1 American Institute of Aerodynamics and Astronautics

GLOBAL STABILITY ANALYSIS OF FLOW

AROUND AN ELLIPSOID AT ANGLE OF ATTACK

Asei Tezuka*, Kojiro Suzuki**

The University of Tokyo

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Abstract

In this paper, the computational study of flowfield

around an ellipsoid at angles of attack is made by

using the global stability analysis. The flow around

an ellipsoid was numerically calculated by MAC

method. The stability of the flow solution is evaluated

by Chiba’s method in the way of the eigensystem

analysis for the velocity disturbance which is distrib-

uted over the computational grids. The

three-dimensional flow around a sphere was analyzed

to check the validity of our numerical code. The

result is in good agreement with the result of Natara-

jan. It is clarified that in the case of an ellipsoid,

non-axisymmetric flow (in the case of zero angle of

attack) and asymmetric flow (in the case of non-zero

angle of attack) become steady and stable in a range

of the freestream Reynolds number around 4000 to

6000 and the angle of attack from 0 degrees to 30

degrees. The effect of the freestream parameters,

Reynolds number, angle of attack, and

length-to-diameter ratio are discussed.

1. Background

For understanding the aerodynamics of bluff-body

at high angles of attack, it is necessary to investigate

the structure of three-dimensional separation around

the body. In the global stability analysis, the perturba-

tion of velocity is set at the whole flowfield and the

most unstable mode of the perturbation is calculated.

Natarajan [1] examined the global stability of the flow

around a sphere and explained the mechanism of the

transition from steady and non-axisymmetric flow to

steady non-axisymmetric flow in the critical Rey-

nolds number regime. It was focused that the flow

around a sphere becomes steady axisymmetric in

such transition regime, i.e. Reynolds numbers based

on the diameter between 210 and 270.

From the result of sphere, it is expected that for a

body with ellipsoidal shape, the flow may also be-

come steady non-axisymmetric in a certain range of

Reynolds number. The angle of attack is also an

important parameter to determine the characteristics

of the flowfield. However, the presence of the critical

angle of attack with respect to the onset of the steady

asymmetric flow has not been clarified yet.

In this paper, the transition mode of flow structure

around an ellipsoid at angle of attack was examined

by the global stability analysis. Chiba’s method[2] was

used to calculate the global stability of flowfield,

since it can be easily coupled with the Navier-Stokes

analysis by the finite-difference method on the gener-

alized coordinates. Three-dimensional flow around a

sphere was analyzed to check the validity of our

numerical code. The result is in a good agreement

with the result of Natarajan [1]. Then, the flow around

an ellipsoid at various incidences was calculated and

the stability of asymmetric flow was analyzed. The

effect of the freestream parameters, Reynolds number,

angle of attack, and length-to-diameter ratio are dis-

cussed.

*Research Student, Department of Aeronautics and Astro-

nautics, Graduate School of Engineering, Student Member

AIAA

** Associate Professor, Department of Advanced Energy,

Graduate School of Frontier Sciences, Member AIAA

33rd AIAA Fluid Dynamics Conference and Exhibit23-26 June 2003, Orlando, Florida

AIAA 2003-4142

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2 American Institute of Aerodynamics and Astronautics

2. Summary of the global stability analysis

The incompressible laminar Navier-Stokes equa-

tions are written as follows:

)(/ ufu =∂∂ t , (1)

where the vector u is the column of the velocity

components at all the computational nodes. If the

number of computational nodes is N, the dimension

of u becomes 3N in the three-dimensional case. Pres-

sure is not included in the dependent variables, since

it is calculated from the velocity. A Taylor series

expansion is applied to Eq. (1) around the stationary

solution u0:

)~(~)()()~( 20

00 uuuuf

ufuuf O+∂

∂+=+ , (2)

where u~

is a small disturbance. Considering that the

second or higher order terms with respect to | u~ | are negligibly small, the linearized stability equation is

given as:

uAuuufu ~~)(~

0 ≡∂

∂=

∂∂

t , (3)

where uufA ∂∂≡ /)( 0 is the Jacobian matrix.

