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Abstract
A three dimensional finite element method is employed for solution of the transonic full potential equation in mass conservation form. The finite element analysis allows the boundary conditions to be treated in simple and exact manner, without the use of a mapping scheme. The full potential equation is based on the assumption of irrotational flow and density of the fluid is calculated using isentropic relation and it is applicable for upstream local Mach number upto 1.6. The nonlinear full potential equation is reduced to a set of nonlinear algebraic equation using Galerkin’s formulation. An iterative scheme in conjunction with the artificial viscosity term is used to solve the set of algebraic equations. The artificial viscosity term stabilizes the numerical scheme and captures the shock very well. An appropriate grid arrangement is selected in the algorithm to maintain proper and continuous linkage between upstream and downstream of the flow at the centroid of the hexahedral element. The surface pressure distribution in the freestream Mach number range 0.80 - 0.95 and at an angle of incidence 5 degree shows a fairly good agreement with the experimental data. The present finite element discretization can be coupled with the commercially available software.
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References
- Kinney, D.J., Gloudemans, J.R. and Hafez, M.M., "The Finite Element Solution of the Full Potential Equation Over Aircraft Configurations Using Unstructured Tetrahedral and Prismatic Grids", AIAA paper 95-1765-Cp, 1995.
- Deconinck, H. and Hirch, C., "Finite Element Methods for Transonic Blade-to-Blade Calculation in Turbomachines", Journal of Engineering and Power, Vol.103, 1981, pp.667-677.
- Krupp, J.A. and Murman, E.M., "Computation of Transonic Flows Past Lifting Bodies and Slander Bodies", AIAA Journal, Vol.10, July 1972, pp. 880886.
- South, C.J. Jr. and Jameson, A., "Relaxation Solution For Inviscid Axisymmetric Transonic Flow Over Blunt or Pointed Bodies", Proceedings of the AIAA Computational Fluid Dynamics Conference, AIAA, New York, July 1973, pp. 817.
- Purvis, J.W. and Burkhater, J.E., "Prediction of Critical Mach Number for Store Configurations", AIAA Journal, Vol. 17, November 1979, pp. 1170.
- Chow, W.L. Bober, L.J. and Anderson, RH., "Numerical Calculation of Transonic Boat Tail Flow", NASA TN-D 7984, June 1984.
- Fig.7 Variation of pressure coefficient (M=0.95, α=5.0 deg) Fig.6 Variation of pressure coefficient (M=0.90, α=5.0 deg) Fig.5 Variation of pressure coefficient (M=0.80, α=5.0 deg) FEBRUARY 2007 TRANSONIC FLOW COMPUTATION OVER A HEAT SHIELD 19 7. Mehta, RC. and Jayachandran, T., "Finite Element Method Applied to Transonic Flow Over a Bulbous Payload Shroud", AIAA Journal, Vol. 27, September 1989, pp. 1298-1301.
- Chima, RV. and Gerhart, P.M., "Finite Element Analysis of Inviscid Subsonic Boat Tail Flow", AIAA Journal, Vol. 20, February 1982, pp. 190-195.
- Holst, T.L., "Transonic Flow Computations Using Nonlinear Potential Methods", Progress in Aerospace Sciences, Vol. 36, 2000, pp.1-61.
- Ecer, A. and Akay, H.U., "Finite Element Analysis of Transonic Flows in Cascades-improving Accuracy and Convergence", NASA CR-3446, July 1981.
- "Buffeting During Atmospheric Ascent", NASA Special Vehicle Design Criteria, NASA SP 8001, November 1970.
- Kobayashi, S., Oh, S-I. and Altan, T., "Metal Forming and the Finite Element Method", Oxford University Press, Inc, New York, 1989.
- Nnow, Y.W. and Bang, H., "The Finite Element Method Using MATLAB", Second Edition, CRC Press, USA, 2000.
- Hodge, B.K. and Koening, A, "A Compressible Fluid Dynamics with Personal Computer Applications", Prentice-Hall Inc., Englewood Cliffs, New Jersey, USA, 1995.
- Deconinck, H. and Hirch, C., "Transonic Flow Calculation with Finite Element", GAMM Workshop on Transonic Flow Calculations with Shock, Vieweg, 1979.
- Hafez, M.M. and Ecer, A., "Artificial Compressibility Methods for Numerical Solutions of Transonic Full Potential Equations", AlAA Journal, Vol.17, August 1979, pp. 883-884.
- Akay, H.U. and Ecer, A, "Finite Element Analysis of Transonic Flows in Highly Staggered Cascades",
- AIAA paper 81-0210, January 1981.
- Whitehead, D.S. and Newton, S.G., "Finite Element
- Method for the Solution of Two Dimensional Transonic
- Flows Cascades", International Journal of Numerical
- Methods in Fluids, Vol. 5, February 1985,
- pp. 115-132.
- Mehta, R.C., "Transonic Flow Simulation for a Bulbous
- Heat Shield", Journal of Spacecraft and Rockets,
- Vo1.34, No.4, 1997, pp. 561-564.
