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Abstract
Eight-nodded quadrilateral finite element is used to generate structured grid for a numerical simulation. The computational domain is sub-divided into number of quadrilateral regions that display the geometry of each sub-domain. The inner and the outer boundaries of the computational domain are described in terms of the surface coordinates. An algebraic homotopy procedure is used to generate grid clustering in order to resolve boundary layer. The structured grid is linked with the flow solver based on finite volume of space discretiztion scheme with multi-stage Runge-Kutta time stepping technique. Examples are illustrated to demonstrate the grid generation procedure and data processing for a forward facing aero-disc spike attached to a hemispherical blunt body at Mach 6 and over a heat shield of a satellite launch vehicle at Mach 0.8 at an angle of attack 5 deg. The present grid generation method is convenient for checking the grid independence test by varying the stretching factor. The quadrilateral grid generated by finite element, vector plot of velocity and contour plots of the computed flow field data are easily drawn with the help of MATLAB.
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References
- Mehta, R. C., "Aerodynamic Design of Payload Fairing of Satellite Launch Vehicle", International Review of Aerospace Engineering, Vol.8, No.3, 2015, pp.167-173.
- Mehta, R. C., "High Speed Flow Field Analysis for Satellite Launch Vehicle and Reentry Capsule", Journal of Magneto-hydrodynamics, Plasma and Space Research, Vol.15, No.1, 2010, pp.1-49.
- Mehta, R. C., "Effect of Geometrical Parameters of Reentry Capsule Over Flow Field at High Speed Flow", Advances in Aircraft and Spacecraft Science, Vol.4, No.4, 2017, pp.453-467.
- Freitas, C. J., "Editorial Policy Statement on the Control of Numerical Accuracy", Journal of Fluid Engineering, Vol.115, 1993, pp.339-340.
- Thomoson, J. F., Warsi, Z. U. A. and Mastin, C. W., "Numerical Grid Generation - Foundation and Applications", North-Holland, Amsterdam, 1985.
- Ushakova, O. V., (Editor), "Advances in Grid Generation", Nova Science Publishers, New York, USA, 2007.
- Thomson, J. F., "Grid Generation Technique in Computational Fluid Dynamics", AIAA Journal, Vol.22, No.11, 1984, pp.1505-1523.
- Armstrong, C. G., Special issue: "Automatic mesh generation", Advances in Engineering Software, Vol.13, 1991, pp.217-337.
- George, P. L., "Automatic Generation of Meshes", Wiley, New York, 1991.
- Piegel, L. G., "On NURBS: A Survey", IEEE Computer Graphics and Applications, Vol.11, No.1, 1991, pp.57-71.
- Farin, G.L., "NURBS Curves and Surfaces from Projective Geometry to Practical Use", First Edition, A. K. Peters Ltd., 1995.
- Thomas, P. D. and Middlecoff, J. F., "Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations", AIAA Journal, Vol.18, No.6, 1980, pp.652-656.
- Caruso, S.S., "Development of an Unstructured mesh/Navier-Stokes Method for Aerodynamics of Aircraft with Ice Accretions", AIAA 90-0758, 1990.
- Zienkiewicz, O. C. and Philips, D. C., "An automatic Mesh Generation Scheme for Plane and Curved Surfaces by Isoparametric Coordinates", International Journal of Numerical Methods in Engineering, Vol.3, 1971, pp.519-528.
- Durocher, L. L. and Gasper, A. A., "A Versatile Two-dimensional Mesh Generator with Automatic Bandwidth Reduction", Computers and Structures, Vol.10, 1979, pp.561-575.
- Segerland, L. J., Applied Finite Element Analysis, First Edition, John Wiley and Sons, 1976.
- Ho, P. T. S., "MSHGEN - A FORTRAN Program on Automatic Two-dimensional and Three-dimensional Triangular Mesh Generation", Advances in Engineering Software, Vol.14, 1992, pp.61-75.
- Suhara, J. and Fukuda, J., "Automatic Mesh Generation for Finite Element Analysis", Advances in Computational Methods in Structural Mechanics and Design, University of Alabama Press, 1974, pp.607- 624.
