A Conservative Pressure-Correction Method on Collocated Grid for Low Mach Number Flows

Affiliation(s)

Department of Applied Mechanics, Indian Institute of Technology, Delhi, India.

Department of Mechanical Engineering, Zakir Husain College of Engineering & Technology, Aligarh Muslim University, Aligarh, India.

Department of Applied Mechanics, Indian Institute of Technology, Delhi, India.

Department of Mechanical Engineering, Zakir Husain College of Engineering & Technology, Aligarh Muslim University, Aligarh, India.

ABSTRACT

A novel extension to SMAC scheme is proposed for variable density flows under low Mach number approximation. The algorithm is based on a predictor—corrector time integration scheme that employs a projection method for the momentum equation. A constant-coefficient Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each time step. The proposed algorithm has second order centrally differenced convective fluxes with upwinding based on Cell Peclet number while diffusive flux are viscous fourth order accurate. Spatial discretization is performed on a collocated grid system that offers computational simplicity and straight forward extension to curvilinear coordinate systems. The algorithm is kinetic energy preserving. Further in this paper robustness and accuracy are demonstrated by performing test on channel flow with non-Boussinesq condition on different temperature ratios.

A novel extension to SMAC scheme is proposed for variable density flows under low Mach number approximation. The algorithm is based on a predictor—corrector time integration scheme that employs a projection method for the momentum equation. A constant-coefficient Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each time step. The proposed algorithm has second order centrally differenced convective fluxes with upwinding based on Cell Peclet number while diffusive flux are viscous fourth order accurate. Spatial discretization is performed on a collocated grid system that offers computational simplicity and straight forward extension to curvilinear coordinate systems. The algorithm is kinetic energy preserving. Further in this paper robustness and accuracy are demonstrated by performing test on channel flow with non-Boussinesq condition on different temperature ratios.

Cite this paper

nullS. Yahya, S. Anwer and S. Sanghi, "A Conservative Pressure-Correction Method on Collocated Grid for Low Mach Number Flows,"*World Journal of Mechanics*, Vol. 2 No. 5, 2012, pp. 253-261. doi: 10.4236/wjm.2012.25031.

nullS. Yahya, S. Anwer and S. Sanghi, "A Conservative Pressure-Correction Method on Collocated Grid for Low Mach Number Flows,"

References

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[3] O. V. Vasilyev, “High Order Finite Difference Schemes on Non-Uniform Meshes with Good Conservative Properties,” Journal of Computational Physics, Vol. 157, No. 2, 2000, pp. 746-761. doi:10.1006/jcph.1999.6398

[4] J. Gullbrand, “An Evaluation of a Conservative Fourth Order DNS Code in Turbulent Channel Flow,” Center for Turbulence Research Annual Research Briefs, NASA Ames/Stanford University, 2000, pp. 211-218.

[5] Y. Morinishi, T. S. Lund, O. V. Vasilyev, P. Moin, “Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow,” Journal of Computational Physics, Vol. 143, No. 1, 1998, pp. 90-124. doi:10.1006/jcph.1998.5962

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[7] B. Lessani, J. Ramboer, C. Lacor, “Efficient Large-Eddy Simulations of Low Mach Number Flows Using Preconditioning and Multigrid,” International Journal of Computational Fluid Dynamics, Vol. 18, No. 3, 2004, pp. 221-223. doi:10.1080/10618560310001654319

[8] A. Majda and J. Sethian, “The Derivation and Numerical Solution of the Equations for Zero Mach Number Combustion,” Combustion Science and Technology, Vol. 42, No. 3-4, 1985, pp. 185-205. doi:10.1080/00102208508960376

[9] R. G. Rehm and H. R. Baum, “The Equation of Motion for Thermally Driven Buoyant Flows,” Journal of Research of the National Bureau of Standards, Vol. 83, No. 3, 1978, pp. 297-308.

[10] B. Müller, “Low-Mach Number Asymptotics of the Navier–Stokes Equations,” Journal of Engineering Mathematics, Vol. 34, No. 1-2, 1998, pp. 97-109. doi:10.1023/A:1004349817404

[11] S. Paolucci. “Filtering of Sound from the Navier-Stokes Equations,” Sandia National Laboratories, Livermore, 1982.

[12] F. Nicoud and F. Ducros, “Subgrid-Scale Stress Modeling Based on the Square of Velocity Gradient Tensor,” Flow Turbulence and Combustion, Vol. 62, No. 3, 1999, pp. 183-200. doi:10.1023/A:1009995426001

[13] M. Germano, U. Piomelli, P. Moin and W. H. Cabot, “A Dynamic Subgrid-Scale Eddy Viscosity Model,” Physics of Fluids A 3, Vol. 3, No. 7, 1991, pp. 1760-1765. doi:10.1063/1.857955

[14] P. Moin, K. Squires, W. Cabot and S. Lee, “A Dynamic Subgrid-Scale Model for Compressible Turbulence and Scalar Transport,” Physics of Fluids, Vol. 3, No. 11, 1991. pp. 2746-2757. doi:10.1063/1.858164

[15] C. Hirsch, “Numerical Computation of Internal and External Flows,” John Wiley & Sons Inc., Hoboken, 1990.

[16] A. A. Amsden and F. H. Harlow, “The SMAC Method: A Numerical Technique for Calculating Incompressible Fluid Flows,” Los Alamos Scientific Laboratory of the University of California, Los Alamos, 1970.

[17] L. Cheng, and S. Armfield, “A Simplified Marker and Cell Method for Unsteady Flows on Non-Staggered Grids,” International Journal for Numerical Methods in Fluids, Vol. 21, No. 1, 1995, pp. 15-34. doi:10.1002/fld.1650210103

[18] A. Jameson, “Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows Past Airfoils and Wings,” Proceedings of 10th Computational Fluid Dynamics Conference, Honolulu, 24-26 June 1991, pp. 91-1596.

