASS(UXS:( I ))+Ae-S(UYS(I ))+ASS(UZS(I))*.79

VM=. 25*UM'IUSQ/k'IVI'l =. 5*(jflM+QJQ/tVf)

* OP =1. -USQVMSR =UX'S( I )*T 4 (1 +UIY3( 1 )*T5( I) +2(I)5()

291 WRITE (6,245) 1,X( I)Y(I),Z( I ) XS(I ) UY3(I),Z:3(I),tl

I8=I8+50IP=I P+50I P~iE-= I FAGE+ 1

- IF(IS-K2 )282, 282, 292C L. CHECK FOP: A FOLURTH FLOW

292 j = 1294 UXI =WO(.JN

__UZ I =WS(J+40)295 IF (U>. I*'+ y +~i.*w-2+1J2 :I *42-') 400, 40 10 .301C301 IS=KI

IP=K 1+49-*320 I F ( IP-K2)330 325,3 25

C N. E[JiT THE UjELOC I TY- AND PRESSURE FOR FOURTH FLOWV325 1IP =K2

3300 WRITE(,2"(RBI)H,5)PEWRITE (5,315 )wXIUYIUZILWRITE (,340) .--

C M1. COMPUTE VELOC I TY AND PRESSURE FOR, FOLURTH FLOW

107

So DO 333 1=15, IF0X4 =-(VXI*VXl(I )+VYI*VX2(I >+VZI*VX3(I))IJY4 (VXJ*uY1(1)+uYI*uY2(I)+uZI*VY3(I)) C024 (LX*VZl(I)+UYI*U22(l)+UZl*LJZS(I))VSQ=0X4 **2tUY4 **2+V24 *+'-A=(ABS(VX4 )+ABS(VY4 )+ABS(024 ) >*.79Vti=UN+USQ /01

IL V=. 2s*VMVrjSo touti = .5*(UM+USQ/Utl)op - l.-(VSQ>/( VYI**2N)ZI**2±LJXI**2>

333 WRITE (6,345) I , X( I )Y( I )Z( I ),X4 ,tY4 ,134 ,Vt

I3=15+50IP=l P450IPRGE=IPRGE+1I F(K2- IS )350, 325, 3210

350 J = JtiGO TO 294

315 FOP;MRT( 19HC ONSET FLOW, UXI=,F6.3,2X,4HVYI=,F6.3,2IX,4H021_=,F6.3 )340 FORtIART(4H PT. , lOX, INX, QX, INY, QX,1HZ-, 13X, 28'JX, SX, 2HYY, SX, 2HV02, lOX,

1 5HABS .0, 7X, 2HC:P)345 FORMAT ( X, 13, 4X,3":F IC 5,4X, 3F 10. 5, IFIS .5, IFI 11.)400 CONT INUE

GO TO 5000300 CONTINUE

WRiTE (6,310) ITAPE8eR,Bs-310 FORrIRT (61H1ITAPE ,12, 17H OLIT OF POSS I TION1 14,5F5. 1)

5000 CONTI1NUESTOP 4END

10

APPENDIX U - XYZIPF SECTION PFS y5

PROGRAM PFPS( INPUT= 128, OUTPLIT= 128, TAPEO3, TAPEO4,1 TA'ES= INPUT, TAPE6=OLITPUT, TAPES=TAPEO3, TAPE@TAPEO4)

CC XYZ POTENT IAL FLOW PROGRAM VERS ION 4 SECT ION 5C COMPUTES VELOCITIES AND PRESSURE COEFFICIENTS FORC OFF BDDY POINTSc

COMMON B(241>, XP(500),YP(SOin)tZp(500),VX1(500>).WS(220)1, VY 1(500I), 1421(500), VX2('500 ), VY2(500),V22(500), UX3(500), UY3(500)2, VZ3(500), VX(S), VY(t), z(S), S 1(650).,S2(650), S(60,V1G I V2(3) U38:3(3) PF;OB(1I5 X(50YY C"~r(50 A65) X604,TY(650), TZ(65C0) ' $1W(EQUIVALENCE (KM.WS(211), (KM, MKD,(NP, WS(2-0 1),, t,W' -410')

INTEGER PAGEI NTEOER SYM

5 FORMAT (49HOXYZ POTENT IAL FLOW PROGRAMl -:T ION 5, tIER'S I ON 4)WRITE (6.,5)

C A. READ I NPUTR EAD -,(5,25>" IjOEP, I EDl I T5, I READ

C A. READ THE OFF BOD'Y POI NTSNOB=NOBPDO 10 I=1,NOB

6REAO(5 215) X(P (I), YP(I ), ZP(I)F (ECF(5) EQ. 0.) GO TO 10

NOE:P= I- IWF;I TE t9) NC0EIP,NOB

9 FOR~iAT(THO1,i5,:31H OFF BODY POINTS SPECIFIED NOT ,13),GO TO 11

10 CONTINUEk11 LIOUT iNUE25 FORMAT (314)246 FORITAT-r(OF 12.5)

P=1.

*90 FORMAT( IHO,184WRITE(b,91) NOEP, EDITS, IREAD

91 FORMAT(SHONOSP =, 14 /SH IEOITS=, 14/SH tRERD =, 14)L4PITE,6 ,92)

92 FOPITAT(1RHO OFF BODY POINTS / 7 4 T, 11)', fHY, 1--, THY, '2'., 1HZ)WRITE(6,93) (IXP'(I ),YP( I ),ZPr;I ), I=1,NOBFP)

923 FOR.!iA'rX.I3.2X,S3FT1O.5)C B. READ THE PARAMETERS , T ARRAY AND SOiURC:,E FROM TAPE 11

READ (03> (IWS(I>,1RI.220)READ (03-) (TX( I ),T'( I ),TZ( I),XN( I),,YN( I),Ztl), TAi( ), 1=1NP)READ (03) SKIP

C FORMERLY:- WS(220) . EQ. 2.C

I F (I12O)0 EQ. 5. ) FEAD(O3) SKIPPEAL (0-3:)' (S1(1), I=1,NP)READ (03) (S2(I>,k=1,NP)READ (03) (S3?( I, Izl,NP)

C C. READ THE F IRST BLOCK OF THE 6 ARRAYREAD (04) (B(I 121,241)K=1

DO 100 '=1,NOBPC D. SET THE PARTIAL VELOCITY TO THE FREE STREAM VELOCITY

109

v ~)-O.VU71 ( f)=O.UX2(I)=O.

UX32)( I )- .UY3(I )=O.O

100 U22(t)=O. r

29'0 IF(B(1i)-P)291 ,295,291291 WRI1TE (6, 90) 82 ), P98 FORMRT(2H0 POINTS OUT OF ORDER B(I)=,1F4.O,4H P=,1F4.O)

STOPc E. START LOOP OUER THE QUADS.

295 J=2c Fl P I CV LrIP QURD. I NFORAT I ON

296 X1=B(J)

X2=B(.J+2)

Y2"=B(J+3)

XC:=T '((K'YCTY'K)Z0=TZ(K)

aA =T~irG:I)xx =e.(3+7)YXBs(J~S>2X=EJ+9,

XY(=S(J+ 10>

ZYv=B(J+ 12-)c F2 C:OMPUTE LEHGTH OF SIDES OF OUPO.