- Bykat, A., "Automatic Generation of Triangular Grids, I - Subdivision of a General Polygon into Convex Subregions, II - Triangulation of Convex Polygon", International Journal of Numerical Methods in Engineering, Vol.10, 1976, pp.1329-1342.
- Imafuku, I. and Kodera, Y., "A Generalized Automatic Mesh Generation Scheme for Finite Element Methods", International Journal of Numerical Methods in Engineering, Vol.15, 1980, pp.713-731.
- Ecer, A., Spyropoulos, J. and Maul, J. D., "A Threedimensional, Block-structured Finite Element Grid Generation Schemes", AIAA Journal, Vol.23, 1985, pp.1483-1490.
- Tu, J., Yeoh, G. H. and Liu, C., "Computational Fluid Dynamics: A Practical Approach", Butterworth-Heinemann, New Delhi, India, 2008.
- Wilson, H. B., Turcotte, L. H. and Halpern, D., "Advanced Mathematics and Mechanics Applications Using MATLAB", Third Edition, Chapman and Hall/CRC Press, USA, 2003.
- Shang J. S., "Numerical Simulation of Wing-fuselage Aerodynamic Interference", AIAA Journal, Vol.22, No.10, 1984, pp.1345-1353.
- Eisemen, P. R. and Smith, R. E., "Mesh Generation Using Algebraic Techniques", NASA CP 2166, 1980, pp.73-120.
- Moitra, A., "An Algebraic Homotopic Method for Generating Quasi-three-dimensional Grids for High Speed Configurations", NASA CR 4242, 1989.
- Zienkiewicz, O. C. and Morgan, K., "Finite Elements and Approximation", John Wiley and Sons, Inc., New York, 1982.
- Peyret, R. and Vivind, H., "Computational Methods for Fluid Flows", Springer-Verlag, Berlin, 1993, pp.109-111.
- Jameson, A., Schmidt, W. and Turkel, E., "Numerical Simulation of Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes", AIAA Paper 81-1259, 1981.
- Mehta, R. C., "Pressure Oscillations Over a Spiked Blunt Body at Hypersonic Mach Number", Computational Fluid Dynamics Journal, Vol.9, No.2, 2000, pp.88-95.
- Mehta, R. C., "Flow Direction Estimation Based on Computed and Flight Measured Surface Pressure", Computational Fluid Dynamics Journal, Vol.12, No.3, 2003, pp.555-561.
- Mehta, R. C., "Computation of Aerodynamic load on Protuberance Over Satellite Launch Vehicle at Supersonic Speed", Sch. J. of Eng. and Technology, Vol.4, No.7, 2016, pp.301-307.
References
Mehta, R. C., "Aerodynamic Design of Payload Fairing of Satellite Launch Vehicle", International Review of Aerospace Engineering, Vol.8, No.3, 2015, pp.167-173.
Mehta, R. C., "High Speed Flow Field Analysis for Satellite Launch Vehicle and Reentry Capsule", Journal of Magneto-hydrodynamics, Plasma and Space Research, Vol.15, No.1, 2010, pp.1-49.
Mehta, R. C., "Effect of Geometrical Parameters of Reentry Capsule Over Flow Field at High Speed Flow", Advances in Aircraft and Spacecraft Science, Vol.4, No.4, 2017, pp.453-467.
Freitas, C. J., "Editorial Policy Statement on the Control of Numerical Accuracy", Journal of Fluid Engineering, Vol.115, 1993, pp.339-340.
Thomoson, J. F., Warsi, Z. U. A. and Mastin, C. W., "Numerical Grid Generation - Foundation and Applications", North-Holland, Amsterdam, 1985.
Ushakova, O. V., (Editor), "Advances in Grid Generation", Nova Science Publishers, New York, USA, 2007.
Thomson, J. F., "Grid Generation Technique in Computational Fluid Dynamics", AIAA Journal, Vol.22, No.11, 1984, pp.1505-1523.
Armstrong, C. G., Special issue: "Automatic mesh generation", Advances in Engineering Software, Vol.13, 1991, pp.217-337.
George, P. L., "Automatic Generation of Meshes", Wiley, New York, 1991.
