Prediction of Vortex Shedding From Circular and ... · PDF fileNASA Contractor Report 3754...

NASA Contractor Report 3754

Prediction of Vortex Shedding From Circular and Noncircular Bodies in Supersonic Flow

M. R. Mendenhall and S. C. Perkins, Jr. LOAN COPY: RETURN TO

AFWL TECHNICAL LIBRARY KIRTLAND AFB, NM. 87117

CONTRACT NAS l- 17027 JANUARY 1984

https://ntrs.nasa.gov/search.jsp?R=19840009061 2018-05-25T09:21:01+00:00Z

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NASA Contractor Report 3754

Prediction of Vortex Shedding From Circular and Noncircular Bodies in Supersonic Flow

M. R. Mendenhall and S. C. Perkins, Jr.

Nielsen Engineering G Research, Inc. Momtain View, California

Prepared for Langley Research Center under Contract NASl-17027

National Aeronautics and Space Administration

Sclentlfk and Technical Information Branch

1983

TABLE OF CONTENTS

Section

LIST OF SYMBOLS .

SUMMARY.....

INTRODUCTION . .

GENERAL APPROACH

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METHODS OF ANALYSIS .

Conformal Mapping

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Analytic tranformation . . Numerical transformation .

Body Model . . . . . . . . . .

Circular bodies . . . . . . Noncircular bodies . . . .

Vortex Shedding Model . . . . .

Equations of motion . . . .

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. Surface pressure distribution Separated wake . . . . . . . Forces and moments . . . . . Vortex smoothing . . . . . . Vortex core . . . . . . . . .

PROGRAM...............

General Description . . . . . . . Limitations . .' . . . . . . . . . Error Messages and STOPS . . . . Numbered STOP Locations . . . . . Input Description . . . . . . . . Input Preparation . . . . . . . .

Panel layouts . . . . . . . . Numerical mapping . . . . . . Integration interval . . . .

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Page No.

V

1

1

3

4

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5 6

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7 8

9

10 12 14 16 18 21

23

23 25 26 28 31 53

53 56 57 58

iii

Section

Sample Cases. . . output . . . . .

RESULTS . . . . . . .

Circular Bodies . Elliptic Bodies . Arbitrary Bodies

CONCLUSIONS . . . . .

RECOMMENDATIONS . . .

REFERENCES . . . . .

TABLES I AND II

FIGURES 1 THROUGH 33

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Page No.

58 60

65

65 70 75

76

77

79

iv

LIST OF SYMBOLS

a

b

cn

cY

5 'rn

'n

C P

cpI

C Y

cN

D

G

e ref L

MX

M Y

MZ

semi-major or horizontal axis of elliptic cross section

coefficients of conformal transformation, Eq. (9)

semi-minor or vertical axis of elliptic cross section

normal-force coefficient per unit length, Eq- (39)

side-force coefficient per unit length, Eq. (43)

rolling-moment coefficient, Eq. (47)

pitching-moment coefficient, Eq. (41)

yawing-moment coefficient, Eq. (45)

pressure coefficient, Eq. (29)

incompressible pressure coefficient, Eq- (31)

side-force coefficient, Eq. (44)

normal-force coefficient, Eq. (40)

diameter of circular body, or equivalent diameter of noncircular body

complex velocity component, Eq. (18)

reference length

total number of Fourier coefficients used to describe transformation,Eq. (9), also model length

rolling moment about the x-axis

pitching moment about the y-axis

yawing moment about the z-axis

V

MC0 P

PC0

%o

r

r'

r C

r 0

Re 5

S

ue

U r

U

v8

VIW

Yv W

XlYIZ

xm

a

a C

B

free-stream Mach number

local static pressure

free-stream static pressure

free-stream dynamic pressure,

radial distance from a vortex to a field point

radial distance to a point on a noncircular body, Fig. 4

vortex core radius, Eqs. (50) and (51)

radius of circle

Reynolds number based on boundary layer run length and minimum pressure conditions, Urns/v, Eq. (37)

reference area

surface velocity in crossflow plane

axial perturbation velocity

local velocity

vortex induced velocity

velocity components in real plane

free-stream velocity

complex potential, Eq. (16)

body coordinate system with origin at the nose: x positive aft along the model axis, y positive to starboard, and z positive up

axial location of center of moments

angle of attack

angle between free-stream velocity vector and body axis

angle of sideslip; also polar angle in c-plane, Fig. 2

vi

8’

Y

Yr

r Ax

3

n

8

V

5

P

u

-c,X

Subscripts

t-1

( )m

( )cp

local slope of body surface, Fig. 4

ratio of specific heats

exterior angles of body segment for numerical mapping, Eq. (7)

vortex strength

axial length increment

-complex coordinate in an intermediate plane, Fig. 2(b)

vertical coordinate in an intermediate plane, Fig. 2(b)

polar angle in v-plane, Fig. 2(a)

complex coordinate in circle plane, Fig. 2(a); also kinematic viscosity

run length, Eqs. (36) and (37); or lateral coordinate in an intermediate plane, Fig. 2(b)

free-stream density

complex coordinate in real plane, Fig. 2(a)

lateral and vertical coordinates in circle plane, Fig. 2(a)

roll angle

velocity potential in real plane

stream function in real plane

conjugate of complex quantity

vortex m

center of pressure

vii

SUMMARY

An engineering prediction method and associated computer code NOZVTX to predict nose vortex shedding from circular and noncircular bodies in supersonic flow at angles of attack

and roll are presented. The body is represented by either a supersonic panel method for noncircular cross sections or line sources and doublets for circular cross sections, and

the lee side vortex wake is modeled by discrete vortices in crossflow planes. The three-dimensional steady flow problem is reduced to a two-dimensional, unsteady, separated flow

problem for solution. Comparison of measured and predicted surface pressure distributions, flow field surveys, and aerodynamic characteristics are presented for bodies with

circular and noncircular cross sectional shapes.

INTRODUCTION

Current missile applications requiring high aerodynamic performance can involve noncircular body shapes in supersonic flow at high angles of attack and nonzero roll angles. The angle of attack range may be sufficiently high to cause formation of body separation vortices, and the body vortex shedding characteristics are directly influenced by the body cross-sectional shape and the flow conditions. It is desir- able to model the body vortex wake by means of a rational method capable of considering a variety of body shapes over a

wide range of flow conditions. It is important that the separation vortex wake induced effects on the nonlinear aerodynamic characteristics of the body be handled properly.

The phenomena of interest are the sheets of vorticity

formed on the lee side of the body at high angles of attack. The vorticity is formed by boundary-layer fluid leaving the

body surface from separation points on both sides of the

'body (Fig. 1) and rolling up into a symmetrical vortex pair.

A method to predict these flow phenomena in the vicinity of circular and noncircular bodies in subsonic flow is described

in References 1 and 2. The extension of the method to

supersonic flow is described in Reference 3.

The purpose of this report is to describe an engineering prediction method and associated computer code developed to

calculate the nonlinear aerodynamic characteristics and flow

fields of noncircular noses at high angles of attack in supersonic flow. The objectives of the method are to use a

three-dimensional, attached flow, supersonic panel method, or

a supersonic line source and doublet method to represent the

body and a two-dimensional, incompressible, separated flow

model to calculate the lee side vortex shedding from the body alone at angle of attack and angle of roll. The predicted

pressure distribution on the body under the influence of the

free stream and the separation vortex wake will be used to calculate the aerodynamic loads on the body. Conformal

mapping techniques are used to transform noncircular cross

sections to a circle for calculation purposes.

The following sections of this report include a discus-

sion of the approach to the problem and a description of the

analysis and flow models required to carry out the calculation.

The prediction method is evaluated through comparison of measured and predicted results on a variety of body shapes,

including circular anh elliptical cross sections. A user's

manual for the computer code is also included. The manual

consists of descriptions of input and output, and sample

cases.

2

GENERAL APPROACH

Bodies at high angles of attack exhibit distributed vorticity fields on their lee side due to boundary-layer fluid leaving the body surface at separation lines. One approach for modeling these distributed vorticity fields has involved the use of clouds of discrete potential vortices. Underlying the basic approach is the analogy between two-dimensional

unsteady flow past a body and the steady three-dimensional flow past an inclined body. In fact, the three-dimensional steady flow problem is reduced to the two-dimensional,

unsteady, separated flow problem for solution. Linear theory for the attached flow model and slender body theory to represent the interactions of the vortices are combined to produce a nonlinear prediction method. The details of the

application of this approach to the prediction of subsonic flow about circular and noncircular bodies are presented in References 1 and 2, and the extension to supersonic flow is

described in Reference 3. Other investigators have also used this approach to successfully model the subsonic flow

phenomena in the vicinity of circular cross section bodies (Refs. 4 and 5). The purpose of this report is to document the extension of the previous subsonic analysis of References 1

and 2 to supersonic flow.

The calculation procedure is carried out in the following manner. The body is represented by supersonic source panels (noncircular cross section) or supersonic line sources and

doublets (circular cross section), and the strength of the individual singularities is determined to satisfy a flow tangency condition on the body in a nonseparated uniform flow at angle of attack and roll. Starting at a crossflow plane

near the body nose, the pressure distribution on the body is

computed using the full Bernoulli equation. The boundary

layer in the crossflow plane is examined for separation

3

using modified versions of Stratford's separation criteria.

The Stratford separation criteria are based on two-dimensional

incompressible flow. At the predicted separation points,

incompressible vortices with their strength determined by the vorticity transport in the boundary layer are shed into

the flow field. The trajectories of these free vortices

between this crossflow plane and the next plane downstream are calculated by integration of the equations of motion of each

vortex, including the influence of the free stream, the body,

and other vortices. The pressure and trajectory calculations

are carried out by mapping the noncircular cross section

shape to a circle using either analytical or numerical

conformal transformations. The vortex-induced velocity con-

tribution to the body tangency boundary condition includes

image vortices in the circle plane. At the next downstream

crossflow plane, new vortices are shed, adding to the vortex

cloud representing the wake on the lee side of the body.

This procedure is carried out in a stepwise fashion over the

entire length of the body. The details of the individual

methods combined into the prediction method are described

in the following section.

METHODS OF ANALYSIS

The development of an engineering method to predict the

pressure distributions on arbitrary missile bodies in super-

sonic flow at high angles of incidence requires the joining of several individual prediction methods. In this section,

the individual methods are described briefly, and the section

concludes with a description of the complete calculation procedure.

4

Conformal Mapping

The crossflow plane approach applied to arbitrary missile bodies results in a noncircular cross section shape

in the presnece of a uniform crossflow velocity and free vortices in each plane normal to the body axis. The procedure used to handle the noncircular shapes is to determine a conformal transformation for mapping every point on or outside the arbitrary body to a corresponding point on or outside a circular body. The two-dimensional potential flow solution around a circular shape in the presence of a uniform flow and external vortices is well known and has been documented numerous places in the literature (Refs. 4 and 6).

Thus, the procedure is to obtain the potential solution for the circular body and transform it to the noncircular body. Conformal transformations used are of two distinct types,

analytical and numerical.

Analytic transformation.- For very simple shapes like an

ellipse [Fig. 2(a)], the transformation to the circle can be

carried out analytically as described in Reference 7. For example,

a2 - b2 u =v+ 4v (1)

where

5 =y+iz (2)

in the real plane, and

V =-c +iX (3)

in the circle plane. The derivative of the transformation

is

da 1- a2 - b2 -= dv I 1 4v2

(4)

5

which is required for the velocity transformation discussed

later.

Numerical transformation.- For complex noncircular shapes, the transformation cannot be carried and a numerical transformation is required.

transformation chosen is described in detail and a brief summary of the conformal mapping

follows.

out analytically The numerical

in Reference 8,

procedure

The sequence of events in the numerical mapping is shown in Figure 2(b). The arbitrary cross section shape of the body in the o-plane is required to have a vertical plane of

symmetry. The transformation of interest will map the region on and outside the body in the u-plane to the region on and outside a circle in the v-plane. The first step is a rotation to the G-plane so that the cross section is

symmetric about the real axis. Thus,

u = y + iz (5)

and

5 = ia

= 5 + iv

A mapping that transforms the outside of the body in the

<-plane to the outside of the unit circle is

m yr

5 1 -rr = 7

IT iv - v,) dv r=l

(6)

(7)

where y, are the exterior angles of the m segments of the

body cross section. In Equation (7), IT denotes a product series. For a closed body

6

i y, = 27T r=l

The final transformation has the form

(8)

(9)

where the AL coefficients are obtained through an iterative scheme described in Reference 8 and r. is the radius of the equivalent circle in the v-plane. The derivative of the transformation is

(10)

which is required for the velocity calculations described in

a later section.

The above numerical mapping procedure has been applied to a wide range of general cross section shapes with good

success.

Body Model

The prediction method described in this report is

applicable to both circular and noncircular bodies. Separate body models are developed for circular and noncircular shapes because arbitrary noncircular shapes require panels and numerical conformal mappings and circular bodies can be represented by more computationally efficient means. Both flow models are described in this section.

Circular bodies.- A potential flow model of an axisymmetric body in supersonic flow is described in References 9

and 10. Such a body is represented by a distribution of line

7

sources and sinks and line doublets,on the body axis to

account for volume and angle of incidence effects, respec-

tively. The strengths of the line singularities are

determined from the flow tangency condition applied at points

on the body surface. A downstream marching procedure from the nose tip to the body base is used for this calculation.

The axis singularity distribution for bodies of revolu-

tion provides an accurate and efficient means to calculate the induced velocity field associated with the body at angle of incidence. The calculation requires much less

computation time than a panel method, and the resulting

velocity distributions are quite smooth. One restriction on the body shape is the requirement that the nose contour lie

inside the Mach cone. This constitutes the major limitation

to the method.

Noncircular bodies.- The configurations considered in this study are bodies alone, without fins or control surfaces.

The body geometry is described by the coordinates of the

cross-sectional shape at a number of axial stations, and linear interpolation between specific points is used. The body is represented by a three-dimensional lattice of super-

sonic source panels distributed on the surface of the body.

The panels account simultaneously for volume and angle of incidence effects. The attached flow solution for a panel

inclined to the flow direction is based on supersonic linear

theory as described in Reference 11 and the method used is a

modified version of the program code presented in

Reference 12. The pitch and yaw angles are related to the

included angle of attack and angle of roll by the expressions

since = sincrc cos$ (11)

sinf3 = sina, sin@ (12)

8

- 0

The flow tangency condition is applied at the control point located at the panel centroid of each body panel. The

result is a set of linear simultaneous equations in terms of the unknown panel source strengths. Solution by an iterative approach produces the singularity strength on each panel and

thus provides a means to calculate the velocity field induced by the body. The nature of the solution is such that the velocity on the body surface is correct only at the control

points; therefore, linear interpolation is used to determine body surface velocity components for use in pressure calculations at points lying between control points. The body panel

induced velocities are calculated one time for the entire body to provide a table of velocity components for interpolation purposes. Using a slender-body theory approximation,

these velocity components are combined with other velocity components for the pressure calculation described in a later

section.

There is a basic limitation in the use of the above

panel method. The angle of inclination between a body source panel and the body centerline must be less than the semi- apex angle of the Mach cone associated with the free-stream

Mach number. Thus, there is a limit to the body nose configurations which can be modeled by the body source panel

method.

Vortex Shedding Model

The body vortex shedding model described in this section is nearly the same as the subsonic model presented in

Reference 2; therefore, the following description will be

brief, and only the differences between the two models will be discussed in detail.

9

Equations of motion.- The equations of motion of a shed nose vortex in the presence of other free vortices in the vicinity of a body in a uniform stream follow. In the circle

W plane, the position of a vortex, rm, is

vm =Tm+ihm r

and the image of rrn is located at

r2 V

0 =- m-r cm

In the real plane, the position of the vortex IYm is

cl mr = ym + izm

The complex potential in the real plane is

W(a) = @+iY

and the corresponding velocity at rm is

dWmW V m - iwm = da = g W,(u) 1 2

U’U, v=vm

(13)

(14)

(15)

(16)

(17)

The complex potential of Tm is not included in Equation (17)

to avoid the singularity at that point. The derivative of

the transformation is obtained from Equation (4) or Equation (10).

The total velocity at rrn in the crossflow plane is

written as

vm - iwm

L = Gcx + G8 + Gn + G, + GT + Gr (18)

10

where each term in Equation (18) represents a specific velocity component in the u-plane. The first term represents

the uniform flow due to angle of attack.

G, =

2 r 0 dv

v, II ’ = u‘=(J m

(19)

The second term represents the uniform flow due to angle of yaw.

GB = m

(20)

The next term represents the influence of all vortices and their images, with the exception of rm.

Gn = i ,b 27rfnv 1

0 m (vm/ro) - (ro/Gm)

1 - (vm/ro) - (vn/ro)

The next term is due to the image of rm.

Gm = i rm 1 271roVo3 (vm/ro)-(ro/3m) m

(21)

(22)

The fifth term in Equation (18) represents the potential

of rm in the a-plane and is written as

(23)

11

The last term in Equation (18) represents the velocity com-

ponents induced by the portion of the body panel singularities representing the body volume effects.

V - iw G,= rv r (24)

00

Since the body panel singularities are three dimensional, they contribute an induced axial velocity, ur.

The differential equations of motion for rm are

dam V m - iw m PC dx VoDcosac + ur

where

a’ m = y, - izm

(25)

(26)

Therefore, the two equations which must be integrated along

the body length to determine the trajectory of rm are

and

dYm V m -= dx v,cosac + ur

dzm W m dx= VoDcosac + ur

(27)

(28)

There are a pair of equations like (27) and (28) for each

vortex in the field. As new vortices are shed, the total number of equations to solve increases by two for each added

vortex. These differential equations are solved numerically

using a method which automatically adusts the step size to provide the specified accuracy.

Surface pressure distribution.- The surface pressure dis- .

tribution on the body is required to calculate the forces on the body and the separation points. The surface pressure coef-

ficient is determined from the Bernoulli equation in the form

12

where

-

:

(29)

and

P - PC0 cp =

+ PVZ

2 2cosa C =

PI l- + - /g I 1 al OJ

(30)

(31)

In Equation (31), U is the total velocity (including V,) at a point on the body. It consists of many of the components described in the previous section. The body induced velocity components from Equation (24) include the total effect of the panel solution at angle of attack and roll. As a consequence of using the full panel solution in this component, the two-

dimensional doublet part of Equations (19) and (20) are not required. The vortex wake contribution is included through Equations (21) and (22).

The last term in Equation (31) represents the axial velocity missing from the two-dimensional singularities

in the flow model. In this case, the shed vortices are the only singularities contributing to this term. The complex

their images in potential of a number of vortices, Tn, and the circle plane is

[

-i Ln(w - un) + i en (32)

13

The "unsteady" term in Equation (31), evaluated on the

body surface, is

3 = Real dw(v) dx dx r=r

0

(33)

Equation (33) becomes

dX dT ('I - Tn) dxn -- (A - XJ 2

(T - -q 2 + (X - An) 2 1

(Tr2 dX 2 dhn drO

,

r n + 2x1, g - o dx - - 2roXn dx I

(rr2 - T,ri) 2 2.

n + (Xr 2 n - An+

dT r 2 d-rn drO

+ (Ari- + 2'rTn --$ - o dx - - 2roTn dx

I

(-crz- -&I 2 2 2

+ (Xr, - h,ri)

(34)

where r2 = r2 + A2 n n n (35)

Separated wake.- The separated wake on the lee side of

the body is made up of a large number of discrete vortices, each vortex originating from one of two possible locations

at each time step in the calculation. The major portion of

the lee side vortex wake has its origin at the primary separation points on each side of the body. The remainder

of the wake originates from the secondary separation points

located in the reverse flow region on the lee side of the

14

body. Both of these points are illustrated in the sketch of a typical crossflow plane of an elliptic cross section body shown in Figure 3. The mechanics of the calculation of the individual vortices follows.

As described in References 1 and 2, the pressure distribution for the primary flow in the crossflow plane is referenced to the conditions at the minimum pressure point,

and a virtual origin for the beginning of the boundary layer is assumed. The adverse pressure distribution downstream of the minimum pressure point is considered with either

Stratford's laminar (Ref. 13) or turbulent (Ref. 14) separa-

tion criterion to determine whether or not separation has occurred. These criteria, based on two-dimensional,

incompressible, flat plate data, are adjusted for three- dimensional crossflow effects in Reference 2. The laminar

separation criterion states that the laminar boundary layer

separates when

$/2 P

= 0.087 sinac

In a turbulent boundary layer, separation occurs when

l/2 -0.1 (ReE ~10~~) = 0.35 sinac (37)

If the criteria indicate a separation point, the vorticity flux across the boundary layer at separation is

shed into a single point vortex whose strength is

r u: Ax -=- Ym 2v2 cosac

co (38)

assuming no slip at the wall,

15

The initial position of the shed vortex is determined

such that the surface velocity at the separation point is

exactly canceled by the shed vortex and its image. When this criterion results in a vortex initial position that is too near to the body surface, certain numerical problems can

cause difficulty in calculating the trajectory of this vortex. If the intial position of the vortex is nearer than five percent of the body radius from the surface, the vortex

is placed five percent of the equivalent body radius.from the

surface. This position represents the approximate thickness of the boundary layer.

