**2. Numerical methods for solving unsteady flow problems sensitive to vortex dynamics**

Numerical dissipation and dispersion are typical drawbacks of the Eulerian computational schemes (e.g., Hirsch, 2007). These drawbacks are partially overcome in the Lagrangian and mixed Eulerian-Lagrangian methods, which describe flow advection by following fluid particles, rather than by considering fixed coordinates on the Eulerian grid (e.g., Dritschel et al, 1999). A remarkable property of the Lagrangian methods is that they are exact for linear advection problems with a uniform velocity field, therefore, in principle, their accuracy is limited only by the accuracy of solving the corresponding Ordinary Differential Equations (ODEs), rather than by the accuracy of solving the full Partial Differential Equations (PDEs), which is the case for the Eulerian schemes. This class of methods can be very efficient for simulations, which involve multiple contact discontinuities, e.g., in the context of multiphase flows and strong shock waves (e.g., Margolin and Shashkov, 2004). However, for the problems where vorticity plays an important role, the standard Lagrangian-type methods have to be adjusted, after not many Lagrangian steps, by some ad-hoc 'repair' or 'contour surgery' procedure. The 'repair' procedure can be actually viewed as a special kind of numerical dissipation that is needed to stabilise the numerical solution.

For the Eulerian schemes, one of the frequently used approaches for improving the numerical dissipation and dispersion properties is based on using central schemes of highorder spatial approximation. The optimized schemes employ a non-conservative form of the governing equations, and, typically, use large computational stencils to replicate the spectral properties of the linear wave propagation in the (physical) space-time domain (Lele, 1992; Tam and Webb, 1993; Bogey and Bailly, 2004). By construction, such methods are particularly efficient in handling linear wave phenomena. The optimized finite-difference methods were developed to overcome typical problems of spectral and pseudo-spectral methods by handling non-periodic boundary conditions and large flow gradients which they handle with the use of hyper diffusion.

On the other hand, there is another popular approach, based on the conservation properties of the governing equations, that forms the basis for the so-called shock-capturing schemes. This is the family of methods based on the quasi-linear hyperbolic conservation laws (Roe, 1986; Toro, 2001; LeVeque, 2002). For improving the numerical properties in this approach, either a second-order or higher 'variable-extrapolation', or 'flux-extrapolation' techniques are used, such as in Method for Upwind Scalar Conservation Laws (MUSCL, Kolgan, 1972; B.van Leer, 1979), for enhancing linear wave properties of the solution away from the largesolution gradients discontinuities. The time stepping is usually treated separately from the spatial approximation and one popular method for time integration is multi-stage Runge-Kutta schemes (e.g, Hirsch, 2007).

To eliminate spurious oscillations of the solution obtained with the second- or higher-order schemes, in the vicinity of the discontinuities, local non-linear limiter functions are suggested, as, for example, in Totally Variation Diminishing (TVD) schemes (Boris et al., 22 Advanced Fluid Dynamics

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

**2. Numerical methods for solving unsteady flow problems sensitive to vortex** 

Numerical dissipation and dispersion are typical drawbacks of the Eulerian computational schemes (e.g., Hirsch, 2007). These drawbacks are partially overcome in the Lagrangian and mixed Eulerian-Lagrangian methods, which describe flow advection by following fluid particles, rather than by considering fixed coordinates on the Eulerian grid (e.g., Dritschel et al, 1999). A remarkable property of the Lagrangian methods is that they are exact for linear advection problems with a uniform velocity field, therefore, in principle, their accuracy is limited only by the accuracy of solving the corresponding Ordinary Differential Equations (ODEs), rather than by the accuracy of solving the full Partial Differential Equations (PDEs), which is the case for the Eulerian schemes. This class of methods can be very efficient for simulations, which involve multiple contact discontinuities, e.g., in the context of multiphase flows and strong shock waves (e.g., Margolin and Shashkov, 2004). However, for the problems where vorticity plays an important role, the standard Lagrangian-type methods have to be adjusted, after not many Lagrangian steps, by some ad-hoc 'repair' or 'contour surgery' procedure. The 'repair' procedure can be actually viewed as a special kind of