1)F( 1,X'2, YV1, Y2, 0. , 0.)0.- uS2F(X2, X3, Y3', .0, .0)[p4 ' "A.,'(. Y3*, Y4, .,0. .)D4 1'-Su2F(CA4. XI.'Y4.Yl. .. .0)

c F3 C:OMPUtTE SLOPE OF S I DES1 -XY '3 ) '30 53?0 030 5

0u TO 310

C 1 23,:=0.

311 C0124 =IGO TO ?20

I315 Cr14=e/Y4-v / x,4-xs)

:3220 F(X4"'A-X 1)3 3,21, 325321 0I41=1.

GO TO 3230

330 IF (XI-X235, 331, 33G5* .~~3 1 i12=1.

GOTO34033v5 CMl2= "'-Y /a2(-X9Y

c F4 CONFUTE OUiROARPOLE MONIENTS340 01XX=B(J'-17)

C IXY=Et(J+ 6S

n110

-s- ' s- t- - ~ ~'- -r- - w'CILYr rr r-Y--r =c.Bnr w - --- v r - .- *--W.'

C F5 OMPUT SINAND COS OF SLOPE ANGLE FOR EACH SIDE

CY23=(Y3-Y2 )/023C'34=(Y4-V3)/3CY4 1=(Y1-Y4 )/D4 1CX 12-(X 1-X2)/D 12CX23=(X2-X3 )/D23CX34=(X3-X4 )/D340X(41-(X4-X I /D41 A

C FE' COMPUITE MAX 0 AG' 1 NALST=SQ2F(X1,X3,,Y1 ,,..)6T2=Si:2F(X2.x4 72. '94,. 0)IF(ST-ST2)341 .342,.342

*341 ST=3T2C 0. START LOOP OVER THE OFF BODY' P0 rATS

342 DO 530 JQ= 1 .NOEF

5s=1XCQ=XPQjc!)VCO=YF '~jO!

345 RP'Q=SC2F7YC, XCQ, YC:, Y:C,2C ZC:O)C H. DETE.TIN METHOD

!F(RPO.-ST*4 )350 ,350,460a350 X= (XCQ-XC )*XX + (YC1,-YC )*yy+ (ZCij-Z )*-ZY

Y= (.XCO-XC )*Xy+ (YCfl-YC )*Y ZCQZ )*&Y2=(XCO-XC )*XN (K )+ (YCQ-YC )*YrAt(K )+(ZCQ-2C x-IKKIF(PQP-ST*" ry552,5,49

C 1. COMPUTE 9 DUKEE VELOC ITY BY EXA:FCT METHOD

R2=502F(XX2,V,Y2,2,0. )R3=S,2(,3,Y, Y32, 0. )P4=S502F (.X,X4, Y9, Y94, 2,0.)IF((R1+Ri2).LE.D12) 00 TO 1000lF(R:3+R2).LE.023) GO TO t000IF((R3*R4).LE.D34) GO TO 1000IF((P1+R;4).LE.041) GO TO 1000

CLA =RLGK R I+F;-D 12)/KR +R2+D 12))U CLA2=RLOO( (R2+R:3 023) /(F2+F3+D2Z))

CLe.4='L'3( (R4',R 1-D41 )/(R4+R 1tD4 1)TVX=CY 1 2t CLP 1+0Y22*CL A2+CY24+CLAS +CY4 1,*CLR44TUY,=CX 12*+CLF I1+CX23-.*CLA2+CXO4*:CLRiStC.X4 1*CLA4

H~~0 FiS() O*ST)3?5,Se251.$S1:

E 1 =ZSO+ (X-X 1E2=ZSzQ+ (X-X2 )**+2E3=23+Y (-y3 )*t')E4=2SQ+' \X-4 )f4

H2l(V-Y'~ 1

H3=(Y-Y;) .X (X,.H4=(Y4Y4)*.*x \4) -.

I F(C1 n6 5Q 4*36:3 WS 1=(CrME IHDI/2tRI),1,

L452=(CM 124E2-H2)/(Z*P2) 4~A RI =ATANI'3.S1 )

AT2=ATAN(WS2)T'~J2AT 1-AT2

*364 1IF(C 123)366, 36,367

366 RTS=ATRN( (Ct'23*E2-R2 )/(Z*R2Z))FiT4=FiTAN( (Ct23*E3-H3 )/(Z*R3)),%~TVZ=TIJZ+AT3-8T4

367 IF(0134)388,388,359

RTb=ATAN!( (0134*E4-H4 /(Z*FR4)iVZ=TUZ.RTS- Rib

369 IF(0141)370,370,375370 RT7=ATAN (C14 1*E4-+I4 )A(2*R4))

ATS=ATRH(011M41*E1-1)(*R1%)Ti.UZ=TkC+RT7-ATS

37 GO TOMP50 INDUSED VELOCITY BY C':!URDRRF'OLE TIETHCD

L4'= X/RPQ'34:2)F

pc tCO+Q..4+ -7

-c,4? *X*QcJ/RPCl?U. ~LS -C:I 'Y:LC3....C , XXL-C IY*4

w'" 1 ='RP Q3~~ ~+~ *Y.CI*'PQi7

T~~~~4 V- -, 4Zi'fW

i'2= FFua l-:3 C I .0*c -5 r!XY-*X*Y+C: I Y')fl5 ) /RFFQ)450 uF S Ii ~~t'~U~NK

13y(~ Z+' I;) TU*\TkY Y+TkiryN (K)

GO0 TO 4?0£ K. COMtPIUTE It-HOI' rED VELOC ITY BY MIONOCPOLE IlETHOD-

(.Jx(IS)= (/CQ XC "*RPF'OIS

I S (YO% F )*9PF'CF 3

c L. FFEFLEL T OFF PODY P0,' HT 1N PLANE OF SYM1ETFA'470 00 TOC4dl 48C 4'O,495.,O0..,O5,51O.515)l,IS4S0 V'3 1JI) 1

11 1

U2(J W1+2 )UZ( 1)U S -I 1+2 )=7 I/F1

IFf SYI 525,525,481481' I5=2

Cj: Ti- '-45485 I Fk.SY M- I17 517 43o5E

486 13=3

GO TO 3454W0 1 S=4

Y00L)-YCCGOT 4

L4Q-5 IF(SY-2)515, 516,4Q65c YZ SYIIETRY 4

495 1S=5XCQ=-XCQ0O TO 345

500 )S=6YCO=-YCOQ0O TO 345hC

505 IS=7200I=-ZOOQGO TO 345 -

510 1 =86Yro=-YCC'

P G0 TO 345M. 5 5 3 O CC'HTF I BUT I ONE" OF FILL REFLECT OtiS

(32(31 )=VU2(31 )-UX(8)+UX(TAU'+X(6)-VX(5)t)3(3J1 )=1,3( J I )-'.y(8 >-J(7 )+ UX(6 )+t~f 5)0 1(3~j1+ 1 )=U 1(W31+1 )-5S:UY( +JY()W()

(.2.+)=22 J+ +1 + y:7 )+I.Y(6 )+U'Y(5)II( 11)1(1+ 1 '' Y7)+kt )-V ;(5

U 32( 1+2 )=U2( 1f 2 J .1 ( 8-iZ (7 )+tjwF ) (5)V2 3(J31+2 )=V312 +~t I + 28 .7 ) + UZ )+6 Z

1)2(31 )=+2( 1 )+r )+,, A) +1 S ) ' I=y3 ( 1 )-U (4 ) U); (3 )

