**1. Introduction**

20 Advanced Fluid Dynamics

Yap, C. (n.d. 1987). *Turbulent Heat and Momentum Transfer in Recirculating and Impinging* 

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

Direct Numerical Simulations of Compressible Vortex Flow Problems 23

1975). By enforcing the TVD property on the solution, the limiter functions introduce implicit numerical dissipation. If the numerical dissipation gets too strong, artificial antidiffusion terms are added to make the method less dissipative (Harten et al., 1987). The nonoscillatory methods are very attractive for computing flows with shocks. For weakly nonlinear flow problems, however, the shock-capturing TVD schemes tend to introduce too much dissipation and for vortical flows, especially in acoustics sensitive applications, the limiters are recommended to switch off (e.g., Colonius and Lele, 2004), i.e., selectively use

One notable exception is the so-called Compact Accurately Adjusting high-Resolution Technique (CABARET) (Karabasov and Goloviznin, 2009). CABARET is the extension of Upwind Leapfrog (UL) methods (Iserlis, 1986; Roe, 1998; Kim, 2004; Tran and Scheurer, 2002) to non-oscillatory conservative schemes on staggered grids with preserving low dissipative and low dispersive properties. CABARET is an explicit conservative finitedifference scheme with second-order approximation in space and time and it is found very efficient in a number of Computational Fluid Dynamics (CFD) problems, (Karabasov and Goloviznin, 2007; Karabasov et al, 2009). In comparison to many CFD methods, CABARET has a very compact stencil which for linear advection takes only one computational cell in space and time. The compactness of the computational stencil results in the ease of handling boundary conditions and the reduction of CPU cost. For non-linear flows, CABARET uses a low-dissipative conservative correction method directly based on the maximum principle. For collocated-grid schemes, the mainstream method of reducing numerical dissipation is to upgrade them to a higher order (typically, by extending its computational stencil). There is a broad range of recommendations on the subject, starting from Essentially or Weighted Essentially Non-Oscillatory schemes (ENO and WENO) (Liu et al, 1994) to Discontinuous Galerkin methods (Cockburn and Shu, 2001). All these methods show significant improvements in terms of preserving the linear flow properties, if compared with the

For illustration of numerical properties of different Eulerian schemes, Fig 1 shows the comparison of phase speed error and the non-dimensional group speed as a function of grid resolution for several semi-discrete central finite differences. E2, E4, E6 denote standard central differences of the second, fourth and sixth-order, respectively, DRP denotes the fourth order Dispersion Relation Preserving scheme by Tam and Webb; and LUI stands for the sixth order pentadiagonal compact scheme of Pade-type. CABARETx stands for the CABARET dispersion characteristic at various Courant number CFL=x. All solutions are shown as a function of the

respectively. Note that the solutions for the second-order discretization are typical of the 'loworder' shock-capturing methods, e.g., the Roe MUSCL scheme, with the limiter switched off. Higher-order central schemes of the 4th and the 6th order are analogues to the high-order shock-capturing methods, such as WENO, in the smooth solution region. The results for two

Note that the dispersion errors of semi-discrete schemes correspond to exact integration in time, which neglects the possible increase of dispersion error due to inaccuracies in time marching. For most Courant numbers and for a wide range of grid resolution (7-20 points per wavelength) the dispersion error of the CABARET scheme remains below that of the conventional and optimised fourth-order central finite differences and close to that of the six-order central schemes. Away from the optimal Courant number range (e.g., for

=/(*k·h*) and the non-dimensional wavenumber, *k·h*,

pseudo-spectral optimised dispersion schemes are also shown.

the non-oscillatory methods only for strong discontinuities.

conventional second-order schemes.

grid refinement parameter, *N*

obtained with unsteady Eulerian schemes are discussed. Part III is devoted to the sound scattering by a slowly decaying velocity field of a 2D vortex. In part IV, the canonical problem of 2D leapfrogging vortex pairs is considered and numerical solutions based on the Eulerian and Lagrangian approach are discussed.