The calculation of secondary separation is carried out

in the same manner as described above for primary separation. It is necessary that a reverse flow region exist on the lee

side of the body and that a second minimum pressure point be

found in this region. Surface flow visualization of secondary separation regions on bodies at high angles of

attack in subsonic flow are shown in Reference 15. For

purposes of this analysis, the reverse flow is assumed to be laminar from the lee side stagnation point to the secondary

separation point, and Stratford's laminar criterion is used

to locate the secondary separation point. Laminar separation in the reverse flow region is expected because of the low

velocities on this portion of the body. The vortex released

into the flow at the

opposite sign of the weaker in strength.

secondary separation point has the

primary vortex and is generally much

Forces and moments.- The forces and moments on the body

are computed by integration of the pressure distribution

around the body. At a specified station on the body, the

normal-force coefficient on a Ax length of the body is

16

- I

An L-l 2lT c' = E =l

n q-D 5 Cpr' COSB' dg

0

where r' is the distance from the axis of the body to the body surface and (3' is the local slope of the body in the crossflow plane. This is illustrated in the sketch in Figure 4. For circular bodies, r' = r. And f3' = 8. The total normal force coefficient on the body is

L N D

cN=9ms=g I cndx

0

and the pitching-moment coefficient is

cm=&=~fcn[~]dx 0

The center of pressure of the normal force is

X C 2=&-e

Similarly, the side-force coefficient on a Ax length of the body is

2T 1. C r' sin8' dB - = - - D J P

0

The total side-force coefficient on the body is

cydx

(39)

(40)

(41)

(42)

(43)

(44)

17

and the yawing-moment coefficient is

The center of pressure of the side force is

X cP

2G 'n xm =

ref r+G Y

(45)

(46)

A noncircular body at an arbitrary roll angle may

experience a rolling moment caused by the nonsymmetry of the loading around the body. The total rolling moment on the body is calculated by summing the moments of the individual

components of normal force and side force around each cross section and integrating over the body length. Using Equations (39) and (43), the rolling moment coefficient is

calculated as

M

cL = qJGref

(YCpr' COSB' - zCpr' sinB')dB dx

3 (47)

Vortex smoothing.- A problem common to discrete vortex

models is the singularity in the vortex induced velocity distribution at the loction of a point vortex and the very

large velocities induced near the vortex. This can lead to errors in the surface pressure distribution when there

are vortices near the body surface and to errors in the

vortex tracking when vortices pass in close proximity.

18

-

The vortex diffusion model described in Reference 2 has the effect of removing the singularity at the vortex, but very large induced velocities are still in evidence near the vortex, thus the noted roughness in the surface pressure distribution continues to be a problem. Several solutions to this problem have been investigated, and a discussion of thesevortex flow models follows.

In Reference 16, discrete vortices are replaced by a distribution of vortices on straight line segments connect- ing the discrete vortex positions. In this way, the algebraic singularity is reduced to a logarithmic singularity. The strength of the uniform distribution of vorticity on each line segment is determined from the strengths of the discrete vortices at the ends of the segment. Defining Gd as the complex velocity induced at vm by the line segment, analogous to Equation (21), the velocity becomes

Gd j

where V, is not on the line segment. The strength of the vortex segment is

rn, = Zj u, + rn+l)

(48)

except when rn or rn+l represents the last vortex in the chain. When this occurs, the remainder of the strength of

the last vortex is included in the last line segment to avoid losing a portion of the shed vorticity.

(49)

This smoothing procedure was investigated for a number of calculations, and it accomplished the smoothing of the induced velocity field to a great extent. The surface

pressure distributions had some residual roughness caused

19

by other effects, but the pressure spikes caused by locally

high vortex-induced velocities were eliminated. The major

disadvantage of the above smoothing technique is cost. The nature of the total calculation procedure is such that

many velocity calculations are,required; therefore, the additional effort to set up the smoothing procedure and calculate velocities at numerous body points causes a

dramatic increase in computation time. For this reason,

alternate smoothing techniques are considered.

The smoothing technique incorporated into the present

version of NOZVTX is one in which a single point vortex

is represented by a simple distribution of several point vortices. The smoothing calculation is carried out only

for vortices which lie within 10% of the local body radius

of the surface of the body, and it is used only for calcu-

lating pressures on the body surface. The vorticity is

distributed according to the procedure shown in Figures 5(a)

and (b). In Figure 5(a), vortex 2 is the only vortex which

lies within 10% of the local body radius of the surface.

The "distributed vorticity" model consists of nine smaller

vortices: one located at the actual position of vortex 2,

four located along the line joining vortices 1 and 2, and

four along the line joining vortices 2 and 3. Each vortex

has a strength equal to one-ninth of the actual vortex

strength. Figure 5(b) shows distributed vorticity models

for vortices which are end pointsofthe vortex cloud;

that is, for the first vortex and the most recently shed

vortex. In the case of the first shed vortex, rl, the

distributed vorticity model consists of five point vortices,

each vortex having a strength equal to one-fifth that of the

actual vortex. The model for the most recent shed vortex,

- -

r3' is shown in Figure 5(b). The vorticity is distributed

along two separate lines. One line is defined by the

actual vortex position and its shed position at the previous axial station, indicated by 3' in Figure 5(b). The second line is defined as before, the end points being the locations of I'2 and r3.

The smoothing technique described above was investigated for a number of geometry and flow condition combinations, and it was found to smooth pressure spikes caused by locally high vortex-induced velocities or values of d$/dx. As expected, use of the smoothing technique increases execution time, but the actual execution time is determined by the number of vortices which lie close to the body surface.

This technique redistributes the vortices and tends to spread their effect over a larger area, but the model is still made up of discrete vortices, each of which has a singularity

at its origin. A viscous core model for the individual vortices is required to remove the singularity in the induced

velocity calculation. This model is described in the

following section.

Vortex core.- The diffusion core model (Ref. 2) for the point-vortex induced velocities removes the singularity at the vortex origin and effectively reduces the velocities near the vortex. The tangential velocity induced by a single point vortex is written as

(50)

21

where r is the distance from the vortex to the field point

and x is a measure of the age of the vortex. The induced velocity from this core model is illustrated in the

following sketch.

Vortex Induced Velocity

The vortex core model represented by Equation (50)

has received considerable attention in the context of body vortex induced effects, and it has a number of shortcomings. Since the exponential term is a function of r, the flow

medium (v), and the age of the vortex (x), the core is constantly changing size as the vortex moves through the

field. Under certain conditions, the core radius, denoted

as r C

in the sketch, can become very small and the induced

22

._. _..-__------ .__. ..- _..__. . _.._ I

velocity becomes unrealistically large. In an attempt to

further modify the core model to keep the induced velocities within physically realistic limits, the following modification was made. The location of the maximum induced velocity,

rc in the sketch, is fixed at a specific radius to be selected by the user. Given a core radius, the vortex induced velocity is

!$ = & [l - exp [-1.256 $11 (51)

The core model described by Equation (51) is included in the version of the code described in this report, and in

a later section, guidelines for the selection of an appropriate core radius are presented. Results indicate that this

simple core model provides adaquate smoothing for the

necessary velocity calculations.

Other investigators of discrete vortex models must use a core model of some type. Although there are many different

core models, all serve the same purpose as those described

above, and nearly all are directed at eliminating the singularity and reducing the maximum induced velocity. The

code described in a later section is easily modified to

incorporate another core model if the user desires to do so.

PROGRAM

General Description

The computer code, NOZVTX, presented in this section

has application as an engineering prediction method for circular and noncircular bodies at high angles of attack in supersonic flow. The code is in effect a combination of

23

two separate codes. The first is the subsonic body vortex

shedding method described in References 1 and 2, and the

second is the supersonic body model for bodies alone described in References 10, 11, and 12. The combined code provides

a means to predict the nonlinear aerodynamic characteristics

and flow fields for bodies with circular and noncircular cross sections. The effect of fins is not included.

The code consists of a main program, NOZVTX, and 50

subroutines. The overall flow map of the program is shown

in Figure 6 where the general relationship between the subroutines and external references can be seen. An alpha-

betical listing of the subroutines is shown in Table I.

It was an objective of this effort to keep the two codes as distinct as possible to maintain their individual identity

so that modifications to either code could be carried out

without major changes in the other code. Note that the

main subroutine of the body vortex shedding calculation,

ELNOSE, includes the body panel code via the calls to

subroutines GEOM, VELCMP, and SOLVE. The line singularity

model for axisymmetric bodies is included through a call to

subroutine BDYGEN. The communication of information between

segments of the program is handled by name common blocks. A cross reference table for the calling relationship

between the program subroutines and the external references

is shown in Figure 7. A similar table for the common blocks

is shown in Figure 8.

The program is written in standard FORTRAN language.

Operation requires four (4) auxiliary files in addition to the usual input and output units. Core storage for

execution is approximately 230,000 octal on a CDC CYBER 760

computer. Execution times on the same computer can vary

from 10 to 400 or more seconds. Execution time depends on

many factors such as the number of panels required for the

24

body r the integration interval (DX), the number of shed

vortices, the use of symmetry, and even the flow conditions.

Limitations

Program NOZVTX is applicable to bodies alone at high angles of attack in supersonic flow, with some limitations.

Certain of the limitations to follow are basically rules-of- thumb for the use of the program that have evolved through experience with the program and verification of the program

by comparison with data. The range of applicability of the program maybeextended by additional verification with appropriate data.

There is a Mach number/geometry limitation in the body model that requires the body surface to lie behind the Mach cone. This has greatest effect at high Mach numbers or on blunt noses at low Mach numbers, and a message is

printed if the geometry condition is violated. It has been the experience of the authors that, when this condition arises, the nose can generally be made slightly more pointed

so that it lies within the Mach cone. Often this modification is required only on the first few percent of body

length, and the effect on the total nose forces and moments

and shed vortex field is minimal.

An associated Mach number limitation involves the quality of the predicted surface pressure distribution at

high Mach numbers. As will be shown in comparisons of experi-

ment and theory, the predicted pressure coefficients on the windward side are much lower than those measured at body points which are near to the shock wave. This is a common

problem for linear body models like the axis singularities and the panels considered in NOZVTX, and it is possible that

the windward pressure can be corrected by use of nonlinear pressure prediction techniques.

25

Another less rigid limitation involves the Mach number/ angle of attack range. It is suggested that the crossflow Mach number (Masinac) not exceed unity. A stricter limitation is that the crossflow Mach number should not exceed the critical value: that is, the local Mach number in the crossflow plane should not exceed unity at any point in the plane.

The root of this limit is the vortex shedding calculation

which uses an incompressible vortex model and an incompress-

ible separation criterion. Also, when the crossflow Mach number exceeds the critical value, a complex shock near the shoulder may contribute to separation. It is important that the user be aware of these limits, particularly when the

program is being used in that range. However, the author has used the program for flow conditions in which the crossflow

Mach number slightly exceeded unity, and the results were

quite resonable. Some of these results are presented in a later section.

Error Messages and STOPS

The program code has numerous internal error messages.

These are generally explicit and they will be described below only if an explanation will provide some insight into

the problem and its correction. There are several execution

terminations at numbered STOP locations within the program. These are described in the table below along with some

suggestions to correct the problem.

The most common message is "CONVERGENCE NOT OBTAINED IN

INTEGRATION." This message, along with the axial station,

XI, and the integration interval, H, is printed when the

trajectory calculation does not converge within the specified

tolerance, E5. One message will terminate execution at

STOP 40 described in the table to follow.

26

The program contains several error messages pertaining to the number of panels used to describe the body. The message "ERROR - NUMBER OF ROWS OF PANELS IN BODY SECTION 1 EXCEEDS 30" is self explanatory and results in STOP 20. A similar message "ERROR - NUMBER OF ROWS OF SINGULARITY PANELS ON BODY EXCEEDS 30" results in STOP 50. The message "ERROR - NUMBER OF BODY PANELS EXCEEDS 600" results in STOP 60. The message "ERROR - BODY HAS MORE THAT 20 CIRCUMFERENTIAL PANELS" refers to the number of panels

describing the body cross section circumference. This error results in STOP 200. All of the errors associated with the body panels are caused by incorrect values being input in items 32 and 41 of the input list.

If a pivot element of the matrix of panel influence coefficients is zero, the message "ERROR - THE MATRIX IS SINGULAR" is printed and the program terminates execution at STOP 300. This is a very uncommon error and is not seen for normal body panel arrangements.

As mentioned in a previous section, the slope of any panel on a noncircular body must be less than the Mach angle. If a panel has a greater slope, the message

"ERROR - BODY PANEL SLOPE EXCEEDS MACH ANGLE" is printed

and execution terminates at STOP 210. A modification of the body geometry to decrease the panel slopes is required to correct this situation.

The surface of an axisymmetric body must lie inside

the Mach cone from the nose tip. If certain body points lie outside the Mach cone, subroutine BDYGEN will make an effort to correct the definition of the body and revise the radius

distribution. If the Mach number is too high to correct the body within its specified length, the message "BODY RADIUS TOO LARGE IN RELATION TO BODY LENGTH" is printed and execution terminates at a STOP. If, in the process of

27

correcting the body points outside the Mach cone, the subroutine cannot locate an appropriate point, the message

"MACH CONE - BODY MERIDIAN INTERSECTION NOT FOUND AFTER 100 TRIALS" is printed and execution terminates at a STOP. These difficulties are corrected by modifying the body

shape or lowering the Mach number of the run.

The numerical conformal mapping procedure can generate

several error messages from subroutine CONFOR. The message "***ERROR IN SUM OF EXTERIOR ANGLES***" is printed if the

routine has difficulty calculating appropriate body angles within a specified tolerance. Another message "***COEFFICIENT

AN(l) = GREATER THAN +- .OOl AFTER ITERATIONS OUT - OF - ' is printed onrareoccasions when the mapping procedure is having difficulty converging on a solution. Both of the above messages are informational, and they do not result in

a program STOP. However, the user is advised to check the transformed shape carefully with the actual shape before

using the results. If there is a difficulty in the numerical

mapping calculation, refer to the Input Preparation section

for assistance.

Numbered STOP Locations

Termination Message Description

STOP Normal program stop or the axisymmetric body

lies inside the Mach cone from the nose tip.

STOP 20 More than 30 rows of panels on body. Revise

input values of NFORX and KFORX in input

items 32 and 41, respectively.

STOP 40 Vortex trajectory integration has failed to

converge at an axial station. This problem is

usually caused by (1) two or more vortices in

close proximity such that they are rotating

28

about each other so rapidly that the integration

subroutine cannot converge on a solution, or (2) a vortex too near the body surface. These situations are often dependent on the body shape and the specific flow conditions, but the following steps should be tried to resolve the

problem. Increase the integration tolerance E5, or increase the size of the combination parameter RGAM so that it is greater than the distance

between the troublesome vortices. It often helps to increase the integration interval DX and restart the calculation from a previous

station.

STOP 41 The vortex trajectory calculation has resulted in a vortex inside the body. This is a rare,

but serious, problem; and it signifies a major

problem with the solution. It usually means that the body paneling scheme is in error. If no obvious problem exists, change the paneling layout to increase the number of circumferential and axial panels.

STOP 44 Total number of shed vortices exceed 200. Combine some vortices and restart the calculation and run to completion. An alternate

solution is to increase the integration interval DX so that fewer vortices are shed.

STOP 45 The number of positive or negative primary separation vortices exceeds 70. Use the same

remedy as for STOP 44.

STOP 46 The number of positive or negative secondary or reverse flow separation vortices exceeds 30. Combine the secondary vortices to reduce their total number and restart the calculation;

29

STOP 50

STOP 60

STOP 200

STOP 202

STOP 210

<STOP 300

increase DX to reduce numbers of vortices shed,

or set NSEPR=O and eliminate the secondary

vortices from the calculation.

More than 30 rows of panels on body. See STOP 20.

More than 600 panels on body. Revise requested number of panels in input items 32 and 41.

More than 20 panels on the body circumference. Revise values of NRADX and KRADX in input items

32 and 41, respectively.

There is an error in the vortex tables in sub-

routine VTABLE. Check input values of NBLSEP, NSEPR, NVP, NVM, NVA, NVR, and RGAM in items

1 and 6.

A body panel has a slope greater than the Mach

angle.

There is a singularity in the panel influence

coefficient matrix. Check panel layout geometry

for errors.

An additional comment is required here about a situation which can occur which results in a fatal execution error and

termination at a TIME LIMIT with no other message. This is

not a usual occurrence, but it has been observed in certain special cases. The typical case is near the end of a very

long body from which a large number of vortices have been

shed. The difficulty is that two differential equations of motion must be integrated for each vortex and the derivatives

involve all vortices in the field. Solution by a Kutta-

Merson scheme can involve many derivative calculations. The

minimum time required for each axial step in the calculation is proportional to the square of the total number of

30

vortices. As the number of vortices increases, convergence becomes more difficult and the time for each step can increase dramatically. If this situation should occur, the best approach is to combine adjacent vortices and restart the calculation with a larger convergence criterion E5. It is sometimes useful to increase the axial interval DX.

Input Description

The purpose of this section is to describe the input to program NOZVTX. The input formats are shown in Figure 9, and the definitions of the individual variables are provided in the followingparagraphs. The input to the body panel and axis singularity portion of the program is maintained nearly identical to the input requirements for program WDYBDY of Reference 10. The purpose for this was to minimize changes in WDYBDY; however, it leads to some redundancy in the preparation. The small amount of redundancy is acceptable in light of the simplification in the use of the current

program. If there is a question about the input in the panel portion of the program, the user is referred to Appendix K of Reference 10 for additional information.

The remainder of this section includes a description of each of the input variables and indices required for the use of program NOZVTX. The order follows that shown in Figure 9 where the input format for each item and the location of

each vafiable on each card is presented. Data input in I-format are right justified in the fields, data input in F-format may be placed anywhere in the field and must include

a decimal point, and data input in E-format are right justified in the fields and must include the decimal point and

the complete exponent designation.

31

Note that all length parameters in the input list are

dimensional variables: therefore, special care must be taken

that all lengths and areas are input in a consistent set of units.

Item 1 is a single card containing sixteen integers,

each right justified in a five-column field. The variables

are defined as follows.

Item Variable

1 NCIR

NCF

ISYM

NBLSEP

NSEPR

Descriution

Cross section shape index.

=0, circular cross section =l, elliptic cross section

=2, arbitrary cross section

Numerical transformation index.

=0, mapping coefficients are calculated

based on body shape input

=l, mapping coefficients are input by user in items 18 through 22

Symmetry index.

=0, right-left flow symmetry

=1, no symmetry (required if $ # 0')

Body vortex separation index.

=0, no separation (required if ac = O"

in item 5)

=l, laminar separation =2, turbulent separation

Reverse flow separation index.

=0, no separation

=l, laminar separation in reverse flow region

32

NSMOTH Vortex induced velocity smoothing index.

=0, no smoothing =l, smoothing of vortex-induced velocities

used in pressure calculation =2, smoothing of vortex-induced velocities

used in pressure calculation and smoothing of 3$/2x calculation

NDFUS Vortex core model index.

=0, potential vortex =l, diffusion core model; preferred (see

RCORE in item 5)

NDPHI Unsteady pressure term index.

=0, omit a$/ax from Cp calculation =l, include a$/ax term; preferred

INP Nose force index.

=0, slender body theory force on portion of nose ahead of starting point (XI)

=l, zero force on nose ahead of XI; preferred value for restart option

NXFV

NFV

Number of x-stations at which flow field

is calculated or special output generated. See item 23.

0 c NXFV 5 8 -

Number of field points for flow field

calculation. See item 24. 0 < NFV < 200 -. -

NVP Number of +r vortices on +y side of body to be input for restart calculation. See

item 26. 0 < NVP < 70 -

33

NVR

NVM

NVA

NASYM

Number of -r reverse flow vortices on +y

side of body to be input for restart calculation. See item 27.

0 < NVR < 30

Number of -r vortices on -y side of body to

be input for restart calculation. See item 28. NVM = 0 if ISYM = 0.

O<NVM<70 - -

Number of +r reverse flow vortices on -y

side of body to be input for restart calculation. See item 29. NVA = 0 if

ISYM = 0. 0 < NVA < 30 - -

Asymmetric vortex shedding index. See

item 7.

=0, no forced asymmetry; preferred

=l, forced asymmetry

Item 2 is a single card defining seven integer output

option indices. Each index is right justified in

column field. The indices are defined as follows.

a five

Item Variable Description

2 NHEAD Number of title cards in item 3. NHEAD > 1 -

NPRNTP Pressure distribution print index.

=0, no pressureoutputexcept at special

x-stations specified by item 23 =l, pressure distribution output at each

x-station

34

NPRNTS

NPRNTV

NPLOTV

NPLOTA

NPRTVL

Vortex separation print index. =0, no output =l, output at each x-station: preferred =2, detailed separation calculation

output; for debugging purposes only

Vortex cloud summary output index.

=0, no vortex cloud output =l, vortex cloud output: preferred

Vortex cloud printer-plot option. =0, no plot =l, plot full cross section on a constant

scale =2, plot upper half cross section on a

constant scale =3, plot full cross section on a variable

scale: preferred

Plot frequency index.

=0, no plots =l, plot vortex cloud at x-stations

specified by item 23 =2, plot vortex cloud at each x-station

Velocity calculation auxiliary output for debugging purposes only (Sub. VLOCTY).

=0, no output; preferred =l, print velocity components at field

points: see items 23 and 24 =2, print velocity components at body

control points during pressure calculations. This option can produce massive quantities of output (not

recommended).