0 1 (JI+1I )=0 1+ I )+VY( 4 )-UYr;2)32(3., 1+ 1 )U(1+ 1)+'(4+.I3)

(3 1+ 1 )U3Q(3 1+ 1 )-VY (4 )+U' !(3 )

5 17 (31+)=I(12 3()(23J3( 12 >2 J(2)V 12 )Z( +Z(

V(3(31+1 l2 ,= J 1+21 )*UZ (2 ",1(3

V W 1+4)=(J (1 ) +U2)t)(((1+1 )=U2(jW1+2 )+U2

J 3<1+ 1U'7 I t-4(

VJ(3 1+ )( 3(J3 Z '+Z

U.)(3C I I VI

'3Y2(3O .'k.2( JO)+'J2(, 1 )4 S2 (LVY2,'.Jil )+(32( 2 )*c2 (L)

(3X (J )=X3(JO )+V.1 2. 1) E3 L)

530 C -N1T INUf-E

*-C N . EiC F LOOIP CUE; CIP ED1Dc PC, ITS5S5 P=P~l

F(K-tiF: 5.86E 5?F 99ih(U, -241x,%C

c 0 - FERLu HiExT ELC L- uP . 2F: RF;F;R', L F HEEDED590 PERO (04.. !. 1 1, 41

113

IlF(B( 1)-P)291,296,291C P. END OF LOOP OVER QURDS

599 CONTINUE -.PAGE = 1IF (lEDlT5 EQ. 1) 00 TO $25

601 FORrIT4H PT..11X,lHWY, 12X, HY. 12X. 1HZ, 14X,2HUX, 11X,2HUlY, 1 1Y,2-H'JZ71, 14X,2HCP)

602 F0RM1AT(7H X FLOW)603 FORI'RT(7H Y FLOW)604 FORIIAT(7H Z FLOW>605 FORMIT ( I H1, 15A4, l OX, 1ISHOFF BODYe PO INTS , 1OX,S5HFRAE ,I)

IF (tlIX.EO.O) GO TO 700WR ITE(b. 605) FROB,PAGE

p WRITE (6,602)WR ITE (6.,601)L INE=lILAST=S3

606 IF(NOBP-LAST)60,610.6510607 LAS-T=NOBP6 10 DO 615 I=LINE,LAST6 611 FOR'iAT (IX.. 113.,2X ,3F 13 .5,23F 13 .5,S3X ,Fl.5)

c Q. COMPUTE PRESSURE AND ED IT 23 E; ,-- I C FLOWSCP1= 1. -QJX I (I )*f*z+(JY I1<1 )*t:2+Il7 f l.A1:-#2

t15 WR ITE (15, t II , XP('I YPI I' ,ZFI UXVIDU'VI tQZ IOFLINE=LA$T+ 1LRS-T=LINE+54FtSE=FR3Et 1I F (* L I NE-NL F )620 6 20, 700i

* 520 L*~~~R I TE- ('r u F E, fO

GO TO 606

WRI TE '6 rn'!

L INE=1ILAiST=55

706 1 F (NOE'-LR?-T )7 07. 710. 7 10'-707 L RST=I6 F

*710 DO 715 1 L INE.,LAST

715 WRiTE (6.611)> I.XF(I0tYP(I),P(I ) UY'2( I ),Y2( 1 07I.'2( 1 ),0F2L I NE=LR;__t+ 1LRST=L I NE+54F~rCE=PR&E± 1I F (1V WF 720. 720,5: 8--

r0WRITE(6,605) PROS,PRiGEWR ITE (6,6501)GO0 TO 705

800 IFP (MlI Z. EO0) 01J TO 8'25FUP! TE(6, 6 5) PF PAGEWRITE (6,6'4)

IITE (6, 60: 1)L I NE= ILROST=55

806 1IF(NOE:P-LAcIT)3107, SlO1l1807 LA2lT=N0E'P8 810 DO 8.15 1I=LIHNE, LA T

CPJ' 1 -> I ): 42+ 1Y I )*+13( 1 )**2)$15 WR ITE (6,611) 1 ,XP( I ),YP( I ),ZPr I ),XS3( I )Y> I, UZ( I COPS

L INE=LPST+ 1LAST=LINEi54

PRGE=PAGE+ 1IF(LlINE-NOBP )820,820,825

820 WRITE(e,50JS) PROB,PRGEWRITE Qa ,6O1) -

GO TO 80b825 J =1I826 IF (IREADEQ.O) GO TO 827

READ(5i,2b) UX4,VY4,UZ4IF (EOF(5).NE. 0.) GO TO QOO K00 TO 828 "

827 UX44JS(J) .4UJY4 = LS(J+ 20)(i74 = WS(J+40)

p828 CP=UX4**2+UY4**2+U24**l.2I F(kCF' )9OO, 900, 830R F. COMiPUTE FOURTH FLOW AND EDIT IT

830 LINE=1LAIST=51WMiTE(65,605) PROB,PAGE

83.1 FORMART( 19HO,1rASET FLOW UX , F?( .315X".4H'JY1F7. 3/ 15 4H-)'~ = .F7.3)WRITE (6,831) UX4, ,UY4,0124WR ITE (6,601)£3510 IF (NSPLST )837,840, 340

S, 3 LRST=AOE'P840 DO 845 I=LINE,Lt3ST

UXP=-U.X4*UX 1 (I )-VY4*VX2( I >VL44UX'30 )UYF'=-UX4*kJY I( I )-UY4*1)Y2 (I )-tUZ4*UYS'( I )

C:P4= 1 . (UXP**2+VYP**:2+U2P**2 )/CP845 WRITE (6,611) I , XP (I ),YP(I ),ZP(I )V P, VYP, VZPF, CP4

LI NE'=LAS'T+ IL LAT=L INEPAGE=PROE+ 1I F(L INE-NOEP )cS0, sO,85sea6

850"- WRITE(6,605) PROE% PAGEWRITE (6,601)rrGr TO 83-,5jcr = j

-32 WRI TE(k6, 1001) JO, L,XP (JQ) YP(JQ) ZP (JQ)100 1 F0Rtr1AT( I6HO0FF: BODY POINT 1 3,23H ON BOUNDARY OF QUAD , 13/

1 3H X=,F12A.5,,5X),2'HY=,F12.' 5, 2H=,P1; .5l'GO TO0 5350

900 CONT I NUEc S, REL4 I NO TAPES AND STOP

REWIND 03REWI-;ND 04STOC1P 5ENDFLI4CT I 0,i SQn2F (X 1, X2, ,Y1, Y2,21,2Z2)

9 -t), 1 -'-'22=21-2RS=2Z*+2.Y**2.X**2

* - =AB(X)+E.SV)+A3(Z+ 1.OE-21R=R+RS /R

A=25*4Ri'RS/U. R= R+RS/R

CQ,4-F= . 25*R+RS-/RRETURNEND

115

APPENDIX VI - XYZPF SECTION PF6

,p NPROGRAM PFFt6( INPUT= 128, TAPE 16, OUTPUT= 128, TAPE03, TAPE04, P,.-". 1 TAPE5= INPUT, TRPE6=OUTPUT, TRPE3=TRPEOS, TRPE4=TRPE04)