The global stability is analyzed by considering the

eigensystem of Eq.(3). However, the size of matrix A

is 3N x 3N, and it is difficult to calculate the eigen-

system of matrix A exactly. Chiba[2] adapted Eriks-

son's method[3] to determine the critical Reynolds

number in the case of a circular cylinder and obtained

reasonable results. In his approach, Arnoldi's method

was used to calculate the eigenvalue analysis ap-

proximately. In the present study, Chiba's method was

used for the global stability analysis, which is briefly

explained as follows;

A series of the vectors ζ i ( Mi ≤≤1 , M<<N),

which are the normalized orthogonal vectors with 3N

elementry, are needed to calculate the eigensystem of

A. At first, the vector ζ1 is given by random numbers.

In order to calculate the stability around the u0, which

is the solution of the Navier-Stokes equations at the

time t0, the Navier-Stokes equations are integrated

until time t0+T, using two types of initial conditions:

u0+εζ1 and u0-εζ2. In the present study, the

non-dimensional time T is set as 1. The parameter ε is

introduced in order to control the magnitude of the

initial disturbance. In the present study, ε is set as

0.01. Let the results of these integrations written as

ui+ and ui-. The vector exp(AT)ζ1 is calculated from

the equation:

exp(AT) ζ1 =( u1+- u1-)/2ε.

The vector ζ2 is introduced to describe the vector

exp(AT) ζ1 by using a set of orthonormal basis unit

vectors ζ1 and ζ2 as follows:

exp(AT) ζ1=c1,1ζ1+c2,1ζ2.

The vectors, ζ3, ζ4, … , ζM are calculated in the

same way as:

First, The vector exp(AT) ζi is calculated from the

equation:

exp(AT) ζi =( ui+- ui-)/2ε.

Next, The vector ζ i+1 is introduced:

exp(AT) ζi =c1, i ζ1+c2, i ζ2+…+ c i+1, i ζ i+1.

By neglecting c M +1, M ζM+1 ( |c M +1, M ζ M +1|<<1) and

assuming:

exp(AT) ζM=c1, M ζ1+c2, M ζ2+…+ cM, M ζ M,

the M x M matrix H with the element cj, k is obtained.

The eigensystem of exp(AT) is approximately calcu-

lated from the eigensystem analysis of the matrix H

of size M x M instead of the matrix exp(AT) of size

3N x 3N. The eigensystem of A is calculated from the

eigensystem of matrix B=exp(AT). The eigenvalue of

the matrix A (λA) is obtained by the relation:

λB=exp(λAT). The eigenvector for the system A is the

same as that for the system B.

The eigenvector of matrix A represents the mode of

disturbance. The stability of the mode is estimated by

the real part of the eigenvalue. When the real part of

eigenvalue is positive or negative, the magnitude of

the disturbance mode is growing (the flow is unsta-

ble) or diminishing (the flow is stable), respectively.

The period of the oscillation mode is evaluated by the

imaginary part of the eigenvalue. The imaginary part

represents the angular velocity of oscillation. When

the imaginary part of the eigenvalue is zero or

non-zero, the mode is steady or oscillatory, respec-

tively. It should be noted that in this method, less

stable mode with larger real part of the eigenvalue is

computed more accurately. For accurate evaluation of

the stability of the flow, M must be large enough. In

3 American Institute of Aerodynamics and Astronautics

the present study, M is set as 40.

The characteristic of Chiba’s method is to calculate

the global stability of flow by adding the disturbance

to the numerical scheme of flow. Therefore, the equa-

tion of perturbation (e.g. Natarajan[1]) is not solved.

The MAC method[4] was used to solve the Na-

vier-Stokes equations. The Poisson's equation for

pressure was solved by the SOR method. The second

order Adams-Bashforth method was used for time

integration. The artificial viscosity is necessary to

calculate the flow in the order of Reynolds number

103. Third-order upwind finite-difference[5] was used

to calculate the convection terms, since the coeffi-

cient of fourth-order artificial viscosity is smaller

than that of Kawakura-Kuwahra’s scheme[6] .

The grid is O-O type around the body and the

number of grid is 51 x 41 x 51. At the outer boundary,

free-stream boundary condition and zero-order ex-

trapolation are applied at the inflow and outflow

boundary, respectively. The value of the velocity

vector u at the inflow boundary is set as freestream

condition and the value of the disturbance velocity

vector u~ is set as zero. At the outflow boundary, the

value of u and u~ are equal to that of the inner mesh

of radial direction.