35

Item 3 is a series of cards containing hollerith informa-

tion indentifying the run. The cards are reproduced in the output just as they are input.


3 TITLE NHEAD cards of identification. Information

to be printed at top of output sheets.

Item 4 is a single card containing reference information

used in forming aerodynamic coefficients. The variables are

defined as follows.

Item Variable

4 REFS

REFL

XM

SL

SD

Item 5 is a

Description

Reference area. REFS > 0

Reference length. REFL > 0

Moment center.

Body length (L).

Body maximum diameter, or equivalent non-

circular body diameter (D). Used in

definition of RE.

single card containing the flow condition

parameters. Each variable is input in a ten column field.

The flow parameters are:


5 ALPHAC Angle of incidence, degrees. O" 2 ac < 90"

If ac = O", NBLSEP = 0 in item 1.

PHI Angle of roll, degrees ($1.

RE Reynolds number (V03D/v).

36

RCORE Ratio of the local vortex core radius to

the local body diameter. Default value is . 025, and maximum allowable value permitted in the code is .05*SD/ro.

XMACH Mach number, MOb;

Item 6 isa single card containing the specification of

the axial extent of the run and certain parameters associated

with the vortex wake. Typical values are shown with the

variables.

Item Variable

6 XI

XF

DX

EMKF

RGAM

Description

Initial x-station. XI > 0

Final x-station. XF > XI -

Increment in x for vortex shedding calcula-

tion. Typical value, DX " D/2 when FDX = 0.0. For variable DX, set DX = 0.0

and FDX > 0.0.

Minimum distance of shed vortex starting

position from body surface.

=l.O,

>l.O,

Vortex =o.o,

>o.o,

vortices positioned such that separation point is a stagnation point in the crossflow plane minimum radii away from body surface for shed vortices. Typical value,

EMKF = 1.05

combination factor. vortices not combined: preferred radial distance within which vortices are combined. Typical

value, RGAM = 0.05 D

37

VRF Vortex reduction factor to account for observed decrease in vortex strength.

=0.6, for subsonic flow

=l.O, for closed bodies, or supersonic flow: preferred

FDX Factor for variable x-increments. When

FDX > 0.0, DX = FDX *(local body radius).

DX should be input as zero when this option

is used.

Item 7 contains only one variable that is of general

use in program NOZVTX, and that is the integration error

tolerance, E5. The variable XTABL is for diagnostic

purposes only. The next three variables concern the use of

forced asymmetry for bodies at very high angles of attack.

The variables are defined as follows.


7 E5 Error tolerance for vortex trajectory

calculation. (Relative error in vortex

position.) Typical range, E5 = 0.01 to

0.05.

XTABL x-location after which a table of corres-

ponding points between the real body and

the transformed circle is not printed. For diagnostic purposes on noncircular

shapes. Typical value, XTABL = 0.0.

XASYMI Initial x-location at which forced

asymmetry of separation points is used. Typical value, XASYMI = 0.0.

XASYMF Final x-location at which forced asymmetry

of separation points is used. Typical

value, XASYMF = 0.0.

38

DBETA Amount of forced asymmetry for separation points on body, degrees. Typical value, DBETA = 0.0.

Items 8, 9, 10, and 11 provide a table of geometry characteristics that must be input for all bodies, circular

or noncircular in cross section.


8 NXR Number of entries in body geometry table (1 z,NXR < 50). -

x-stations for geometry table (NXR values, 8 per card).

9 XR

10 R rO’ circular body radius at x-stations, or radius of tranformed Circle (NXR values, 8 per card). For an ellipse,

rO = (a + b)/2.

11 DR dro/dx, body slope of transformed shape at x-stations (NXR values, 8 per card).

Items 12 and 13 are included only if NCIR = 1; that is, for elliptical bodies.


12 A.E a, horizontal half-axis of elliptic cross section (NXR values, 8 per card).

13 BE b, vertical half-axis of elliptic cross section (NXR values, 8 per card).

39

Items 14 through 22 are included only if NCIR=2; that

is, for a body with arbitrary cross section that must be handled with the numerical transformation and conformal

mapping procedures.


14 MNFC Number of coefficients describing trans-

formation of arbitrary body to a circle (1 5 MNFC < 25). -

MXFC Number of x-stations at which transforma-

tion coefficients are defined (I~MXFCL~O).

For a similar shape body at all cross sections, MXFC = 1.

15 XFC x-stations at which transformation coef-

ficients are defined. For a similar cross

section body, MXFC=l, XFC(l) 5 XI (MXFC values, 8 per card).

Items 16 and 17 are included when NCF = 0. This block

of cards is repeated MXFC times, once for each x-station input in Item 15.


16 NR Number of coordinate pairs defining the

body cross section at the axial station

defined by XFC(J) (2 I_ NR 5 30).

17 XRC(I,J) Coordinates of right-hand side of body. YRC(I,J) The convention for ordering the coordinates

from bottom to top in a counter-clockwise

fashion, as shown in Figure 10, is observed. Right/left body symmetry is required in

the numerical mapping. (NR cards with one

set of coordinates per card.)

40

-

Items 18 through 22 are input in lieu of Items 16 and 17 when NCF > 0. The values of these variables must be obtained from a previous run of the numerical mapping portion of the code. The only purpose for providing the option to input the mapping parameters is to eliminate the need to recompute the numerical mapping in subsequent runs and save computation time. This set of items is repeated MXFC times, once for each x-station input in Item 15.

Item

18

19

20

21

22

Variable Description

ZZC(J) Axis shift and radius of mapped cross RZC(J) section (one card with two values).

NR Number of sets of body coordinates at a

particular axial station.

XRC,YRC Coordinates of right side of body defining cross section starting at f3 = O" and ending at B = 180" (NR cards with one set of coordinates per card). See Figure 10.

THC Angle of defined body points in circle

plane (NR values, 8 per card).

AFC Mapping coefficients (MNFC values, 6 per card).

Items 23 and 24 are included only if NXFV > 0. Item 23

specifies the axial stations at which additional output or plots are requested. Item 24 is included only if NFV > 0.

Each card of this item contains the nondimensionalized y,z-coordinates of a field point at which the velocity field is calculated at each axial station specified in Item 23.

41


23 XFV x-station at which field point velocities

are calculated or at which additional output is printed (NXFV values).

Omit Item 24 if NFV = 0.

24 YFV,ZFV Y/r0 I z/r0 -coordinates of field points at

which velocity field is calculated, expressed as a fraction of the local body radius (NFV cards with one set of coordi-

nates per card). It is important that the field points lie outside the body surface.

Items 25, 26, 27, 28, and 29 are included for a restart

calculation only, and the condition NVP + NVR + NVM + NVA# 0 must be satisfied. It is not necessary to input all four

types of vorticity in a restart calculation.

Item 25 is a single card containing the nose force and

moment coefficients at the restart point; that is, X = XI. These values may be set equal to zero, but the resulting

forces and moments from the current calculation will be in

error.

Item Variable

25 CN

CY

CM

CR

CSL

Description

Normal-force coefficient.

Side-force coefficient.

Pitching-moment coefficient.

Yawing-moment coefficient.

Rolling-moment coefficient.

42

Item 26 is a block of NVP cards to specify the positive separation vorticity on the right side of the body. Omit this item if NVP = 0. The variable XSHEDP associated with each vortex is used only to identify individual vortices and permit the user to follow individual vortices during the calculation.


26 GAMP VVm I positive separation vorticity on right side of body.

YP,ZP coordinates.of discrete vortices on right

side of body at starting point (XI).

XSHEDP x-location at which individual vortex was shed (may be 0.0).

(Item 26 consists of NVP cards.)

Item 27 is a block of NVR cards to specify the secondary or additional vorticity on the right side of the body or at any other position in the field. This array is a con-

venient way to put an arbitrary cloud of additional vorticity which is to be maintained separately from the other body vorticity in the field. Omit this item if NVR = 0.


27 GAMR VVD3 I reverse flow or additional vorticity

on right side of body.

XR,ZR coordinates of discrete vortices on right


XSHEDR x-location at which individual vortex was shed (may be 0.0).

(Item 27 consists of NVR cards.)

43

Item 28 specifies the negative separation vorticity on the left side of the body, analogous to Item 26. Omit this item if NVM = 0 or if ISYM = 0. In the case of a symmetric flow field (ISYM=O), this block of vorticity is automatically defined as the mirror image of the positive vorticity input

in Item 26.

Item Variable

28 GAMM r/V- I negative

side of body.

YM,ZM Coordinates of discrete vortices on left

Description

separation vorticity on left


XSHEDM x-location at which individual vortex was

shed (may be 0.0).

(Item 28 consists of NVM cards.)

Item 29 specifies the secondary or additional vorticity

on the left side of the body or at any other position in the

field. This block of vorticity is analogous to Item 27, and it is omitted if NVA = 0 or if ISYM = 0. As with the previous

item, this block of vorticity is automatically defined as

the mirror image of the negative vorticity input in Item 27 if symmetry is required by ISYM = 0.


29 GAMA VV03 I reverse flow or additional vorticity on left side of body.

YA,ZA Coordinates of discrete vortices on left


XSHEDA x-location at which individual vortex was

shed (may be 0.0).

(Item 29 consists of NVA cards.)

44

The above 29 items make up the body vortex shedding portion of the input deck.

Items 30 through 43 are associated with the paneling

of bodies with noncircular cross sections and are taken from the input specifications for program WDYBDY (Ref. lo).. Item 44 is used to describe bodies with circular cross sections. As described above, it was desired to keep the

panel code input as unchanged as possible from the original WDYBDY code; however, the resulting input deck requires the specification of several variables over which the user has

no control. The input deck considered herein contains only those variables which are optional to the user. Other nonoptional variables have been hard coded with their appropriate value, and these should not be changed by the user.

Omit Items 30 through 43 if NCIR = 0.

Item Variable DescriDtion

30 TITLE1 One card of identification information to be printed ahead of the paneling output.

Item 31_ is a single card containing two indices, each index is right justified in a three column field. The

purpose of IXZSYM is to define the manner in which the body paneling is carried out for different flow conditions. A symmetric body in a symmetric flow (0 = O") can benefit from the IXZSYM = 0 option, which panels only one half of the

body and then utilizes the symmetry characteristics to reduce computation time. The same symmetric body in an asymmetric

flow condition ($ # O") must be paneled in total, IXZSYM = 1, to pick up the nonsymmetry in the loading around the body. The body symmetry is used to reduce the input required as described later.

45

Item Variable Description . 31 IXZSYM Panel symmetry option.

=0, panel symmetric half of body for symmetric flow ($I = O")

ql, panel complete symmetric body using

+y side geometry for a symmetric body in asymmetric flow ($ # O")

=-1, panel complete configuration using

geometry of both sides

ITBSYM =O, configuration has top/bottom symmetry

=l, no top/bottom symmetry (e.g., see Fig. 10)

Item 32 contains indices to specify the body geometry.


32 J2

J6

Body geometry specification option. =0, no body (not recommended)

=l, geometry for arbitrary shaped body

=-1, circular body defined by cross- sectional area at XFUS stations (not

recommended)

=-2, circular body defined by radius at

XFUS stations =-3, elliptic body defined by both semi-

axes at XFUS stations

=0, cambered body - not available =l, body symmetrical with respect to

XY-plane (uncambered body)

=-1, circular or elliptic body with J2 # 0 (preferred value)

46

NRADX Number of points used to represent the body segment about the circumference. If the configuration is symmetric (IXZSYM = 0, l), NRADX is input for the half section. If the entire configuration is input (IXZSYM = -l), NRADX is input for the full section. The half section or full section is divided into NRADX-1 equal

angles, and the y- and z-coordinates of the panel corner points are defined by the intersections of these meridian angles with the body surface (3 < NRADX < 20). - -

It is the experience of the authors that the best panel representation of the body is obtained when NRADX is an even number; that is, there should be an odd number of panels on the half-body. It is very important for the pressure calculation that there be a panel control point

on the B = 90° meridian.

NFORX Number of axial body stations used to define geometry (2 5 NFORX 5 30).

The next block of input specifies the geometry of the body for purposes of laying out the panels. This input is

somewhat redundant with the input in Items 8 through 13; however, it is important that the paneling geometry be separate from the previous geometry but consistent with it. In many cases, the paneling geometry may be specified in a

more coarse grid than that required by the separation portion of the program.

47

Item 33 is a block of cards containing the axial stations

at which body geometry is input for paneling purposes. The values are input ten to a card in seven column fields with a

decimal point required.


33 XFUS x-stations at which body paneling geometry

is input (NFORX values).

Item 34 is inlcuded only if 52 = -1 (Item 32). The

circular body cross-sectional areas are specified at the XFUS stations defined in Item 33.


34 FUSARD Circular body cross-sectional areas at XFUS stations (NFORX values).

Item 35 is included only if J2 = -2 (Item 32). The

circular body radii are specified at the XFUS stations

defined in Item 33.


35 FUSRAD R, circular body radii at XFUS stations

(NFORX values).

Items 36 and 37 are included only if J2 = -3 (Item 32).

The elliptic body horizontal and vertical semi-axes are

specified at the XFUS stations defined in Item 33.

48


36 FUSBY Elliptic body horizontal semi-axes at XFUS stations (NFORX values).

37 FUSAZ Elliptic body vertical semi-axes at XFUS stations (NFORX values).

Item 38 is included only if 52 = 1 (Item 32). The

y,z-coordinates of points on an arbitrary cross section body are specified in Item 38, and there is one set of Item 38 for each of the axial stations defined by XFUS in

Item 33. The convention for ordering the coordinates from bottom to top is observed (e.g., Fig. 10). If the full

cross section is to be specified (IXZSYM = -1, Item 31) the ordering continues counter-clockwise from the top of the body back to and including the bottom point to close the cross section.


38 YJ,ZJ y,z-coordinates of arbitrary body at XFUS

stations (NRADX cards).

Item 38 is repeated NFORXtimes, once for each XFUS station.

The following portion of the input deck is described as

Auxiliary Input Cards in Appendix K of Reference 10. The

purpose of this input is to provide a simple means to adjust the actual panel layout keeping the geometry specified above unchanged.


39 TITLE2 One card of identification information for paneling geometry.-

49

Item 40 contains an index to specify the output for the

body panel portion of the program. Because of the quantity

or output possible, it is recommended that [IPRINT/ f 1 be the highest level of output specified. Note that the

values of the first two indices on this card are fixed.


40 IPRINT Panel output option.

=0, minimum output (recommended)

=l, output panel corner coordinates, final source strengths, and velocities at

panel control points (recommended)

=2, same as IPRINT=l and source strengths from previous iteration (not recommended)

=3, print source strength for each iteration (not recommended)

=4, same as IPRINT= and complete aero-

dynamic influence matrix (not

recommended).

If IPRINT < 0, the panel geometry is included with the output

specified by [IPRINT]. This option provides a useful means

to check the paneling arrangement.

Item 41 is a single card containing two indices in three

column fields. These indices provide a conventional means to

change the panel layout without changing the geometry input.


41 KRADX Number of meridian lines used to define

panel edges on the body segment (see note for NRADX in Item 32).

50

KRADX =0, meridian lines are defined by NRADX

in Item 32

'0, meridian lines are calculated at KRADX equally spaced points

(3 c KRADX < 20) - - co, meridian lines are calculated at

specified values of PHIK in Item 42

(3 5 IKRADX] 5 20)

For symmetric configurations (IXZSYM= O,l), KRADX is the number of meridians on the half section. For full configurations (IXZSYM = -l), KRADX is the number of

merdians on the full section including

meridians at 0" and 360°.

KFORX Number of axial stations used to define leading and trailing edges of panels on

the body. =0, the panel edges are defined by NFORX

and XFUS in Items 32 and 33,

respectively

'0, the panel edges are defined by XFUSK in Item 43 (2 < KFORX 5 30) -

Item 42 is included if KRADX < -3 in Item 41. This - item provides a means to easily modify the panels by changing the body merdian angles which specify the panel corner

points. The purpose of this option is to allow the user to pack additional panels into a region of the body which is changing rapidly.


42 PHIK Body merdian angles (degrees) with PHIK = 0' at the bottom of the body and PHIK = 180° at the top (IKRADXI values).

51

Item 43 is included,only if KFORX is nonzero. The variable XFUSK defines the axial positions of the panel edges.

If this item is not included, the panels are specified by XFUS in Item 33. The purpose of this item is to simplify the changing of a panel layout without changing the entire geometry input deck.


43 XFUSK Axial stations of leading and trailing

edges of body panels (KFORX values).

Item 44 is included only if NCIR = 0; that is, for bodies with circular cross sections.


44 NXBODY Number of line sources/sinks and line

doublet singularities distributed along

body centerline (1 5 NXBODY < 100). - Default value is NXBODY = 51.

NCODE

XNOSE

XLBODY

Integer control index for specifying fore-

body shape over length of XNOSE.

NCODE LO, Parabolic =l, Sears-Haack

=2, Tangent ogive

=3, Ellipsoidal =4, Conical

Length of nose part of body, measured from

nose tip.

Length of body.

52

This concludes the input deck for a single run of program NOZVTX. The program is not set up to stack input for multiple cases because of the sometimes long and generally unpredictable. run times.

Input Preparation

In this section, some additional information is provided to assist the user in the preparation of input for certain

selected problems. The previous sectiononthe input description must be used to understand the individual variables which go into NOZVTX, and this section will permit the

user to select the appropriate input to get optimum results

from the code.

Panel layouts.- The choice of panel arrangement to represent the noncircular body shape is perhaps the most important part of the input selection as the quality of the panel layout can determine the quality of the predicted results. Simply using the largest possible number of panels is one solution: however, this will not guarantee an optimum panel layout, nor will it guarantee the best results.

In the following paragraphs, some guidelines are presented

for various situations to optimize both results and computation time whenever possible.

The elliptic cross section will be considered first

because the code has been checked out for a number of different elliptic bodies. In each case to follow, right/

left symmetry of the panels (IXZSYM = 0 or 1 in Item 31) is

assumed: therefore, only one side of the body will‘be shown to illustrate the panel layout.

A 2:l elliptic cross section is shown in Figure'11 where

several different panel arrangements have been selected. In Figure 11(a), nine program-determined panels (KRADX = 10

53

in Item 41), each specified by a 20' central angle, are shown.

The only problem area is near the shoulder of the ellipse

where the outer three panels miss the actual surface by a considerable amount. Increasing the number of panels to lo (KKADx = 11) produces the arrangement shown in

Figure 11(b). Even though an additional panel has been included, the region near the shoulder is not modeled

better than before. In fact, this layout is missing an

important physical effect because no panel has the same orientation as the shoulder (dz/dy = a), and the entire

effect of the shoulder is lost.

An improved panel layout is shown in Figure 11(c) where

nine panels are specified by their meridian angles (KRADX = -10). In this layout, the panels are concentrated

in the region of highest curvature near the shoulder and

spread out over the lessor curved region. Note that an odd number of panels is specified so that the outer panel

nearest the shoulder has the proper slope. Top and bottom

symmetry of the panels is shown, and this is a recommended

arrangement for cross sections with top/bottom symmetry. For this particular cross section, additional panels can

be specified for better resolution at the cost of increased execution time. It is important that the user check the

panel layout used by the code to ensure reasonable modeling

of the actual surface.

Different panel arrangements on a 3:l elliptic cross section body are shown in Figure 12. The sharper curvature

of the shoulder of this shape creates a bigger modeling

problem than the 2:l ellipse shown previously. In Figure 12(a), a 13-panel arrangement (KKADX = 14) with equal

spacing is shown, and it is obvious that a large portion

of the shoulder region is not modeled properly for the prediction of flow field characteristics near to the body.

54

For this particular shape, more panels are required for

better resolution. Additional panels are added in the

region 69O < B < lll" to produce the.19-panel layout (KRADX = -20) shown in Figure 12(b).

An example of reasonable panel arrangements on a lobed

cross section body without top/bottom symmetry is shown in Figure 13. The shape shown in this figure is that described in References 17 and 18. The panel edges do not

lie precisely on the actual body shape, a problem which is discussed in the numerical mapping section to follow. The

g-panel layout in Figure 13(a) is a resonable representation for preliminary calculations; however, certain detail is missing in the regions of greatest curvature. Additional panels are included in these regions, and the resulting layout

is shown in Figure 13(b). Custom panel layouts are shown

for both lobed-body examples to take advantage of the fewer panels required on the flat portion of body.

It is difficult to specify rigid guidelines for panel

layout on arbitrary cross sections. The examples shown

above were determined by this author after several trials, and proper panel selection becomes easier as the user gains experience with the code. The user should remember that

linear interpolation of panel characteristics is used between panel control points in the current version of the code.

Based on the experience of the authors, it is always useful to compare the paneling layout with the actual body shape, and generally, good engineering judgement will allow the

user to select a reasonable panel layout.

Nothing has been said about axial panel layout (NFORX or KFORX) in the above discussion. Axial spacing can be

critical to the predicted results, and as mentioned above,

good engineering judgement will usually produce an adequate

55

layout. Linear interpolation is also used in the axial

direction to determine panel characteristics between panel

control points. This procedure can be used to great advantage for a specific class of body, the cone. Since the predicted linear results on a cone are the same at each

axial cross section, fewer panels are required in the axial direction. The major requirements are the following.

1. There should be one row of panels sufficiently

close to the nose tip or starting point (XI) for the

calculation that a control point can fall between the starting point and the nose tip.

