A New Discretization Method for the Convective Terms in the Incompressible Navier-Stokes Equations

Abstract

In this contribution we present the use of local one-dimensional boundary value problems (BVPs) to compute the interface velocities in the convective terms of the incompressible Navier-Stokes equations. This technique provides us with a better estimate for the interface velocities than linear interpolants.

Flux Approximation Scheme for the Incompressible Navier-Stokes Equations Using Local Boundary Value Problems

“…For example, the nonlinear term u 2 in f u,x is linearised as U u, where U is the approximation of the interface velocity. The details regarding the iterative computation of the interface velocities using local BVPs are given in [4,5]. Thus, for the approximation of the flux f u,x i+1,j we solve the linearised local BVP…”

A Sub-cell Discretization Method for the Convective Terms in the Incompressible Navier-Stokes Equations

“…The linearized equation is then solved iteratively, in order to account for the nonlinearity of the problem. The details for solving the linearized local BVP under the assumption that F u,y = 0 can be found in [1]. In this paper, we briefly outline the method used in [1] and then extend it by including a constant cross flux term F u,y .…”

“…The details for the computation of J(σ ) and J(1) can be found in [1]. At this point we rewrite u(σ ) as a sum of components arising from terms in the RHS of equation (7),…”

“…These velocities can also be computed using the iterative local BVP method. Further details for the iterative computation of these interface velocities can be found in [1].…”

“…Using standard methods for computing the interface velocities we tend to ignore the nature of the flow in most cases. In the sub-cell computation, we solve a reduced momentum equation locally over an inter- In [1], we presented the idea of including a piecewise linear pressure gradient in the local BVP for the computation of interface velocities. In the present paper we further extend the method, by including the cross-flux term to the righthand side (RHS) of the local BVP.…”