C XYZ POTENTIAL FLO4 PROGRAM VERS ON 4 SECTION 6 -

C COMPUTES VELOCITIES AND PRESSURE COEFFICIENTS FORC OFF BODY STREAMLINESC

COMMON XP(100), YP( 100 , ZP( 10,'X 1( 100 ), WS(220)1, U', 1 ( 100), Z I ( 100), UX2( 10'0 ), UY2( 100 ), 'Z2,' 100 ), UX3( 1C00), VY3( 100)2,VZ3( 100),VX(8), UY(S),UZ(8),SI(650),S2(650),53(650),U1(3), U2(3 ),U33(3) ,PROB(15) ,XN(650),YN(650).,ZN(650)..TX(650),TY(e5),.TZ(650)1, 8( 13000), XT(100,YT(100),ZT(100),AP(5),,Gt(4), SK"( 100),1 6K2 ( 100",SKX l0 .DX 1(00),DY ,100), 0Z(100), CP(100) TA< (65)EQUIVALENCE (KMuS(211)),(K1.lK),(rP,4.(201 ) (.""t-S(210).1. (Y2, Y3) .C-"

INTEGER SYM PC A. READ INPUT

WRITE (6,5)tb FORM1AT(314..4F 12.5)8 FORMtIT(3F12. 55 FORMAT(49HO'YZ POTENTIAL FLOW PROGRAM SECT I ON 6, VERS ION 4 )7 FORWIAT (1X,9F12...)

20 FORR'!T( IH!-. 14..31H STREAMLINES TO BE COMPUTED AT 14, 1OH STEPS OFI ,F8.4,28H T FOR AN ONSET VELOCITY OF ,3F8.4)

21 FORMAT ( 1X, 15HSTART I NO PO I MTS/, 3.)r, Hr-, 5X, 1HX, 11X, 1HY, Ii X, 1HZ"22 FORMAT( IX, 14,3F 12.5)

READ (03) (PROE( ) ).,I= 1, "15)WRITE (6,90 ) (PROB( I), =1, 15)-

§0 FORMAT( IHO, 1584)C . READ THE PR A. TER, T RRA",' AND SOURCE FROM TAPE 31

READ (03) (W:I), 1=1, 220)REA' 0) 0 TYT! ),TY(I ),T-71 ),X( l >,YN(I >,Zt(l >,TFI(I >, I=1,NP)READ (03) SKIPIF ( WS,(220) EQ. 2. ) READ(<0) SKIP

READ (03) (SI(I), =1, NP)READ <03, <S2 I ,=1, NP)READ (03) ' -' ' (:" I ) =I , NP

SC. READ THE E; ARRAY•z = NFW2=':4Z+1 .0)/12.0NE = ,Z,

-" lIS = "2

!F=241D0 12 IP = IW:.- .READ (04) P, (BI).II5, IF)2F=IF+240

APKI)= .5AP(2) =A F,, = I1.AF, (4 )0.

GM(I) = 1./6.

e3 READ < 5,6> NOBP,NS.-T, IEND,DT,VXI ,UYI , 'U''JfS=UX I **2+UY I **2+UZ I*'2 -

NOE:=NOBPDO 10 I=,NOBREAD(5, 8) XP(1),YP(I),ZP(1)

i116

.. . . . . . . . . . . . . . .. . . . . . . . .....-.-. ............. ,-.-- ".

IF (EOF'(5) .EQ. O.). Gu TO 10 -lNOBP= I-iW.RITE(6,9) NOBP,NOB

§ FORMAT(IHO, IS, 28H STREAMLINES SPECIFIED NOT 13G)GO TO 11

10 CONTINUE11 CONTINUE

WRITE(1t6) NOBP,NST,IEND,UXI,VVI,UZIWRITE(6,20) NOBP',NSTDT,UX1,UVI,UZIWRITE(.6,21)

*WRITE(6,22) (I,XP(I),YP(I),ZP(I), k1I,NOPeP)C NOBP - NUMBER OF STREAMLINES TO BE TRACED.C NST - NUMBER OF STATIONS AT WHICH STREAMLINES SHOULD BE COMPUTED.

DO 15 I=1,NOBP'a T(I) =XF(I)YT( I) = VP( I)ZT( I) =Z7( I)SKX(I)0O.

-'SKY(I) = 0.*15 SK1(() = 0.

I TC=O'IPRK=5

98SK=1P = 1J= I00 100 I=1,NOBP

S ~C D. SET THE PART IAL VELOCI TV TO THE FREE STREAM" UELOC ITVUX1(I )=-1,O

VY1(I )=0.

VY2(1 #-1.OUXS(I #0.0

UZ2(I )=-1.0,100 U22(l1#O.

*C E. START LOOP OVJER THE QUADS.

C Fl P'I CK UP OURD. I NFORMRT ION*29t, ' BJ

wpwYC=T:'(K ")YC=TY(K)

A =TA(iK)

* Y-E(J+S1)

YY=9"J+1 1)

C F2 COMFUTE LENGTH OF SIDE'S 0r' QUAD.0 12= c'O2F(01 ,X2, Y 1, Y2, 0O., 0.)D23*rL2F(X2x3Y2,Y3,.0,.0)03 4=SQ2F X3, X4, Y3, Y4, .0, .0)0'41=z11F(X4,X1,V4,Y1, .0. .0)

C F3 COMPUTE SLOPE OF SI DES

1 17

'--w - Irr --- --- rrxrwi- rwr vrrrNir.- tR t- U Nup'w, rgr v

IF (X2-X3 )305, 300, 305300 C123=1.

e. GO TO 310305 Cr23=Y2-)/2X

C123=0. #

310 IF(X3-X4)315,311..31531C 134-1.31 G TO 320

315 CM34=(Y4-Y3)/(X4-X3)C 134=0. %a %

A 320 IF(X4-XI)325,321,325C.321 C141=1.

G0 TO 3ZO325 0t41= (YI1-Y4'I/ (YX1 -K"4

C14 1=0330 IF(X'-X2)a35, 331,335r.J331 C1 12=1.

00 TO 340335 Ctl 12=,Y2-Yl1)/(X2-X 1)

F4 COMPLITE QUAOF;FPOLE MOMENTS'30CIXX=6(J+ 17)C I XYT=P(J+ 19)

oY78=F (J3+ 19)C F5 COMPUTE St N AND COS OF SLOPE ANGLE FOR: EACH StD

CYI2=(Y2-Y1)/D12mY 023= (y2Y;7 I022

CX 2=(.X 1-x2 )/0 12,:C:X23= (X2-XS .)!/0230X34= (X3Z-X4j ~C*X1=(X4-Xl1/4

FF, COMPUTE MAX 01 MALST=SO2F(X1,X3,Y1,Y3,O. ,O.)SrL=50F (X2,X4, Y2,V4,.0, .0)?FST-5231,4 4

:41 ST=Sf'2c 173. Hr1F, LOOr OvEFi THE OFF BODYf POtINTS

*I On- 03 .20 .30=1, NOBE

IS 1= 1

.:49 FFO PQ=S:C Fr - C. Y OOZC, Z0O )l C H. DLTLF4ItN METHOD

I FcPQ-ST-4 )?50, 3253, 450250 U (XC0 XC )$ f t+'YCO-YC )*:'YX+( C 0-20)*7Y

Y=XO-'.L *.'QYvC -Y )-YY(CO2 K.+Z-Q7C)*Z

1F(PP-ST2.5)35-, 35-5, 400c COMPUTE I t1Z"JSED VJELOr.I TY E'Y EXACT METHOD

Ri2=$SO2 F('X ,X2 "7w', Y2,Z, 0.'P R3=S 02F ( X Y-" V, 'Y2,,0.R4=Sc.YF02,X 4, Y, Y4,2,O0.) I

CLA=LORI +R-0,,12,)/(A P+212)hCLA -2=ALOC(;2+3-2 )/(P2+R3+023 ))