[19] J. Kim, P. Moin and R. Moser, “Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number,” Journal of Fluid Mechanics, Vol. 177, 1987, pp. 133-166. doi:10.1017/S0022112087000892

[20] B. Lessani and M. V. Papalexandris, “Time-Accurate Calculation of Variable Density Flows with Strong Temperature Gradients and Combustion,” Journal of Computational Physics, Vol. 212, No. 1, 2006, pp. 218-246. doi:10.1016/j.jcp.2005.07.001

[21] F. C. Nicoud, “Numerical Study of a Channel Flow with Variable Properties,” Center for Turbulence Research Annual Research Briefs, 1998, pp. 289-310.

[1] A. J. Chorin, “On the Convergence of Discrete Approximations to the Navier-Stokes Equations,” Mathematics of Computations, Vol. 23, No. 106, 1969, pp. 342-353. doi:10.1090/S0025-5718-1969-0242393-5

[2] A. J. Chorin, “Numerical Solution of the Navier-Stokes Equations,” Mathematics of Computations, Vol. 22, No. 104, 1968, pp. 745-762. doi:10.1090/S0025-5718-1968-0242392-2

[3] O. V. Vasilyev, “High Order Finite Difference Schemes on Non-Uniform Meshes with Good Conservative Properties,” Journal of Computational Physics, Vol. 157, No. 2, 2000, pp. 746-761. doi:10.1006/jcph.1999.6398

[4] J. Gullbrand, “An Evaluation of a Conservative Fourth Order DNS Code in Turbulent Channel Flow,” Center for Turbulence Research Annual Research Briefs, NASA Ames/Stanford University, 2000, pp. 211-218.

[5] Y. Morinishi, T. S. Lund, O. V. Vasilyev, P. Moin, “Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow,” Journal of Computational Physics, Vol. 143, No. 1, 1998, pp. 90-124. doi:10.1006/jcph.1998.5962

[6] F. Nicoud, “Conservative High-Order Finite-Difference Schemes for Low-Mach Number Flows,” Journal of Computational Physics, Vol. 158, No. 1, 2000, pp. 71-97. doi:10.1006/jcph.1999.6408

[7] B. Lessani, J. Ramboer, C. Lacor, “Efficient Large-Eddy Simulations of Low Mach Number Flows Using Preconditioning and Multigrid,” International Journal of Computational Fluid Dynamics, Vol. 18, No. 3, 2004, pp. 221-223. doi:10.1080/10618560310001654319

[8] A. Majda and J. Sethian, “The Derivation and Numerical Solution of the Equations for Zero Mach Number Combustion,” Combustion Science and Technology, Vol. 42, No. 3-4, 1985, pp. 185-205. doi:10.1080/00102208508960376

[9] R. G. Rehm and H. R. Baum, “The Equation of Motion for Thermally Driven Buoyant Flows,” Journal of Research of the National Bureau of Standards, Vol. 83, No. 3, 1978, pp. 297-308.

[10] B. Müller, “Low-Mach Number Asymptotics of the Navier–Stokes Equations,” Journal of Engineering Mathematics, Vol. 34, No. 1-2, 1998, pp. 97-109. doi:10.1023/A:1004349817404

[11] S. Paolucci. “Filtering of Sound from the Navier-Stokes Equations,” Sandia National Laboratories, Livermore, 1982.

[12] F. Nicoud and F. Ducros, “Subgrid-Scale Stress Modeling Based on the Square of Velocity Gradient Tensor,” Flow Turbulence and Combustion, Vol. 62, No. 3, 1999, pp. 183-200. doi:10.1023/A:1009995426001

[13] M. Germano, U. Piomelli, P. Moin and W. H. Cabot, “A Dynamic Subgrid-Scale Eddy Viscosity Model,” Physics of Fluids A 3, Vol. 3, No. 7, 1991, pp. 1760-1765. doi:10.1063/1.857955

[14] P. Moin, K. Squires, W. Cabot and S. Lee, “A Dynamic Subgrid-Scale Model for Compressible Turbulence and Scalar Transport,” Physics of Fluids, Vol. 3, No. 11, 1991. pp. 2746-2757. doi:10.1063/1.858164

[15] C. Hirsch, “Numerical Computation of Internal and External Flows,” John Wiley & Sons Inc., Hoboken, 1990.

[16] A. A. Amsden and F. H. Harlow, “The SMAC Method: A Numerical Technique for Calculating Incompressible Fluid Flows,” Los Alamos Scientific Laboratory of the University of California, Los Alamos, 1970.

[17] L. Cheng, and S. Armfield, “A Simplified Marker and Cell Method for Unsteady Flows on Non-Staggered Grids,” International Journal for Numerical Methods in Fluids, Vol. 21, No. 1, 1995, pp. 15-34. doi:10.1002/fld.1650210103

[18] A. Jameson, “Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows Past Airfoils and Wings,” Proceedings of 10th Computational Fluid Dynamics Conference, Honolulu, 24-26 June 1991, pp. 91-1596.

[19] J. Kim, P. Moin and R. Moser, “Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number,” Journal of Fluid Mechanics, Vol. 177, 1987, pp. 133-166. doi:10.1017/S0022112087000892

[20] B. Lessani and M. V. Papalexandris, “Time-Accurate Calculation of Variable Density Flows with Strong Temperature Gradients and Combustion,” Journal of Computational Physics, Vol. 212, No. 1, 2006, pp. 218-246. doi:10.1016/j.jcp.2005.07.001

[21] F. C. Nicoud, “Numerical Study of a Channel Flow with Variable Properties,” Center for Turbulence Research Annual Research Briefs, 1998, pp. 289-310.