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
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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 repre- sent 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 pro- cedure 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 axisym- metric 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 calcula- tions 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 interpola- tion 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 con- figurations 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 dis- tribution 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 modifica- tion 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 appro- priate 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 sub- routines 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 modifica- tion 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 limita- tion is that the crossflow Mach number should not exceed the critical value: that is, the local Mach number in the cross- flow 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 follow- ing steps should be tried to resolve the
problem. Increase the integration tolerance E5, or increase the size of the combination param- eter 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 calcula- tion 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 justi- fied 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.
Item Variable Description
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:
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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
Item Variable Description
23 XFV x-station at which field point velocities
are calculated or at which additional out- put 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.
Item Variable Description
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 second- ary 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.
Item Variable Description
27 GAMR VVD3 I reverse flow or additional vorticity
on right side of body.
XR,ZR coordinates of discrete vortices on right
side of body at starting point (XI).
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
side of body at starting point (XI).
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.
Item Variable Description
29 GAMA VV03 I reverse flow or additional vorticity on left side of body.
YA,ZA Coordinates of discrete vortices on left
side of body at starting point (XI).
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.
Item Variable Description
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 calcula- tion 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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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 itera- tion (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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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.
Item Variable Description
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 vari- ables 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 computa- tion 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 specifica- tion 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 cross- flow 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 mini- mum 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 down- wash 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 agree- ment 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 distribu- tions 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
e cy
linde
r bo
dy
case
w
ith
line
sour
ces
and
doub
lets
an
d va
riabl
e D
X;
surfa
ce
pres
sure
ca
lcul
atio
n an
d no
se
para
tion.
An
ellip
tic
cros
s se
ctio
n fo
rebo
dy
with
pa
nel
sym
me-
try
; su
rface
pr
essu
re
calc
ulat
ion
and
vorte
x fie
ld
plot
s at
ea
ch
axia
l st
atio
n.
Sam
e as
Sa
mpl
e C
ase
6 w
ith
body
pa
nels
ta
ilore
d to
th
e sh
ape
by
spec
ifica
tion
of
the
cent
ral
anql
es
of
the
pane
l ed
ges.
An
ellip
tic
cros
s se
ctio
n co
nica
l fo
rcho
dy
with
pa
nel
sym
met
ry,
Flow
sy
mm
etry
, an
d va
riabl
e in
tegr
atio
n in
terv
al;
surE
ace
pres
sure
ca
lcul
atio
n w
ith
vorte
x sm
ooth
ing.
:xec
utio
n Ti
me
CD
C-C
YBER
76
0)
6.0
set
28.1
se
t
28.1
se
t
61.8
se
t
Tabl
e II
- C
oncl
uded
Sam
ple
Cas
e
10
11
Ref
. Fi
g.
18
14(j)
22
14 (
kl
Body
C
ross
Se
ctio
n
Lobe
d
2:l
Ellip
se
1.7
1.5
20.3
6O
15O
* G
ener
al
Com
men
ts
O0
O0
A lo
bed
cros
s se
ctio
n fo
re-
body
ca
se
with
le
ft/rig
ht
pane
l sy
mm
etry
, flo
w
sym
me-
try
, va
riabl
e D
X,
and
num
eric
al
map
ping
: su
rface
pr
essu
re
calc
ulat
ion
with
vo
rtex
smoo
thin
g.
An
ellip
tic
cros
s se
ctio
n m
issi
le
body
w
ith
pane
l sy
mm
etry
, flo
w
sym
met
ry;
surfa
ce
pres
sure
ca
lcul
a-
tion,
pe
riodi
c vo
rtex
field
pl
ots.
Exec
utio
n Ti
me
(CBC
-cYB
EI
760)
47.0
se
c.