CCLFIS=RFLOG(( Fe+R4-03-,4 )/(R.3+P4+034))CLA4=RLOO C(F:4+R 1-04 1V(R4+R1+04 1))

- ~ 4.IC. -.*C--

y)~ 4 L~w 1CY3't LR+CY4~lieCY ~L1

TUY=C'X12*CLR 1+CX2 3*CLA'2+CX34*CLRS+CX4 1*CLR4TVZO0.JF(ABS(2)-.O01*ST)375,361.361

a361 ZSQ=Z**2E frZSQI+(X-X? )**2

E2260Q+(X-X2 >**2

E4='SO+(X-X4 )**2Hfr(Y-YI )*(X-X1)H2=(Y-V2 )*(X-X2)

H4=(Y-V4 )*(X-X4)I F(CI1 12 )353.363,3154

p 363 WS1=(Ct112*E1-H1)/(Z*R1)WS2=(Ctl124*E2-H2 )/(Zt:R2)AT 1=AT.AN"WS 1)AT2ARTAN(L4S2) .-

TVZ=AT 1-AT2

166 RT3=RTAN( (0t123*E2-H2)/ZP)AT4=ATRt'( (CM23.*E3-H3 )(Z*R3)TUZ7=TU:'+AT.-P14

367 F (Ct34)36 369,36

RTS=RTRN( (*Ct-134*E4-H4 )/(2*F;4))0.1 TVZ=TUZ+RTS-AT6

3T C! 41)270,3-70, 375.70 RT7=AT~Rtl(CM1*E4-H4)/(Z*R,4))

ATE=FT&FNYr C-t4 ;:*E *-H )/?L$1

37 5 GCO TO 450J COMPUTE M NUSED VELOC ITV BY CU!P"RRPOL-E MiETHOD

400 PQ =,:O*''

F;='YSQ+Z30i-4. *XS0* C$XySQ +ZlSQ-4. *Y SO

LS2* .~*FS+30 '*XS0)/FlPQ7

W4=3 <c R0

T .JX F' L- I X 4 32C-Y*3W4S2=t~ b + *''SJ )/RPR'?

TUY=R*WS$1 -C 1~ ; CIYY4 W$4-C: IYYfl3T2TI4:=*:(+-F(C: krI XX*PS-5 . IC X'Y-*X*YC IYY'*CS )/PP',iT)

* ~ ~ 45D UX( IS) TWX+TY +TX(K)

$Z -=Tk. *Z.)+kJV*"7Y+TVZ*Z-N (K)GO TO 470

C K. C:OMPUtTE I NOLISED IJELOC I TV BY MONO$fPOLE tETHC'0t460 F$CK=R/(ARFFO 4t3

'~ ~ Z r- Z . YC-C *RFC1PQ32

C L AU;LEjr ' OFF 600OY POINT IN FLP!iE OF cYfIETRY

470 0TO (48.0, 485,490, 4Q 51 rvI, 505, 511oC, 515)

02(J1)=JX 1)IQ I )=L'x"O 1)

1197

U1(J1+1 1Y~)

i\3( U(1+ 1 )=Y(1I)U1(J1+2>=IJZ(1)V2(311+2)=UZ( 1)U3(31+2)=UZ( 1)IF(SYM~) 525,525,481

481 19=2c XZ SYMETRY

YCQ--YCQ0G O 3 45

485 IF(SYI- 1>517,517,486 5.;

C XY SYIIETRYp485 9S=3

ZCO=-ZCOGC TO 245

490 19=4YCQ=-YCQ0O TO 345

495 IF(SYI-2)516,515,496,£ YZ SYTIETRY

4K 1=5XCO=-XCOGO TO 3:45

5 Q 09I=5

YCQ=-YCQ0 0 TO0 3.45

505 9S=?

K, TO 345510 I9='o- I

ycrr=-yr.f:GO TO 345

C M + ADD COHTR I BUT I OtiS OF ALL REFLECT I OhS515 U 1 (J1 )=U 1(J1 )+UX(Sl' 8 Q+Y* 1''7 )+U): (6 )+jX (5 )

i123W( ) 2,- W I )-L)X (8S )+-t'Y (7 )+14X (E5 )+U( (5 )

j31+1 )U1' 3 1 +1 +.!( )tjI.A 7 +LJ ()l I 5V42(31+ 1 )=V2 (J1 1 >+UY k' )+IJY(7 )+UY (e. )+U.Y(5>USl (3I+ 1 )U3 (3 1 + 1 )+V' (S -VY (7 )+UY C S )-UY (5)'1(31+2 )=Ul,' 1(12)U()U?)U 6U()

V3(,31+2 )=143(3 1+2 )+VZ(3I )+U2(7 )+UZ(5 )+UJZ(5)5 16 U1(311i=UI(31 )+UYc4)+UX'(3 t we

'32(3 1#32 31)+14l(4 )-IyX( (3) r;

U 1(31I+ 1 )=0I(31I+1I )+VY("14 -QY( S)V42 (31+ 1 )=U2(W1 +1 )+t.,'(4 )+UY (3)

3 J 1433+ 1 )=U?3(J3 1+1 )-UY( 4)+UJY (2)

12 (+2 )=I '32 (3+2 $142 (4 )-+.17 (313(1+2 :'-U J 1+2 +24 +23

142(31 +1 )=V43W 1+1 )+UYZ(2)+V51)3(31+ )=U 11 )1Y (2

141(312)=4(3 )+<2)

Q12(31+21 )=U2(3 1+2 1 32(2)

V33(J 1+2 )=V2,3(J1+2 )+Q( 2),=V IIY(Jf%)+!I1 1 )*S1 (P

1277'

I VI(JQ)=UYI(JQ)+U1(2)*St1(P)VZ1(JQ)=VZ1(JQ)+V1(3)*S1(P)VX2(JQ )=LJX2(JQ )+V2( 1)*S2(P)UY2(JQ)='JV2(JQ )+t.2(2 )*S2(P)0Z2(JQ )4)22(JQ >0)2(3 )*82(P)MX$ (dc )UX (JO )+V3 ( 1 )*$3 (P)UY3(JQJ)=VY3(JO >0)3(2 )*S53(p)a5 Z0 V330(J)=Z3(O)+3(3)*8:3(P) PIT

K=K+1J=J1+20IF (K-NP )2%6,295,t599

*C P. END OF LOOP OVER QUADS599 H=RP(IRK)*DT

DO 7310 I = 1,NOBP63 FORr1RT(2X, 13, 3F12.5,9X.4F12.:.5)

DXC I )=(X*j1( I * IyQ ( I )+VY I *',Y2i) U *)-( I)DW W( I )=-(dYiI( I )+kUY 1*IjY2 ( 1 )+UZI *VY3( I

730 DZ( I)=-(UX I*UZ1I( I)+Y I *UZ2(I )+UZ I*U3I)IF(1RK.EO.5) 00 To 900IF(IRK.EQ.4), GO TO 800

00u 750 IiNoe~PXF' )XT)+D::,(( I ):

