The main aim of this research project is to develop high order gasdynamic solvers for simulating three-dimesional viscous flows featuring shock waves. In particular, both an unstructured and a structured solver will be described: the former is based on a recently proposed shock-fitting technique, which will be further developed in order to deal with mutually interacting shocks. The latter will combine the shock-fitting technique for unstructured meshes with some ideas coming from the embedded boundary techniques in order to overcome the constraints related to the rigid topology of the structured grids, which has prevented the development of shock-fitting structured solvers for computing complex flows.
The use of a shock-fitting approach for the numerical simulations of complex 3D viscous flows will allow to achieve more accurate solutions than those provided by the most common shock-capturing solvers used today in DNS and LES computations or in direct acoustic simulations of flows characterized by the presence of shock waves.
By the turn of the century nearly all high-speed flows were simulated using the shock-capturing approach. Despite the widespread use of shock-capturing codes and the continuous efforts made over the last 20 years, shock solutions obtained by means of shock-capturing schemes are plagued by a number of problems concerning the stability, the accuracy and the quality of the solutions that sometimes give anomalous results. Indeed, these methods are characterized by different numerical problems such as an excessive shock width, spurious oscillations arising along the captured discontinuity and the reduction of the order of accuracy within the entire shock-downstream region, to name just a few. Some researchers [1] have found that the troubles encountered with shock-capturing discretizations are intrinsic to the numerical details of the capturing process, in particular they are caused by the existence of intermediate shock points (located in between the pre- and post-shock states), the state of which is unphysical. It follows that a definite cure to the shock-capturing anomalies is unlikely to be ever found and remedies such as anisotropic mesh adaptation, beside having their own computational cost, can only alleviate some of the aforementioned drawbacks. Therefore the analysis of the limits connected to the shock capturing nature encourages the study of alternative approach which are able to overcome the aforementioned numerical problems, such as the shock-fitting technique.
The development of a shock-fitting technique for computing 3D flows with shocks on both unstructured and structured grids would make it possible to calculate higher quality solutions on grids coarser than those required for the shock capturing technique, allowing to decrease the computational cost and time necessary to perform a simulation: this is most important for viscous flows numerical simulations, which are computationally expensive.
Reliable flow field solutions on coarse grids are also required in order to improve components' design and optimization. For example, shock fitting approach may provide a precise estimation of the drag on a transonic airfoil from the computation of the entropy variation across a shock wave, because the upstream and downstream states of each shock point are given by the R-H relations. Moreover, shock-fitting could provide important results for turbulent flows: for example, the knowledge of the shock position allows to apply different numerical models for upstream and downstream turbulence regions in LES simulations. Besides that, the shock capturing solvers commonly used in DNS and LES computations must include stabilization mechanisms against numerical instability, which have a negative effect on flow physical instability in turbulent regime.
The shock fitting technique allows to reduce the presence of these tools where the shock is captured. Due to this feature, the extension of shock fitting technique to 3D viscous flows could open to various possibilities in the field of DNS and LES computations.
References
1. D. Zaide and P. Roe, 'Shock capturing anomalies and the jump conditions in one dimension' in Fluid Dynamics and Co-located Conferences, American Institute of Aeronautics and Astronautics, June 2011. Paper 2011-3686.