Direct Numerical Simulations of Compressible Vortex Flow Problems

Vortical flows are one of the most fascinating topics in fluid mechanics. A particular difficulty of modelling such flows at high Reynolds (Re) numbers is the diversity of space and time scales that emerge as the flow develops. For compressible flows, in particular, there are additional degrees of freedom associated with the shocks and acoustic waves. The latter can have very different characteristic amplitudes and scales in comparison with the vorticity field. In case of high Re-number flows, the disparity of the scales becomes overwhelming and instead of Direct Numerical Simulations (DNS) less drastically expensive Large Eddy Simulations (LES) are used in which large flow scales are explicitly resolved on the grid and the small scales are modelled. For engineering applications, examples of unsteady vortical flows include the interaction of wakes and shocks with the boundary layer in a transonic turbine and vorticity dissipation shed due to the temporal variations in blade circulation that can have a profound loss influence and affect the overall performance of a turbomachine (e.g., Fritsch and Giles, 1992; Michelassi et al, 2003). Another example is dynamics and acoustics of high-speed jet flows that is affected by the jet inflow conditions such as the state of the boundary layer at the nozzle exit (e.g., Bogey and Bailly, 2010). The computational aspects involved in the modelling of such complex flows, typically, include the issues of high-resolution numerical schemes, boundary conditions, non-uniform grids and the choice of subgrid scale parameterization in case of LES modelling. Stepping back from this complexity to more idealised problems, two-dimensional (2D) vortex problems are a key object for testing different modelling strategies. Such reducedorder systems play an important role in the understanding of full-scale flow problems as well as in benchmarking of computational methods. One example of such important idealised systems is isolated vortices, their interaction with acoustic waves and also nonlinear dynamics when interacting with each other. In particular, such vortical systems are a classical problem in the theory of sound generation and scattering by hydrodynamic non-uniformities (e.g., Kreichnan, 1953; Howe, 1975) The structure of the chapter is the following. In part I, an outline of unsteady computational schemes for vortical flow problems is presented. In part II, the test problem of a stationary inviscid vortex in a periodic box domain is considered and a few numerical solutions