Piegel, L. G., "On NURBS: A Survey", IEEE Computer Graphics and Applications, Vol.11, No.1, 1991, pp.57-71.
Farin, G.L., "NURBS Curves and Surfaces from Projective Geometry to Practical Use", First Edition, A. K. Peters Ltd., 1995.
Thomas, P. D. and Middlecoff, J. F., "Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations", AIAA Journal, Vol.18, No.6, 1980, pp.652-656.
Caruso, S.S., "Development of an Unstructured mesh/Navier-Stokes Method for Aerodynamics of Aircraft with Ice Accretions", AIAA 90-0758, 1990.
Zienkiewicz, O. C. and Philips, D. C., "An automatic Mesh Generation Scheme for Plane and Curved Surfaces by Isoparametric Coordinates", International Journal of Numerical Methods in Engineering, Vol.3, 1971, pp.519-528.
Durocher, L. L. and Gasper, A. A., "A Versatile Two-dimensional Mesh Generator with Automatic Bandwidth Reduction", Computers and Structures, Vol.10, 1979, pp.561-575.
Segerland, L. J., Applied Finite Element Analysis, First Edition, John Wiley and Sons, 1976.
Ho, P. T. S., "MSHGEN - A FORTRAN Program on Automatic Two-dimensional and Three-dimensional Triangular Mesh Generation", Advances in Engineering Software, Vol.14, 1992, pp.61-75.
Suhara, J. and Fukuda, J., "Automatic Mesh Generation for Finite Element Analysis", Advances in Computational Methods in Structural Mechanics and Design, University of Alabama Press, 1974, pp.607- 624.
Bykat, A., "Automatic Generation of Triangular Grids, I - Subdivision of a General Polygon into Convex Subregions, II - Triangulation of Convex Polygon", International Journal of Numerical Methods in Engineering, Vol.10, 1976, pp.1329-1342.
Imafuku, I. and Kodera, Y., "A Generalized Automatic Mesh Generation Scheme for Finite Element Methods", International Journal of Numerical Methods in Engineering, Vol.15, 1980, pp.713-731.
Ecer, A., Spyropoulos, J. and Maul, J. D., "A Threedimensional, Block-structured Finite Element Grid Generation Schemes", AIAA Journal, Vol.23, 1985, pp.1483-1490.
Tu, J., Yeoh, G. H. and Liu, C., "Computational Fluid Dynamics: A Practical Approach", Butterworth-Heinemann, New Delhi, India, 2008.
Wilson, H. B., Turcotte, L. H. and Halpern, D., "Advanced Mathematics and Mechanics Applications Using MATLAB", Third Edition, Chapman and Hall/CRC Press, USA, 2003.
Shang J. S., "Numerical Simulation of Wing-fuselage Aerodynamic Interference", AIAA Journal, Vol.22, No.10, 1984, pp.1345-1353.
Eisemen, P. R. and Smith, R. E., "Mesh Generation Using Algebraic Techniques", NASA CP 2166, 1980, pp.73-120.
Moitra, A., "An Algebraic Homotopic Method for Generating Quasi-three-dimensional Grids for High Speed Configurations", NASA CR 4242, 1989.
Zienkiewicz, O. C. and Morgan, K., "Finite Elements and Approximation", John Wiley and Sons, Inc., New York, 1982.
Peyret, R. and Vivind, H., "Computational Methods for Fluid Flows", Springer-Verlag, Berlin, 1993, pp.109-111.
Jameson, A., Schmidt, W. and Turkel, E., "Numerical Simulation of Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes", AIAA Paper 81-1259, 1981.
Mehta, R. C., "Pressure Oscillations Over a Spiked Blunt Body at Hypersonic Mach Number", Computational Fluid Dynamics Journal, Vol.9, No.2, 2000, pp.88-95.
Mehta, R. C., "Flow Direction Estimation Based on Computed and Flight Measured Surface Pressure", Computational Fluid Dynamics Journal, Vol.12, No.3, 2003, pp.555-561.
Mehta, R. C., "Computation of Aerodynamic load on Protuberance Over Satellite Launch Vehicle at Supersonic Speed", Sch. J. of Eng. and Technology, Vol.4, No.7, 2016, pp.301-307.