## Manchester Applied Mathematics and Numerical Analysis Seminars## Spring 1999 |
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Lecture Theatre OF/B9 Oddfellows Hall (Material Science)

Disturbance development in incompressible boundary-layers can often be viewed as a quasi-parabolic process. This means that good approximations can be obtained by neglecting the upstream propagation of information. Unfortunately, the validity of any parabolic approximation cannot usually be determined prior to its application to a particular flow-configuration. Moreover, there are many situations where significant upstream influence may be anticipated on physical grounds. Thus fully elliptic numerical simulations remain necessary, even though they can incur considerable computational cost.

In order to minimise the expense of such simulations we have developed a new velocity-vorticity formulation of the Navier-Stokes equations. It will be shown that it is possible to derive a system that is comprised of only three governing equations for three unknown quantities,but which remains fully equivalent to the usual primitive variables formulation. A key feature of the new formulation is the division of the six components of the velocity and vorticity fields into two distinct sets. Only three primary variables, namely the wall-normal velocity component and the two horizontal components of the vorticity, need to be computed. The remaining three components can be identified as secondary variables, since they are defined explicitly in terms of the primary variables. A further novel feature of the formulation is that the no-slip conditions are translated into vorticity integral constraints, via the definitions of the secondary variables.

We shall illustrate the utility of the new formulation by presenting the results of numerical simulations that have been conducted for non-parabolic disturbance development in boundary-layers involving (i) steady suction or blowing slots, (ii) interactive MEMS devices, (iii) compliant surfaces, and (iv) absolute instability.

For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk)or the seminar secretary (Tel. 0161 275 5800).