N/A
, z
o=y+iz
REAL PLANE
W=T+ii
CIRCLE PLANE
(a) Analytical transformation procedure
Z
u=y+ iz v= T+ iX
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
I I-- CONFOR ---- TAPE6 I 2 ---- CABS
I 1:: SORT
I I I-- AlANE
I f I-- cos I-- CEXP
1 I I-- CLOG
-- SIN I I-- cos I
t -- ASIN
I -- SORT I-- ELNOSE ---- TAPE6
I -- F ---- SHAPE
I I-- BHAP OS-- AMAp --s- 2 ---- CABS I I I-- cos
I I I-- SIN
I I I I-- ATAN I-- SPRT
I I-- ATAN I I-- CHAP ---- OZONU ---- CABS I I I-- 2 ---- CABS
I f I-- CSQRT I-- SORT
I I I-- ATAN
I I I-- CEXP I-- CABS
I I-- VELCAL ---- SQRT
I t I-- ATAh I-- SOURCE ---- TAPE6
I I I I-- SORT
I I I-- ALOG I-- OOUBLT ---- TAPE6
I I I-- SQRT
I I I-- ALOG I-- cos
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
I I-- SIN I-- cos I-- SORT I-- SHAPE I-- BHAP ---- AMAP I I I I I 1 I-- SORT I-- ATAN I-- CMAP ---- DZDNU
I I-- z I-- CSQRT
I I-- SQRT
1 I-- ATANE I-- CEXP
I I-- CABS
-- BDYGEN ---- TAPES
I I-- EXP
-- ASUW ---- CABS
-0-o z ---- CABS I-- cos I-- SIN I-- ATAN
---- 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
I-- ATAN VELCMP ---- TAPE10
I-- TAPE6 I-- tAPE7 I-- ~APEB I-- TAPE9 I-- BODVEL ---- TAPE10
I-- TAPE6
I I-- SOLVE ---- TAPE10 I I-- tAPE6
I-- TAPE7 I-- CAPES I-- SIN I-- cos I-- ASIN
I I I
I I
I I-- l-- I-- l
t I
I
I
I I-- l I I
I
6 E
I-- l
I l--
I I-- l I-0.
BODrR ---- SRFVEL ---- VLOCTY ----
l-- I-- l-- l-- l--
I
I I--
VELCAL ---- I-- l--
I I--
I-- TAPE8 I-- TAPE9 I-- SIN I-- cos I-- TAN I-- SORPAN ---- TAPE6
I-- SORT I-- ATANE I-- ALOG I-- ACOS
DIAGIN ---- I -0 I-- l--
I TRITE .----
1::: PARTIN ----
l-- VOLUME ---- SQRT ATANL TAPE6 SUM ---- SQRT EXP ATAN SMOOTH ----
;::
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
Figure 6.- Continued.
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--
I--
ATAN SORT SIN’ cos EXP
TAPE6 PLOTAB ---- PLOTA ---- ALOGlO
I-- PLOTAS PLOTA
I-- FPVEL ---- TAPE6
I I-- ATANE I-- CHAP ---- DZDNU ---- CABS
I I I
I-- 2 ---- CABS I -- CSQRT
I I I-- SQRT
I I I -- ATANt I I-- CEXP
I I I-- CABS
I I-- VELCAL ---- SORT
-- ATANE I I I -- SOURCE ---- TAPE6
I I I I -- SQRT I I-- ALOG
I I I-- DOUBLT ---- TAPE6 I I I I-- SQRT
I I I I-- ALOG
-- cos
I I I-- SIN I--, FLDVEL ---- TAN
I I I-- cos
I I I-- SIN I-- SORPAN ---- TAPE6
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
Figure 6.- Continued.
94
PROGRAM EN0
I
I
l-- I--
OTABLE VCENTR COHBIN
Q 1 Q 3'
I
I -- ATAN I-- CHAP
I I
Q K -- SIN
I I-- ATAN I-- SQRT
I-- VELCAL I
I
I
I I-- FLDVEL I I
---- DZDNU ---- CABS I-- 2 ---- CABS I-- CSQRT I-- SQRT I-- AfAN I-- CEXP I -- CABS
---- SQRT I-- ATAN I -- SOURCE ---- TAPE6
; I -- SQW I-- ALOG
I-- OOUBLT ---- TAPE6 I I-- SQRT I I-- ALOG I-- cos I-- SIN
-c-w T&N I-- cos I-- SIN I-- SORPAN ---- TAPE6
I -- SQRT I-- ATAN
I -- ALOG -- ACOS
---- TAPE6 I I-- VLOCTY
I-- SUM ---- DZONU ---- CABS I-- SQRT i -- EXP I-- ATAN I-- SMOOTH ---- ATAN I I-- SQRT I I-- SIN t I-- cos I I-- EXP I-- ASUM ---- CABS
---- TAPE6 I-- SORT
(e) Page 5
Figure 6.- Concluded.