3. Results and discussions

3.1. Flow around a sphere

The flow around a sphere is known to generate the

non-axisymmetric flow when the Reynolds number is

larger than 210. The results of our numerical code

shows that the critical Reynolds number is around

226. This value is larger than the above-mentioned

Reynolds number. The result of critical Reynolds

number using fine grid (71 x 41 x 51, clustering in the

wake region) is 213. The difference of the value of

critical Reynolds number is considered to be caused

by the numerical viscosity, which depends on the size

of grid spacing, since the numerical viscosity of fine

grid is lower than that of coarse grid. However, the

pattern of the most unstable mode (the vectorfield of

the eigenvalue) is almost same as the result of Nata-

rajan[1]. In this paper, the qualitative discussion of

asymmetric flow is focused on and the effect of the

grid spacing is supposed to be trivial in qualitative

discussion.

Figure 1 shows the definition of the planes and

forces considered in the following discussion. The

streamline of flow at Re=200 (Re represent the Rey-

nolds number based on the diameter) is plotted in

Fig.2. In this case, the flow is axisymmetric. The

streamlines converge to one thread in the wake. Fig-

ure 3 shows the streamline of flow around a sphere at

Re=250. The flowfield is not axisymmetric and the

streamlines converge to two parallel threads in the

wake(double-thread wake).

The global stability of the flow around a sphere is

numerically calculated. The steady state solution of

the Navier Stokes equations at Re=200 is selected as

the reference flow. The disturbances are added to the

reference flow, and its stability is numerically exam-

ined.

Figure 4 shows the eigenvalues plotted on the

complex plane. The mode pointed by an arrow in the

figure has the largest real part of the eigenvalue and is

considered as the most unstable mode. The imaginary

part of the most unstable mode is zero. Therefore, this

mode is non-oscillatory mode. Figure 5 shows the

disturbance vectors of the most unstable mode. The

cross sectional view of the disturbance vectors on the

plane of symmetry (corresponding to Plane A in

Fig.1) is plotted in Fig.5(a). All the components of the

disturbance vector are parallel to this plane. Figure

5(b) shows the contours of the vector component in

parallel to the plane A on the plane B. All the com-

ponents of the disturbance vector are normal to the

plane B. Thus, the magnitude of the disturbance

vector is plotted by contour lines. The

three-dimensional plot of disturbance vector is plotted

in Fig. 6. In the vicinity of the surface of a sphere

(one mesh away from the surface), the streamline of

disturbance vector is plotted. The component of dis-

turbance vector which is parallel to the surface is

connected in order to draw the surface streamline.

This pattern is not axisymmetric.

If this mode is added to the flowfield and the real

4 American Institute of Aerodynamics and Astronautics

part of the eigenvalue becomes positive, the flowfield

changes from axisymmetric to non-axisymmetric.

The imaginary part of the eigenvalue is zero and the

mode is non-oscillatory. Therefore, in the case that

the Reynolds number is larger than the critical Rey-

nolds number, the transition of flow from

non-oscillatory axisymmetric to non-oscillatory

non-axisymmetric occurs.

To illustrate the transformation of flowfield caused

by the most unstable mode, the disturbance vectors

are superposed to the reference flow as illustrated in

Fig. 6. The streamlines of the wake converge to two

threads. As Ghidersa[7] pointed out, the phenomena

that the streamline converges to double-thread wake

is explained by growth of mode.

3.2. Flow around an ellipsoid

3.2.1. Angle of Attack of 0 degrees

First, the stationary solution u0 was calculated from

the Navier-Stokes equations and this solution is used

as the reference flow of the stability analysis. The

reference length of the Reynolds number is the major

axis of the ellipsoid. The angle of attack is defined as

the angle between flow direction and the major axis.

The length and diameter of the ellipsoid is 1 and 0.25,

respectively. The definition of the planes and forces

used in the following duscussion is shown in Fig.8.

Figure 9 shows the surface streamlines at Re=3500,

and the pattern of surface streamline is steady sym-

metric. Figure 10 shows the surface streamlines at

Re=5000, and the pattern is still steady but not sym-

metric. The critical Reynolds number from symmetric

flow to asymmetric flow seems to exist between these

Reynolds numbers.

The global stability of the flow around an ellip-

soid was calculated. Figure 11 shows the eigenvalues

on the complex plane. The most unstable mode has

the largest real part of the eigenvalue. The imaginary

part of eigenvalue of the most unstable mode is zero.

