Citation:
D. Drikakis, M. Hahn, S. Patel and E. Shapiro. High resolution methods for
incompressible, compressible and variable density flows. ERCOFTAC Bulletin No.
62, 2004, p 35-40.
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].