95
SUBR
OU
TIN
E N
AME
----
----
--
EXTE
RN
AL
- R
EFER
ENC
ES
-
-r--r-
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25
6781
1
H
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S AL
OG
I 0
ALO
G
AHAP
AS
IN
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M
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BH
AP
BOO
PAN
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BOD
YR
CAB
S C
EXP
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G
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CO
WIN
C
ON
FIG
C
ON
FOR
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s
CSQ
RT
DFE
OKM
D
IAG
IN
DO
UBL
1
OPH
IDT
DTA
BLE
DZD
NU
EL
NO
SE
EOF
EX
P
(a)
Page
1
Figu
re
7,-
Subr
outin
e C
ross
R
efer
ence
Ta
ble
for
Prog
ram
N
OZV
TX.
.-----
------
.-----
------
~----
------
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------
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------
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-- ---
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r--r-r
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rrrrr+
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BCC
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SOM
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lLEH
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AIER
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--
----
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1
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l256
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1 H
l
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ENC
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-I 1
I ---
------
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--1-1
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FL
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L FO
RC
EP
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L
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IN
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ATE
NEY
RAO
OU
TPU
T PA
NLL
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TA
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TA
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TAB
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E SI
N
SMO
OTH
SO
LVE
SOR
PIN
SOU
RC
E SO
RT
SRFV
EL
SUM
TA
N
(b)
Page
2
Figu
re
7.-
Con
tinue
d.
l ---
------
--*---
------
--+---
-----*
----*
---
------
--*---
------
--*-
------
----*
------
-----*
--r--r
rr---+
lAAG
G6I
9BC
CC
lC0D
DD
l0O
EFFl
FFG
lIl~N
NPP
lPPP
PPlP
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l lM
SOH
OlO
OM
OO
lOFI
OPl
TZLL
OlP
EN
NlT
EOAA
lLLL
iLlL
EHM
Ol
SUBR
OU
TIN
E IA
UYA
DID
DAM
NIN
EAU
HlA
DN
DR
lV
OPV
lRU
ZNR
lOO
GG
OlO
PAO
Lt
NAM
E lP
MG
PPlV
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A
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IIl
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LDlL
USE
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R~T
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AETE
~ -s
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----
--
1 N
N
I
L N
GIR
HN
TTIE
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PI
1 IE
DX
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5678
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REF
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(c)
Page
3
Figu
re
7.-
Con
tinue
d.
l --a
------
--*a
------
----+
------
-----*
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------
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svlv
vvvv
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EOF
EX
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(d)
Page
4
Figu
re
7.-
Con
tinue
d.
.-----
------
,-----
------
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------
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------
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---*-
------
----+
(d)
Page
4
Figu
re
8.-
Con
tinue
d.
4----
s ---
---*-
------
----*
------
-----+
------
-----
*----
------
-*---
--r---
--*---
--r---
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------
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Isss
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EL
RIL
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ON
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KS
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HAP
NTH
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APN
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APP
MA
PX
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ATC
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N
EWC
OM
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Aid
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POIN
T PR
ESS
PRIN
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EF
R22
i SE
G
SHPE
SM
OO
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STAC
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x x
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I I
x I
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x X
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l ---
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t x
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(e)
Page
5
Figu
re
8.-
Con
tinue
d.
+---
----
----
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----
----
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----
--*-
--4-
----
----
--4
------
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------
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lsss
svIv
vvvv
I2
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CIE
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TI
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TIN
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KS
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l I
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4
(f)
Page
6
Figu
re
8.-
Con
clud
ed.