2. There should be a row of panels with control points

located just aft of the last axial station of the calcula-

tion (XF).

3. If a conical nose is attached to the cylindrical

afterbody, there should be enough axial stations specified

near the junction of the nose and afterbody that adequate body characteristics are available for interpolation

purposes.

An example of a panel arrangement for a cone forebody

is presented in the sample cases.

Numerical mapping.- The specification of the appropriate

numerical mapping parameters (Items 14 through 22) depends on

the shape of the cross section. In the interest of optimum

computation time, the fewest possible terms in the transforma-

tion series (MNFC in Item 14) should be used which will

produce an adequate mapping. Each different shape should be

checked by the user to determine the proper number of

coefficients. For example, a simple cross section which is

similar to a circle may require as few as 10 coefficients. The shape shown in Figure 13 required 20 coefficients for

an excellent mapping. If a large number of calculations are

56

to be made for a noncircular shape, it is worthwhile for the user to try different numbers of coefficients in an effort to find an optimum number. Not only should the shapes be compared carefully, the unseparated flow pressure coefficients should be compared to evaluate the quality of the mapping.

The numerical mapping now included in NOZVTX has been used for a variety of cross section shapes, and very few problems have been observed. One general problem area is

bodies with concave sides. The lobed body in Figure 13 has a moderate concave section on each side, and the mapping procedure had difficulty converging on a set of coefficients.

When the body shape was modified slightly, as shown in Figure 13(a), to give the concave region a flat or slightly convex shape, the mapping converged easily. This small

change in the shape probably has very little effect on the predicted pressure distribution. This is an example of the

type of problems to which the user should be alert.

Integration interval.- Choice of the appropriate

integration interval (DX in Item 6) depends to a large extent on the type of results of interest. The code permits the

interval to be a constant (FDX = 0 in Item 6) or a function

of the local radius (FDX > 0). If the user is,interested

in detailed nose loads and separation on the nose, where the radius is changing rapidly, the latter option should be

used with 0.5 < FDX < 1.0. This option can lead to a wide - - range of vortex strengths which, on occasion, has been

determined to be the cause of certain vortex tracking difficulties. The user should be alert to this possibility.

The constant interval option (DX = constant) is the best

choice when the results of a major interest are the vortex clouds near the end of the body.

57

.-

Vortex core.- The core size specification is referenced'

to the local body diameter (RCORE in Item 5). The default

value is 0.025, and this value is adequate for most cases. The larger the core radius specified, the more the vortex

effects are reduced. To eliminate the possibility of

negating the vortex effect entirely, a maximum vortex core

of . 05 times the maximum body diameter is built into the

code. Except in unusual situations, the default value is

recommended, and this is the value used in the comparisons with data shown in a later section.

Sample Cases

In this section, 11 sample input decks are described

to illustrate the input preparation and use of the program

for a wide range of configurations and flow conditions.

Insofar as possible, each major option in the program is

illustrated in the set of sample cases. The major features

of each sample case are identified in Table II to assist the

user in finding an appropriate example to suit a specific

need.

Sample Cases 1 through 11 shown in Table II are designed

to exercise the most common options of program NOZVTX, and Sample Case 1 iS computer run for

sample cases are sample cases are

included specifically to provide a short

testing purposes. The input decks for the

shown in Figures 14(a) through (k). The

based on typical bodies used in the data

comparisons to follow in a later section, and the reference

in which the configuration is defined is noted on each

sample case.

Sample Case 1 is a test case based on the 3:l elliptic

missile body in Reference 19; however, only a portion of

the nose is considered to keep the execution time and output at a minimum. The purpose of this run is to provide the

58

user with a short run to check out the code. The output from this case is described in the next section. Sample Case 2 in Figure 14(b) is a restart of the previous run, and it

illustrates the continuation of a calculation. Sample Case 3 in Figure 14(c) is the input deck for a complete run of this 3:l elliptic missile. The first three sample cases

do not necessarily use the optimum panel arrangement in an effort to minimize execution time. For production runs, the authors recommend a panel layout similar to that shown

in Figure 12(b).

Sample Case 4, Figure 14(d), illustrates the missile body in a rolled condition (4 = 45"). The lack of flow

symmetry and consequently the lack of right/left panel symmetry dictate the use of a smaller number of panels because the entire cross section must be paneled. This

run is typical of any noncircular body in a rolled flow

condition.

Sample Cases 5 and 6, Figures 14(e) and (f), illustrate

input decks for circular cross section bodies (Refs. 20 and 21). Since axis singularities are used in place of

panels, the input is shorter and easier to prepare. The

primary objective of Sample Case 6, an axisymmetric body atcx =O",

C is the prediction of the surface pressure

distribution with no separation effects included.

Sample Cases 7 and 8, Figures 14(g) and (h), are input

decks for a 2:l elliptic cross section forebody from Reference 18. The first case is for a standard panel layout determined by the code. The panels for the second

case are tailored to the cross section shape by specification of the central angles of the panel edges.

Sample Case 9 is a 3:l elliptic cross section conical

forebody (Ref. 17). The input specifies a variable axial

59

integration interval, and the axial panel distribution is determined by the conical shape. Because of the character of conical flow, a very small number of axial panels is required as discussed in a previous section on input preparation.

Sample Case 10, Figure 14(j), is a lobed cross section forebody from Reference 18. This is an arbitrary noncircular body and numerical mapping is required for solution as indicated in the input. Since the cross section is changing at each x-station, the cross section shape must be input at a number of stations thus accounting for the large quantity of input. The number of axial stations must be sufficient

to provide acceptable accuracy in the body shape using linear

interpolation between input stations.

Sample Case 11, Figure 14(k), is set up for the 2:l

elliptic cross section missile body in Reference 22. The panel layout for this body is the same as that shown in

Figure 11(c).

output

A typical set of output from program NOZVTX is described

in this section. In general, the output quantities are labeled and each page is headed with appropriate descriptive

information. The actual output obtained is determined by the output options selected by the user. For purposes of this description, representative pages from Sample Case 1,

described in a previous section, are presented in Figure 15.

The total output from Sample Case 1 is contained on 30

pages I including pages which are continuations of another

page - The first page shown in Figure 15(a) is headed by

the title cards input as Item 3. The remainder of this

60

page t the following two pages shown in Figures 15(b) and (c),

and the top of the fourth page shown in Figure 15(d) contain all the input data with appropriate labels. Body source panel geometry is shown on pages 5 through 13, Figures 15(e) through (m). Page 5, shown in Figure 15(e) and the following two pages shown in Figures 15(f) and (g) indicate the panel corner coordinates. Pages 8, 9, and 10, shown in Figures 15(h),.(i), and (j), contain a table of the body panel centroid coordinates. This is followedbya table of body panel areas and inclination angles. The inclination angles DELTA and THETA orient the panels in the body coordinate system. Detailed descriptions of these angles are given in Reference 10. This table continues onto page 12 and the top of page 13, shown in Figures 15(L) and (m). The next item on page 13, GB(N), is the list of source strengths for the body at angle of attack. This is followed by a table of velocity components at the control point of each panel. Each component is labeled, and the table continues onto pages 14 and 15 shown in Figures 15(n) and (0). The last item on

page 15, under the heading MODIFIED SOURCE STRENGTH, is the source strength on each panel of the body with no crossflow influence. This panel strength represents the volume

solution of the body for purposes of tracking vortices. This list of source strengths continues onto page 16 shown in Figure 15(p).

The next two pages, shown in Figures 15(q) and (r),

contain the predicted pressure distribution around the body at the first axial station. A summary of the geometric characteristics and the Mach number is printed at the top

of page 17, followed by the pressure distribution. The'

columns are labeled, but a brief explanation of the printed

items follow. The first three columns, Y, Z, and BETA,

locate the specific points on the body cross section. u/v0

is the local axial velocity referenced to the free-stream

61

velocity. V/V0 and W/V0 are the local lateral and. vertical velocities, respectively. These velocities include all perturbation effects including vortex induced velocity com-

ponents. The column VT/V0 is the total velocity in the crossflow plane given as the vector sum of the V/V0 and W/V0 velocities. A positive VT/V0 represents flow in a counter-

clockwise direction when viewed from the rear. CP(M) is the compressible pressure coefficient as calculated from

Equation (29). DPHI/DT is the unsteady part of the incompressible pressure relation as represented by the last

term in Equation (31). CPZ is the steady part of Equation (31), and CP(1) is given by Equation (31) in its

entirety. VCP/VO is the local two-dimensional crossflow

plane velocity whose positive sense is the same as that for

VT/VO.

The local force and moment coefficients are printed at

the middle of page 18. CN(X) and CY(X) are local force coefficients calculated by Equations (39) and (43),

respectively. The remainder of the forces and moments

represent total quantities on the nose between the origin and the local axial station. The normal force and pitching moment coefficients, CN and CM, are calculated with

Equations (40) and (41), and the axial center of pressure of the normal force, XCPN, is given by Equation (42). XCPN is not normalized by the reference length on page 7. The

corresponding side force and yawing-moment coefficients and center of pressure of the side force are given by

Equations (44), (45), and (46), respectively. The rolling-

moment coefficient, CSL, is positive in a counterclockwise

sense when viewed from the rear of the body. The small value shown in this output is due to numerical round-off

error during the calculation. The same is true for all

lateral coefficients which appear as very small, nonzero,

62

values when a symmetric body and symmetric flow conditions

dictate that they are identically zero.

Page 19 of the output from Sample Case 1, Figure 15(s), summarizes the important points in the pressure distribution and the separation calculation. The last two lines of output

contain specific information on the vortices shed at this station. The vortex numbers, NV, assigned here are the

cumulative numbers of vortices in the flow field. The

vortex strength, GAM/V, is obtained from Equation (38) where the boundary layer edge velocity ue/Va is shown as VT/V in this line of output. M(K) is the actual distance of the

vortex from the body surface. The position of the vortex in the body plane is given by Y, Z, and BETA. The position

of the vortex in the circle plane is given by the coordinates

YC, zc. The radial position of the vortex in the transformed circle plane is RG/R. The latter quantity is useful in

providing a quick estimate of the relative positions of new

vortices being shed into the wake.

Page 20, shown in Figure 15(t), summarizes the vortex field after the trajectories have been calculated over one axial interval. The x-station at the top of this page is

exactly DX downstream of the previous x-station. Information

to locate each individual vortex in both the body plane and the circle plane is provided. One additional piece of

information for each vortex is XSHED, the x-station at which the individual vortex originated. This is printed to allow

the user to keep track of any.individual vortex throughout the trajectory calculation. Each vortex maintains its

individual identity and the right and left side vortices are grouped together on this page. Centroids of the individual

groups of vorticity also show on page 20. The only exception

to the above discussion of individual vortices occurs when vortices are allowed to combine through use of the parameter

63

RGAM in Item 6 of the input list. When vortices are combined, the particular vortices are identified in a special line of

output.

This completes one cycle of output. The next page of output is the pressure distribution at the new axial station, and the format is the same as page 17 shown in Figure 15(q).

At each axial station, the output repeats through the cycle shown in Figures 15(q), (r), (s), and (t). At specified axial stations, certain additional output is available. A simple plot of the body cross section and the vortex field

at x = 4.2 is shown in Figure 15(u). This particular output is on page 23 of the actual output from Sample case 1. Each type of vortex is shown with a different symbol in this figure.

Postive body vorticity shed from the right side of the body is shown as an X, and negative vorticity from the left side

is represented by an 0. If it is requested, the calculated

velocity field appears as shown in Figure 15(u). This output appears on page 24 of the actual output. The coordinates and velocities shown on this page are in the

body plane. The normalizing diameter, D, in the last two columns is the equivalent diameter of the noncircular body.

The last page of output is shown in Figure 15(w). The

top portion of the output is the summary page for the separa-

tion calculation at the last x-station, X = XF. This part of the output is similar to that shown in Figure 15(s).

The lower part of the page is printed only at the completion

of the calculation, and it contains a summary of the forces and moments and the vortex strengths and centroids at the

end of the body.

64

RESULTS

For purposes of evaluating the accuracy and range of

applicability of the engineering prediction method and associated program code NOZVTX described above, comparisons of measured and predicted aerodynamic characteristics of

various configurations have been made. The objective of

the method is to predict the characteristics for noncircular

cross section bodies, but selected data for bodies of revolu-

tion will also be considered to illustrate certain features of the prediction method. Typical results from the predic-

tion method are presented in the following sections.

Circular Bodies

The prediction method was applied to an ogive-cylinder model (Ref. 21) in supersonic flow to evaluate the ability

of the prediction method to calculate pressure distributions. The configuration has a circular cross section and a three caliber ogive nose followed by 3.67-caliber cylindrical

afterbody. Circumferential pressure distributions at a large number of axial stations are available for a range of angles of attack and supersonic Mach numbers. The free-stream

Reynolds number, based on diameter, is 0.5~ lo6 for the

results to follow.

The model is represented by 51 line sources on the

body body axis for the calculation at cc = O". The measured

and predicted pressure distributions at three Mach numbers are compared in Figure 16 where the multiple data symbols

at each axial station represent the range of scatter in the data. The purpose of this comparison is to verify the body model and the pressure calculation technique under conditions

in which there are neither crossflow nor separation effects.

65

In Figure 16(a), the agreement is very good at M, = 1.6 except in the region very near the nose. As seen in Figures 16(b) and (c), the agreement between measured and predicted pressures on the nose gets progressively poorer as the Mach number increases. It is possible that nonlinear effects caused by the close proximity of the shock wave are responsible for this disagreement. For purposes of this investigation, a small error in the pressure very near the

nose will have very little effect on the vortex shedding from the remainder of the body. As a further check on the body model, a panel distribution of 13 rings of 14 equally

spaced circumferential panels was used for the same calcula-

tion. The pressure results are virtually identical to those

obtained from the line singularity model.

The prediction method was next applied to the same model

at UC = 20" angle of attack and Mm = 1.6. At this angle of

incidence, the separation vortex wake on the lee side of the

body is well developed. Measured and predicted pressure distributions at two axial stations on the nose and one

station on the cylinder are compared in Figure 17. The

agreement between experiment and theory at x/D = 0.8 is very good except on the windward side of the body upstream of

the minimum pressure point. Similar results are illustrated

in the lower portion of Figure 17(a) for a station near the end of the nose (x/D = 2.8). The predicted minimum pressure point is upstream of the measured value, but the general

trend of the predicted pressure distribution is correct. The

slightly irregular behavior of the predicted curve near

B = 130° is caused by local interference of the lee side

separation vortices.

Measured and predicted pressures on the cylindrical

portion of the body (x/D = 5.1) are shown in Figure 17(b).

A large vortex wake, also shown in the figure, has developed

66

on the lee side, and it has a dominant effect on the pressure distribution. The solid curve, representing the predicted pressure distribution in the presence of the vortex wake, is in good agreement with the measured results near the minimum pressure point and on the lee side. The predicted pressures are lower than those measured on the windward side of the body. There is also some disagreement between experiment and theory near B = 130" where it appears that some of the vortex induced effects are missing from the predictions. Contrast the solid curve with the dashed curve representing a predicted pressure distribution with no

separation effects included.

The predictionmethod was next applied to an ogive- cylinder model for which detailed lee side flow field measurements are available (Ref. 20). This model is an ogive-cylinder with a two-caliber nose and a thirteen-caliber

cylindrical afterbody. The tests were conducted at Mach numbers of 2 and 3 at a free-stream Reynolds number, based on diameter, of approximately 2x 106. The data consist of

lee side velocity components and separation vortex strengths and positions. Neither pressure nor force and moment data are available from these tests.

The model is represented by 51 line source and doublet singularities on the body axis. The predicted vortex cloud pattern at x/D = 10 is shown in Figure 18 for Mco = 2 and a = 15".

C The vortex shedding is symmetric on both sides of

the body. Also shown in this figure are the separation

points located at approximately 86 degrees measured from the windward stagnation point. The measured and predicted vortex strength and centroid of vorticity are shown to be in

reasonable agreement in this figure. The predicted vortex wake is approximately 9 percent stronger than the measured value, and the centroid of the predicted vortex cloud is

67

--

slightly lower and more outboard than the measured results.

Again, as a further check on the body model, the body was

represented by 15 rings of 14 equally-spaced circumferential panels. The predicted results were nearly identical in each

case; however, the execution time for the. line singularity

model was only 20 percent of that required for the panel solution.

Measured and predicted body vortex strengths on the

same ogive cylinder at Mach numbers 2 and 3 and angles of

attack 10, 15, and 20 degrees are compared in Figure 19.

At Mco = 2, the predicted results are slightly higher than

those measured. At Mm = 3, the predicted results are lower

than those measured on the forward portion of the body and in good agreement on the rear portion. The slopes of the

predicted curves are in good agreement with the experimental

results, an indication that the predicted shedding rate and strengths of the shed vortices are correct. The difference

in magnitude between the measured and predicted vortex

strength in Figure 19 is attributable in part to the location of the onset of separation. If separation is predicted to

start further aft on the body than the actual start of

separation, the predicted vortex strength will be less than

measured. The maximum error in vortex strength exhibited

in Figure 19 represents an error in the predicted axial

position of onset of separation of less than one body diameter

except at the highest angles of attack.

Measured and predicted downwash velocity fields in the

plane of symmetry of the model at Mm = 2 and ~1~ = 15' are

shown in Figure 20 for three axial stations. The results

are generally in good agreement, and the predicted velocity

distribution exhibits the correct trend in every case.

The vortex field used to calculate the induced velocities at x/D = 10 is the distribution shown in Figure 18.

68

Additional comparisons of measured and predicted downwash velocities on the lee side of the body at x/D = 10 are

shown in Figure 21. In Figure 21(a), the velocities are calculated along a line at z/D = .75 which passes through the predicted centroid of vorticity. The agreement is only fair

in the vicinity of the centroid (y/D : 0.4), an indication that the present discrete vortex cloud model is not completely reliable inside the cloud. In Figure 21(b), the measured and predicted velocities are compared along a line at

z/D = 1.05 which passes above both the measured and predicted centroid of vorticity. These results indicate the correct trend in the predicted velcoity distribution; however, since

the strength of the wake vorticity is nearly correct, it appears that a cloud model that is higher off the body is required to bring the predicted velocities into better agree-

ment with measurements. The need for a higher cloud is verified in Figure 18 where it is shown that the actual centroid of vorticity is approximately .15D above the pre-

dicted centroid. As shown in Figure 19, the strength of the vortex cloud is in good agreement with experiment.

Similar comparisons of measured and predicted downwash

velocity fields on the lee side of the ogive cylinder at

MO3 = 3, cLc = 15' are shown in Figures 22 and 23. The results on the plane of symmetry on the model lee side are shown in Figure 22 where it is obvious that the comparisons are not as good as those shown in Figure 20 for MW = 2; The general trends are correct, but the magnitude of the velocity components is in error. This is likely due to a combination

of circumstances. For example, in Figure 19, the predicted

vortex strength corresponding to M, = 3, CL~ = 15' is less than that measured at x/D = 7 and in good agreement with

measurements at x/D = 13. As seen in Figure 22, the vortex

induced velocity is less than that.measured at x/D = 7 and nearly correct at x/D = 13. The fact that the predicted

69

velocity profiles are not in agreement may be due, in part, to the problem discussed previously, the error in the predicted centroid of vorticity. The velocity profiles

indicate that the predicted centroid of the separation vorticity may be too low, a possible problem at higher Mach numbers.

Analogous to the Moo = 2 results in Figure 21, comparisons

of measured and predicted velocity profiles on the lee side of the body at x/D = 10 are shown in Figure 23 for Mm = 3.

The results look very good in Figure 23(a) at z/D = .75 where

the region of comparison is in the vortex cloud and near the vertical centroid of vorticity. In Figure 23(b), the agree-

ment between the velocity profiles is poor -with the predicted

velocities being much less than those measured.

Elliptic Bodies

The prediction method has been applied to a number of

different elliptic cross section bodies. A variety of

results are presented for several configurations under a range of flow conditions, the major objective being to

examine a large number of configurations to identify the

strengths and weaknesses of the code.

The 3:l elliptic cross section missile of primary

interest in these comparisons is the sharp-nose body described

in References 19, 23, and 24. The tests were conducted at

MC0 = 2.5 and a free-stream Reynolds number of 2x lo6 per foot.

The sharp-nose body considered in the comparisons to follow

was modeled with 18 circumferential panels around the cross

section and 29 axial rings of panels. Sample Cases 3 and 4

describedinTable II and Figures 14(c) and (d), respectively,

produce the results to follow.

70

Flow visualization via vapor-screen photographs (Ref. 23) and pressure data (Ref. 24) are available for

comparison with the theoretical results. For better visualization of the vortex development along the body, the photographs were digitized. These results are presented in the upper left corner of Figure 24 for the missile at

MC0 = 2.5, ~1~ = 20°, and 4 = O" and 45O. Cross sections through the body at three axial stations are shown. The extent of the vortices from the vapor screen photographs

are shown by the solid curve, and the predicted discrete vortex wake is shown by the individual vortices connected by a dashed line. The unrolled flow condition ($ = 0') shown in Figure 24(a) illustrates very good agreement between the predicted vortex cloud and the vapor screen results. The strength of the predicted vortex cloud is shown at each axial station, and because of symmetry, the left and right vortices are of equal magnitude and opposite sign.

