Gertjan van Zwieten

Site of the project:

Habanera

Kluyverweg 6

2629 HT Delft

Hoekstede Building

The Netherlands

start of the project: September 2005

In December 2005 the Interim Thesis has been appeared.

The Master project has been finished in August 2006 by the completion of the Masters Thesis and a final presentation has been given. For working address etc. we refer to our alumnipage.

The prediction of interactions between compressible turbulent flows and flexible structures is of great importance in Aerospace Engineering. Such problems have been historically intractable, however, due to the enormous difference in scale between turbulent fluctuations and fluid-structure interactions. A promising approach is to make use of Large-eddy simulation (LES) techniques, in which only the largest scales of turbulence are simulated, while the effects of the unresolved scales are modelled. Recently, a new approach to LES, known as the variational multiscale (VMS) method, has been developed [Hughes00,01a,01b]. This approach avoids some of the inconsistencies of traditional LES techniques, and can be directly incorporated into h-p finite-element discretisations [Collis02a]. The latter are advantageous in that they can be locally adapted in heterogenous flows to minimize the total number of degrees of freedom required for accurate solutions. For problems with moving domains, space-time h-p finite-element discretisations are particularly advantageous, as they inherently incorporate mesh movement.

Unfortunately, space-time VMS discretisations of the compressible Navier Stokes equations produce non-linear systems which can be quite difficult to resolve. Firstly, space-time discretisations are necessarily implicit, and introduce twice as many unknowns when time-discontinuous formulations are used. Secondly, VMS discretisations involve multiple basis functions per element, each of which can behave quite differently. Finally, the compressible Navier-Stokes equations are themselves a coupled system of equations which introduce five unknowns per shape function. The resulting non-linear system is thus large and heterogenous.

A typical LES simulation normally requires tens of thousands of time steps to establish reliable turbulence statistics. For each time step, an outer Newton iteration loop is used to resolve the non-linear system, while an inner iteration loop is used to solve the associated sparse linear Jacobian system. For space-time VMS discretisations, the Jacobian matrix tends to be ill-conditioned, particularly at larger time steps. Numerical experimentation has shown, however, that often the Jacobian matrix can be re-used, not only for several Newton iterations, but also for several time steps. One is therefore motivated to store the Jacobian matrix to avoid re-computation, and to construct and store an effective preconditioner to minimize the number of inner iterations. For heterogenous systems, ILU preconditioners are convenient, as they can be generated automatically. Unfortunately the effectiveness of ILU preconditioning drops rapidly when increasing numbers of parallel processors are used. This currently limits application of the store-and-reuse strategy to relatively small problems.