Thus, this mode is non-oscillatory. The real part of

the eigenvalue for the oscillatory mode in which the

imaginary part of eigenvalue is not zero is smaller

than the non-oscillatory mode. Therefore, the oscilla-

tory mode attenuates faster than the non-oscillatory

mode. Figure 12 shows the disturbance vector of the

most unstable mode. The disturbances of Fig.12(a) is

a plane of symmetry and every disturbance vectors on

the plane A (see Fig.8) are shown in Fig.12(a). These

vectors are parallel to the plane A. On the plane B,

every disturbance vector is normal to this plane, so

the normal component of the vector is represented by

contour lines in Fig.12(b). This result is similar to the

result of the flow around a sphere (Fig.5).

When the Reynolds number increases, the real part

of the non-oscillatory mode becomes zero and the

flow is neutral stable. The critical Reynolds number

is determined from this Reynolds number. This value

is about 3840. If the Reynolds number becomes lar-

ger than the critical Reynolds number, the

non-oscillatory mode becomes unstable, and the

transition from non-oscillatory axisymmetric flow to

non-oscillatory non-axisymmetric flow occurs.

The rearward view of the surface streamline (plot-

ted same way as that of Fig.6) is plotted in Fig.13.

The pattern of Fig.13 is similar to the pattern of Fig. 6.

If this mode of disturbance vector is added to the

flowfield, the pattern of surface streamlines change

from axisymmetric to non-axisymmetric. This pattern

is similar to that of the flow around a sphere. The

non-oscillatory mode of the flow around an ellipsoid

at zero angle of attack and the non-oscillatory mode

of the flow around sphere are considered to be essen-

tially the same phenomena.

3.2.2. Angle of attack of 10 degrees

When the angle of attack is not zero, the flow also

becomes steady asymmetric in a certain range of

Reynolds number. Figure 14 shows the surface

streamlines of the flow at Re=3900 and an angle of

attack of 10 degrees. The pattern of the surface

streamline is symmetric. Figure 15 shows the surface

streamlines of the flow at Re=5500. The pattern of the

streamlines is asymmetric. The asymmetry of the

flowfield is also confirmed by the fact that the side

force is generated on the body. The magnitude of the

side force is about 5% of the drag force.

5 American Institute of Aerodynamics and Astronautics

The stationary solution of the flow at Re=3900 is

selected as the reference flow for the global stability

analysis. Figure 16 shows the eigenvalues on the

complex plane. The imaginary part of the eigenvalue

whose real part is the largest is zero. When the Rey-

nolds number becomes larger than the critical value,

which is estimated as 4050 in this case, the

non-oscillatory asymmetric mode becomes unstable,

and the flow changes from non-oscillatory symmetric

flow to non-oscillatory asymmetric flow.

The flow at Re=5500 was also calculated and the

result is shown in Fig.17. The stationary solution,

which shows the asymmetric pattern in the surface

streamline (see Fig. 15), is selected as the reference

flow. The result of the stability analysis shows that

the real part of the eigenvalue is smaller than zero for

all the modes and the asymmetric flow at Re=5500 is

stable.

The real part of the most unstable mode is also

smaller than zero. The imaginary part of the most

unstable mode is not zero (oscillatory mode). The

period of oscillatory mode is calculated as follows:

Tperi=2 π / Im(λ), (4)

where Tperi and λ represent the period and the eigen-

value of the mode, respectively. The period of the

most unstable mode is 8.8. The flow changes from

steady flow to oscillatory flow when the Reynolds

number becomes larger than the critical Reynolds

number which is estimated around 5700. Figure 18

shows the power spectrum of the oscillatory side

force at Re=6000. Peaks of the spectrum are observed

at frequency 0.1, 0.2 and 0.3. This fact indicate that

oscillatory flow at Re=6000 include several mode of

oscillation. The frequency whose amplitude is largest

is 0.2. This value is not consistent with the period

estimated by eq.(4) for the most unstable. The effect

of non-linearity is expected to appear.

3.2.3. Angle of attack of 30 degrees

The Navier-Stokes equations were also solved at

angles of attack 30 degrees. The flow also becomes

steady asymmetric in a certain range of Reynolds

number. Figure 19 shows the surface streamlines of

the flow at Re=2800 and an angle of attack of 30

degrees. The primary and secondary separation lines

exist on the surface of ellipsoid in this case. This

result indicates that the separation pattern changes

according to the change in the angle of attack. Figure

20 shows the surface streamlines of the flow at

Re=5500. The pattern of the streamlines is asymmet-

ric. The magnitude of the side force is 6% of the drag

force.

