The usage of Discontinuous Galerkin spectral techniques in the numerical simulation of complex problems in fluid dynamics

Abstract

The main objective of the present project is the numerical simulation of complex problems in fluid dynamics through the usage of high-order spectral techniques. The approximation approach is known in the literature as the Discontinuous Galerkin (DG) method. This is, in a way, a special application of the spectral tools for the approximate solution of differential equations. The DG numerical scheme borrows some good ideas from the finite volume and finite element approximation methods, with the advantage of not carrying together some inherent difficulties of those classical techniques. Therefore, the DG offers the user a series of definite advantages, most of which will be discussed in the following. One of those advantages deserves to be mentioned and is related to the high-order that the method is able to reach. Under certain conditions the precision is exponential, and the exponent is proportional to the highest degree of the polynomial basis functions. To improve precision for classical methods one has to resort to grid refinement, what corresponds to algebraic convergence and in consequence to a much lower convergence speed.Especially for this reason, the DG approach has spreaded widely in the fluid-dynamical numerical simulation community. In the present project the following problems will be tackled: (i) Shock tube, to be simulated via one- and two-dimensional approaches; (ii) Supersonic quasi-one-dimensional nozzle; (iii) Supersonic two-dimensional nozzle; (iv) Supersonic channel flow with forward-facing step; (v) Supersonic flow about a diamond profile which is positioned between parallel walls; (vi) Subsonic and supersonic flow about a circular cylinder; (vii) Transonic flow about airfoils; (viii) Interaction between a shock wave and a boundary layer. The main drive behind the choosing of these problems is the fact that they contain the majority of the most difficult situations that the analyst normally encounters in physical problems. Namely, (a) Shock waves, expansion waves and contact surfaces, both stationary and in movement; (b) Reflections of shock and expansion waves from solid walls; (c) Crossing of system of waves of the type shock/shock. shock/expansion, expansion/expansion; (d) Formation of slip-streams after some kind of mechanism of entropy unbalancing; (e) Geometrical discontinuities, as for examples sharp corners; (f) Crossing of the exit plane by systems of waves; (g) Shock wave/Boundary layer interaction. Most of these cases are used by the computational fluid dynamics community due to the stringent challenges that they pose to a new numerical code. The mathematical model to be used is represented by the Navier-Stokes equations, the grids will be, without exemption, of the nonstructured type, and the code will be parallelized according to the OpenMP/MPI ("Open MultiProcessing", "Message Passing Interface") technique. The only financing item as requested in this project corresponds to the purchasing of a clustered machine, without wich the practical usage of a DG code type is not viable. The team involved in this effort is composed also of under-graduate, master of science, and PhD students; this attends the always necessary goal of professional formation and development. (AU)