P 0 a3
ITEM
5
10
IS
20
25
30
35
50
45
sn
55
60
65
70
75
a0
-_-
NcI
R
1 N
CF
1 IS
YH
1 I
NBL
SEP
INSE
PR
~NSI
KJTH
IND
FUS
IND
PIII
IINP
INXF
V I
NFV
IN
VP
I N
VR
I N
VM
1 N
VA
~NAS
YM
1 I
I I
I I
I I
I I
I I
I I
(1)
(2)
(3)
(4)
(5)
(6)
(7)
5 10
IS
20
75
30
3s
NR
EM
INPR
NTP
~PR
NTS
~N
PRN
TV IN
PW
INPL
OTA
IN
PRTV
L I
I I
I I
I
TITL
E (N
IIEAD
ca
rds)
In
20
30
CO
SO
REF
S IJ
!&EI
L I
XM
I SL
I
SD
I
10
20
30
50
50
ALPH
AC
I PH
I I
FE
KC’0
P.E
I XM
ACH
I
I I
10
20
30
4"
so
60
70
I
XI
I XF
I
DX
I EH
KF
I R
GAM
I
VRF
I FD
X
IO
20
30
'IO
5"
I
E5
1 XT
ABL
I- Ix
nsyH
F I
I I
(a)
Page
1
Figu
re
9.-
Inpu
t fo
rms
for
Prog
ram
N
OZV
TX.
ITEM
_.
._
(1))
(9)
(10)
(11)
1 N
XR
"1
10
70
30
I,” (
NXK
Vi1
1 LI(-
s)
‘.”
Ffl
70
80
1 XR
(1)
I XR
(2)
I XR
(3)
I I
I I
I I
I I
I I
I I
I I
I
IO
70
xl
,” (N
SH V
.alv
!:)
I ,,D
A0
70
00
R
(1)
R(2
) R
(3)
I I
I I
I I
la
20
30
,,. I (N
XR V
alw
S)
DR
(1)
DR
(2)
DR
(3)
5. I
60
70
80
I I
I I
I I
Valu
es)
IO
70
30
,,. (N
XR
~”
60
70
80
AE(1
) I
AE(2
) x5
(3)
I c1
21*
I I
I I
I 10
20
30
4.
(N
XR V
alnc
s)
5.
GO
70
80
R
E(1)
1
RE(
2)
BE(3
) I
I (1
3)*’
I I
I I
I l
Om
it ite
ms
(12)
an
d (1
3)
if N
C~R
#
1
(b)
Page
2
Figu
re
9.-
Con
tinue
d.
ITEM
--
(14)
‘1
M
NFC
51M
XFC
lo
,
ZZC
(J)
10
(lEl)'
t 1
RZC
(J)
20
I
XRC
(I,J)
10
1 YR
C(I,
J)
20
(20)
l
‘e
I N
R C
ards
l f%
it ite
ms
(14)
th
roug
h (2
2)
if N
CIR
'
2 :,(
nnit
item
s (1
6)
and
(17)
if
NC
F #
0 ar
mit
itrm
s (1
8)
thro
rlgh
(22)
if
Ncf
: =
0
(cl
Page
3
Figu
re
9.-
Con
tinue
d.
ITKM
(N
R "
a I I
II':.)
_-
-. _
TIIC
(1.J
) '('
TI
IC(2
.J)
20
‘TIIC
(
3 ,.I
1
w
4)
50
Gl)
70
80
I (2
1)
l o
I I
I I
(IIN
IT
"a11
1r:i)
XFC
(1)
l2
XFC
(2)
24
XFC
(3)
3G
48
GO
72
(22)
+0
(NXF
V va
l11e
s)
XFV(
1)
lo
1 XF
V(2)
2"
IX
FV(3
) 30
40
50
60
70
80
(23)
+ I
I I
I I
I I
YFV
10 ,
ZFV
CM
)+*
I I (N
FV c
ards
)
10
ICY
20
CM
70
40
5n
1
1 C
R
ICSL
I I
I I
I
lr’ IYF
20
I:
I zr
3n
I
XSIII
:DP
40
NW
ca
rds
(26)
r
I I
I rh
uit
item
(Z
?(,)
if N
W
7 II
AOm
it ite
ms
(25)
th
roug
h (2
9)
if N
VP +
NVR
+ N
VM +
NVA
= 0
+O
mit
item
s (2
3),
(24)
if
NXF
V =
0 *O
mit
item
(2
4)
if N
FV =
0
*Om
it ite
ms
(21)
, (2
2)
if N
CIR
-c
2 oO
mit
item
s (2
1),
(22)
if
NC
F =
0
(d)
Page
4
Figu
re
9.-
Con
tinue
d.
