SUMMARYIn the context of LES of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. Designing such an algorithm, that is, as less dissipative as possible while being simple, for the resolution of variable density Navier–Stokes equations is the aim of the present work. The developed numerical scheme, based on a pressure correction technique, uses a Crank–Nicolson time discretization and a staggered space discretization relying on the Rannacher–Turek finite element. For the inertia term in the momentum balance equation, we propose a finite volume discretization, for which we derive a discrete analogue of the continuous kinetic energy local conservation identity. Contrary to what was obtained for the backward Euler discretization, the dissipation defect term associated to the Crank–Nicolson scheme is second order in time. This behavior is evidenced by numerical simulations. Copyright © 2013 John Wiley & Sons, Ltd.

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree non-conforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second-order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions. Keywords: Incompressible flows, unsteady Stokes problem, projection methods, RannacherTurek finite elements, Crouzeix-Raviart finite elements IntroductionWe consider the time-dependent incompressible Stokes equations, posed on a finite time interval (0, T ) and in an open, connected, bounded domain Ω in R d (d = 2, or 3), which is supposed to be polygonal (d = 2) or polyhedral (d = 3) for the sake of simplicity. The system under consideration reads:where u stands for the (vector-valued) velocity, p for the (scalar) pressure, and f for a (vectorvalued) regular known forcing term. The boundary Γ of Ω is supposed to be split in Γ = Γ D ∪ Γ N , and the measure of Γ D is assumed to be positive. The velocity is prescribed over Γ D while Neumann boundary conditions are imposed over Γ N :This system must be supplemented by the initial condition u = u 0 on {0} × Ω. The vector fields u D , f N and u 0 are given and supposed to be regular. We present in this paper a discretization of System (1.1) with non-conforming low-degree Rannacher-Turek [19] or Crouzeix-Raviart [6] finite elements. The time discretization is performed by an incremental projection method [5,23, 9,24]. Since the pressure is approximated by piecewise constant functions, the projection step must be left as a Darcy system. We thus choose to use a lumped discretization for the time -derivative terms, which allows us to obtain the elliptic problem for the pressure by an explicit algebraic process. Extended to variabledensity Navier-Stokes equations, this scheme is used in the open-source ISIS code [14] developed at IRSN for the computation of low-Mach-number turbulent reactive flows, and extensively used for simulation of fires.Our results are twofold. First, we are able to lay down the scheme in a variational setting, with mesh-dependent inner-products, operators and norms, which allows us to adapt for the problem at hand the error analysis performed in the semi-discrete time setting [20, 11] or for conforming elements [10]; we thus obtain, for homogeneous Dirichlet boundary conditions, a second-order estimate (with respect to the time step) for the splitting error. Second, we derive an explicit expression for the d...

No abstract