A Compact Momentum Interpolation Procedure for Unsteady Flows and Relaxation

Fully-Coupled Balanced-Force VOF Framework for Arbitrary Meshes with Least-Squares Curvature Evaluation from Volume Fractions

“…The general form of the advecting velocity u n f is derived for an equidistant Cartesian mesh using a first-order temporal discretisation. The temporal discretisation can be extended to higher-order 224 F. DENNER AND B. G. M. VAN WACHEM methods without difficulty [32]. The semidiscretized x component of the momentum equation, Eq.…”

“…Inside matrix A, A i j represents the coefficient submatrix for primitive variable j of the ith equation, with superscripts x, y, and z denoting the three momentum equations and c denoting the fourth equation derived using the momentum interpolation method presented in Section 2.2. Solution vector / is constituted by the solution subvectors of the three velocity components u, v, and w, and pressure p. Several studies have successfully applied similar fully coupled frameworks based on the momentum interpolation method [29,[31][32][33][34].…”

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

“…The approach taken by Barton et al is however so different that it cannot be considered a scheme close to PICTURE. Cubero et al [13] proposed a consistent scheme separating the temporal term from the underrelaxed one, i.e., they did not underrelax the complete A The writing of Eq. 29 is not unique and another variant can lead to a different expression for u * e , in fact the one proposed by Choi.…”

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 this approach discretized momentum equations are formulated for the convective (or face) velocities which appear in the discrete continuity equation [1]. The result of the interpolation technique is an expression for the face velocity which consists of the standard linear interpolation of the cell-centered velocities with the addition of a pressure smoothing term [8,6,9]. It can be shown, with certain assumptions, that the pressure smoothing term is a third order artificial dissipation term, which accordingly results in the introduction of a fourth order dissipation term in the continuity equation [6,9].…”