American Institute of Aeronautics and Astronautics

1

The Application of CFD to the Prediction of Body Vortices for Cone-Cylinder Configurations in Supersonic Flow

Muhammad Asif1 and R. A. Khuram2

CFD-Chapter, Computational Modelling, Control & Simulation Society of Pakistan

S. Zahir3 CFD-Chapter, Computational Modelling, Control & Simulation Society of Pakistan

Mahmood A. Khan4

CFD-Chapter, Computational Modelling, Control & Simulation Society of Pakistan

This study has been carried out to ascertain the CFD, PAK-3D code capabilities to the prediction of body vortices. The flow analysis has been performed on Ogive -Cone Cylinder with and without Controlling Surfaces and on Blunted-Cone Cylinder Configurations. PAK-3D is a 3D Navier-Stokes Solver based on Beam and Warming algorithm. Computational results were obtained using Baldwin-Lomax algebraic turbulence model. Cp distribution was compared at different X/D locations and it was found that up to the separation point, computational results are within 10 % of experimental data [1] but after the separation point which is a highly separated flow region, computational results differ more than 10 % with the experimental data although follow the same qualitative trends. This study has been extended to Ogive -Cone Cylinder Configuration with controlling surfaces . Since controlling surfaces, Fins are on cylinder portion where CFD Cp prediction is not in good match, where choice of turbulence model is vital, therefore quantitatively results differ from the WIND CFD code data [2]. The aberration of data in highly separated flow regions indicates the further development of the predictive technology is required. However, Normal force coefficient has been compared with the experimental data and computational results are within 4 % of the wind tunnel data, which is within acceptable design requirement.

Nomenclature M = Mach number α = Angle of attack [degree] Re/L = Reynold’s number per unit length Cn = Normal force coefficient Ca = Drag force coefficient Cm = Pitching moment coefficient Cp = Pressure coefficient XCp/L = Location of centre of pressure per unit length X/D = Axial distance per unit diameter L = Length of the body [m] Φ = Circumferential angle [degree] D = Diameter of the body [m] p = Pressure pl = Local non-dimensional static pressure p∞ = Non-dimensional free stream pressure q∞ = Non-dimensional free stream dynamic pressure E, F, G = Cartesian fluxes Et = Total Energy H = Flux tensor ___________________________ 1E-mail: masif_aziz@hotmail.com 2E-mail: rakhuram@hotmail.com 3E-mail: cfdpak@apollo.net.pk 4E-mail: mkhan@uwindsor.ca

22nd Applied Aerodynamics Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-4833

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

American Institute of Aeronautics and Astronautics

2

Q = Conservative variable vector in Eq. (2) t = time u, v, w = Cartesian velocity components V = Volume τ = Stress tensor x, y = Coordinates in the crossflow plane z = Coordinate in the body axis direction Subscripts i, j, k = Indices in three transformed computational coordinates t = Reservoir or total values ∞ = Freestream condition Superscripts i = Inviscid v = Viscous

I. Introduction

N complex flow situations with highly separated flow regions where there are adverse pressure gradients results in the spinning of constituent fluid particles and are said to possess the vorticity. In practical measurements on flow around

bodies, data is generally presented in terms of the pressure coefficient rather in the form of pressure alone. Pressure coefficient gives the information about pressure distribution around the bodies. Pressure coefficient, Cp is defined [3] as,

∞∞−= qppCp /)( 1 (1) Cp data is superior to integrated parameters like Cn, Ca and XCp, because it describes local phenomenon whereas these parameters show integrated effect. It depicts local flow separation, for example on wing, fins and at the interface of cone-cylinder. Because of local phenomenon, it is used in wing loading and wing fluttering cases. Cp distribution also helps in the selection of turbulence models. It is also used in aerodynamic loading and pitching moment coefficient calculations. In this study the predictive capability of computational Navier-Stokes technique has been evaluated. PAK-3D, is a 3D Navier-Stokes solver, which is based on Beam and Warming algorithm [4]. This algorithm is an implicit scheme, which solves the set of equations produced by central differencing of the Navier-Stokes equations on a regular grid. This scheme incorporates the ADI style formulation [5]. In this algorithm each sweep involves the inversion of a set of scalar pentadiagonal matrices. These flow analyses have been made with Baldwin-Lomax algebraic turbulence model [6]. This model has its application in wall bounded viscous flows.

