Vector and scalar penalty-projection methods - for incompressible and variable density flows

This work deals with the solution of Navier-Stokes equations governing incompressible flows with variable density, e.g. [10], or dilatable flows for which the Boussinesq approximation is no more valid like low Mach number flows, e.g. [5]. In these cases, the divergence constraint makes the fully-coupled system to solve at each time step for the velocity and dynamic pressure very ill-conditioned and efficient preconditioners are required, e.g. [4]. Falling into the class of fractional-steps method, the projection methods have become very popular and useful since the pioneering papers of Chorin (1968) and Temam (1969), see the recent overview in [6] and the references therein. In the case of incompressible flows for instance, the predicted velocity is corrected through a pressure-correction step involving the solution of a Poisson equation for the incremental pressure scalar to obtain a divergence-free end-of-step velocity. Here, we present the recent penalty-projection methods, where the prediction step includes a penalty term corresponding to the lagrangian of the constraint in the same spirit as the augmented lagrangian iterative method, e.g. [9]. Then, the scalar correction step is modified consistently for the incremental pressure, see [7]. This method can be also generalized for dilatable and low Mach number flows as in [8]. We shall show through some numerical examples that these methods are really efficient, with only moderate values of the augmentation parameter, to supress the artificial pressure boundary layers obtained by usual projection methods with Dirichlet boundary conditions on the velocity. They also increase largely the convergence rate of both velocity and pressure for outflow boundary conditions, the results being slightly better for the pressure with a rotational variant, see [7]. The nice convergence properties are also proved by deriving theoretical estimates of the splitting errors for the velocity and pressure in adequate energy norms [1]. Moreover, a new class of projection methods will be proposed where a vector correction step for the velocity replaces the standard scalar one, see [2]. In this method, the constraint on the discrete divergence of the end-of-step velocity is only satisfied approximately and the augmentation parameter is increased until the resulting error is made negligible compared to the time and space discretization errors. For very large values of the augmentation parameter, our method still keeping well-posed prediction and correction steps, is shown to have a link with the so-called vector projection method first introduced in [3] which uses a singular operator for the velocity correction step. Finally, the vector penalty-projection method introduced in [2] has several nice advantages : the Dirichlet or open boundary conditions are not spoiled through a scalar pressure-correction step and it can be generalized in a natural way for variable density flows.