The global stability analysis of the flow at angle of

attack 30 degrees was conducted. The stationary

solution of the flow at Re=2800 is selected as the

reference flow. Figure 21 shows the eigenvalues on

the complex plane. The imaginary part of the eigen-

value whose real part is the largest is zero. When the

Reynolds number becomes larger the critical value

Re= 2930, the non-oscillatory mode becomes unsta-

ble, and the flow changes from non-oscillatory sym-

metric flow to non-oscillatory asymmetric flow.

The global stability of flow at Re=5500 was calcu-

lated. Figure 22 shows the eigenvalues plotted in

complex plane. The real part of the eigenvalue is

smaller than zero for all the modes, thus the asym-

metric flow is stable in this Reynolds number. The

critical Reynolds number from steady flow to oscil-

latory flow is estimated around 5600.

3.3. Case of length-to-diameter ratio 6:1

The case where the length-to-diameter ratio is 6:1

is also calculated. When the angle of attack is 30

degrees, the flow becomes steady asymmetric in a

narrow range of Reynolds number around 7100.

However, the side force is negligibly small, far

smaller than the case of Fig 20. When the

length-to-diameter ratio becomes larger, the range of

Reynolds numbers in which the steady asymmetric

flow is observed becomes narrower and the flow

tends to change directly from steady symmetric pat-

tern to oscillatory pattern when the Reynolds number

becomes larger than the critical Reynolds number.

3.4. Experimental Results

In order to confirm the existence of steady asym-

6 American Institute of Aerodynamics and Astronautics

metric flow around ellipsoid at a certain range of the

Reynolds number and attack angle, experimental

research also held in Ref.[8].

The experimental model is supported by the sting.

The flow around an ellipsoid with a sting was also

calculated. The angle of attack is set as 10 degrees.

The result of numerical calculation shows that the

flow at Re=4000 is symmetric. The cross sectional

view of plane that is normal to the symmetric plane

and includes the axis of ellipsoid (see Fig.23) is plot-

ted in Fig.24. The experimental result of the flow

visualization by using a smoke generator is shown in

Fig.25. The Reynolds number is about 3300. The

pattern of the re-circulation region seems symmetric.

The computed streamlines at Re=6000 is asymmetric

as shown in the cross sectional view of Fig.26. The

experimental result is shown in Fig.27. The Reynolds

number is about 6500. The pattern of the

re-circulation region is asymmetric.

In the experiment, several pictures are taken at

difference timing. The smoke pattern is always simi-

lar to the pattern shown in Fig.27. From Figs.26 and

27, the onset of the steady asymmetric flow around an

ellipsoid at attack angle 10 degrees is confirmed both

numerically and experimentally.

Conclusions

l A computational code for the global stability

analysis of three-dimensional incompressible

flows has been developed. Code validation is

made by analyzing the critical Reynolds number

for a sphere and comparing the results with

those by the results of Natarajan.

l When the Reynolds number increases, steady

axisymmetric, steady non-axisymmetric and os-

cillatory patterns successively appear at zero

angle of attack.

l At zero attack angle case, the imaginary part of

eigenvalue of the most unstable mode is zero.

The most unstable mode is non-oscillatory and

the real part of this mode become zero according

to the increase of the Reynolds number. There-

fore, the transition of flow from non-oscillatory

axisymmetric to non-oscillatory

non-axisymmetric occurs and the steady

non-axisymmetric flow observed.

l In the case of an ellipsoid at angle of attack, the

critical Reynolds number for the transition from

steady symmetric flow to steady asymmetric

flow exists and it is reduced when the angle of

attack increases.

l The imaginary part of the eigenvalue of the most

unstable mode is zero at angle of attack case.

The transition from steady symmetric flow to

steady asymmetric flow occurs not only in zero

attack angle case, but also in non-zero attack

angle case.

l The Reynolds number range, in which the

steady asymmetric flow becomes stable, de-

pends not only on the angle of attack but also on

the length-to-diameter ratio.

l The onset of steady asymmetric flow around an

ellipsoid at angle of attack is confirmed by the

flow visualization in the low speed wind tunnel

experiment.

Bibliography

(1) Natarajan, R. & Acrivos, A, J. Fluid Mech., 254

(1993), pp. 323-344.