II. Governing Equations The law of conservation of mass, momentum, and energy over a volume V bounded by a surface S can be expressed in an integral form as

( )∫ ∫ =⋅+∂∂

v SdSnHdVQ

t0 (2)

For steady flow as in these paper analysis, reduces to ( )∫ =⋅S dSnH 0 (3)

The flux tensor H can be written in terms of Cartesian fluxes as

( ) ( ) ( )kGGjFFiEEH vivivi −+−+−= (4)

I

American Institute of Aeronautics and Astronautics

3

( ) ( )

( )

−++

=

−++

=

+

=

+

+=

+

+

=

=

yyzyyxy

yz

yy

xyv

xxzxyxx

xz

xy

xxv

t

i

t

i

t

i

t

qwvu

F

qwvu

E

wpE

wv

wu

w

G

vpEwv

p

vu

v

F

upEwu

vu

P

u

E

Ew

v

u

Q

w

v

u

τττ

τ

τ

τ

ττττ

τ

τ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

0

0

2

2

2

(5)

−++

=

zzzyzxy

zz

yz

xyv

qwvu

G

ττττ

τ

τ

0

III. Geometry

The model configurations of interest for these studies are a 3-caliber ogive with a 10–caliber cylindrical after body as shown in Figure 1 and Figure 2 without and with controlling surfaces respectively. Geometry is based on the cylinder of diameter 0.094 m.

Figure 2: Ogive -Cone Cylinder with controlling surfaces configuration.

Figure 1: Ogive -Cone Cylinder configuration.

American Institute of Aeronautics and Astronautics

4

IV. Grid Independence Study

For accurate CFD results, grid plays a vital role. The final solution should be grid independent that is the qualitative and quantitative results do not change with the change in grid. The grid depends on the geometry, flow conditions and it also depend on the nature of study. So the grid independence has to be done for every problem. In these three cases different topologies and grids had to be generated. As a sample case, for the first case of Ogive-Cone Cylinder Configuration, grid independence study has been shown. This study is intended to select a suitable grid for an Ogive-Cone Cylinder body and to highlight some salient features, which are necessary to select the final grid. It is also intended to show the order of error, which may be encountered by selecting an unsuitable grid. The geometry of Ogive-Cone Cylinder body was initially analyzed using PAK-3D Code. The geometry was discretized using PAK-GRID; results were post scripted using PAK-VIEW. For CFD analysis of Ogive-Cone Cylinder body, quantitative analyses were made at Mach 2.5 and Angle of attack kept at 14 degree using Baldwin Lomax model for viscous flows. Ogive-Cone Cylinder body was analyzed at standard supersonic conditions. Four types of grids were used for this purpose. The grid dimensions were as follows:

1. 80 x 80 x 80 2. 160 x 80 x 80 3. 160 x 160 x 80 4. 200 x 200 x 200

(in all of these cases wake was modeled with 80 x 80 x 80) In order to select the grid for final analysis, Cp distribution for all the grids at X/D = 4.5 were plotted against Φ as shown in Figure 3. It is depicted from the figure that GRID 3 that is 160 x 160 x 80 results are more close to experimental qualitatively as well as quantitatively. Therefore GRID 3 has been selected for complete analysis purpose.

V. Flow Conditions Case 1 and 2: Ogive -Cone Cylinder with and without Controlling Surfaces respectively: Calculations are based on the following flow conditions: Mach No. =2.5 Angle of attack, α =14 degree Specific heat ratio, γ =1.4 Reynold’s Number, Re/m = 5.25x 106 Reynold’s Number, Re/m = 1.46x 106 (for controlling surfaces case)

VI. Grid Generation The Ogive-Cone Cylinder with and without controlling surfaces were modeled using a 3-dimensional structured grid. In these studies, there is no side slip angle. Hence only 180-degree grid is needed to simulate the flow conditions. A symmetry plane is applied through the center of the bodies. Grid is densely packed towards the body to capture the boundary layer and also at the regions where flow separation was expected. The computations were done using multi-block grids. Figure 4 shows a typical structured grid around Ogive-Cone Cylinder Configuration and Figure 5 shows grid around fins and the portion of body.

X/D = 4.5

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 30 60 90 120 150 180

Phi (deg)

Cp

GRID1 (80*80*80)

GRID 2 (160*80*80)GRID 3 (160*160*80)

GRID 4 (200*200*200)

Experimental

Figure 3: Grid independence study (case 1).