YP( I)=YT( I)+DY( I)ZP( I =ZT( .I >DZ( I>SKvX( I V)X ,+rim. I' )*rt(I

SYI )=SKY( I )+GM1 I RK )MY ( I )7" C! )=CKZ( I )+Gfl( IFK )*DZ ( I )I RK = IRK,, + 1GO TO 98

80'H=DTDO0 830 1=1,"N0Ep

c")' ( )=- C *1)5Y i I )+'J *:W/2 I )i+ k7 I 2H'( 1DZ ( I )=-( U'I.UZIK I )+WU'/i 12( 1)WtZ I *UZS(XFP( I ')=XT( I )HO(I )6+SK.-( )*H'(T( I )X(IYP C I T. H+ DW I) Y6. + SKY *:

Z'P(I )=ZT I )+ 1D( )/6.+SKZI )*H

1 I) = 0 .

a-GO TO 9P900)C I RK: = I

DO 905 I =1,NOEPD, 30=0C I '$*2D (I

905 CONTINUELJrI!TE'5, f, I ITC

61 FORM tRT':rHv STEr, 14/)JR I TE( E1 6, 62''

6' FocATC'Y 4H' I E, 5X, 1 HY, I IX., I'Y, 1 I,12X, IH UX, IOX2H'YI1 1 Ox' 2ZH 1O),2HCF)I4PIITE(5636 (I YP( I ),YP( I ), ZP( I ), OX ( I )D'' I f' I)cp'(. I) 1 1, HOBP)L4R iTE( 6) k P ) I': ), YF' 1 ), I) 1- 1, ICEIF(ITO*.EO.NSlT) GO TO 910I TC= ITCt 100 TO 599

in 121

IOFCIEND.EQ.O GO TO 23REWIND 03 raREWIND 04ENOF ILE 16REWIND 16STOP 6ENDFUNCTION SQ2F(X1,X2,V1,Y2,Z1,Z2)rX=X 1-X2Y=Y 1-Y2

I PS=2**2+Y**2+X**2F='ie$-(X)tAEBS(Y)+AES(Z)t 1 .QE-2'

P R= .3422*(R+(RS+FS)/R'R = P+FR:S:eRSQ2'F= .25*lR+RS/--RRETUFiEND,

122S

-~~j 7-. lip -. - --

APPENDIX Ulf - XYZPF SECTION PF?

PROGRAM PFP7(TAPE7, INPUT= 128, OUTPLIT= 128, TAPE5= INPUT, TAPE 17, -

ITAPE6=OUTPUT, TAPEO3, TAPEO4, TAPE3=TAPEO3, TAPE4=TAPEO4)

C XYZ POTENT I AL FLOW PROGRAM UERS I ON 4 AND VERS I ON 5 SECT ION 7C COMPUTES VELOCITIES AND PRESSURE COEFFICIENTS FORC ON BODY STREAMLINESC

COMMON X(658>,Y(658>,Z(658),XN(650),YN(650)%

2, U22(650 ), UXS:(650 ), UYS(650 ), 1)23(650 ), XC (658 ), YC 1(656 )3, XC2(658),YC2(6,58),XC3(6)58),XC4(658),YC4(656),442(3 (65c-), YS(6k'E58).Z3(656"-:),X4(658), Y4(658)2Z4(658)*.15)DIMENSION XL( 150), .YL( 150),2ZL( 150), UX(( 150) LIYI 150O> LIZ50

5, DjmX(65 ,PRS 15, YC3 (658), SF(S5), XC(5).YO0F:(;")7NSP(SO),WS(220) ,XST(50 ),YST(50),ZST(5Q)

EO!UIUIALENCE (WS(20 1), NP), (YC3,YC2)READ c'S3(PROE (I ), = 1, 15)

READ~~ ( ') S(), 1 1.,220)RiEAD i) K) I), Y (I 2(I )XN( I )YN( i )ZN I)

ISKIP, 1=1,'NP)

U0EF 4W-2(20)W~iTE(5,5% 1 UER

5 F OR M AT (4 6 HOX)YZ POITENT? A L FLOW PROGRAM SECT I OIN 7, VERS! QU 2) -.

IF (IUJEF;EG 9) REROKOS) SKIPREAO(CTA)ScL IFP

READ 02 >4I PI I L'1< I) UZi 1 1I1i,NP)

NE=K(NP+I1 12

C40 Y- - I (Jr4 (rl X 'I1YU2(J),XCSG(J) XC4KJ ), YC4 (J), X3(J ), YQKJ),2-23,3JX4 (J), Y4 kI\Z, 4 ( J' SK IP.,K= 1, 7) 1=S, I FN>

I FKNC. NE. 9)c;' GO 70 450I

R;EWIND 04DO 91-- 1= I NP

02=('XC2(I )**2.+Y02(I)*)?1OJ=K0 3K( I 14*2 +YC:3 I. 1 *2 1 1) 1r4=! 4 + 't

90 OfM ( I )=M1 AN kI (D 1, 02 D 3 4)

M ID=75100 READ (5, 11> UX I UI k7 I NL IN, MAYAJ, IWR I TEAtIAcH -

IF (Er'S' -5,,NE.Cr NL I ti=-V

rI F ( NR". LE 0 .R+ MAXJ OT 07 F 72' = HFMNJN=M IO-MRXJMAXJ=MID.?lAXJIF (MHXt) GT.MID1*2) MAXU=MID*2

123

-. . . . . . .. . . . . . . .%

LIF CMINd LT.1 MINd =I

WRITE(7 ) NLIN%WRITE( 17) NLIN,UXI ,UYIJZIIF(NLIN.LE.O) GO TO 550WRITE(6,30) (PR02.(I),Iz1,15)WRITE(6.34) VXI,UYIUZI,NLIN,IIXJ,ILJRITE,ARC:H

34 FORMAT(34kOON EBODY STREAMLINES - INPUT DATA /6k UXI r,F1O.5/16H VI =,F1O.5/61- VI =.F1O.5/6H NLIN=,I1Oi/2 6.H JMAX=, 1 10, /,SH I LJI~TE=,110, /,§QH MACH NO=,FI11.S)WRITE(6,38)

33 FO0 ." RT 'c 2H .: F0TRE A.'L I NE ST A RT ItGPW N TS/15H L I N E, 1X , lH), , 2X, 1IHY1 12), lkH2, 1X ,S-1HNSP)L IN=NLI N

P EQ 45 i<lLWREAD(5, 12 ) XST (I ) YST(I ), ZTKI )i NF1)IF (EOCF(5X.E0.0. ) CGO TO 4NL I N= I- 1lIRIT&k6,42* r4LIN,LIN

42 FOR:$iT( IHO,-, I5, 25SH STREAiL I NES SPEC IIED NOT ,13)

IF(NLIN.LE C) GO TO 550CGO TO! 41c

4~ 5* T E iE 4;5 ~CT:! ~ 7, H:46 FOFRMATJ I 1-,3. F 135, 1g)48 CONTINUE

USQ=XI**'&IJI~k+H~**t2I F k-' F H En0 C' 'C TOC 1130

Uso 0DCO 110 Qfl 1 1

= 1..)ty 4f'C': + 1 *-C ' 11 ' *4 -. +

I TI NR kLOL-

D1110 I

C. r, 4; = H " 'c Ccrripr0MN: . ,- Nq+C ! j', , E I 10 Ctl'H

112'0 FORM1-Ttz21k CPITICAL MAC-H NO0. =,F5.3)1 113 0 C:OIT11 ; I

SRT LOPOER SCTR:EWiL IEDr. 4 CC L='!