The rolled flow condition (a = 45O) is shown in Figure 24(b). To be consistent with the output from the code, the body is shown unrolled with the flow angle rolled

45 degrees. The general character of the predicted vortices is in fair agreement with the experimental results: however, there are certain areas of disagreement. The theoretical vortex on the right side has rolled up much tighter than the vapor screen indicates, and the left theoretical vortex has not moved out away from the body as far as the actual

vortex, nor has it rolled up as tightly as the actual vortex.

Comparison of measured and predicted pressure distri-

butions at x/L = 0.60 on the body are shown in Figure 25 for

the unrolled and rolled condition. The unrolled results shown in Figure 25(a) illustrate some of the same problem areas discussed for circular bodies. The predicted

71

stagnation pressure on the windward side is in good agree-

ment with experiment, but the predicted pressure around the shoulder region of the cross section is much lower than the measurements. The pressure on the lee side is also under- predicted, possibly due to vortex interference effects. As

shown in Figure 24(a), there is a strong concentrated vortex field developed on the lee side at this axial station

(x/L = 0.60). The irregularities in the predicted pressures near the shoulder region are caused by local vortex inter-

ference effects. As noted in the discussion for the circular

bodies, the discrepancy in the pressure comparisons on the windward side of the body near the shoulders may be caused

by nonlinear effects induced by the close proximity of the

shock wave.

The measured and predicted pressures on the 3:l elliptic body in a rolled flow condition are compared in Figure 25(b).

The qualitative agreement is very good, but a problem on the

windward side of the body is in evidence. The vortex induced effects cause the irregularities in the predicted pressures,

and it appears that the induced effects are too large on the

positive lee side (0 = 320'). In Figure 24(b) it is obvious that the close proximity of the predicted wake will have a

large effect on the pressure distributions, and it is likely

that the predicted wake is causing the lack of agreement in

this region.

NOZVTX was also applied to a conical forebody with a

3:l elliptic cross section shape (Ref. 17). The major axis, in planform, is defined by a 20-degree half angle, and the total length of the forebody is 14 inches. The test was conducted at a free-stream Reynolds number of 2x lo6 per foot.

The panel model representation consists of 17 circumferential panels on the half body (34 panels on the total cross section)

and four axial rings of panels. The limited number of axial

72

panels is permissible because of the similar characteristics

on a conical configuration in supersonic flow described in the previous section on input preparation. The detailed

panel layout is described in Sample Case 9.

Comparisons of measured and predicted pressure distri-

butions on the conical forebody are shown in Figure 26 for two axial stations. Except for small differences on the lee side of the body, the measured pressure distributions are

nearly identical at the two axial stations. In Figure 26(a), the theoretical pressure distribution is not in good agreement with experiment on the windward side of the body,

particularly near the shoulders. On the lee side, two pre- diced curves are shown; the dashed curve represents no separation vortex effects, the solid curve has vortex effects . included. There is considerable difference in these results,

all of the difference attributable to the vortex induced effects. The vortex effects improve the predicted pressures

in the region around B = 120' on the body, but they detract from the agreement near the lee side stagnation point, 150° < B < 180". This may be an indication that the actual

lee side vorticity is rolled up tightly around f3 = 120°,

whereas the predicted vortex cloud is distributed over the entire lee side of the body as illustrated in Figure 27.

The prediction method has been applied to the elliptic

cross section body described in Reference 22 and illustrated in Figure 28. The major axis of the cross section is specified by an ogive distribution on the nose, and the cross section is a 2:l ellipse over the entire body length. The equivalent base diameter of this body is 1.125 inches. Circumferential pressure measurements at numerous axial

stations are available at M, = 1.5 and a range of angles of

incidence and angles of roll. The tests were conducted at a

free-stream Reynolds number, based on the equivalent

diameter, of approximately 0.3x 106.

73

This 2:l elliptic cross section body is modeled with 18 circumferential panels around the complete cross section

and 15 axial rings of panels. No smoothing is used in any of the calculations to follow. The general layout of the geometry is illustrated in Sample Case 11 in Figure 14(k).

Measured and predicted pressure distributions on the

elliptic cross section body at M, = 1.5, ~1~ = lS", and several roll angles are compared in Figure 29 at two axial

stations on the nose. The agreement between experiment

and theory for $ = O" is better away from the tip of the nose as shown in Figure 29(a). This is consistent with the circular body results described previously. The sparse nature of the measurements make it difficult to fully

evaluate the prediction method other than to observe that the basic trends of the pressure distribution are predicted.

At x/D = 2.1, the separation vorticity has grown in strength

and scope, and the major region of vortex wake influence covers the region between B = 90' and 13OO.

The predicted pressure distributions on the same body

at roll angles of 22.5O and 45O are shown in Figures 29(b)

and 29(c), respectively. The predicted results are generally

in good agreement with experiment, and the correct trends of

the data are modeled by the theory in each case. The local irregularities in the predicted curves are due to the

influence of the separation vortices.

NOZVTX was next applied to a 2:l elliptic cross section cone (forebody 4 in Ref. 17) in an unrolled flow condition.

The pressure results for Mm = 1.7 and ac = 20.35O are shown

in Figure 30 where it is obvious that agreement is poor over much of the surface. The lack of agreement on the wind-

ward side and near the body shoulder is typical of previous

results on elliptical cross section bodies. On the lee side,

there is a large effect of the predicted vortex field which

74

I I-

is spread out over much of the lee side. The effects of the

vortices on the measured pressures appear to be more limited in scope. These results are the same as those shown on the 3:l elliptic cone forebody in Figure 26(a).

The final 2:l elliptic cross section body considered

in these comparisons is forebody 2 from Reference 18, and the predicted results are from Sample Case 8 in Figure 14(f). The comparison.of measured and predicted pressure distributions in Figure 31 are in good agreement over the entire

body surface, even on the windward side. The pressure

distribution on the body is such that separation does not occur over most of the length of the forebody. Separation

begins at approximately x/L = 0.8, but in contrast to the elliptic cross section bodies with their major axis horizontal, the vortices are very weak and located near

the lee side stagnation point.

Arbitrary Bodies

The arbitrary noncircular body option in NOZVTX has been checked out for a number of different cross section shapes to verify the numerical transformation. Some of the shapes

considered are a square cross section with rounded corners,

a diamond cross section with rounded corners, and a lobed or pear shape cross section. The latter cross section is the

only shape for which pressure data are available, therefore, results on this configuration are described as follows.

The configuration selected for these comparisons is the

lobed cross section body, forebody 3 from Reference 18, shown

in Figure 13. The panel arrangement shown in Figure 13(b)

is used for the predicted results included herein, and the

specific calculations were made using Sample Case 10. Comparison of measured and predicted pressure distributions

75

at two axial stations are shown in Figure 32. At x/L'= .14

in Figure 32 (a), the agreement is very good on the windward

side of the body, only fair in the region of minimum

pressure, and poor near the lee side stagnation point. This

particular location near the nose of the body has minimal

vortex interference effects. At the mid-point of the

forebody length, x/L = 0.5, shown in Figure 32(b), the agree-

ment is only fair over the entire circumference even though

,thetrendof the data is predicted well. One unusual result in Figure 32(b) is the overpredicted pressure on the wind-

ward side of the body, a situtation which has not occurred

on any other configuration. The roughness in the predicted pressure distribution on the lee side of the body is caused

by local vortex interference effects. The vortices are

distributed along the body surface, and they have not started

to roll up.' A sketch of actual predicted vortex positions is

shown in Figure 33.

CONCLUSIONS

An engineering method and the program code NOZVTX to

predict the vortex shedding from circular and noncircular

bodies in supersonic flow at angles of attack and roll are described. The coupling of a supersonic panel method, an

axisymmetric source and doublet model and a two-dimensional

crossflow vortex shedding method has proved to be a successful

means to predict the general flow characteristics about slender, circular and noncircular bodies at large angles of

incidence and moderate Mach numbers. Comparisons of

measured and predicted pressure distributions and associated flow fields on a variety of bodies indicate that the

principal features of the flow phenomena are modeled. The

predicted vorticity distributions in the wake of a body of

76

revolution in supersonic flow are in reasonable agreement

with experiment. This.also results in reasonable agreement between measured and predicted velocity components in the flow field of the body. The ability to model the correct

flow field near the body leads to the capability to calculate wake induced interference effects on fins and other control surfaces as well as surface pressure distributions. The vortex shedding into an overall

analysis described herein can be incorporated computation method for missiles and aircraft.

RECOMMENDATIONS

In the course of development of the program code NOZVTX, several specific areas for needed improvements to the method

were discovered. These improvements were beyond the scope of the present study, but they are noted here as specific recommendations for future work.

The first, and possibly the most important, recommenda-

tion involves a thorough testing of program NOZVTX. Limited numbers of comparisons with experimental results on a

variety of bodies are included herein, but many other comparisons are needed. The program should be tested over a wide range of angles of attack and roll and Mach numbers to better define the operational limits of the method. The

program should also be applied to a wide range of noncircular body shapes for which data are available.

A critical weakness in the prediction method appears to be the difficulty in calculating accurate pressure

distributions on the windward side of certain bodies. For

example, from the comparisons with experiment described in

a previous section, the method has problems with conical

77

forebodies at all Mach numbers and most shapes at high Mach

numbers. These difficulties may be caused by nonlinear

,.effects induced by the shock wave when it is relatively near

to the windward body surface. Further investigation into

this problem is recommended.

Comparisons of measured and predicted velocity components on the lee side of an axisymmetric body at high angle of

attack indicate the method has a potential capability to

handle this problem. However, some additional work on the details of the vortex cloud are required. This may involve

a different procedure to blend the discrete vortices when

velocities in the midst of the cloud are to be calculated. With regard to the cloud itself, qualitative results on a

3:l missile body are in excellent agreement with experiment;

but based on one set of experimental results, predicted vortex clouds on an axisymmetric body are typically lower

than those measured. Additional investigation into the

mechanics of the vortex tracking procedure may explain this disagreement.

The final set of recommendations involve improvements

to various parts of the program. The Stratford separation

criteria have proved to work very well even though they are based on incompressible flow; however, similar criteria

or corrections to the existing criteria should be developed

for compressible flow. In this same area, it has been

observed in surface flow visualization studies that the

separation line on a body exhibits characteristics of laminar,

turbulent, and transitional flow. At the present time, the

crossflow plane separation may be either laminar or turbulent but not transitional or a combination of laminar and turbu-

lent. It may be possible to use surface flow visualization information to correlate the three separation regions on the

body and build this feature into the prediction method.

78

REFERENCES

1. Spangler, S. B. and Mendenhall, M. R.: Further Studies of Aerodynamic Loads at Spin Entry. Report ONR CR212- 225-3, U.S. Navy, June 30, 1977.

2. Mendenhall, M. R., Spangler, S. B., and Perkins, S. C., Jr.: Vortex Shedding from Circular and Noncircular Bodies at High Angles of Attack. AIAA Paper 79-0026, Jan. 1979.

3. Mendenhall, M. R.: Predicted Vortex Shedding from Noncircular Bodies in Supersonic Flow. J. Spacecraft & Rockets, Vol. 18, No. 5, Sept.-Oct. 1981, pp. 385-392.

4. Marshall, F. J. and Deffenbaugh, F. D.: Separated Flow Over Bodies of Revolution Using an Unsteady Discrete- Vorticity Cross Wake. NASA CR-2414, June 1974.

5. Deffenbaugh, F. D. and Koerner, W. G.: Asymmetric Wake Development and Associated Side Force on Missiles at High Angles of Attack. AIAA Paper 76-364, July 1976.

6. Mendenhall, M. R. and Nielsen, J. N.: Effect of Symmetrical Vortex Shedding on the Longitudinal Aerodynamic Characteristics of Wing-Body-Tail Combina- tions. NASA CR-2473, Jan. 1975.

7. Nielsen, J. N.: Missile Aerodynamics. McGraw-Hill Book Co., Inc., 1960.

8. Skulsky, R. S.: A Conformal Mapping Method to Predict Low-Speed Aerodynamic Characteristics of Arbitrary Slender Re-Entry Shapes. J. of Spacecaft, Vol. 3, No. 2, Feb. 1966.

9. Dillenius, M. F. E., Goodwin, F. K., and Nielsen, J. N.: Prediction of Supersonic Store Separation Character- istics. Vol. I - Theoretical Methods and Comparisons with Experiment. AFFDL-TR-76-41, Vol. I, May 1976.

10. Dillenius, M.. F. E. and Nielsen, J. N.: Computer Programs for Calculating Pressure Distributions Includ- ing Vortex Effects on Supersonic Monoplane or Cruciform Wing-Body-Tail Combinations With Round or Elliptical Bodies. NASA CR-3122, Apr. 1979.

79

--- -

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

Woodward, F. A.: An Improved Method for the Aerodynamic Analysis of Wing-Body-Tail Configurations in Subsonic and Supersonic Flow. Part I - Theory and Application. NASA CR-2228, Part I, May 1973.

Woodward, F. A.: An Improved Method for the Aerodynamic Analysis of Wing-Body Tail Configurations in Subsonic and Supersonic Flow. Part II - Computer Program Description. NASA CR-2228, Part II, May 1973.

Cebeci, T., Mosinskis, G. J., and Smith, A. M. 0.: Calculation of Viscous Drag and Turbulent Boundary- Layer Separation on Two-Dimensional and Axisymmetric Bodies in Incompressible Flow. McDonnell Douglas Rept. MDC JO 973-01 (Contract N00014-70-C-0099), Nov. 1970.

Stratford, B. S.: The Prediction of Separation of the Turbulent Boundary Layer. J. of Fluid Mech., Vol. 5, 1959, pp. l-16.

Keener, E.: Oil-Flow Separation Patterns on an Ogive Forebody. NASA TM-81314, Oct. 1981.

Barger, R. L.: A Distributed Vortex Method for Computing the Vortex Field of a Missile. NASA TP-1183, June 1978.

Townsend, J. C., Collins, I. K., Howell, D. T., and Hayes, C.: Surface Pressure Data on a Series of Conical Forebodies at Mach Numbers From 1.70 to 4.50 and Combined Angles of Attack and Sideslip. NASA TM-78808, Mar. 1979.

Townsend, J. C., Howell, D. T., Collins, I. K., and Hayes, C.: Surface Pressure Data on a Series of Analytic Forebodies at Mach Numbers from 1.70 to 4.50 and Combined Angles of Attack and Sideslip. NASA TM-80062, June 1979.

Graves, E. B.: Aerodynamic Characteristics of a Mono- planar Missile Concept With Bodies of Circular and Elliptical Cross Sections. NASA TM-74079, Dec. 1977.

Oberkampf,.W. L. and Bartell, T. J.: Supersonic Flow Measurements in the Body Vortex Wake of an Ogive Nose Cylinder. AFATL-TR-78-127, Nov. 1978,

Landrum E. J.: Wind-Tunnel Pressure Data at Mach Numbers from 1.6 to 4.63 for a Series of Bodies of Revolution at Angles of Attack From -4' to 60'. NASA TM X-3558, Oct. 1977.

80

22. Goodwin, F. K. and Dyer, C. L.: Data Report for an Extensive Store Separation Test Program Conducted at Supersonic Speeds. AFFDL-TR-79-3130, U.S. Air Force, Dec. 1979.

23. Allen, J. M. and Pittman, J.L..: Analysis of Surface Pressure Distributions on Two Elliptic Missile Configurations. AIAA Paper 83-1841, July 13-15, 1983.

24. Allen, J. M., Hernandez, G., and Lamb, M.: Body- Surface Pressure Data on Two Monoplane-Wing Missile Configurations With Elliptical Cross Sections at Mach 2.50. NASA TM-85645, 1983.

81

TABLE I - NOZVTX SUBROUTINE LIST

Subroutines

NOZVTX" AMAP

ASUM

BDYGEN

BMAP

BODPAN

BODVEL

BODYR

CMAP

COMBIN CONFIG

CONFOR

DFEQKM DIAGIN

DOUBLT DPHIDT

DTABLE

DZDNU

ELNOSE

F

FLDV-EL

FORCEP

FPVEL

GEOM

INPT

INVERT

Subroutines

ITRATE

NEWRAD PANEL

PARTIN

PLOTA

PLOTAS

PLOTA

PLOTA

PLOTA

PLOTV

SEPRAT SHAPE SMOOTH

SOLVE

SORPAN

SOURCE SRFVEL

SUM

VCENTR

VELCAL

VELCMP

VLOCTY

VOLUME

VTABLE

Z

* Main program.

82

TABL

E II

- SA

MPL

E C

ASES

FO

R

PRO

GR

AM NO

ZVTX

~-

Sam

plei

R

ef.

Cas

e /

1 :

19

Fig.

14 (a

)

14(b

)

14 (c

l

14(d

)

14(e

)

--

Body

M

_ :

ac

' $

( G

ener

al

Com

men

ts

Exec

utio

n C

ross

I

Tim

e Se

ctio

n ~

(CD

C-C

YBER

3:l

; 2.

5 El

lipse

3:l

Ellip

se

3:l

Ellip

se

3:l

Ellip

se

Circ

le

2.5

2.5

2.5

2.0

I 20

° 1

o"

20°

2o"

2o"

15O

O0

O0

4s" O0 -c

I I I

An

ellip

tic

mis

sile

bo

dy

case

w

ith

flow

sy

mm

etry

, pa

nel

sym

met

ry:

surfa

ce

pres

sure

ca

lcul

atio

n,

vorte

x fie

ld

plot

s at

ea

ch

axia

l st

atio

n,

and

velo

city

fie

ld

calc

ula-

tio

n at

se

lect

ed

axia

l st

atio

ns

(TES

T C

ASE)

.

Res

tart

of

Sam

ple

Cas

e 1

Full

run

for

mis

sile

bo

dy

in

Sam

ple

Cas

e 1

Sam

e el

liptic

m

issi

le

body

fro

m

Sam

ple

Cas

e 1;

no

flo

w

sym

met

ry,

surfa

ce

pres

sure

ca

lcul

atio

n an

d vo

rtex

field

pl

ots

at

each

ax

ial

stat

ion.

An

oyiv

e cy

linde

r bo

dy

case

w

ith

line

sour

ces

and

doub

lets

, flo

w

sym

met

ry:

surfa

ce

pres

sure

ca

lcul

atio

n an

d vo

rtex

field

pl

ots

at

each

ax

ial

stat

ion,

flo

w

field

ca

lcul

atio

n at

se

lect

ed

axia

l st

atio

ns.

760)

.i

8.6

sec.

N/A

-- N/A

15.5

se

c.

al

W

a3

P

Tabl

e II

- C

ontin

ued

Sam

ple

Cas

e :e

f.

18

18

Fig.

14 (f

)

14 (9

)

14 (h

)

14(i)

Body

C

ross

Se

ctio

n

Circ

le

2:l

Ellip

se

2:l

Ellip

se

3:l

Ellip

se

1.6

1.7

1.7

1.

l-t C

O0

20.3

6'

20.3

6"

20.3

5'

$ O0

O0

0"

O0

Gen

eral

C

omm

ents

An

ogiv

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REAL PLANE

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CIRCLE PLANE

(a) Analytical transformation procedure

Z

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T

(b) Numerical transformation procedure

Figure 2.- Conformal mapping nomenclature.

87

Secondary separatibri

point

- Primary separation

point

t’ ‘l&sin cxc

Figure 3. - Sketch of crossflow plane separation points.

88

Z

t t

C n 3 0 +r

Figure 4. - Body crossflow plane nomenclature.

89

. ---. - i----Y

i I I i

(a) Single vortex

(b) Multiple vortices

Figure 5.- Vortex smoothing procedure.

90

PROGRAM NOZVTX

I-- INPUT I-- OUTPUT I-- TAPE10 I-- TAPE5 I-- TAPE6 I-- TAPE7 I-- TAPE8 I-- TAPE9 I-- INPT ---- TAPES

I I-- TAPE6 I-- EOF

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I I I-- SQRT

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I I-- SIN I-- FLDVEL ---- TAN

I I-- cos I-- SIN I-- SORPAN ---- TAPE6

(a) Page 1

Figure 6.- Flow map of pr‘ogram NOZVTX.

91

0 B I I-- SQRT I I-- ATAN I I-- ALOG I I-- ACOS I-- VLOCTY ---- TAPE4

I-- SUM ---- DtDNU ---- CABS I-- SQRT I-- EXP I-- ATiN I-- SMOOTH ---- ATAN I I-- SQRT

I I-- SIN I I-- cos

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-- BDYGEN ---- TAPES

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---- CABS ---- CABS

I I I I I I I I-- GEOH

I-- TAPE6 I-- SIN I-- SQRT I-- ASIN I-- BDDYR ---o. SmT I-- SOURCE ---- TAPE6 I I-- SQRT I I-- ALOG I-- DOUBLT ---- TAPE6

I-- SORT I-- ALOG

---- TAPE10 I-- TAPES I-- TAPE6 I-- TAPE7 I-- EOF I-- CONFIG ---- TAPES I I-- TAPE6 I I-- TAPE9 I I-- SORT I-- NEYRAD ---- TAPE10 I I-- TAPE5 I I-- TAPEb

I I-- SIN I-- cos

I I-- SQRT I

0 D I-- ATAtit

(b) Page 2

Figure 6.- Continued.

