A compact monotonic discretization scheme for solving second-order vorticity-velocity equations. (English) Zbl 0997.76060
Summary: This paper presents a numerical method for solving steady-state Navier-Stokes equations for incompressible fluid flows using velocities and vorticity as variables. The method involves solving a second-order differential equation for velocity and a convection-diffusion equation for vorticity in Cartesian grids. The success of the numerical simulation depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity. In this paper, we present a monotonic advection-diffusion multi-dimensional scheme, and describe an implementation of vorticity boundary conditions. Lid-driven cavity problem and backward-facing step problem are selected for comparison and validation purposes.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
lid-driven cavity flow; second-order vorticity-velocity equations; steady-state Navier-Stokes equations; incompressible fluid; convection-diffusion equation; Cartesian grids; vorticity transport equation; monotonic advection-diffusion multi-dimensional scheme; vorticity boundary conditions; backward-facing stepReferences:
| [1] | Bernard, R. S.; Kapitza, H., How to discretize the pressure gradient for curvilinear MAC grids, J. Comput. Phys., 99, 288-298 (1992) · Zbl 0743.76053 |
| [2] | Wienan, E.; Liu, J.-G., Finite-difference methods for 3D viscous incompressible flows in the vorticity-vector potential formulation on non-staggered grids, J. Comput. Phys., 138, 57-82 (1997) · Zbl 0901.76046 |
| [3] | Clercx, H. J.H., A spectral solver for the Navier-Stokes equations in the velocity-vorticity formulation for flows with two non-periodic directions, J. Comput. Phys., 137, 186-211 (1997) · Zbl 0904.76058 |
| [4] | Quartapelle, L., Numerical solution of the incompressible Navier-Stokes equations (1993), Birkhauser: Birkhauser Basel · Zbl 0784.76020 |
| [5] | Gatski, T. B., Review of incompressible fluid flow computations using the vorticity-velocity formulation, Appl. Numer. Math., 7, 227-239 (1991) · Zbl 0714.76033 |
| [6] | Speziale, C. G., On the advantages of the vorticity-velocity formulation of the equations of fluid dynamics, J. Comput. Phys., 73, 476-480 (1987) · Zbl 0632.76049 |
| [7] | Fasel, H., Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations, J. Fluid Mech., 78, 2, 355-383 (1976) · Zbl 0404.76041 |
| [8] | Wu, J. C.; Thompson, J. F., Numerical solution of time-dependent incompressible Navier-Stokes equations using an integral-differential formulation, Comput. Fluids, 1, 197-215 (1973) · Zbl 0334.76013 |
| [9] | Daube, O., Resolution of the 2D Navier-Stokes equations in velocity– vorticity form by means of an influence matrix technique, J. Comput. Phys., 103, 402-414 (1992) · Zbl 0763.76046 |
| [10] | Giannattasio, P.; Napolitano, M., Optimal vorticity conditions for the node-centred finite-difference discretization of the second-order vorticity-velocity equations, J. Comput. Phys., 127, 208-217 (1996) · Zbl 0868.76057 |
| [11] | Bertagnolio, F.; Daube, O., Solution of the div-curl problem in generalized curvilinear coordinates, J. Comput. Phys., 138, 121-152 (1997) · Zbl 0888.65115 |
| [12] | Gatski, T. B.; Grosch, C. E.; Rose, M. E., The numerical solution of the Navier-Stokes equations for 3-dimensional unsteady incompressible flows by compact schemes, J. Comput. Phys., 82, 298-329 (1989) · Zbl 0667.76051 |
| [13] | Y. Huang, V. Ghia, G.A. Osswald, K.N. Ghia, Velocity-vorticity simulation of unsteady 3D viscous flow within a driven cavity, in: M. Deville, T.H. Le, Y. Morchoisne (Ed.), Notes on Numerical Fluid Mechanics, vol. 36, Vieweg, Braunschweig, 1992; Y. Huang, V. Ghia, G.A. Osswald, K.N. Ghia, Velocity-vorticity simulation of unsteady 3D viscous flow within a driven cavity, in: M. Deville, T.H. Le, Y. Morchoisne (Ed.), Notes on Numerical Fluid Mechanics, vol. 36, Vieweg, Braunschweig, 1992 · Zbl 0758.76041 |
| [14] | G.A. Osswald, K.N. Ghia, U. Ghia, A direct algorithm for solution of incompressible three-dimensional Navier-Stokes equations, in: AIAA Eighth Computational Fluid Dynamics Conference, Honolulu, HI, 1987, pp. 408-421; G.A. Osswald, K.N. Ghia, U. Ghia, A direct algorithm for solution of incompressible three-dimensional Navier-Stokes equations, in: AIAA Eighth Computational Fluid Dynamics Conference, Honolulu, HI, 1987, pp. 408-421 · Zbl 0683.76027 |
| [15] | Dennis, S. C.R.; Ingham, D. B.; Cook, R. N., Finite-difference methods for calculating steady incompressible flows in three dimensions, J. Comput. Phys., 33, 325-339 (1979) · Zbl 0421.76019 |
| [16] | Gatski, T. B.; Grosch, L. E.; Rose, M. E., A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity varriables, J. Comput. Phys., 48, 1-22 (1982) · Zbl 0502.76040 |