American Institute of Aeronautics and Astronautics

5

Figure 5: A sample of mesh on Ogive -Cone

Cylinder body and fins (case 2).

VII. Boundary Conditions

At the upstream of Ogive-Cone Cylinder, supersonic inflow condition was fixed, while the condition of extrapolation was imposed at the downstream of the bodies. The adiabatic viscous wall condition was used on the surfaces of the bodies. Symmetry condition was imposed on the symmetry plane. Baldwin-Lomax turbulence model was used in these CFD analyses.

VIII. Results and Discussions As mentioned earlier that Cp is more important than other static aerodynamic force coefficients like Cn, Ca and Xcp/L

because it depicts the qualitative flow trends also. Case1: Ogive -Cone Cylinder Configuration Table 1 shows the comparison of normal force coefficient with the experimental data. It is found that the prediction of normal force coefficient is within 4%, which is an acceptable range in aerodynamic design calculations.

Table 1: Comparison of Normal Force Coefficient

S. No.

Aero. Coefficient

Wind Tunnel

PAK-3D % age Difference

1. Cn 1.91 1.84 3.7 Figure 6 shows the Cp distribution at different X/D locations and compared with experimental data. Computational and experimental results indicate separation on the lee-side for X/D>3.5. At the wind-side computational results and experimental data are in close match. Up to the separation point PAK-3D data is within 10 % of the experimental data. Computational data differs from the experimental data in the separation region which is a complex flow region. In separation region, qualitative flow trends are strong function of turbulence, therefore proper choice of turbulence model is vital. In these computational analyses using PAK-3D Baldwin-Lomax turbulence model has been used. At X/D = 2.4, which is located on the Ogive-Cone, PAK-3D results within 10 % of experimental data. This is the region located on the cone where there is no separation; both the data are in close match. PAK-3D predicts the same qualitative trends as experimental data. At X/D = 3.5, both at the wind-ward and lee-ward sides, PAK-3D results are within 10 % of the experimental data. PAK-3D predicts the similar suction peak as the experimental data. Up to the X/D = 3.5, flow is not separated from the body. At X/D = 4.5 and 5.5, flow separation starts, up to the separation point both the results, PAK-3D and experimental are within 10 %, but as the separation point reached both the results differ significantly. One of the possible reasons, as mentioned earlier that is the choice of turbulence model. Experimental data shows two suction peaks but computational results do not predict second suction peak. At X/D = 7.5 and 11.5, at the wind-ward side and at the lee-ward side up to the separation point PAK-3D results are within 10 % with the experimental data. After the separation point, PAK-3D results differ significantly from the experimental data.

Figure 4: A mesh around an Ogive -Cone Cylinder (case 1).

American Institute of Aeronautics and Astronautics

6

The first suction peak picked by the computational results is at higher circumferential angle as compared to the experimental results. This may be due to the Baldwin -Lomax turbulence model, which over predicts the location of first suction peak. Also computationally second suction peak was not picked because in the separated flow region, the vorticity from the second source is intermixed with the wall bounded flow which causes the turbulence model to incorrectly predict the second peak. Another reason for this may be because first suction peak is at higher angle it delays the second suction peak to be appeared. In the separation region physics of the flow has been changed, therefore changes in computational modeling and algorithm must be made to incorporate this physics along with the choice of proper turbulence model. Figure 7 shows the comparison of pressure contours with the experimental data at X/D = 11.5. This figure depicts the body vortices. Figure 8 shows the pressure distribution around the Ogive-Cone Cylinder. Figure clearly depicts the complexity of the separated flow at high angle of attacks. Case2: Ogive -Cone Cylinder Configuration with Controlling Surfaces As discussed in the above case of without controlling surface that CFD prediction need improvement in the separated flow region. Figure 9 compares the pressure contours from PAK-3D with the WIND code at X/D = 11.5. This figure shows that two codes PAK-3D and WIND show the same qualitative trends and formation of vortices. Although quantitatively the location and strength of the vortices may differ but both codes pick the same qualitative trends. Figure 10 shows the pressure contours just down stream of fins, which is the high separating flow region and clearly depict the formation of fin vortices. In Figure 11, pressure distribution has been shown. This figure shows the separation of flow at the lee-ward side of the body. Figure 11 shows the comparison of Cp distribution at X/D = 8.79 with the WIND code data. PAK-3D data is quite aberrant from the WIND code data. This may be because of the reason that this is the location where there are fins, and this is highly separated flow region, where choice of turbulence model is of foremost importance. Although both the code used the same turbulence model Baldwin-Lomax but in case of this model the turbulent length scale is determined by a search to the flow field outer boundary. In WIND code may be these things are possible but with PAK-3D code this is not possible presently that might be the one of the reason of this deviation in prediction of Cp by the two codes .