ITEM
_-
-- 10
20
30
40
G
AMR
YR
ZM
xs
III1D
l~
NVR
ca
rds
(27)
* 0G
.t ite
m
(27)
if
NW
? =
0
1
10
2(L
30
40
GAM
M
YM
ZM
XS
IIED
M
NVM
ca
rds
(28)
* (k
nit
item
(2
R)
if N
W
= 0
or
if IS
YM =
0
GM
A lo
YA
20
1 zn
30
1 10
XS
IIIC
DA
NVA
car
ds
(29)
* I
I C
hit
item
(2
9)
if N
VA =
0
or
if IS
YM
= 0
80
(3O
)A
TITL
E1
(NFO
RX
valu
es)
7l
1 XF
US(
3)
21
28
35
42
49
56
63
70
XFlJ
S(l)
XFU
S (2
) I
(33)
A I
I I
I I
I I
I
AOm
it ite
ms
(30)
-(43
) if
NC
IR
= 0
*Om
it ite
ms
(27,
(28)
,(29)
if
NVP
+ N
VR +
NVM
+ N
VA =
0
(e)
Page
5
Figu
re
9.-
Con
tinue
d.
ITEM
(N
FOR
X “a
IIll
’S)
--
Fusn
R”(1
: F”
snR
D(:;
, 21
28
35
42
43
56
63
70
1
I I
I I
I (3
4)
*A
I I
I 7
14
valu
es)
21
28
(NFO
;; 42
43
56
FU
SRAD
(l)
FUSR
AD(2
) 1
I 63
70
I
I (3
5)
+A
_ I
I I
I I
7 U
SBY
(1)
FUSB
Y (2i
4,
21
20
~NFo
jy
Ival
ur?n
) 42
4.
56
63
70
I
I I
I I
I I
(36)
)h
I I
I I
I I
I I
I I
07)p
I.
YJ
10
w
20
(38)
xA
NR
ADX
card
s
l Cm
it ite
m
34)
if J2
#
-1
b0m
j.t
item
( 3
5)
if J2
#
-2
pnit
item
s (3
6),
(37)
if
52
# -3
x )rh
it ite
m
( 3R
) if
J2
# 1
I R
epea
t ite
m
(38)
N
FOR
X tim
es
IOni
t itr
ms
(34)
-(43)
if
NC
TR
= 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
) PH
IK(2
) PH
IK(3
) (4
2)*A
1
I I
I I
I (
(KV'
OR
X "a
Lue
s)
XF"S
K(1:
14
1
XFU
SKO
) 21 1
28
3.
, 42
49
56
63
70
XF
USK
(2)
(43)
tA
I I
I I
I I
I
NXB
OD
?IN
CO
D?
