High resolution methods for incompressible, compressible and variable density flows.

Abstract

The uid dynamics community has dealt with a number of numerical challenges since the 1950's. These include the development of numerical methods for hyperbolic conservation laws with particular interest in capturing shock wave propagation and related phenomena, solution algorithms for the solution of the incompressible Navier-Stokes equations - a numerical challenge arises here due to the absence of the pressure term from the continuity equation - methods/ techniques for the acceleration of the numerical convergence, modelling of turbulence and grid generation techniques. Within each of those areas di erent numerical approaches have been pursued by various researchers aiming to achieve higher accuracy and ef ciency of the numerical solution. Continuous interest exists in relation to the development of accurate and e cient numerical methods for the computation of instabilities, transition and turbulence. It has been observed for more than a decade that high-resolution methods can be used in (under-resolved) turbulent ow computations without the need to resort to a turbulence model, but this approach has only recently gained some theoretical support and structural explanation for the observed results [3, 15]. Because of this there is a necessary overlap between the classical modelling of turbulence and its computation through high resolution methods [5]. These methods are currently used to simulate a broad variety of complex ows, e.g., ows that are dominated by vorticity leading to turbulence, ows featuring shock waves and turbulence, and the mixing of materials [23]. Such ows are extremely dif cult to practically obtain stably and accurately in under-resolved conditions (with respect to grid resolution) using classical linear (both second and higherorder accurate) schemes. Further, new applications at micro-scale, e.g. micro uidics, microreactors and lab-on-a-chip, have raised a number of challenges for computational science methods. In this paper we provide a brief overview of highresolution methods in connection with some of the above problems. An extensive description of these methods for incompressible and low-speed Flows can be found in [5].