Operator-splitting schemes for the stream function-vorticity formulation

Abstract The paper deals with new implicit finite-difference schemes for solving the time-dependent incompressible Navier-Stokes equations in the stream function-vorticity formulation. New skew-symmetric second-order approximations are developed for the convective terms. The fully implicit implementation of both (no-slip and no-permeability) boundary conditions is constructed on the basis of the operator-splitting technique. For the discrete solution, there does exist an a priori estimate of the kinetic energy integral, free of any restrictions on grid parameters. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To validate the new algorithms, a lid-driven cavity flow of an incompressible fluid has been considered for a range of Reynolds numbers.

[1]  P. Roache,et al.  The bid method for the steady-state Navier-Stokes equations , 1975 .

[2]  Studies of numerical methods for the plane Navier-Stokes equations , 1981 .

[3]  P. N. Vabishchevich,et al.  Operator‐splitting methods for the incompressible Navier‐Stokes equations on non‐staggered grids. Part 1: First‐order schemes , 1995 .

[4]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[5]  A. Arakawa.  Computational design for long-term numerical integration of the equations of fluid motion: two-dimen , 1997 .

[6]  H. Wagner,et al.  A reliable solver for nonlinear biharmonic equations , 1990 .

[7]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[8]  P. Gresho.  Incompressible Fluid Dynamics: Some Fundamental Formulation Issues , 1991 .

[9]  S. Mikhlin,et al.  Variational Methods in Mathematical Physics , 1965 .

[10]  A. A. Samarskii,et al.  Numerical Methods for Grid Equations , 2018 .

[11]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[12]  H. B. Keller,et al.  Driven cavity flows by efficient numerical techniques , 1983 .

[13]  T. A. Zang,et al.  On the rotation and skew-symmetric forms for incompressible flow simulations , 1991 .

[14]  Rolf Rannacher,et al.  Numerical methods for the Navier-Stokes equations , 1994 .

[15]  T. Taylor,et al.  Computational methods for fluid flow , 1982 .

[16]  R. Mittal,et al.  Fast finite difference solution for steady-state Navier-Stokes equations using the BID method , 1987 .

[17]  V. Chudanov,et al.  A NUMERICAL STUDY ON NATURAL CONVECTION OF A HEAT-GENERATING FLUID IN RECTANGULAR ENCLOSURES , 1994 .

[18]  R. Glowinski,et al.  Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows , 1987 .

[19]  Oleg Iliev,et al.  Performance of certain iterative methods in solving implicit difference schemes for 2-D Navier-Stokes equations , 1992 .

[20]  G. Marchuk.  Methods of Numerical Mathematics , 1982 .

[21]  F. Thiele,et al.  A numerical method to solve the steady-state Navier-Stokes equations for natural convection in enclosures , 1982 .