1 20
30
(4
4)x
XNO
SE
1 XL
BOD
Y I
I I
l Cnn
it ite
m
(42)
if
KRAD
X 1
0
rcan
it ite
m
(43)
if
KFO
RX
= 0
/.chi
t ite
ms
(33)
-(43)
if
NC
IR
= 0
xCkn
it ite
m
(44)
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
30 0.0 7.700 15.400 22.400 0.0 1.465 7.179 2.202 .2eo .12A .055 -.047 n.00
2.198 3.269 3.303 n.00
,733 1.090 1.101
4.20 .2s .25 .50 .50 .75 .75 1.0
I!;: 1.25
5.6 0.0
16.8 666463. 1.4
0.0
28.0
A” 0.0
4.n 2.5 0.0 1.0 0.0
0.0
.420 .700 2.100 3.500 4.900 8.400 9.100 i0.5no 11.500 12.hOO
16.800 17.500 18.200 19.0*0 19.700 23.100 24.50n 25.QOO 26.600 2h.noo
.iie .19b .507 .850 1.078 1.550 1.631 1.781 1.91b 1.977 2.252 2.279 2.299 2.309 2.3114 2.165 2.086 2.006 1.370 I.‘123 .2en .280 .234 ,175 .15.8 .119 .llO .102 .093 .00S .047 .034 .n2n .004 -.017 -. 055 -.057 -.055 -.lJ3LI -.034 .I7b .294 .ii81 1.275 1.617
2.326 2.447 ?.b72 r.n74 .?.Yb5 3.37R 3.418 3.4-H 3.4a4 3.Sk.b 3.240 3.128 3.009 2.vss 2.8H5 .059 .n9e .294 .425 .539 .7?5 .R16 .tl91 .95b .9na 1.126 1.140 1.149 I.155 1.152 1.083 1.043 1.003 . 9us .vo2
.75 1.0 .75 1.0 .50 1.0 .50 1.0 .50 1.0
3:l ELLIPTICAL CROSS-SECTION UWY 0 n
-3 -1 10 30 0.0 .420 .700 2.10 3.50 4.9n 5.60 6.30 7.70 0.*0 9.10 10.50 il.9n 12.60 13.30 lS.70 15.40 lb.60 17.50 lb.20
19.040 19.70 7n.30 21 .OQ 22.40 73. In 24.50 25.uo Cb.60 28.00 0.0 .17h ,294 .aAl 1.27s l.bl7 1.776 1.922 Z.lYH 2.3c’b
2.447 2.672 2.87* 2.9h5 3.051 3.203 3.2bY 3.37l.4 3.41b 3.*48 3.464 3.456 3.432 3.397 3.3u3 3.2*R 3.12n 3.uo9 2.V55 C.HA5 0.0 .05Q .09A ,294 .425 .53v .591 .brl .733 .775 .I416 .R91 .95A .98e 1.017 l.ObU 1.090 l.li’b I.160 I.169 1.155 1.152 1.144 1.112 1.101 l.Uh3 1.043 l.oUJ .‘+m5 .962
3:l ELI IPTICAL CtiOSS-SECTION hllL)y - &DUIllWJAL PAIJELII~~ INIJUl f-l n -1 n A
5.600 6.300 13.300 14.700 20.3UO 21.000
1.182 1.2dl 2.034 2.136 2.2G0 2.265
.1*5 .135
. 076 .069 -.030 -.O*l
1.774 1.922 3.051 3.203 3.432 3.357
.591 .641 1.017 l.obCI 1.144 1.132
(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
i 2.56588 20.0 G.6 .@l
30 n.0 7.700 15.400 72.000 n.0 1.465 1.179 ?.202
240 :12n .n55 -.04-l ".OO
2.19n 7.2e9 3.303 7.00
.733 1 .OQO 1.101
. 42n .7no 2.100 3.500 l3.400 s.ino 10.500 11.900
16.80-O i7.5nn lY.200 19.040 23.100 2n.5on 25.~00 26.400
.119 .!9b .id7 .H50 1.55’) 1.631 1.7n1 1.9lh 2.252 2.279 2.299 2.33V 2.165 2.046 2.506 1.970 ..?I30 >en
.119 :11a .23r .175 0132 .093
.047 . n3* .020 .oo* -.F55 -.057 -.055 -.039 .176 .?51 .*a1 1.27';
2.326 ?.&I.7 I.672 c.r;7* 3.378 3.416 3.496 3.rbl 3.24P 3.12P, 7.009 2.q5.5 ,059 .n99 .2Ul . 42S .775 .a16 .i91 .95a 1.126 l.lb*J l.lL9 1.155 1.043 1.063 1.003 .si5
1.20011 0.0 0.0 0.G 1.02A64 .9Qb61 2.an 1.59082 .77324 4.203 l.Sbll .nq70 z.40
4.9no 12.600 19.700 2b.000
1.078 1.977 2.30~ 1.923 .15i! .Od5 -.017 -.33* 1.617 2.555 3.456
2.885 .!Y39 .VOR
1.152
,591 .6&l 1 . 017 1.*766 1.146 1.132
. 952 . 362’1 . ~5032 . 42231 .00346
ELLIPTICAL CaUSS-SECTIOQ d30Y - mDldI,‘l Gt>*tTey Ih.'uT 3:1 0 0
-3 -1 0.0 v.10
19.040 0.0
2.~47 3.464 0.0 . ‘lb 1.155
3:1 0 0 0 0
0.0
5.600 6.300 13.300 14.700 20.300 11.000
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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.
