Solving high Reynolds-number viscous flows by the general BEM and domain decomposition method

Abstract

SUMMARYIn this paper, the domain decomposition method (DDM) and the general boundary element method (GBEM) are applied to solve the laminar viscous ow in a driven square cavity, governed by the exact Navier-Stokes equations. The convergent numerical results at high Reynolds number Re = 7500 are obtained. We ÿnd that the DDM can considerably improve the e ciency of the GBEM, and that the combination of the domain decomposition techniques and the parallel computation can further greatly improve the e ciency of the GBEM. This veriÿes the great potential of the GBEM for strongly nonlinear problems in science and engineering.

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

“…While other linearized versions of Equation (28) exist, the use of the solution of Equation (30), albeit other factors, results in the most rapid convergence of the series solution (22). Equation (22) may readily provide the solution in terms of the nonlinear operator and initial approximation defined by Equations (28) and (29), respectively. The series still contains the auxiliary parameterh, which results in a family of solutions with varying convergence rates and regions.…”

“…The nonlinear nonlinear water waves [24], generalized Hirota-Satsuma coupled KdV equation [25], thin film flows of a fourth-grade fluid [26], and even the valuation of American put options [27], and produced highly accurate numerical and analytical solutions to the governing partial different equations. In addition, it has been implemented with other numerical techniques such as the boundary element method for the solution of nonlinear problems [28,29]. The following section provides a summary for the nonlinear shallow-water equations and the associated Riemann problem and describes the implementation of the homotopy analysis method to provide a series solution to the exact Riemann problem.…”

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

“…From these we can count the studies of Zhao and Liao [13] and Wu and Liao [14], which are general BEM solutions in a driven cavity with Re values up to 7500. A parallel computation is used in [13], whereas both parallelization and domain decomposition are used in [14] to reduce the CPU time. There are some other numerical schemes used for solving Navier-Stokes equations at high Reynolds numbers for lid-driven cavity problem and also for two-dimensional natural convection flow in a square cavity.…”

“…These obtained values of u and v will be used as constants in the solution to vorticity transport equation. (iv) Solve the vorticity transport equation (14). Since Equation (14) involves *w/*t, the vorticity is approximated by using the same coordinate function f j (x, y) as…”

“…Thus, an iterative procedure is necessary to solve them. The iterative procedure proposed here reduces Equation (14) to a set of ordinary differential equations in time and Equation (15) to a system of linear algebraic equations in each iteration.…”