92

0 C I I

I I I I-- l I I I

f I

I I

I I

I

0 D

I-- BODPAN ---- TAPE10 I-- TAPES I-- TAPE6 I -- TAPE7 I-- PANEL ---- SORT

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I-- TAPE6

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I TRITE .----

1::: PARTIN ----

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;::

1:: ASUM ---- SORT ATAN SOURCE ----

l-- I--

DOUBLT ---- I-0

0 G

TAPE 10 TAl’E7 TAPE9 INVERT ---- TAPE6 TAPE10 TAPE6 TAPE9 TAl’E9 INVERT ---- TAPE6 TAPE6

DZDNU ---- CABS

ATANL SORT SIN CDS EXP CABS

TAPE6 SORT ALOG TAPE6 SORT

(12) Page 3


93

0 E I I

I-- l I

I

I-- l -0 I I

F Q 0 G I-- ALOG

I-- cos I-- SIN

DPHIDT ---- ATAY2 I-- SMOOTH ----

l-- I-- l-- l--

FORCEP PLOTV o-o- pLoTA OS--

l--

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ATAN SORT SIN’ cos EXP

TAPE6 PLOTAB ---- PLOTA ---- ALOGlO

I-- PLOTAS PLOTA

I-- FPVEL ---- TAPE6

I I-- ATANE I-- CHAP ---- DZDNU ---- CABS

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-- ATANE I I I -- SOURCE ---- TAPE6

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I I I-- cos

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I I I-- SORT I I-- ATAN I I -- ALOC I I I -- ACOS

; I-- VLOCTY ---- TAPE4

I I I -- SUA ---- OZONU ---- CABS I-- SQRT

I I I-- EXP I I I-- ATAN I I I-- SMOOTH ---- ATAN I I -- SQRT

I I I I-- SIN I I I-- cos

I I I

I -- EXP I -- ASUM ---- CABS I I-- SORT I-- SEPRAT ---- TAPE6 I I-- SQRT I-- VTABLE I-- DFEQKH ---- TAPE6

I-- F ---- SHAPE I-- BHAP o--o AMAP -o-o 2 ---- CABS

(d) Page 4


94

PROGRAM EN0

I

I

l-- I--

OTABLE VCENTR COHBIN

Q 1 Q 3'

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I -- ATAN I-- CHAP

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---- TAPE6 I-- SORT

(e) Page 5

Figure 6.- Concluded.

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P 0 a3

ITEM

5

10

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20

25

30

35

50

45

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55

60

65

70

75

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1

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Page

2

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re

9.-

Con

tinue

d.

ITEM

--

(14)

‘1

M

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3

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ITKM

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(d)

Page

4

Figu

re

9.-

Con

tinue

d.

ITEM

_-

-- 10

20

30

40

G

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(28)

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if

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(e)

Page

5

Figu

re

9.-

Con

tinue

d.

ITEM

(N

FOR

X “a

IIll

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--

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snR

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, 21

28

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42

43

56

63

70

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5)

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6),

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if

52

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epea

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m

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N

FOR

X tim

es

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t itr

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-(43)

if

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= 0

(f)

Page

6

Figu

re

9.-

Con

tinue

d.

ITEM

80

(3

9)b

KRAD

X KF

OR

X

(41)

b (IK

RAD

XI

valu

es)

r '

'41

2 28

35

42

49

56

63

70

PH

IK(1

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IK(2

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1

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28

3.

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49

56

63

70

XF

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(2)

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I

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D?

1 20

30

(4

4)x

XNO

SE

1 XL

BOD

Y I

I I

l Cnn

it ite

m

(42)

if

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X 1

0

rcan

it ite

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(43)

if

KFO

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= 0

/.chi

t ite

ms

(33)

-(43)

if

NC

IR

= 0

xCkn

it ite

m

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if

NC

IR

# 0

(9)

Page

7

Figu

re

9.-

Con

clud

ed.

Plane of syrmtry

1 I I I

I / [m 0% J) , = (N% J) 1

Figure lO.- Convention for ordering coordinates for a noncirctiar cross section at X = m(J).

115

(a) 9 panels, equal spacing

(b) 10 panels, equal spacing

--- Body surface

- 9 panels (KRADX = 10)

--- Body surface

- 10 panels (KRADX = 11)

--- Body surface - 9 panels

(K~Dx = -lo>

(c) 9 panels, custom spacing

Figure ll.- Panel layout on a 2:l elliptic cross section.

116

(KtiDX = 14) (a) 13 panels, equal spacing

--- Body surface - 19 panels

(KRADX = -20) (b) 19 panels, custom spacing

Figure 12.- Panels layout on a 3:l elliptic cross section.

117

e-w Body surface --- Body surface

- 9 panels, custom spacing = -10)

(a ) 9 panels, custom spacing lb) 1 7 panels, custom spacing

Figure 13.- Panel layout on a lobed cross section.

- 17 panels, custom spacing

(KRADX = -1 8)

118

Item

(1) (2) (31

(4) (5) (6) (7) (8) (9)

(10)

(11)

(12)

(13)

(23) (24)

(30) (31) (32) (33)

(36)

(37)

1 n 0 :

0 1 0 1 10 0 0 0 0 0 4 1 1 3

NIELSEN ENGINEEkfING 5 RESEAWCn. INC. PROGRAM NOZVTA SAMPLE CASE 1 3:l ELLIPTICAL CROSS-SECTION IWOY REF : NASA TM-74079 BY GHAVES MACH NO = 2.5~ ALPrlAC = 20 OE~EESI PHI = 0 DEGREES

12.56588

;.a .Ol

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3:l ELI IPTICAL CtiOSS-SECTION hllL)y - &DUIllWJAL PAIJELII~~ INIJUl f-l n -1 n A

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1.774 1.922 3.051 3.203 3.432 3.357

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(39) (40) (41) (43) 0.0 .420 .700 2.10 3.50 4.Yn 3.bO b.30

(a) Sample Case 1

Figure 14.- Sample input decks for program NOZVTX.

119

1 0 0 0 1 1 0 0 0 3 0 3 0 0 4 1 1 : 3 0

NIELSEv ENGINEERING K. GESEAXH- INC. PROirUAM NI)ZVTX SAMPLE CASE 2 3:1 ELLIDTICAL CPOSS-SECTION JODY QEF: NASA TM-74079 BY GQPVES MACH NO r 2.5. ALP-AC = 20 OFG*EFSa PHI = 0 tEGclEE5

26.0 a.n 0.0 2.5

28.0 l.c. 1.05 0.0 1.0 0.0 0.0 0.U 0.0

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ELLIPTICAL CaUSS-SECTIOQ d30Y - mDldI,‘l Gt>*tTey Ih.'uT 3:1 0 0

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1.1er 1.281 i.u3* 2.136 2.286 2.265

,165 .135 .07b ,059 -.030 -.041

1.77* 1.922 j.651 3.203 3.~32 3.397

10 30 ,420 .700 2.10 3.=0 a.90 5.60 0.30 7.70 A.40

10.50 11.9r) 12.40 13.30 iti.7n 15.60 Ib.F30 17.50 18.20 19.7'l 20.30 21.00 22.40 23.10 24.50 25.90 26.bO 28.00

,176 .294 aa1 1.275 1.617 1.774 1.922 2.148 2.326 2.672 2.874 i.965 3."51 3.203 3.269 3.378 3.*16 3.448

U'," 3.432 ,098 .2Q4 3.397 3.303 .425 3.248 .539 3.128 .591 3.ou9 .641 2.755 .733 2.0Q5 .775 . A91 .950 .qaa 1.017 l.Oh8 i-090 1.126 l.l*O 1.149 1.152 1.144 1.132 1.101 l.OU3 1.043 1.003 .V85 .562

ELLIPTICAL CROSS-SECTIOY dOOy - A@DlTIOkAL PAlrELIlYCI INPUT -1

(b) Sample Case 2


120

1 0 0 1 U 0 0 0 0 0 0 0 ‘ 1 1 :

NIELSEN ENGIWEQING 5 QESEAQCH. INC. PRJWAH hOZVTX SAMPLE CASE 3 3:l ELLIPTICAL CQOSS-SECTION BOOY WF: NASA TM-74079 dY GEAvES MACH NO = 2.5. ALP+AC = 20 DEG9EES. !-WI = 0 DEWEES

1?.565RB 4.n 20.0 0.0 E.8 2R.0 . 01 0.0

30 n.0 7.7no 15.*00 ?Z.AOO n.O 1.445 2.179 2.2n2 .280 .1Z!a .055 -.047 n.00

2.19n 3.2'9 3.3')3 an.00

.733 I.090

.420 P.400

16.800 23.100

.110 1.550 2.252 2.165 .Pe.O ,119 .047 -.955 .174

2.326 3.378 3.24R .059 ,775 1.126

14.R Mh463. 1.1

0.0

.700 4.109

17.500 2*.5on .196 1.631 2.279 2.086 .20n .11n .n3* -.057 .294

2.d.L7 3.418

3.12R .n9= . nl6 i.len

?8.0 0.0

1.05 0.0

2.100 lfil.500

ld.200 25.000 .537 1.7n1 2.209 2.006 .234 .102 .020 -.055 .Pdl

2.672 3.443

3.9ns .294 .A91 l-1*9

i.lni 1 .oi33 I.003 1.003 3:1 ELLIPTICAL CPOSS-SECTIUK dODy -

0 0 -7 -1 10 30

4.0 2.5 0.0 0.0

1.0 0.0

3.500 4.900 5.400 11.900 12.bOO 13.300

19.osn 19.700 20.300 26.600 2.5.000 .a50 1.079 1.182 l.Vl6 1.977 2.J34 2.309 2.3112 i.286 1.970 1.923 .175 .15s .1*5 .093 .OR5 .w76 .004 -.017 -.030 -.039 -.034 1.275 l.bl7 1.776

2.c74 2.045 3.Lf51 3.&b* 3.45b 3.&32

2.955 2. .tiq .025 .53Q .591 .950 .9rla 1.017

1.155 1.152 l.lUL . Y85 . -02

0.0 .420 ,700 2.10 3.50 b.9t.l 5.60 0.31) 7.70 8.-O 9.10 10.50 11.90 I%.60 13.30 lb.70 15.40 l?.i+O 17.50 16.20

19.040 19.70 20.30 21.00 22.&O 23.10 2b.50 25.~0 Cb.50 is.00 0.0 ,176 .294 .?kll 1.275 l.bl7 1.77r. 1.922 2.136 2.326

2.~47 2.672 2.07~ 2.965 3.p51 3.2n3 3.269 3.37H 3.*id 3.44d 3.464 3.656 3.432 3.397 3.303 3.208 3.128 3.ofiv 2.355 2.995 0.0 .059 .09@ .2QL .L2S .539 .591 .5*1 .733 .775 ..'16 .a91 .95R , PRA i.nl7 l.Oh@ 1.990 1.126 i.i*n 1.1-Y 1.15% 1.152 1.144 1.132 1.101 1.0r3 1.043 l.Oti3 ,985 .YbZ

3:l ELLIPTICAL CQOSS-SECTION HODY - aDCITIONAL *r\NE~lhut IWLIT n n -1 n 0

b. 300 14.700

21.000

1.261 2.136 2.265

.I35

.oa9 -.O*l

l..J22 3. r03 3.357

.4&l 1. .)48

I. 132

(c) Sample Case 3


121

1 0 t :

0 Ii

1 1 0 0 0 0 0 0 4 1 3 0

NIELSEQ ENGINEERING L. REzEARCH* INC. PROGfiAh NOZVTX SAMPLE CASE A 3:1 ELLIPTICAL CQOSS-SECTION 8ODy I?EF : NASA TM-74079 dY MAC* NO.= 2.5. ALPHAC = %b DEGSEES. %I = 05 DtWEES

lZ.S658A 70.0 ?.B .Ol

30 0.0 7.700 \5.40@ 22.400 0.0 1.405 2.179 7.202 .2eo .12P .055 -.nA7 n.00

2.19n 3.269 3.3?3 0.00

'.733 1.090

4.0 16.8 2d.0 45.0 666463. 0.0 20.0 l*A 1.05 0.0 0.0 0.0

.uzo . 700 R.40@ 9.100

16.800 17.SOF 23. lo.0 24.5on

.118 .I96 I.550 1.631 2.242 2.279 2.165 2.096 .200 .2er! .119 .llO .047 . n34 -.055 -.057 ,176 .290

?. 326 2.147 3.370 3.bl.S 3.2&R 3.128 .059 . n9a .775 . 916

2. Ino 10.500

lR.200 25.+00 ,587 1.791 2.299 2. a!!‘4 .23c. .102 ,020 -.055 .@I31

2.472 3.4&d

3.oi)9 .29& .991 1.119

1.003 1.126 1.140 1.093 1.043 1.101

3:1 ELLIPTICAL CROSS-SECTION -lOOY - 1 n

2.5-- 0.0 1.0 0.0

3.500 4.voo 11.900 lC.600

19.040 19.700 26.600 28.000 .650 i.07e 1.915 1.577 2.31)') 2.31~4 1.97n 1.023 .175 .15? .093 . l)d5 .ilO4 -.017 -.035 -.O?* 1.275 1.61 7

2.P7C 2.465 3.464 3.-50

2.955 ;1.ta5 . *25 .539 .95R .3ee

1.155 1.152

0.0

5.600 6.300 13.300 14.700 2u.300 21.000

l.ld2 1.281 2.034 2.136 C.2i3d 2.265

.lC5 .135

.07b .069 -.030 -.O*l

1.772 I.Y22 3. US1 3.2b3 3.432 3.397

.5Yl 1.a17 1.1&k

.641 1. soa

1.132 . Yes . ib2

cCY3or r,EDrETqr INPUT

-3 -I in 30 0.0 ,420 .70$l 2.10 3.50 4.9n 5.60 4.3ti 7.70 b.40 9.10 10.50 11.9fl 12.60 15.30 14.70 15.-o lb.:0 17.55 16.20

19.040 19.70 20.30 21.00 22.40 23.10 2*.50 25.90 26.60 re.oa 0.0 .176 .294 .ee1 1.275 1.617 1.774 I.922 2.lqh 2.326

2.447 2.672 2.074 2.965 3.551 3.203 3.249 3.378 3.w19 3.440 3.464

f;;;" 3.432 3.397 3.303 3.2&R 3.12Q 3.0ns 2.355 2.835

0.0 .OV8 .204 .r25 ,539 .591 .bel .733 .775 .a16 A91

i.152 .95P .9e.A 1.017 I.069 1.09n 1.125 1.140 l.lLY

1.159 1.144 1.132 1.101 l.lJY3 1.0&3 1.003 .3e5 .562 3:1 ELLI"TlCAL CqOSS-SECTIOY YODY - ACWITIcJkAL PAiuELIW IluVuT

0 0 -1 n 0

0 0

GFtAvES

(d) Sample Case 4


122

7

0 0 0 :

0 1 0 3 109 .o 0 0 0 0 4 1 1 3

NIELSEN ENGINEERING r. REXARC** INC. P~OGI?AM Wi!vTa SAMPLE CASE 5 CIRCULAR OGIVE CrLINOER REF : AFATL TR-78-127 tlr OBER*PMPF MAC+ NO.= 2.0. ALWAC = 15.-EGREES. OhI = 0 ‘tGREES

.Obe56 3.0 39. 3.0 15.0 3.0 .Ol

24 n.0. 7.00 4.on 4.00 0,. 0

.a5630 1.34216

1.500 . c3333 .33041 .15883

n.0 21.0 0.0 0.0 0.0 0.0 r1.0 n.0 0.0 .3n .3 .3 .3 .3 .3 .3 .45 .45 .45 .45

0.0 39. 0.0

1.0 0.0

-25 2.25 4.25 36.0

.129Al

.93606 1.37933 1.500 .50526 .30773 .13857 0.0

30. 1.2 1.50

I. rn 2.10 2.40 2.7n

3.00 1.2 I.50

1..9n 2.1n 2.40 2.70

3.00 ‘1.05

1.10 1.2 1.30

l.cn 1.50 1.60

1.70 1.90 1.90

2.00 2.10

2.20 2.30

2.40 2.70

3.00 1.05 1.10

1.2 1.30 1.40

1.50 I.hrl

.=I0 2.50 r.5n

.25272 1.n102o I.41144

0.0 2.0 .05 0.0 0.0 0.0

-75 1.00 2.75 3.00 a.75 5.00

36095 .47d70 .07@@3 1.14203 .43RSd 1.461772

45195 42630 26361 :24209 09851 .07867

1.25 1.50 3.25 3.50 5.25 ea.50

.58216 , bi'948 I.19090 1.25250 1.47792 l..bYO14

.‘lUl4S .37722

.22OP9 .13440

.05393 . U3525

.77(182 1.29990 1 .ri9755

. 47815

.2e548

.I1847

.35355

.I7920

.01441

. 45

.45

.45

. 45

.45

.45

.45

.45

.45

. 4s

.45

. 45

.45

.bO

.40

.60 .60 .60

.6

.h

3G.

(e) Sample Case 5

1.75 3.75 5.75


123

id i.70 .6 1.80 ;i 1.90 .6 2.m .6 2.10 .h 2.20 .6 2.3n

.6 2.4r)

.6 2.70 .6 3.00 .75 1.2 . 75 1.30 .75 1.4 .75 1.5n .75 175 75

.75

.75

.75

.75

. 75

.75

.75

.75

.Q

.9

.9

.Q

.9

.9@

.9

.9

.9

.9

.9

.9

.Q

.Q

.Q

.9

.Q

.Q

1.6 1.7 1.l3n 1.9

2.0 2.1n 2.zn 2.3 2.4n 2.7n

3.00 .75

.n

.Q 1.1 1.1

1.2 1.3 1.4

I .5n 1.6 1.7 1.a 1.9 2.0 2.1.) 2.2 ?.3 2.4”

.9 2.711 .Q 3.00 1. n5 l.@C :; l.ns .d 1.05 .9 1.05 1.9 1.05 1.1 1.05 1.2 1.05 1.3 1.05 1.6 1.05 1.50 1.05 1.6 1.05 1.7 1.05 1.90

1.05 1.05 ::;: 1.05 2.10 1.05 2.2 1.05 2.3 1.05 2.41l 1.05 2.70 1 .nr. 7.nn

(e) Sample Case 5, Continued.


124

.__- 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.2n 1.20 I.20 1.2n 1.20 1.20 1.2 1.2 1.2 I.2 !.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3= 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 I.35 1.35 1.35 1.35 1.35' !.35 1.35 1.35 !.35 1.35 1.35 1.35 1.35 1.35 1.3s 1.35 1.35 1.35 1.5n 1.5n I.50 1.50 1.50 1.50 1.50 1.50 1.50 1.5n 1.50

51

0.0 .lO .2

.3 .4 .5

.;”

.A .Q

1.0 1.1 1.2

1.3 1.4

1.5n 1.6 .7 1.Q 1.9

2.n 2.1n

2.4n 2.70

3.on 0.0 .l .2 .3

.;' .b

.7 .8 .9 1 . 1 1.1

1.2 1.3 1.4 1.5n 1.6

1.7 1.80

1.9 2.0 2.10 2.2

2.3 2.20 2.70

3.no 0.0 .3 .b .9

1.2 l.SO

1.80 2.10 2.40 2.70

3.00 z 6.n -49.

(e) Sample Case 5, Concluded.


125

0 0 0 l-l 0 Cl 11 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0

NIELSE~J ENGiNEEAiNG L WsEWH:INC. SAMPLE CASE 6. CIQCLILAR OGIVE MACH NO.= 1.6.

7.07 20.0 n.0 0.0 .50 19.5 .Ol 0.0

11 .-I . 1. 2. 9.

CYLINDER PEF: NASA TW h-3551) 9Y LAivDRUW ALPWAC = r) DEGYEESt ~1 = 0 DEGNEES

9.0 20.0 3.0 ~00000. 0.0 1.6

0.0 1.05 0.0 1.0 1.0 0.0 0.0 0.0

2. ZP.

3. 4. 5. b.

n. .32 .60 .!?.r 1.04 1.22 1.34 1.ca 1.50 1.5n .343 .301 .?61 .221 .lt)3 .14h .109 .E3h 0.0 0.0

51 2 9.0 2n.

7.

1.42

. 0 I2

(f) Sample Case 6


126

1 0 0 1 0 1 1 0 0 0 0 0 0 0 0

NfELSEN 1 ENGINEEYING 1 1 5 3 RESEA%H .‘INC. DQOGirAP NOZVTX SIMPLE CISE 7 2:1 ELLIPTICIL CROSS SECTION FOREBODY REF: N4S4 TM-80062 MY TOWNSEND MACH NO.= 1.7. AL+rt4C = 20.36 DEGREES. PHI = 0 DEGREES

13.17679 7.0 3.60342 20.36 .50 .05

39 0.0 p.on 4.00 7.00 11.0

0.00000 0.50696 0.93592 1.43313 l.n2310 0.27300 0.23398 0.19498 0.136&Q 0.05R49 0.00000 0.33797 0.62395 0.95542 1.215~0 0.00000 0.67594 1.24790 1.91nsc

14.0 0.0 14.0 0.0

900855. .SO 0.0

14.0 0.0

1.05 0.0

1.7 - 0.0 0.0

1.0 0.0

025 2.25

4.25 7.50 11.50

0.06764 0.564@4 n.994ob l.L9r+94 1.84991 0.26310 0.22910 0.19012 0.1267* 0.04R75 0.0*509 0.37656 0.65604 ?. 99929 1.23327 0.0~019 0.75319 1.31208 1.39P59

. SO 2.50

4.50 4.00 12.0

0.13405 q.62151 l.n3n9fi 1.55997 1.n71a5 Il.?6322 n.22424 0.19524 0.11690 r).o39on n.n8937 0.61434 0.68732 1.n3991 I.?!*790 0.17874 n.a2868 l.-,7cb4 2."7983 ?.*9TF.-

.75 2.75

4.75 8.50

12.so n.19925 (3.67696 1.37669 1.51593 1 .&MY1 0.25836 0.21936 O.l8036 n.10724 0.02924 0.13283 0.45131 n.71779 1.07729 1.25927 'l.2b567 0.90241 1.*>557 2.15057

2.*30RO 2.rh455 2.51+5* ;r:1 ELLTPTICAL CaOSS-SECTION ;ORkiODr n 0

-3 -1 18 29 0.0 .50 5.0 5.5 k",

10.0 10.5 11. n.0 .na937 .17549

. 74744 .a0431 .05793 1.169901.19~281.215~0

. 17076 .35(197 1.:9:&60~621.715ft6 2.'.39#12.38A552.430~0

2:1 ELLIPTICAL CROS n 0 -1 n 11 n.O 1.0 2.0 14.