(2) Chiba S., Ph.D. thesis, The University of Tokyo,

(1997), (In Japanese).

(3) Eriksson L. E. & Rizzi A., J. Comp. Phys. 57

(1985), pp. 90-128.

(4) Harlow, F. H. & Welch, J. E., Phys, Fluids, 8,

(1965), pp. 2182-2189.

(5) Greenspan, D., Comput. J., 12, (1969), pp. 89-94.

(6) Kawamura T. & Kuwahara K. AIAA Paper

84-0340, (1984)

(7) Ghidersa, B. & Dusek, J. J. Fluid Mech. 423,

(2000), pp. 33-69.

(8) Tezuka A., Ph.D. thesis, The University of Tokyo,

(2003), (In Japanese).

7 American Institute of Aerodynamics and Astronautics

Plane of Symmetry

Axis

Normal to the Planeof Symmetry

Drag

Lift

Figure 1. Definition of the planes and forces

(Flow around a sphere)

Figure 2. Axisymmetric flow around a sphere

(Re=200)

Figure 3. Non-axisymmetric flow around a sphere

(Re=250)

Figure 4. Eigenvalues of flow around a sphere on a

complex plane (Re=220)

(a) Cross sectional view of plane A

(b) Cross sectional view of plane B

Figure 5. Cross-sectional view of the disturbance

velocity (Re=220, Most unstable mode)

(Plane A)

(Plane B)

8 American Institute of Aerodynamics and Astronautics

Figure 6. Three-dimensional view of the disturbance

velocity and the surface streamlines of the distur-

bance velocity (Re=220, Most unstable mode)

Figure 7. The streamline of velocity that the most

unstable mode is superposed to the base flow

Plane of Symmetry

Axis

Normal to the Planeof Symmetry

Drag

Lift

Flow

Angle of Attack

Side Force

Figure 8. Definition of the planes and forces

(flow around an ellipsoid with angle of attack)

Figure 9. Axisymmetric pattern of computed stream-

lines past an ellipsoid (Re=3800 AOA=0deg)

Figure 10. Non-axisymmetric (planer symmetric)

pattern of computed streamlines past an ellipsoid

(Side view, Re=5000, AOA=0deg)

Figure 11. Eigenvalues of flow around an ellipsoid on

a complex plane (Re=3800, AOA=0deg)

Plane A

Plane B

∞U

(Plane A)

(Plane B)

∞U

9 American Institute of Aerodynamics and Astronautics

(a) Cross sectional view of plane A

(b) Cross sectional view of plane B

Figure 12. Cross-sectional view of the disturbance

velocity (Re=3200, AOA=0deg)

Figure 13. Rearward view of the surface streamlines

of the disturbance velocity. (Re=3200, AOA=0deg)

Figure 14. Leeside view of surface streamlines past

an ellipsoid (Re=3900, AOA=10deg)

Figure 15. Leeside view of surface streamlines past

an ellipsoid (Re=5500, AOA=10deg)

Figure 16. Eigenvalues of flow around an ellipsoid on

a complex plane (Re=3900, AOA=10deg)

∞U

∞U

10 American Institute of Aerodynamics and Astronautics

Figure 17. Eigenvalues of flow around an ellipsoid on

a complex plane (Re=5500, AOA=10deg)

Figure 18. Frequency analysis of the side force

(Re=6000, AOA=10deg)

Figure 19. Leeside view of surface streamlines past

an ellipsoid (Re=2800, AOA=30deg)

Figure 20. Leeside view of surface streamlines past

an ellipsoid (Re=5500, AOA=30deg)

Figure 21. Eigenvalues of flow around an ellipsoid on

a complex plane (Re=2800, AOA=30deg)

Figure 22. Eigenvalues of flow around an ellipsoid on

a complex plane (Re=5500, AOA=30deg)

11 American Institute of Aerodynamics and Astronautics

Viewing Plane (Fig.24-27)

Freestream Direction

AOA=10deg

Plane of Symmetry

Model

Sting

Figure 23. Experimental model of the flow around

ellipsoid

Figure 24. Numerical result of the flow around ellip-

soid with sting (Re=4000)

Figure 25. Experimental result of the flow around

ellipsoid supported by sting (Re~3500)

Figure 24. Numerical result of the flow around ellip-

soid with sting (Re=6000)

Figure 25. Experimental result of the flow around

ellipsoid supported by sting (Re~6500)