a101 JI=1

t>'I1)=oU Ill MCI 0LI(M IEI C, -;

10 2 N.0=NSEF ''a. LNO=NC

xL'; D=)ST(ILL)',£ YL (MO1 '='ST (L)

ZL M1IE=ZS-T(fLL')

124

JL=JC SEPARATE CALCULATItON OF SECONDC POINT FROM MAIN LOOP

XL T=XL(J)-X (NO))X3(NQ )+ (YL (J )-YNQ)A*Y3(NQ)+(ZL Wj)-Z (NQ )*3 NO)

YL-T=(XL(J)-xuiQ ) ,)*)(4(9Q )+(Y.L(d ),-Y(CI i ) )*y14c"jN+(ZL<J)-Z(NQ))*Z4(NO) V

XL(J )XLT*X3(NQ )+YLT*X4(NQ )+X (NQ) o47YL (3 )=XLT*YS( NQ )+YLT*Y4 (NO )+Y(NQ)ZL(J )=X-LT*23(NQ )+YLT*Z4(NQ )+Z(IIQ)

105 1dGO TO (6-20,600,610,620) IO.T

600 NR=NQ+ 1

GO TO 107t 10 NR=riQ+2

NU=NQ- IGO TO 107

t'20 NR=NO-2N! !W'c+ I

GT r,.N 1 1

UYQ= ed)' I *43Y I (NO )+Y i *VJ\2 (NO ) Id *4jY3(tNQO'i=- V, Ux I*41(C!)U I *07"2(NQ )+UZ I *QZSIHQ ) )

L iF U '-)'1(rP;)UY',22(F )3 I *J):3(r;) )I , UY I (NR )+UjY I *2( NR )4U2JZ I *UY)(P)

UYRF-rtrx I *'1 1 (tE; )+YI *1jZ2(NF;)+YZL I *LJZS"lF;) )UX=(XI IV' (Nd ),+l.Y I *Q1jX2(1L)+tZ I *LVj3(NLI) )UY= NI *:UYI (NLI )+UY 'I *IJ,'2(NLI )+'J2Z I *13Y(NU,)

Ijy'fl=-(IJY !y 'V'1 (NU )+UVY I *VYz2 (rIM )+ UZ *IiZ3 (NU)

C TR CPORM' 'ELOC IT IES TO QUAD SYSTEMlUQ=UO*X3(NO +Lly~y3 NO )IJ2*3 (to)

J=Lflf4 (NO )+L!YC'Y(P; )+LIZO*24 (NO!CSF;= 1. /(XN(NO )'XN(4R )Y'N("NQ )*Yi''k'p,)+Z'N(NC!)*ZN(NP))

'~UfP*(3( IP)+UYRVYO (NR )+LIZR*23:,( R)U7 -r >4(f; )LIPX 4 (NF: )+UIZR,*2-4(MR ) *S

~ F, (XONP 'VY(r~i )YS~lF;)*Y3(NO )1+ZSN R )*:Z3(Nj! Op (4(rP;)VX NX+Y (P)'Y (NC' )+Z4 (NE; )*7': (O)-Ic=T;#) ? ; \~T'YR

* V f 'T*'4 NO'F

O=61'Ot 4' IML' rI-UY'Y4,(NO ,+ UZLI*24 (NO. ))/CE;UF!N2 ; ELPT1I"E Cfl0ROD!NATE3- OF NEIGHB(EORING CURDS

iD-i'1

*(YN4)

YT=(-4*:SflRT' 'y'T-*v4+T*:4 *- +YTT'CSF;P+'YTT )'VCSR4. 166EE.Ef,'7

H~Y*X+TY'

125-.

- - -. 7 7- -: .77W,.

E ZT=XD*XN(NO );YOIYN(NQ')+ZOD*ZN(NQ) ).1666

C FIN' COEFFICIENTS OF VELOCITY FUNCTIONSDEN= 1. /rXR*YIeuXut*VR)U 1 ((LIP-LIO)*YU- (LUj-LIQ)*YR) *DENIj2=-(W(IF-UC! *XUl-(UI -LIG *XR )*DENU 1=( (UR-VO )*YLI-(U-UO )*YR) *EDEN'32=-( (VR-UQ )*XU-(U-UO)*Xt. )*OEN

C FIND UELOCITY AT STREAMLINE POINTLISL=UQ+U 1*XLT+L12*YLT

UXP=LISL*X3(NO )+USL*X4(NO)LIFIJLYE NO (NQ:*Y II

U L:~SL ,z.:-,zND+S L *Z 4 (NC F INDA GEODES IC CURUATUPES Oil G1,2

VUSODL'=SL**2+USL**2'DEN=USCJO*SORTUS.QD)Gl< P= ULj L (1.1:L*4 2-LIS3L*U2, OES4 ( 4U1ULU1

O F;.i43 L 1 RL STR EAM FUNOT IOIN

, =Tf, T*J!C- *

TZ:RM FLV:CT;ON RT CORtiER P'OiTS

XCR F (-SL'NO)C F'. r,4 .. C 4 r N,4

SC .$ 2: '17,0 Nu)YCR 4)V(49 ~ 'D- 110 N=1,4

112 Z~t)= o+I X CF (N )+lU!ctYO(N)XMXO'N)*CF 1?YYOR '2

IF ( ~'~t+) GE. 0. GO TO12

0 FitiO tiTEF;" -E' -T; OH W! TH S I-ECF Cll-oua:.

T *r~ F'; B- 24 7$f+5 1

F <iF' LE 1 Ft TF, GE 0 >Gn Ti) 115

GO TO 115 -

113 IF <PC .EO. 0 )GO TO 120TF'=SF<N>/F'fl

115 XN'=( I .-TF' )*XOP.(N )+TF'),XOF;(11N+ 1)Y I=(. -TP >*YORcjj )TP 4 'YOFHR 1)

TESTP=" (YHP-XLT )*H-C;+(YtAF'-YLT >421 )*O I PTIF (TES.TE LE. TEST) GO TO 120TES-T=TES:_TP)IH.T=.XNP

72

_7 . - -.

120 YONTIYNUEATVLCIYADCRRUEu

IF( TEST EQ.. 0) GO TO 25'u

0K2(J)=(6K2J )+GK2P)F4'A

CP(J )=.-(U(J )4''2UYJ)*2+I(J%*42 /GKE1(J)=SQRTI(J)G 1 .-C:F(J)

U S=UYT I I <YNT*U

CPJ =1 I- I 2* . : :7

c ~ i~l',jrD <i TYA00) T C iN

't RZ Ik ' ILL )=NO *I2IIL~J N".;*Ic;4~if __~i

r''IJ i!.fl=*i+I4+hJi . : *ItJl42,

ClSL=-+'fNT ( Y} T KL )42+ 'N -'T)*2 )'S

THI rNT 1 ___

IR LCL)

.4-' UT+-.TU +vi4X ~i~(i

' 31 -- ,e? l)+YNit*Y4f NO )+%Yr (ri)

:;7 L~ IHi O E. IIR ) GE Ft Ti

F :U IND TI*1 Ola

I IJ++ kY

4o I , I 1 '+44 r2

o ~ ~ ~ ~ ~ -C I -LI~*~ l 1:

05 4 t*.X- 4 '-'C I( 1 I -,

-L J )1-7 1+'J +2: (JLr4 ' .J ~ 1 4I + . ~ ki4 I- r, ~ '

1 -4 <( YL Q :~'()-Y' J ! i4'I

TE T= --- i 4'ty~

IF(T'E$T. GT.0. ) GtO TO 280PIO =S 0JRI(Z-SO+(X LT-V.C 1: 1 4 42+YLT-YC 1Ic

127

RC3=SCIRT(ZSQ+(XLT-XC3(< 1) )*+YTY(1 ) )**4)RC4=SQRT(ZSO.+(XLT-XC4(lI>)**2+(YLj-YC4( ) )**2)

*TEST= ((RC1+RC2)**2)-OS1 *1.21IF(TEST.LT.0.) 3010O 105TEST= ((R02tR03 )**2 )-0S2 * 1.21IF(TEST.LT.0.) G0 TO 105TEST- ((R03+R04 )**2 )-DS3 * 1.2 1IF(TEIST.LT.O.) 0O TO 105TEST= (C4+RCI)**2)-0S4 *1.21IF(TEST.LT.O.) G0 TO 105

280 1=1+1IF(I.LE.NP) GO TO 250

282 IF (DtRT LT. 0.) 30 TO 285

31 ;'=-1

yip"x=jGO TO 102

285 JI'IN=J'SS=STNL(JN IN)00 29;0 J=311I N, .JNAX

290f ST711L (J )=STNL1 (,1 )-SSJt1lN=Xt I N+ 1

IF= 1.

L4PI TE<(F, 30)(PROB( 1), 1=1, 15)30FOF:IAT("I 1 15R4)

WR4P I T Er O 6 % " L 2' ,hi , VUZ I.-2'FORMT 18HCIt ONSET FLOW*, VX 1,F 32,HJIF'32)4'Z= r3

W4R!TEr ~cij\ LL NSP(LL), XST(LL,$(LS(L5O F0RfAT~ 11 rO LI HE NO. , 12, 3 1H PASS ING THROUGH QUARRILPITERqL ,IS

I 28H WITH STARTING POINT, XF2.,2,2Y 1.5,X

U IF (JlHEUI .0R. JUY S.tR)WRI TE(F6R6 5 FORMRl~T (35-H PROBABLE ERROR - L I NE I S VERY LONG)

00 3:30 J=Jr1N,JriXIF ( (STML(J+2)-STIIL(L-1)).LT. 8.*(ST1L(J+1)-STlL (L))) GO TO 3,20

';F;,IiTi4 esl: ELETELLD _____

XL (L )(AF*XL(L )+XL (34 1 ) )/(AF+ 1 .)YL (L fAP*YL (L'>YL (. 1I) )/(AF+ 1.)

UY(L )=,'A +I, L (Ja+ 1 J))(F + 1

0K2(LN)..PCF*(L )+Ci(3± ))(F+ . )

H2(L )=(ciF*H2 (~L *,+H2 (J+ 1) )(AP+ 1.CP ' )= 1Ii . U'.L "4* 2+1_Y (L )*2*,Ij 2(L )2)/US;O

LIRE:3(L; FQRk -09(L)RF=AF+l.

L=L~1- K=,'+I

STM-XL )=SoTtLr.K)XL(L)=XL(K)YL (L )='YL (K)ZL(L )=ZL(K)UX(L)LIX(K)UY(L)=UY(K)

128

UZK'L)=4JYK)

0K2(L)=GK2(K)H2(L )=H2(K)t.CP(L)WCP(K)UAB6S(L )=UABS(K) N

30NQURD(L )NQURD(K)30CONTINUE

L=L+1STML(L >-STtIL(JMAX)XL(L )=XLGJtIA)

N YL(L )='YL(Jr1RX)ZL(L )=ZL(JMrItX) 'f

*U U(L )UXGJMRX)LIY (L )=LIY(JF-IRM")UZ:L )=UZ(Jt1RX()GK I(L)=GK1I rIAX)9K2(L )=CiK2 (JMAX>H2(L )H2l(J'RX)CP(L )=CP(JMIRX)LIAES (L )=LIRS(JIF'Y

HEIURE ( JIM)=NOLIAEI ( JtIRX- 1)NiQURO3(-JM IN ) 10UAD (JMl U+ 1)W-;! TE (5, 5 1)

51 FORt1AT(4H0 1 ,6X, 1HX.9X, ,iHY,S iX, 1HZ-, 0QX, 2H:,,, 2HkJY, 8X, 2HVZ, 0lXY

22HCP, SX,2HKI, SM(, 2f-2, SM4,2H82-, S-$, 2H'-, SM, lJk, gM, IHP-IF (A11RCH EQ. 0.) GOTE' 110

C **CMUECONPRESSI1IL ITY CORRECTIONH£10 1 150 J=JM'IHJMi'R\'LISO = (IUX(J 1)4*2+U j )*2+UI2(j )*2 /SQ

I ~SDA = AUCD<

DO 114£' 1=1,3

IF (F; .LT. .000001) R = 00000 1

* IF (R .LT. .000001) R = 000001USD0 USO /R,**2 .5

1 140 USDA9 (VSDC*US£IR-USDB**2 >/(USDC+USE'A-2 . *Ij$EB)R. = 1 2*SM(i'I: D)IF (R .LT. .000001) P = 000100 1

F:'- = **1.2)5/R

LIYkJ) = UY(J)/RUZ(J) =UZ(J)/R

1150 CF( J) =(*281)(7S1

1150 GOUT I flUE

CO t3 =JrilI N, JIRXK=K+ 1

-53 WRI1T E(6, 60 )KXLI ,YL IZLa ),UX(I )UY(I ),UZ(I ),F'I0W) 93K(H2 () Su~I UR OJRK

50 POFRURT(1X,IS,SFIO.5, IX, SF10.5, 1,6F10.5,1(E)8 FOP4IFT(3F 12.5)

WRI TE( 17.)K, (XL-(I ),YL( I ),ZLKIl ), NOLIFD( I ), I =JtrI ,Jfl~x )EC (WRItTE LE. 0 -- WRITE SL,U, H2,KI'2

C I WRI TE GE. 2 - WR I TE X, Y,Z., CFC IWRITE EQ.. 1 -- WRITE SL,U,H2,K2 AND X, Y,7, CP

r 129

IF (iWRITE.GT.1) GO TO 340WRITE(7 K, (STrL(I>,URBSU ),H2(I>,5K2(I ), I=JtIIN,JtIAX>

C'340 IF I=UtiINJMRY) 00TO40

30WRITE(6,65>

'pG 3010282400 CONTINUEK

00 TO 100 C

c REA NEX SET OF STREANL INES450 LRITE(6,451)IS,NQ45 OMR 4 TAFE 0 EF;FOR, 214)

*550 ENOFILE 7REW INHE 7ENDFILE 17REWIND 17REWIND 04STOP 7END

1370

APPENDIX U1It - TRIAXIRL ELLIPSOID INPUT FILE

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* (EOF:)

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RD-A1I68 W6 FORNULATION OF NUNERICAL METHODS USED IN THE XYZ 3/'3I THREE-DIMENSIONAL POTENT..(U) TEXAS A AND N UNJYI COLLEGE STATION COIL OF ENGINEERING N J BERRY NAY 86

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