I.5 6.5 11.5

-25835 .9083n

1.23327 .51671

l.Rl66il 2.J.6655 5-5ECTI

3.0 4.0 5.0 6.0 8.0 10.0 12.

1.00 3.00 5.00 9.00

13.0 d.26323 0.73119 I.12116 l.bb711 1.90109 0.25349 0.214&r! 0.17426 0.09769 II.01950 0.17549 0.4074h 0.74744 1.11141 1.267*0 0.35097 0.97*9? 1.49-69 2.22282 2.53*7Q

1.25 3.25 5.50 Y.50 13.50

0.32599 0.76~20 1.2064b 1.71342 1.90941 0.248bO 0.20960 o.lb;73 O.UH775 ti.OOQYl cl.21733 0.52280 0.60431 l.li228 1.27227 0.*3&65 1.04560 1.609b2 2.28-56

1.50 3.50 b.00 10.0 16.0

0.38753 o.f!3599

KEZ . l.Yll 0.24372 0.20474 0.15599 0.07800 0.00000 @.25835 0.55733 0.*5793 1.16996 1.27.+rJO 0.51671 1.11*66 1.71586 2.33981

Z.>rrS* 2.54ano - -Dr=CY GEWETAY JNgdT

2.0 2.5 3.0 3.5 4.0 7.0 7.5 8.0 a.5 9.0 Y,:i'

12. 12.5 13. 13.5 14. . 33797 .*1434 .487&b .55733 .62395 .6t1732 .95542 .~99291.039911.077291.111411.1~22’? .2~7901.259271.267c01.272271.274 . h7594 .eze4a .97~2~1.1144hl.247901.374b4 . 9lOA*l.9995Y2.079932.15~572.222822.28~56 .495802.51P542.534792.5k~542.5*B N FOYEtdODY - ADDITIONAL JnNELINlj INPClT

1.75 3.75

b.50 10.50

0.447d5 r).@4bS7 1.3b245 1.79142

0.2368b 0.149tlb 0. I*624 0.04Ll2.l

'1.29B57 0.59105 0.901330 1.19LZd

0.5-.7lr 1.18209 1.81bbU 2.3n855

(g) Sample Case 7


127

-

1 0 0 : i

11 0 0 0 0 0 0 0 0 4 1 1 0

NIELSELi EMGINEERING & RESE4RCH. 1%. PROGSA.CI NOZVTX S4uPLE CASE P 2:1 ELLIPTICAL CROSS SECTION F5REGODr REF: N4SA TM-80062 BY TOJNSEND r4CH NO.= 1.7. ALPUAC = 20.3h JEGREESt PHI q 0 DEG;cEES

13.17679 20.36 .50 .05

39 n.0 2.00 4.00 7.00 11-n

0.00000 0.50h96 0.93592 1.43313 1.82310 0.27300 0.23398 0.19*9s 0.13649 0.05Rb9 0.00000 0.33797 0.62395 0.95542 1.21540 0.00b00 o.fl759a 1.247QO l.9lne4

l*.O 7.0 14.0 3.60342 0.0 900855. 0.0 1.7 14.0 .50 1.05 0.0 0.0 0.0 0.0 0.0

.25 2.25

4.25 7.50 11.50

0.06764 0.564@* 0.98406 1.49894 i .e49Yl 0.26810 0.22910 0.19012 O.ltS74 o.ora75

.GO 2.50

o.w a.00

. 12.0 0.13405 n.62151 l.n3nsq 1.55987 l.Fl7185 0.2b322 0.22424 n.18524 0.11599 o.n39on 0.00Y37 n.41434 O.h873> i.n399i i.2479n O.l7@74

.7j 2.75

L.75 8.50

12.50 0.19925 n.b7h96 l.n7668 1.61593 1.aen91 0.25R36 0.21936 n.id034 O.lU724 0.02Y24 n. 13293 0.65131 0.71779 1;07729 1.25927 0.26567 n.90261 1.43557 2.15457

0.04509 g.37656 0.65604 Il.99929 1.23327 0.09018 n.75319 fl.R296R 1.31208 1.374b4 1.99859 2.17987

2.*3080 2.46555 2.&958n 2.51954 7:l ELLIPTICAL CROSS-SECTION coCE90Dy !l @

-3 -1 19 29 n.0 5.0

10.0 9.0 .

. 74744 . 1.169901

n.0 . i.r94nei . 2.339912 .

2:1 EL L n 0 - 1

.50 1.n 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.5 6.0 6.5 7.0 7.5 8.0 e.5 Y.0 9.5

10.5 11. 11.5 12. 12.5 13. 13.5 14. r)0Y37 .l7509 .25R35 .33797 .41434 .48746 .55733 .h2395 .btr732 90431 .e5793 .90830 .95542 . 999291.039911.0772Y1.111411.14228 194281.215401.233271.2~790l.259271.267401.272271.274 7074 . 35097 .51671 .6759* . 82868 .974211.11*661.247901.37*b4 608621.715~61.~16601.910~41.99R592.07Y~32.154572.222~22.2845b 79~552.430~02.466552.4~5PO2.51$5~2.5347~2.5~~542.S4~ IPTICAL CROSS-SECT!ON FOREsOOY - ADDITIONAL ?ANELI~~ INPUT

- -IQ 11 n.0 5. 10. 1s. 26. 30.5 45. 58.75 72.5 107.5

121.25 135. 1*4.5 154. 162. 170. 175. 140. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 6.0 10.0 12.

1.0 0.0

1.00 1.25 1.50 3.00 3.25 3.50 5.00 5.50 b.00 9.00 9.50 10.0

13.0 13.50 14.0 0.26323 0.32599 0.30753 0.73119 0.78420 o.e3599 1.12116 1.203-b 1.2Rod9 1.66711 1.71342 1.75466 l.YOlOY l.YLiC41 1.911 0.25348 0.2&860 0.2*372 0.21448 U.209bo 0.20479 0.17426 O.lb573 0.15599 0.09749 0.01775 0.07800 0.01950 0.009Yl 0.00000 0.175&9 U.21733 0.254-15 0.68746 0.52280 0.55733 0.7474r 0.60431 O.tj57P3 1.11141 1.1*228 l.lb990 1.267~0 1.27227 1.27riIO 0.35097 O.rhb5 O.Slb71 0.97492 1.04540 1. llub6 1.49&P9 l.bfi*52 1.71584 2.22282 2.25456 2.339al 2.53&79 2.5rr5r 2.54hOO

14.

1.75 3.75

6.50 10.50

n.427a5 n.B6657 1.36245 I.74142

0.23666 0.1996b 0. l-624 O.ObU24

0.29837 0.55105 0.90330 l.lYC2ti

0.59714 1.182OY 1.81660 2.3Rb55

(h) Sample Case 8


128

1 n 0 2 1 1 4 1 1 : -P 2 0

3 0 0 0

SAMPLE CASE 9 3:l ELLIPTICAL CqOSS SECTION CONE REF: NASA r*-78808 MACH NO.= 1.7. ALPMAC =

2T.lW6 6.794 7.0 20.351;E;REE5. . PHI = 0 DEGREES 6.794

20.35 0.0 1130000. .25 1.70 .50 7.0 0.0 1.05 0.0 1.0

.Ol 0.0 0.0 0.0 0.0 2

9.0 14.0 0.0 3.397 .2*27 .242? n.O 5.0455 0.0 1 .b9R5 3:l CONICAL ELLIPSE - dDY9Dy GEOMETRY IhPUT n 0

0 0 0 0

BY TOWNSEND

.50

-3 -1 1P 2 0. lC.0

n.O 5.0955 0.0 1.6995

3:l CONICAL ELLIPSE - aDDITIONAL PPNtlLlhG INPUT -1: 5 n -1

n.0 13.q5 27.69 b1.56 55.38 69.50 77.00 81.50 e7.00 93.00 9a.50 103. 110.5 126.62 13P.46 152.31 166.15 130. 0.0 .50 6.25 12.0 lb.0

(i) Sample Case 9


129

-- ..----.--.- I

2 0 0 1 0 2 11 0 0 0 0 o..o 0 0 s i 1 i 3 2 0

NIELSEN ENGINEERING ir ~ESE4PCnr INC. PROGRAM NOZVTX SAMPLE CASE IO LOAED CROSS SECTION FDaEqODY eiEF : NASA TM-80062 bY TOWNSEND M4Cti ND.= 1.7. ALPHAC = ?0.3/, DEGQEE5. PHI = 0 DEGREES

15.05941 14.0 7.0 14.3 4.37884 to.34 0.0 728764. .075 1.7 .50 7.0 0.0 1.05 0.0 1.0 1.0 .05 0.0 0.0 0.0 0.0

29 n.0 -0

4.0 25 1.0 5.0

::5’ 2.0 2.5 3.0 3.5 6.0 7.0 7.5

Q.0 0.5 0.0 9.5 10. 1::: 11.0 11.5 12.0 12.5 13.0 13.5 14.0

J:r;i237 1.19128 .15359 . 1.2n461 .3nibo .44403 1.36235 J .58087 .47*51 1.5bJO8 .71212 1.64207 .63779 .95787 1.71747 1.70728 2.14474 .31 .?2341 .I3404 .34468

20 1 .50

24 .oooon .0155R .031os . II4629 .oblln . 07524 . OR840 .1002h .JlO49 .llRA2 .1250R .12926 .I3147 . J 3194 .13097 .12891 .12607 .Jl’OO .1ooon .08365 . 07847

1.85151 1.01014 1.96322 2.31069 2.05258 2.08P89 2.11961 2.16429 2.17925 7.14663 2.18042 . 30160 .2St44 .27927 .268’.!9 .25692 .2~575 .23&5d .21224 .2n1’J7 .18990 .17873 .16756 .15639 .1*52l ,122.M .11J71 .1nus3 .06936 .07820 .06703 .05585 .03351 .02234 .OllJ7 0.0

-.17874 -. 17809 -.17612 -.I7275 -.Jb7Ah -.16134 -.15312 -. Jr319 -.13JbR -.117@2 -.10&9h -.09051 -.07590 -.‘I6152 -.Or767 -.r.345* -.n2223

.01900

.05800

. 09970

.11206 .a7212 .12492 .06431 .1379J . cl5476 .15044 .0433s .i6171 . 03011 .17077 .OlS*b . J76bR .oooon .1187r

LOBED CROSS SECTION PODY - vDYHDY GEOHETHY INPUT l-l 1 1 1 20 29

5:; 5.5 .5 6.0 1.0 6.5 J.5 2.0 7.0 2.5 7.5 8.0 3.0 8.5 3.5 4.0 9.0 4.5 9.5 1n. 10.5 11.0 11.5 12. 12.5 13. 13.5 14.

0.0 0.0 n.n n.n

Cj) Sample Case 10


130

. __ n.O 0.0 0.0 n.O 0.0 0.0 n.0 0.0 0.0 0.0 n.O 0.0 0.0 (1.0 0.0 l-I.0 0.0 n.0

.ooooo

.n155n

.031ns

. n4b29

.06110

.osa*a

.11040

.125n,P

.13147

.I3097

.124n7 114ao

: 10000 .00345 . 07212 .05*7t+ .n4333 .03Gll .-I1546 .ooooo .ooooo .0305Q .06@9S .09069 .I1997 .17359 .2169h .245hl .25814 .25719 .24756

.2239 .I944

.lb427

.14142

.I0752

.08509

.O5913

.03035 o.oooon

.ooooo

.045n4

.ne970

.I3301

.17642

.x557

.31942

.341hn

- . . 0.0 0.0 0.0 0.0 0.0 0.0

X:i 0.0

8:; 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-. 17674 -.17809 -. 17612 -. 17275 -. 16786 -.1c312 -.1316P -.10*94 -.q759n -.I34751 -. G2223

.OlQ

.nsn .09970 .12492 .lSOLO .14171 .17077 .I7461 .17r??i

-. 35097 -.3497n -. 145n4 -. 33922 -. 32942 -.30047 -.2585? -.20609 -. l&905 -.n9341 -.0*345

.n373 .1139

.1957?

.24530

.29541

. -41755

. 33534

.3r4v3

.35097 -.51471 -.5l*R4 -.5c915 -.4994n -.c8527 -. 44245 -. 3.3047 -. .3034?

(j) Sample Case 10, Continued.


131

.38004 -.21943

.3?Bh3 -. 13781

.36447 -.06427 .72956 .05493 .2@909 .16-r&?

.24184 .ZRAZI

.2085n .36114

.1563n .43*92

.12527 .4475n

.08705 .a9349

.04469 .51076 0.n0000 .51671

.oooon -.47594

.05@92 -.4735n

.11744 -. bbhn4

.lms -.65331

.231n4 -.63*i?2

. 33432 -.57907

.41786 -.4979n

. 47304 -. 39492

.49719 -.287A5

.49532 -. lP?ZP

.6767r( -.08407 .-3113 .?I7185 .37a1a .21935

.31637 .377n3

.27274 .47243

.207na .54895

. 14387 .4115@

.i13nn .54593

.CS8&4 .bb817 0.00000 .67594

.ooooo -. 82bbR

. n7224 -. 82549

.143Q9 -.RlbSr,

.21*41 -.POO93

.29327 -.77027

. LO907 -.70991

. =122a -.c1051

.57992 -.48541

.40953 -.35191

158.52 60724 -.22102 -. 10307 . 52.356 .08809 . 44344 .24891

.38785 .Pb222

.33439 .5791Q

.25387 .49751

.20090 .74977

.13941 .79l77

. 07147 0.00000

.ooooo

.81915

.0206A

. 09499

.14939

.2524R

.3332q

.4@22n

.40269

.48224

.71710

-.97492 -.97139 -.QbObb -.94227 -.91541 -.p3519 -.71025 -.572&Q -.4i4n2 -.24002 -. 12125

.10344

.31436 .54-i70

. 71440

. 60767 .A2102 .<4545

. h5hln



132

I., I .I. I ..-... ._..._._.. -..-..- -. _ --._...-. . ..-. . - ., - . m.. I I I m

. ..-. _. __ .39340 .6bIJQ 8 29867 -82060 w 23b35 .882QR .I6425 .9314Q .n8431 .96371

0.00000 .9?492 .ooooo -1.11466 .09717 -1.11063 .I9367 -1.09635 .26867 -1 .n7733 .3PlOZ -1.ndm5 .55131 -.95491 .68906 -.a2120 .78005 -.65454 .e19flq -.47335 .PlbcJtI -.29729 .79424 -. 13863

.71OPS .11849

.12364 .34171 .5217n .62174 .44979 .779nh .341*a .93822 .27023 1.00951 .I8779 ). n650n .09440 1.101P4

0.00000 1.11444 .ooooo -1.24790 .1087c. -1.24339 .21ae2 -1.22941 .32317 -1.20610

r2b.57 :41722

-1.17199 -1.04905

.77143 -.9193*

.p733n -.732?q

.91789 -.52094

.91444 -.332.&3

.0002? -. 15521 . ‘9593 .1324’S .m3ia

.59404 -, 40*95

.69405 .5n355 .P721n .3923n I.05036 .30253 1.12904 .21020 1.19231 .10792 1.23355

0.00000 1.24790 .ooooo -1.37464 .11983 -1.3bVb7 .231X04 -1.35453 .35bOo -1.32bbo .64989 -1.29101 .4799n -1.17762 .R497g -1.ni273 .94199 -.90721

1.01111 -.%3374 I .00731 -.36463

.94961 -.17097 . a7676 .14613 .7b909 .44407

.44338 .?4675

.55*70 .96076 042113 1. IS704 .33326 1.243?3 .23159 1.31340 .1168P l-35883

n.nnnnn I. 37rPl*



133

_.-___. __-. - .ooooo -1.494RA .13031 -1.40947 .25973 -1.47301 .3871b -l.-r44EI1 .51099 -1.40393 .73937 -1.28963 .924ll -1.10131

1.04bl4 -.87781 l.nq955 -.63403 1.09542 -. 39870 1.05*43 -. 18592

.95345 .I5891 ‘.a3436 .Ctl509

.49945 .83362

.40322 1.044ao

.45796 1.25825

.36241 1.35.252

.25164 1.4282R

.I2929 l.4776Q 0.00000 1.494dFl

.ooooo -1.60862

.14023 -1.~0290

.27949 -1.58509

.41459 -1.55474

.54907 -1.51076 .79543 -1.37807 .99442 -1.18511

1.12573 -.964bO 1 .l8321 -.h.3313 1.17877 -.429n4 1.13446 -.20007

1.02hOO .17100 . ~0000 .52200

.75209 .R9724 .649ll I.12430 .*9281 1.35398 .3899@ 1.45543 .27101 1.534% .I3912 1.59012

0.00000. 1.40642 .ooooo -1.71584 .1495R -1.70965 .29813 -1.59076 .44*37 -1.45839 .5ab53 -1.61147 .04067 -1 .a6994

1.04072 -1.24411 1.20078 -1.00758 1.2421@ -.72867 1.25735 -.45764 1.2103n -.21341

1.09400 -1a240 .94000 .55680

.a0309 .95708

.49239 1.19925

.52544 1.44425

.4159r! 1.55246

.28907 1.43942

. i4e39 1.49412 0.00000 1.71584

.ooooo -1.81660

.15836 -1.61003

.31563 -1.79003

.47045 -1.75576

.62096 -1.70409

. aoc15n -1.55c170

(j> Sample Case 10, Continued.

134


-V.--w ____.-

1.12299 -1.33833 1.2712n -1.06673 1.33620 -.77145 1.33117 -.4a451 I.28136 -. 22594

1.15965 .19311 1.01636 .58949

.F5023 1.01327

.73304 I;24946

.55653 1.529n4

.44040 1.44361

.30405 I. 73567

.lS?lO 1.79571 0.00000 1.815bn

.ooooo -1.9lOAu

.16457 -1.9’)393

.332nn -l.BAPRP

.494.36 -1.fl4b85

.6531! -1.79459

.94511 -1.4349c1 1.18125 -1.*0776 1.33723 -1.122n7 1.~0551 ~~.A1147 1.40023 -.509br 1.34783 -.23764

1.2lR74 -20313 1.04909 .42007

.09434 1. P5S93

. 771n7 1.33s53

.5854n 1.50937

.44325 1.72U88

.32192 1.62572

.16525 1.888a7 0.00000 1.91094

.ooooo -1.99359

.I7422 -1.99135

.34725 - I. 94935

.5175a -1.93145

.68317 -1.87700

.9ne51 -1.71215 I.23540 ‘-1.47241 1.39854 -1.1735” 1.47005 -.a6674 1.44*53 -.53304 1.40972 -.24057

1.27473 .21245 1.11818 .44855

.93541 1.1147R

.A0647 1.39405

. hl22R 1.48222

. 4f3452 1 .PO824

.33471 1.90955

.I7281 1.975bil 0.00000 1.99859

.ooooo -2.07993

.18130 -2.07231

.34137 -2.04940

.53fl62 -2.01017

.71094 -1.9533n 1.02849 -1.7P175 1.28572 -1.53226 1.45540 -1.221311 1.52981 -.88324 1.52406 -.55471 1.44703 -.25948

1.72655 .27in9



135

__e---- ---_. 1.16364 .6?491

.9?343 1.16009

.R3926 1.45364

.63?17 1.75061

.5n422 1.88177

.35039 1.98718

.179El7 2.05591 0.00000 2.07983

.ooooo -2.15457

.113782 -2.1467R

.37435 -2.123ns

.55798 -2.08241

.73449 -2.32350 1.06566 -1.84578 1.33192 -1.50732 i.507an -1.26519 1.50479 i .57a83 1.51975

1.37422 1.20545

i.ooari .a4942 .44007 .52234 .34299 .I9433

0.90000 .ooooo .I9377 .38421 .57545 .75902

1.09941 1.37411 1.55556 1.43499 1.42884 1.56780

1.41775 1.24344

1.04036 .99696 .6809? .5300P .37448 .19223

0.00000 .ooooo .I9915 .39494 .59145 .79093

1.12995 1.4122.9 1.59877 l.bao*o 1.674n9 1.61144

1.45713 1.27818

1.06925 .92167 .09909 .55385 .794b39

-.9149a -.57445 -.26797

.22904

.49914 1.20178 1.5nsa4 l.Fl352 1.9494n 2. nsa59 2.12979 2.15*57

-2.22252 -2.21478 -2.19030 -2.14837 -2. n8759 -1.90~24 -1.637611 -1.30527

-.94396 -.59285 -. 27644

.23629

.72131 1.2398~ 1.55357 1 .a7096 2.filll4 2.12379 2.19725 2.22292

-2.2P456 -2.2743n -2.25114 -2.20805 -2.1455a -1.95714 -1.4e3n9 -1.34153

-.970lR -. ho932 -.28414

.24205

. 74135 1.27429 1.59673 1.92293 2.04701 9. IA?70



136

-- -.-- -___-. .19x7 2.25029

0.00000 2.2845b .00000 -2.33981 .20397 .40654 .60595 .79981

1.1572R 1.44643 1.63743 1.72104 1.71457 1.65041

1.4923b 1.30909

1.09511 .o441r, .71681 .56725 .3Q419 .20235

0.00000 .ooooo .2nfi22 .41501 .hlt!5@ .nlb47

1.18139 1 .A7457 1.67154 1.75609 1.75029 1.60470

1.523~5 1.33636

1.11793 . 96383 .73175 .77906 .40200 .20657

0.00000 .ooooo .21190 .A2235 .62952 .c33091

1.2022p 1.50268 1.70111 1.70797 1.7R125 1.71459

1.55040 1.36000

1.13770 .9POOA .74469 .59931 .4n952 .21022

0.00000 .ooooo .21501 .42A5+ .h-4A77

-2.33135 -2.30559 -2.26145 -2.1974b -2.00447 -1.72370 -1.37397

-.993b4 -. b2*n5 -.29101

.24073

.75927 1.30510 1.63534 1.96943 2.11699 2.23557 2.31290 2.339nl

-2.3895~ -2.37992 -2.35361 -2.3085& -2.24324 -2. ‘)4623 -1.7597n -1.63259 -1.n1434

-.63705 -.29707

.253Ql

.77509 1.33229 1.66941 2.n104b 2.1611ll 2.28215 2.36108 2.30055

-2.430130 -2.42201 -2.39524 -2.34939 -2.24292 -2.08242 -1.79083 -1.42740 -1.03228

-.64032 -.30233

.25840

.78860 1.35586 1.6989L 2.04602 2.19932 2.32251 2.40204 2.43ORO

-2.46655 -2.45763 -2.43046 -2. PA794

Cj) Sample Case 10, Continued.


