Cell face velocity alternatives in a structured colocated grid for the unsteady Navier–Stokes equations

Abstract

The use of a colocated variable arrangement for the numerical solution of fluid flow is becoming more and more popular due to its coding simplicity. The inherent decoupling of the pressure and velocity fields in this arrangement can be handled via a special interpolation procedure for the calculation of the cell face velocity named PWIM (Pressure Weighted Interpolation Method). In this paper a discussion on the alternatives to extend PWIM to unsteady flows is presented along with a very simple criterium to ascertain if a given interpolation practice will produce steady results that are relaxation dependent or time step dependent. Following this criterium it will be shown that some prior schemes presented as time step independent are actually not, although by using special interpolations can be readily adapted to be. A systematic way of deriving different cell face velocity expressions will be presented and new formulae free of ∆t dependence will be derived. Several computational exercices will accompany the theoretical discussion to support our claims.

An Alternative Finite Volume Discretization of Body Force Field on Collocated Grid

“…The dependence on ∆t is not completely removed because ∆t is still contained in a α P . The complete removal the dependence on ∆t was proposed, for example, by Yu et al (2002) and recently by Pascau (2011). The latter work shows that this topic is still actual.…”

The Numerical Multi-Physics project (NUMPS) at VTT Technical Research Centre of Finland

“…The under-relaxation and time-step dependencies were first reported by Majumdar (1988) and Choi (1999), respectively. Recently, a well written article with a thorough and elegant derivation for the face velocity equation, which deals with both the under-relaxation and time step dependencies, was presented by Pascau (2011). In addition, the possible choices made in the interpolation of the terms in the face velocity equations and the different methods that result are well illustrated in Pascau (2011).…”

“…As a result, the form of the face velocity equations is also different. The under-relaxation and time step dependencies have been removed according to Pascau (2011).…”

Analysis and application of high order implicit Runge–Kutta schemes to collocated finite volume discretization of the incompressible Navier–Stokes equations

“…A recent review of the literature on this topic is presented in [10]. To the authors' knowledge, currently a few time-consistent interpolation schemes are available in the literature for Backward Euler (BDF1); they are the interpolation schemes proposed by Lien and Leschziner [9] or similarly Yu et al [11], Cubero and Fueyo [8], and Pascau [10]. It follows that in applying high order implicit time schemes, one of the three interpolation procedures should be adopted.…”

“…In computing the solution to the unsteady INS equations (in particular using a class of segregated solution algorithms such as SIMPLE-like and PISO-like approaches), care should be taken in selecting the interpolation method, otherwise it will result in a time-step dependent steady-state solution, temporal order reduction, and/or temporal inconsistency. A recent review of the literature on this topic is presented in [10]. To the authors' knowledge, currently a few time-consistent interpolation schemes are available in the literature for Backward Euler (BDF1); they are the interpolation schemes proposed by Lien and Leschziner [9] or similarly Yu et al [11], Cubero and Fueyo [8], and Pascau [10].…”