X/D = 2.4

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK-3D

Experimental

X/D = 3.5

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK-3D

Experimental

X/D = 4.5

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK3DExperimental

X/D = 5.5

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK3D

Experimental

American Institute of Aeronautics and Astronautics

7

Figure 6: Comparison of Cp Distribution on Ogive -Cone Cylinder.

Figure 7: Comparison of Pressure Contours at X/D = 11.5.

PAK-3D, BL Turbulence Model

X/D = 7.5

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK3D

Experimental

X/D = 11.5

-0.15

-0.1

-0.05

0

0.05

0.1

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK3D

Experimental

American Institute of Aeronautics and Astronautics

8

Figure 8: Pressure Distribution on Ogive -Cone Cylinder.

Figure 9: Comparison of Pressure Contours at X/D = 11.5 (case 2).

American Institute of Aeronautics and Astronautics

9

Figure 10: Pressure contours at the fins downstream (showing vortices).

X/d = 8.79

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 30 60 90 120 150 180

Phi (deg)

Cp

PAK-3D

WIND Flow Code

Figure 11: Comparison of Cp Distribution on Ogive -Cone Cylinder with Controlling Surfaces.

American Institute of Aeronautics and Astronautics

10

IX. Conclusions 1. PAK-3D predicts the Normal Force Coefficient, Cn within the acceptable range. 2. Cp prediction using PAK-3D at different X/D locations up to separation point are within 10 % with the experimental

data. 3. After the separation point, which is the region of highly separated flow, Cp predicted by PAK-3D is more than 10 %

with the experimental data. Except the second suction peak PAK-3D predicts the same qualitative trends, which may be because of the reason that vorticity from the second source is intermixed with the wall bounded flow which causes the turbulence model to incorrectly predict the second suction peak.

4. Experimental data shows two suction peaks, which are regions of low pressure (vortices initiation region), whereas PAK-3D successfully picks the first suction peak but not the second one. First suction peak picked by PAK-3D is circumferentially at higher angle than the experimental data. This may be because of the reason that Baldwin-Lomax turbulence model over predicts the first suction peak. This delay in second suction peak may also be the cause of not picking of second suction peak.

5. From design point of view, it can be concluded that PAK-3D can be used with confidence to predict the Cp distribution for bodies like cone or power law profiles (increasing slope) but for cylindrical bodies especially at higher angles of attack (where separation initiates) it should be used with special care.

6. PAK-3D predicts the separation location which is close to the predicted by wind tunnel. 7. As mentioned earlier that Cp distribution after separation point differs significantly from the experimental data, where

choice of turbulence model is of prime importance. Therefore, it is suggested that this study may be extended to encompass other turbulence models as well. Also since the physics of the problem also changes and infact this is more complex flow situation, the changes in algorithm and modeling is suggested to encompass these highly separated flow situations.

References

1. Sturek, W. B., Birch, T., Lauzon, M., Housh, C., Manter, J., Josyula, E., and Soni, B., “The Application of CFD to the Prediction of Missile Body Vortices”, AIAA 97-0637, 35th Aerospace Sciences Meeting & Exhibit, January 6-10, 1997 / Reno, NV.

2. Sturek, W. B., and Haroldsen, D., J., “Calculation of Vortices for Finned Missiles using the WIND Flow Solver", AIAA 2000-0752, 38th Aerospace Sciences Meeting & Exhibit, January 10-13, 2000 / Reno, NV.

3. Houghton, E. L. and Carpenter, P. W., “Aerodynamics for Engineering Students”, Edward Arnold, a division of Hodder Headline PLC, Euston Road, London.

4. Beam, R. M. and Warming, R. P., “An Implicit factored scheme for the compressible Navier-Stokes equations”, AIAA Journal, Vol. 16, pp.393-402, 1978.

5. Hoffmann, K. A. and Chiang, S. T., “Computational Fluid Dynamic”, Volume 1, Engineering Education SystemTM, P. O. Box 20078, Wichita, KS 67208-1078, USA.

6. Baldwin, B. and Lomax, H., “Thin-layer Approximation and Algebraic model for separated Turbulent flows”, AIAA paper 78-257, 1978.