137

.---. _.__ -. .04313 -2.31649

1.21996 -2.11304 1.52478 -1.01716 1.72612 -1.44839 l.Al426 -1.04746 1.80744 -.65706 1.73981 -.3Ob77

1.57320 .26220 1.38000 .80040

1.15443 i.37san .99531 1.72392 .75564 2.07611 .59?97 2.23166 .41554 2.35667 .21331 2.43818

0.00000 2.46655 .ooono -2.49580 .21756 -2.40677 .433b4 -2.45925 .60635 -2.41221 .n5313 -2.34396

1.23443 -2.13411’ 1.54286 -1.83871 1.74659 -1.46556 1 .a3577 -1.05988 1 .a2897 -.66566 1.76046 -.31041

1.59195 .26531 1.39636 -.80989

1.16812 1.39211 1.00711 1.74436

.764bn 2.10073

.60506 2.25813

.42047 2.38461

.21500 2.46709 0.00000 2.49580

.ooooo -2.51854

.21955 -2.50944

. 43759 -2.48170

.65224 -2.43419

.R6091 -2.36532 1.24568 -2.15759 1.55692 -1.85547 1.76251 -1.47092 1. P5251 -1.06955 I.#4554 -.67172 1.77bCA -.31324

1.60636 .2b773 1.40909 .a1727

1.17877 1.40480 1.01629 1.76026

.77157 2.11987

.blOSR 2.27871

.42430 2.40635

.21781 2.48959 0.00000 2.51854

.ooooo -2.53P79

.22096 -2.52563 .44041 -2.49771 .65645 -2.44990 . ebb44 -2.380513

1.25372 -2.17150 1.5bb97 -1.86744 1.77389 -1.48846 1.86446 -1.07645 l.A97CG -. h7hfIcI

Cj) Sample Case 10, Continued.


138

i I7;;9; -:;;iii 1.61673 .26945 1.4lAlEl .a2255

1.19637 1.41384 1.02284 1.77162

.77655 2.13355

.61452 2.29341

.42704 2.42107

.21922 2.5r)5b4 0.00000 2.53479

.ooooo -2.54454

.221Al -2.53534

.44211 -2.51-1732

.65097 -2.4593?

.Pb979 -2.38974 1.25854 -2.17986 1.573nn -1.87462 1.79n7n -1.49419 I.+‘7163 -1.08059 1 .Rb459 -.h78bb 1.79*c)2 -.3164a

1.42295 .2704q 1.42364 .R2571

1.19093 1.4193n l.n267n 1.77443

.7795& 2.1417b

. hl hOa 2.30223

. 4286n 2.63119

.22noh 2.51529 o.ooono 2.54454

. nonon -2.54779

.222ln -2.53858

.44267 -2.51.‘52

.45981 -2.46246

.n7n91 -2.39279 1.26315 -2.18264 1.47500 -1.87702 1.7929R -1.49610 1 .a7402 -1.n8197 1 .R4694 -.67957 1.79711 -.3149R

1.62502 .27004 1.42545 .a2676 ’

1.19245 1.42111 1.02809 1.78070

.79053 2.14449

.61767 2.30517

.42923 2.43429

.22034 2.51849 0.00000 2.54779

LOBED CROSS SECTIOY 4ODY - nooITI0td1~ PANELING INPUT -,I: 14 0 -1

0.0 5.5 11. IQ.5 28. 36.5 45. 53.5 62. 71. Brl. 97.5 115. 145. 154.5 164. 172. 180.

n. .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6. 7. R. 9. 10. 12. 14.

Cj) Sample Case 10, Concluded.


II

139

1 0 1 1 0 1 0 0 0 0 0 0 4 1 : f 0

NIELSEW ENGINEERING 4 RESEARCH* I*cC. PROGRAM NOZVTX SAMPLE CASE 11 2:] ELLIPTICAL CD055 SECTION BODY REF : AFFDL TR-79-3130 BY GOODiIN MACH NO.= 1.51 ALPHAC = 15 DEGREESe ?HI = 0 DEGREES

.a636 .5625 6. 10. 1.025 15. 0.0 100@000. 0.0 1.5

21 2.

20 1.05 0.0 1.0 0.0

a.0 0.0 0.0 14

n. 1 .b if8 2” ;P2

.I3 1.0 1.2 1.4 2.4 2.6

n. .0935 .176b .2530 .3149 .3714 .*200 .4611 .49;r9 .5218 .5AlA .555i! .5617 .5625 .4740 .4280 .3a5b 3443

:0515 .3044 .2659 .2205 .1919

.1561 .1210 '.OabO .01714 0. 0. .I246 .2354 .3335 .4199 .4952 .SbOO .blCP . 6599 .6957 .7224 .7roo .7409 .75no

r. . .0623 .l I77 .I668 .2099 .2*76 .ZYOO .3074 .33on .3479 .3612 .3700 .3744 .3750

1 .6 2:l ELLIPTICAL CROSS SECTION ilODY - hDYEiDY GEOHET6Y INPUT

n 0 -3 -1 8 12 0. .2 .4 .h .9 1. 1.2 1.4 1.8 2.2 2.6 6.2 0. .1246 .2354 .3335 .*200 .4952 .5600 .4148 .5957 .74n1 .7500 .7500

0. .0423 .1177 .1668 .2100 .247b .28011 .307u .3479 .3700 . 7750 .3750

2:1 ELLIPTICAL CYOSS SECTION d)oD~’ - ADDITlOhAL PANELJNG IW=UT -1: 11 0 -1

0. 20. 53. 7?. 05. 95. 108. 127. 160. 180.

0. .2 .* .h .a 1.0 1.2 1.4 1.7 2.0 7.5

(k) Sample Case 11


140

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1

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put

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1.

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Page

2

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tinue

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3:

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IC

AL

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FOR

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Page

3

Figu

re

15.-

Con

tinue

d.

l *

GEU

H

l e

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l *

3:l

ELLI

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AL

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4.

9000

5.

6000

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(d)

Page

4

Figu

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tinue

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-

1 AN

D

3 IN

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1 2 3 4 5 h 7 H 9 ;: I.?

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20

21

22

23

24

25

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29

30

31

32

33

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Page

30

Figu

re

15.-

Con

clud

ed.

083

02

5 OJ

0

T

4 bo = 116

a, = O0

- z 0 EXPEWWT (REFi21)

-THEORY

x/D

Figure 16. - Measured and predicted pressure distribution on an ogive-cylinder at CL=OO.

164

003

OB2

011

0

8

M,= 213

Q = 0’ C

0 EXPERIKENT (REF, 21

-THEORY

-011 0 1 2 3 4 5 6 7

x/D

(b> M,= 203

Figure 16. - Continued.

165

013

082

Oal

0.

-011

8 M,= 2196 Q = O"

C

0 EXPERIFiENT (REF,21)

-THEORY

0 1 2 3 4 5 6 7

x/D

Cc) Mm= 2m96

Figure 16. - Concluded.

166

. 8

. 6'

.4

C P

. 2

0

-. 2

~

t

i -

I I I I I I I

MO3 = 1.6

aEXPERIMENT (Ref. 21) -Theory

1--- - . 4 -~----I I 1 I 1 x/D = 2.8

0 EXPERIMENT (Ref. 21) . 2 0 -Theory

0 20 40 60 80 100 120 140 160 180 8, degrees

(a) x/D = 0.8 and 2.8

Figure 17.- Measured and predicted circumferential pressure distribution on an ogive-cylinder body

at K, = 1.6, cxc = 20°.

167

.

. 1 21) / ,= j

x/D = 5.1

0 EXPERIMENT (Ref. - Theory, with separation / /

--- Theory, no separation ,'

-I - --e-

-. 3 I I 0

I 1 I I

0 20 40 60 80 100 120 140 160 180 8, degrees

(b) x/D = 5.1


168

c

c

C .

c .

c .

CX

c .

c .

Centroid of vorticity

OEXPERIMENT

r nDV,

I V,.sincl,

Figure 18.- Predicted vortex wake on the lee side of an ogive cylinder.

169

r -- TDV,

.4

. 3

.2

. 1

0 8

x/D 10 12

(a) M, = 2.0 . 4

. 3

r .2 TDV,

0

M, = 3

El

4 6 8 10 x/D

(b) M, = 3.0

12 14

Figure 19.- Measured and predicted body vortex strength on an ogive cylinder.

170

I

w I

t -Location

v I of data

x/D = 7

-Theory I I I I I I I

W v, -* 2

-. 4 0 EXPERIMENT (Ref. 20) - Theory

-. 6 I I I I I I I

-.

W

v,

-.

-.

O- I I

2-

x/D = 13

4- 0 EXPERIMENT (Ref. 20)

- Theory 6 I I I I I I I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.8

z/D

Figure 20.- Measured and predicted downwash distribution on the lee side of an ogive cylinder

at M, = 2, oLc = 15'.

171

.6

.4

.2

W -

VW. 0

-. 2 (

Location of data

-. 4 - I I I !. A

0 .1 .2 .3 .4 .5 .6 .7 .8

z/D = .75

OEXPERIMENT (Ref. 20) - Theory

Y/D

(a) z/D = .75 .6 1 I I I I I 1 I

Location W ,cDf data 0

. 4 t 0

- 0

0 o-

W .2 - - z/D = 1.05' VW 0 EXPERIMENT (Ref. 20)

0 vortex

centroid 0

I 0 0 -. 2 I 1 +

1 1 I I

0 .1 .2 .3 .4 .5 .6 .7 .8 Y/D

(b) z/D = 1.05


at M, = 2, ac = 15', x/D = 10.

172

W

v,

. 1

0

-. 1

-. 2

x/D = 7

0 EXPERIMENT -Theory (Ref.

/

0 20)

0 W

t 1 Location I of data I I

Z

0 -L Y

0 W

VW

-. 2

-. 4 I I I I I I I I

.2 , I 1 I I I I I 1

0 EXPERIMENT (Ref. 20) 0- Theory

W - x/D = 13 VW 0

-. 2 -

-. 4 I I I I I I I 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6

z/D


at M, = 3, ccc = 15O.

173

Location of data

.

.

.

.

W -

VW

-. x/D = 10

z/D = .75

0 EXPERIMENT (Ref. 20) -Theory

(a) z/D = .75

Figure 23.- Measured and predicted downwash distribution on the lee side of an ogive cylinder at

M, = 3, crc = 15', X/D = 10.

174

.5

.4

. 3

.2

w .1 - VW

0

-. 1

-. 2

(

-. 3

e----

Location of data

.

x/D = 10

z/D = 1.05

0 EXPERIMENT (Ref. 20)

0 - Theory

vortex 0 centroid

I v I I I I 0 .2 .4 .6 . 8 1.0 1.2

Y/D

(b) z/D = 1.05


175

Vapo

r sc

reen

x/L

= 0.

32

EXPE

RIM

ENT

(Ref

. 23

) -

m-c

;‘- -,

Theo

ry

x/L

= 0.

64

83

(a)

$I =

00

x/

L.

Y 1.

0

261

Figu

re

24.-

Mea

sure

d an

d pr

edic

ted

vorte

x pa

ttern

s on

a

3:l

ellip

tic

cros

s se

ctio

n bo

dy

at

M,

= 2.

5 an

d c1

c =

2o".

Vapo

r sc

reen

-0

.036

2

X/L

= 0.

32

- EX

PER

IMEN

T (R

ef.

23)

--c-

Theo

ry

0.18

3

x/L

= 0.

64

2 (b

) Q

=

45"

x/L-

" 1.

0 \

Figu

re

24.

-Con

clud

ed.

-,

0 EXPERlMENT

0 40 80 120 160 200 240 280 320 360 8, degrees

(a) 4 = 0'

x/L =

EXPERIMENT Theory

0.60

4 I I I -.

I I I I I

0 40 80 120 160 200 240 280 320 360 8, degrees

(b) + = 45'

Figure 25.- Measured and predicted circumferential distribution on a 3:l elliptic cross section

missle at M, = 2.5, ac = 20°, x/L = 0.60.

178

1

.6 (

.4

.2

cP

O

-. 2

-. 4

x/L

= .2

5

. a

EXPE

RIM

ENT

(Ref

. 17

)

- Th

eory

, w

ith

sepa

ratio

n

---

Theo

ry,

no

sepa

ratio

n

U

LU

4u

60

8U

100

120

140

160

180

8,

degr

ees

(a)

x/L

= 0.

25

Figu

re

26.-

Mea

sure

d an

d pr

edic

ted

circ

umfe

rent

ial

pres

sure

di

strib

utio

n on

a

3:l

ellip

tic

cone

fo

rebo

dy

at

M,

= 1.

7,

CQ

=

20.3

5O.

.6 (

-. 6

0 20

40

60

80

10

0 12

0 14

0 16

0 18

0

P,

degr

ees

(b)

x/L

= 0.

50

Figu

re

26.-

Con

clud

ed.

1

Figu

re

27.-

Pred

icte

d vo

rtex

patte

rns

on

a 3:

l el

liptic

co

ne

fore

body

at

M

, =

1.7,

~1

~ =

20.3

5O,

X/L

= 0.

25.

--

---

--w

.- X

TO

P VI

I3

Figu

re

28.

- 2:

l el

l.ipt

ic

cros

s se

ctio

n bo

dy.(R

efer

ence

22

)

0,4

b

on2

Cl

-9,s

-aa- I

bo = la5

cr, = 15O

c3 = o"

x/D = 087

0 EXPERIKEFjT (REFi22)

---THEORY

I f x/D -' 291

2- Y 0 EXPERIKEVT

- THEORY

20 40 60 80 - 1(

& DEGREES

30 120 140

(a) 4 = O"

160 180

Figure 29.- Measured and predicted circumferential pressure distribution on a 2~1 elliptic cross section body

at M, = 1.5, lxc = 15O.

183

--.... _..- --- - ,.-.- -- -. . . . . ,--- --. ,..,,.,., . . . . . . . . . . . - . _ .._---. I

on4

CP

O&2

0

-0,2

0 r, .L

L = 1.5

a, = 15'

0 = 229

-- I I I I I

x/D= 0,7

0 EXPERIKENT (REF,22)

-THEORY

-0 EXPEJIMT

200 ,240 280 320 36!l 6, DEGREES

(b) $I = 22.5"


184

- -

flco = 1*5

a, = 15"

0 EXPERIKENT (REF, 22)

~I- I I I I I I I 40 80 120 160 200 240 230 320

B, DEGREES

(c) cp = 45O


185

. 8 .4

cP

.2 0

-. 2

-.4

x/L

= 0.

5

a EX

PER

IMEN

T (R

ef.

17)

-- Th

eory

, w

ith

sepa

ratio

n

--'Th

eory

, no

se

para

tion

0 20

40

60

80

10

0 12

0 14

0 16

0 18

0

13,

degr

ees

Figu

re

30.-

Mea

sure

d an

d pr

edic

ted

circ

umfe

rent

ial

pres

sure

di

strib

utio

n on

a

2:l

ellip

tic

cone

fo

rebo

dy

at

M,

= 1.

7,

~1~

= 20

.35"

, x/

L =

0.5.

. 6

I I

I I

I I

I I

x/L

= 0.

5

EXPE

RIM

ENT

(Ref

. 18

) -

Theo

ry

@

I

B

t V,si

ncxc

0 20

40

60

80

10

0 12

0 14

0 16

0 18

0

8,

degr

ees

Figu

re

31.-

Mea

sure

d an

d pr

edic

ted

circ

umfe

rent

ial

pres

sure

di

strib

utio

n on

a

2:l

ellip

tic

fore

body

at

M

, =

1.7,

C

+ =

20.3

6",

x/L

= 0.

5.

.8

Ji

( . .

6 .4

C P

. 2 0

-. 2

I I

I I

I I

I I

x/L

= -1

4

0 EX

PER

IMEN

T (R

ef.

18)

Theo

ry

Q

B

t V,s

intic

0

Figu

re

>

20

40

60

80

100

120

B,

degr

ees

(a)

x/L

= .1

4

32.-

Mea

sure

d an

d pr

edic

ted

pres

sure

di

strib

utio

n se

ctio

n bo

dy

at

M,

= 1.

7,

ac

= 20

.36'

.

140

160

180

on

a lo

bed

cros

s

. .

-. -.

6 6 I I

I I I I

I I I I

I I I I

I I 1

x/L

= 0.

5

EXPE

RIM

ENT

(Ref

. 18

) -

Theo

ry

4 4 1

I I I I

I I I I

I I I I

I I I I

I

0 0 20

20

40

40

60

60

80

10

0 80

10

0 12

0 12

0 14

0 14

0 16

0 16

0 18

0 18

0 8,

8,

de

gree

s de

gree

s

(b)

x/L

= .5

Figu

re

32.-

Con

clud

ed.

Figure 33.- Predicted vortex wake on a lobed body at M, = 1.7, ac = 20.36', x/L = 0.50.

190

1. Rewrt No. 2. Govcrnmcnt Accarion No. 3. iiecipcnt’s Gtalog No.

NASA CR-3754 4. Title and Subt;tlt 5. Report Date

PREDICTION OF VORTEX SHEDDING FROM CIRCULAR January 1984

AND NONCIRCULAR BODIES, IN SUPERSONIC FLOW 6. Performing Organization Code

815/C 7. Author(s)

M. R. Mendenhall and S. C. Perkins, Jr. 8. Pepfarming Orpnization Report No.

NEAR TR 307

g. Performinq Organization Name and Address

10. Work Unit No.

Nielsen-Engineering & Research, Inc. 510 Clyde Avenue Mountain View, CA 94043 I I I. ~onrrac: or Grant No.

NASl-17027

12. Sponsoring Agency Name and Address

13. Typ of Repon and PC tried Cavered

Contractor Report National Aeronauticsand Space Administration 14.SponsoringAgcncyC;ode Washington, DC 20546 t-

I I 5. Stippkmcntary Nores

I Langley Technical Monitor: Jerry M. Allen I Final Report

15. Absrract

An engineering prediction method and associated computer code NOZVTX to predict nose vortex shedding from circular and noncircular bodies in supersonic flow at angles of attack and roll are presented. The body is represented by either a supersonic panel method for noncircular cross sections or line sources and doublets for circular cross sections, and the lee side vortex wake is modeled by .discrete vortices in crossflow planes. The three-dimensional steady flow problem is reduced to a two-dimensional, unsteady, separated flow problem for solution. Comparison of measured and predicted surface pressure distributions, flow field surveys, and aerodynamic chararacteristics are presented for bodies with circular and noncircular cross-sectional shapes.

17. Key Words (Suggested by Author(s) 1

Supersonic Flow 18. Distribution Statcmenl

Unclassified - Unlimited Vortex Shedding Elliptic Bodies

Subject Category ‘02

Missile Aerodynamics

19. Sxurity Oassif. (of thlr report1 20. Security Classif. (of this pge) 21. No. of Pages 22. Rice

Unclassified Unclassified 200 A09

For yle by the Nation4 la&tier! Informrtign Slrvicr. Sprinpfirld. Virgima 22161

aU.S.COVERNMENTPRMTINGOFFlCE:1984 -739-01CU 47 REGION NO. 4

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