Figure 14.- 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.
Figure 14.- Continued.
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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
Figure 14.- Continued.
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.
Figure 14.- 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
(j) Sample Case 10, Continued.
Figure 14.- Continued.
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*
(j) Sample Case 10, Continued.
Figure 14.- Continued.
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
Figure 14.- Continued.
-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
(j) Sample Case 10, Continued.
Figure 14.- Continued.
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
(j) Sample Case 10, Continued.
Figure 14.- Continued.
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.
Figure 14.- 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.
Figure 14.- 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.
Figure 14.- Continued.
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
Figure 14.- Concluded.
140
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Page
2
Figu
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15.-
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tinue
d.
3:
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N
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Page
3
Figu
re
15.-
Con
tinue
d.
l *
GEU
H
l e
l *
CO
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G
l *
3:l
ELLI
PIIC
AL
CR
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(d)
Page
4
Figu
re
IS.-
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tinue
d.
-
1 AN
D
3 IN
nlC
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20
21
22
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24
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30
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7
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Page
18
Figu
re
15.-
Con
tinue
d.
SlR
ATFO
IID
QPA
f2Al
lnlJ
C
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1~
lW4
(CAM
lflAU
O
6 fs
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III
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Al
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9
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rnn
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nn
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T5
INIT
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POSI
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llYD
ST
REN
GTH
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VUH
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Al
x
= r.f
lon
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229
(s)
Page
19
Figu
re
15.-
Con
tinue
d.
DC
P’/D
X
0.0n
O
.S25
DC
P’/D
x
0.00
0 .5
25
SUH
MAU
Y O
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WlL
X Fi
t! I)
Al
& q
4 .
/ I1
n
H 5
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4 0
0 0
II
NV
GAW
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A5H
F.U
H
tlA
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62
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0.61
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1 2
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r-42
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1696
2 .h
v2n2
z.
rloou
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9..3
R9
CEN
TRD
ID
OF
VOR
TIC
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GhH
/V
Y z
l Y
9“D
Y:
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032
I.lhV
h2
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lIL
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2 -1
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02
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zc
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R
G/R
I .3
2024
1.
3695
4
1.32
024
1.36
954
(t)
Page
20
Figu
re
15.-
Con
tinue
d.
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(u)
Page
23
Figu
re
15.-
Con
tinue
d.
VF L
OC
I
1 Y
CO
~Whr
f
J ”
1 .2
rln
2 .2
41n
3 .
4A7.
l
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8
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nsn
10
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NT5
Al
W
tCltl
tlJ
f It.
Ln
WIN
15
Al
X =
4.2n
2 (l/
V0
v/v0
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Page
24
Figu
re
15.-
Con
tinue
d.
~TR
4TFO
Ull
5EPa
Rbl
lnN
C
UIT
FOIU
N
11,4
’.‘lN
dr)
F 15
1
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MAI
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JUT:
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ll SE
PAU
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l Y
qlnE
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4GN
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PT
. o.
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M
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Wt
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1.77
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on
5EP4
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DE:
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= .II
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a =
5.60
CP
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01
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000
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?10
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INIT
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POSl
llON
S AN
D
STR
ENG
THS
OF
SHED
VU
RTl
CllY
AT
X
= 5.
600
NV
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/V
HIK
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2 BE
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VT/V
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011
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oo
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on
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CL
= .3
59
cn
= .I3
1
(w)
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
Figure 17.- Concluded.
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
Figure 21.- Measured and predicted downwash distribution on the lee side of an ogive cylinder
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
Figure 22.- Measured and predicted downwash distribution on the lee side of an ogive cylinder
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
Figure 23.- Concluded.
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"
Figure 29.- Continued.
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
Figure 29.- Concluded.
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 aero- dynamic 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