| [17] | Gatski, T. B.; Grosch, L. E.; Rose, M. E., The numerical solution of the Navier-Stokes equations for 3-dimensional unsteady incompressible flows by compact schemes, J. Comput. Phys., 82, 298-329 (1989) · Zbl 0667.76051 |
| [18] | Napolitano, M.; Catalano, L. A., A multi-grid solver for the vorticity-velocity Navier-Stokes equations, Int. Numer. Meth. Fluids, 13, 49-59 (1991) · Zbl 0724.76060 |
| [19] | Napolitano, M.; Pascazio, G., A numerical method for the vorticity-velocity Navier-Stokes equations in two and three dimensions, Comput. Fluids, 19, 3/4, 489-495 (1991) · Zbl 0733.76046 |
| [20] | Orlandi, P., Vorticity-velocity formulation for high Re flow, Comput. Fluids, 15, 2, 137-149 (1987) · Zbl 0616.76034 |
| [21] | Gui, G.; Stella, F., A vorticity-velocity method for the numerical solution of 3D incompressible flows, J. Comput. Phys., 106, 286-298 (1993) · Zbl 0770.76045 |
| [22] | Huang, H.; Li, M., Finite-difference approximation for the velocity-vorticity formulation on staggered and non-staggered grids, Comput. Fluids, 26, 1, 59-82 (1997) · Zbl 0889.76046 |
| [23] | Daube, O.; Guermond, J. C.; Sellier, A., Sur la formulation vitesse-toubillon des equations de Navier-Stokes en ecoulement incompressible, C.R. Acad. Sci. Paris, 313, serie II, 377-382 (1991) · Zbl 0735.76016 |
| [24] | Patankar, S. V., Numerical Heat Transfer and Fluid Flow (1980), Hemisphere: Hemisphere Washington DC · Zbl 0595.76001 |
| [25] | Shay, W. A., Development of a second-order approximation for the Navier-Stokes equations, Comput. Fluids, 9, 279-298 (1981) · Zbl 0453.76022 |
| [26] | T. Meis, U. Marcowitz, Numerical solution of partial differential equations, in: Applied Mathematical Science, vol. 32, Springer, Berlin, 1981; T. Meis, U. Marcowitz, Numerical solution of partial differential equations, in: Applied Mathematical Science, vol. 32, Springer, Berlin, 1981 · Zbl 0446.65045 |
| [27] | T. Ikeda, Maximal principle in finite-element models for convection-diffusion phenomena, in: Numerical and Applied Analysis, vol. 4, North-Holland, Kinokuniya, Amsterdam, Tokyo, 1983; T. Ikeda, Maximal principle in finite-element models for convection-diffusion phenomena, in: Numerical and Applied Analysis, vol. 4, North-Holland, Kinokuniya, Amsterdam, Tokyo, 1983 · Zbl 0508.65049 |
| [28] | Schneider, G. E.; Zedan, M., A modified strongly implicit procedure for numerical solution of field problem, Numer. Heat Transfer, 4, 1-19 (1981) |
| [29] | Lage, P. L.C., Application of the optimized modified strongly implicit procedure to nonlinear problems, Numer. Heat Transfer B, 30, 423-435 (1996) |
| [30] | Lage, P. L.C., Adaptive optimization of the iteration parameter in the modified strongly implicit procedure, Numer. Heat Transfer B, 30, 255-270 (1996) |
| [31] | Dennis, S. C.R.; Hudson, J. D., Compact \(h^4\) finite-difference approximation to operators of Navier-Stokes type, J. Comput. Phys., 85, 390-416 (1989) · Zbl 0681.76031 |
| [32] | Dennis, S. C.R.; Hudson, J. D., An \(h^4\) accurate vorticity-velocity formulation for calculating flow past a cylinder, Int. J. Numer. Methods Fluids, 21, 489-497 (1995) · Zbl 0846.76059 |
| [33] | M. Deville, T.H.Lê, Y. Morchoisneceds, Numerical Simulation of 3D Incompressible Unsteady Viscous Laminar Flowss, vol. 36, NNFM, 1992; M. Deville, T.H.Lê, Y. Morchoisneceds, Numerical Simulation of 3D Incompressible Unsteady Viscous Laminar Flowss, vol. 36, NNFM, 1992 |
| [34] | Ghia, U.; Ghia, K. N.; Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48, 387-411 (1982) · Zbl 0511.76031 |
| [35] | K. Morgan, J. Periaux, F. Thomasset (Eds.), Analysis of Laminar Flow over a Backward-Facing Step, A GAMM-Workshop, Friedr Viewly and Sohn, Germany, 1984; K. Morgan, J. Periaux, F. Thomasset (Eds.), Analysis of Laminar Flow over a Backward-Facing Step, A GAMM-Workshop, Friedr Viewly and Sohn, Germany, 1984 |
| [36] | Armaly, B. F.; Durst, F.; Pereira, J. C.F.; Schönung, B., Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech., 127, 473-496 (1983) |
| [37] | Gartling, D. K., A test problem for outflow boundary conditions - Flow over a backward-facing step, Int. J. Numer. Fluid, 11, 953-967 (1990) |
| [38] | G.A. Osswald, K.N. Ghia, U. Ghia, Study of incompressible separated flow using an implicit time-dependent technique, in: AIAA, Sixth CFD Conference, vol. 686, Denver, MA, 1983; G.A. Osswald, K.N. Ghia, U. Ghia, Study of incompressible separated flow using an implicit time-dependent technique, in: AIAA, Sixth CFD Conference, vol. 686, Denver, MA, 1983 · Zbl 0683.76027 |
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