- Mehta, R.C., "Flow Field over Bulbous Heat Shield
- in Transonic and Low Supersonic Speeds", Journal
- of Spacecraft and Rockets, Vol. 35, No.1, January-
- February 1998, pp.102-105.
- Segerlind, L.J., "Applied Finite Element Analysis,
- First Edition, John Wiley & Sons, U.K.,1976.
- HyperMesh® Software, Version 7, Altair Engineering,
- Inc., USA.
- Ahmed, S., "Pressure Measurements in the Heat
- Shield at Transonic Speeds", National Aerospace
- Laboratories, Bangalore, India, 1984.
References
Kinney, D.J., Gloudemans, J.R. and Hafez, M.M., "The Finite Element Solution of the Full Potential Equation Over Aircraft Configurations Using Unstructured Tetrahedral and Prismatic Grids", AIAA paper 95-1765-Cp, 1995.
Deconinck, H. and Hirch, C., "Finite Element Methods for Transonic Blade-to-Blade Calculation in Turbomachines", Journal of Engineering and Power, Vol.103, 1981, pp.667-677.
Krupp, J.A. and Murman, E.M., "Computation of Transonic Flows Past Lifting Bodies and Slander Bodies", AIAA Journal, Vol.10, July 1972, pp. 880886.
South, C.J. Jr. and Jameson, A., "Relaxation Solution For Inviscid Axisymmetric Transonic Flow Over Blunt or Pointed Bodies", Proceedings of the AIAA Computational Fluid Dynamics Conference, AIAA, New York, July 1973, pp. 817.
Purvis, J.W. and Burkhater, J.E., "Prediction of Critical Mach Number for Store Configurations", AIAA Journal, Vol. 17, November 1979, pp. 1170.
Chow, W.L. Bober, L.J. and Anderson, RH., "Numerical Calculation of Transonic Boat Tail Flow", NASA TN-D 7984, June 1984.
Fig.7 Variation of pressure coefficient (M=0.95, α=5.0 deg) Fig.6 Variation of pressure coefficient (M=0.90, α=5.0 deg) Fig.5 Variation of pressure coefficient (M=0.80, α=5.0 deg) FEBRUARY 2007 TRANSONIC FLOW COMPUTATION OVER A HEAT SHIELD 19 7. Mehta, RC. and Jayachandran, T., "Finite Element Method Applied to Transonic Flow Over a Bulbous Payload Shroud", AIAA Journal, Vol. 27, September 1989, pp. 1298-1301.
Chima, RV. and Gerhart, P.M., "Finite Element Analysis of Inviscid Subsonic Boat Tail Flow", AIAA Journal, Vol. 20, February 1982, pp. 190-195.
Holst, T.L., "Transonic Flow Computations Using Nonlinear Potential Methods", Progress in Aerospace Sciences, Vol. 36, 2000, pp.1-61.
Ecer, A. and Akay, H.U., "Finite Element Analysis of Transonic Flows in Cascades-improving Accuracy and Convergence", NASA CR-3446, July 1981.
"Buffeting During Atmospheric Ascent", NASA Special Vehicle Design Criteria, NASA SP 8001, November 1970.
Kobayashi, S., Oh, S-I. and Altan, T., "Metal Forming and the Finite Element Method", Oxford University Press, Inc, New York, 1989.
Nnow, Y.W. and Bang, H., "The Finite Element Method Using MATLAB", Second Edition, CRC Press, USA, 2000.
Hodge, B.K. and Koening, A, "A Compressible Fluid Dynamics with Personal Computer Applications", Prentice-Hall Inc., Englewood Cliffs, New Jersey, USA, 1995.
Deconinck, H. and Hirch, C., "Transonic Flow Calculation with Finite Element", GAMM Workshop on Transonic Flow Calculations with Shock, Vieweg, 1979.
Hafez, M.M. and Ecer, A., "Artificial Compressibility Methods for Numerical Solutions of Transonic Full Potential Equations", AlAA Journal, Vol.17, August 1979, pp. 883-884.
Akay, H.U. and Ecer, A, "Finite Element Analysis of Transonic Flows in Highly Staggered Cascades",
AIAA paper 81-0210, January 1981.
Whitehead, D.S. and Newton, S.G., "Finite Element
Method for the Solution of Two Dimensional Transonic
Flows Cascades", International Journal of Numerical
Methods in Fluids, Vol. 5, February 1985,
pp. 115-132.
Mehta, R.C., "Transonic Flow Simulation for a Bulbous
Heat Shield", Journal of Spacecraft and Rockets,
Vo1.34, No.4, 1997, pp. 561-564.
Mehta, R.C., "Flow Field over Bulbous Heat Shield
in Transonic and Low Supersonic Speeds", Journal
of Spacecraft and Rockets, Vol. 35, No.1, January-
February 1998, pp.102-105.
Segerlind, L.J., "Applied Finite Element Analysis,
First Edition, John Wiley & Sons, U.K.,1976.
HyperMesh® Software, Version 7, Altair Engineering,
Inc., USA.
Ahmed, S., "Pressure Measurements in the Heat
Shield at Transonic Speeds", National Aerospace
Laboratories, Bangalore, India, 1984.