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

Abstract

SUMMARYThe two-dimensional time-dependent Navier-Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first-order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two-dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two-dimensional flow in a cavity when a force is present, the lid-driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers.

An efficient algorithm based on the differential quadrature method for solving Navier–Stokes equations

“…Differential quadrature method has also been used in different fields of fluid mechanics . In hydraulic and free surface water flow fields, the work completed by Kaya et al to solve a flood propagation problem in open channel, Kaya and Arisoy to solve the Saint‐Venant equations for linear long wave propagation in open channels, Hashemi et al to modeling long waves in shallow water and tidal and surge using incremental DQM, and Hashemi et al to solve the Saint‐Venant equations for numerical simulation of unsteady open channel flow can be pointed out.…”

“…In solving NS equations in the form of vorticity‐stream function using DQM, the work of Shu et al and Lo et al and Bozkaya and Tezer‐Sezgin could be pointed out. They used DQ to discretize space derivatives to simulate flow in the cavity.…”