A bridge between projection methods and SIMPLE type methods for incompressible Navier–Stokes equations

Abstract

SUMMARYA bridge is built between projection methods and SIMPLE type methods (Semi-Implicit Method for Pressure-Linked Equation). A general second-order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non-linear update of pressure term aswhere n is a coefficient matrix, which may depend on the grid size, time step and even velocity. It includes three-and four-step projection methods. The standard SIMPLE method is written in a concise formula for steady and unsteady flows. It is proven that SIMPLE type methods have second-order temporal accuracy for unsteady flows. The classical second-order projection method and SIMPLE type methods are united within the framework of the general second-order projection formula. Two iteration algorithms of SIMPLE type methods for unsteady flows are described and discussed. In addition, detailed formulae are provided for general projection methods by using the Runge-Kutta technique to update the convective term and Crank-Nicholson scheme for the diffusion term.

Simple, coupled algorithms for solving creeping flows and their application to electro‐osmotic flows

“…K. SUH AND S. KANG All the terms within [ ] after combination vanish in the interior p-cells reproducing Equation (7) or the following in an explicit form:…”

Recent advances in computational techniques for MHD flow and application to fusion

“…The incompressible Navier-Stokes equations are discretized using a four-step projection method, which is summarized in Refs. [26,27]. A sketch of the cross-section is shown in Fig.…”

The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations

“…Secondly, the pressure variable does not appear in the continuity equation. Therefore, some additional techniques are necessary to be included to bridge velocity and pressure fields, including penalty methods [12,13], artificial compressibility methods [14,15], projection methods [16,17], SIMPLE-like methods [18,19]. Nevertheless, the employments of those artificial techniques do not totally improve the weak pressure-velocity coupling existed in these schemes.…”

“…To solve Navier-Stokes equations, many formulations are presented, including primitive variable formulations [12][13][14][15][16][17][18][19], vorticity-based formulations [20][21][22] and stream function formulations [23,24]. In general, the primitive variable (velocity-pressure) formulation is the most popular approach for simulating the flow fields.…”