For the Eulerian schemes, one of the frequently used approaches for improving the numerical dissipation and dispersion properties is based on using central schemes of highorder spatial approximation. The optimized schemes employ a non-conservative form of the governing equations, and, typically, use large computational stencils to replicate the spectral properties of the linear wave propagation in the (physical) space-time domain (Lele, 1992; Tam and Webb, 1993; Bogey and Bailly, 2004). By construction, such methods are particularly efficient in handling linear wave phenomena. The optimized finite-difference methods were developed to overcome typical problems of spectral and pseudo-spectral methods by handling non-periodic boundary conditions and large flow gradients which

On the other hand, there is another popular approach, based on the conservation properties of the governing equations, that forms the basis for the so-called shock-capturing schemes. This is the family of methods based on the quasi-linear hyperbolic conservation laws (Roe, 1986; Toro, 2001; LeVeque, 2002). For improving the numerical properties in this approach, either a second-order or higher 'variable-extrapolation', or 'flux-extrapolation' techniques are used, such as in Method for Upwind Scalar Conservation Laws (MUSCL, Kolgan, 1972; B.van Leer, 1979), for enhancing linear wave properties of the solution away from the largesolution gradients discontinuities. The time stepping is usually treated separately from the spatial approximation and one popular method for time integration is multi-stage Runge-

To eliminate spurious oscillations of the solution obtained with the second- or higher-order schemes, in the vicinity of the discontinuities, local non-linear limiter functions are suggested, as, for example, in Totally Variation Diminishing (TVD) schemes (Boris et al.,

numerical dissipation that is needed to stabilise the numerical solution.

they handle with the use of hyper diffusion.

Kutta schemes (e.g, Hirsch, 2007).

Eulerian and Lagrangian approach are discussed.

**dynamics** 

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 the non-oscillatory methods only for strong discontinuities.

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 conventional second-order schemes.

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 grid refinement parameter, *N*=/(*k·h*) and the non-dimensional wavenumber, *k·h*, 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 pseudo-spectral optimised dispersion schemes are also shown.

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

Direct Numerical Simulations of Compressible Vortex Flow Problems 25

The analytical solution of the problem is trivial: at all time moments the solution remains equal to the initial conditions. From the viewpoint of unsteady computational schemes, however, preserving the vortex solution on a fixed Eulerian grid that is not specifically

To illustrate the point we consider numerical solutions of this problem obtained with two high-resolution Eulerian methods mentioned in the introduction. These are the Roe-MUSCL scheme with and without TVD limiter (MinMod) and the CABARET method. The former method is based on the third-order MUSCL variable extrapolation in characteristic variables and the third order Runge-Kutta scheme for in time. The latter is based on a staggered space-time stencil and is formally second order. Note that the MinMod limiter used with the Roe MUSCL scheme is more robust for vortical flow computations in comparison with more 'compressive' limiters, e.g., SuperBee, that are better tailored for 1-D shock-tube problems. This is because the former is less subjected to the 'stair-casing' artefacts in smooth solution regions (e.g., see Hirsch, 2007). The Euler equations with the initial conditions (1) are solved on several uniform Cartesian grids: (30x30), (60x60), (120x120) and (240x240) cells. These correspond to the grid density of 1.5, 3, 6 and 12 grid spacings per the vortex core radius, respectively. Figs 2 show the grid convergence of the vorticity solution obtained with the CABARET method at control time t=100. The shape and the peak of the vortex is well preserved on all grids including the coarsest one. For qualitative examination, the kinetic

> , ( ) *i i x y K t uu*

linear decay with the grid size: it is 0.011 for grid (30x30), 0.0061 for grid (60x60), and 0.003 for grid (120x120). For the Roe-MUSCL scheme, the solution of the vortex problem is much more challenging. The activation of the TVD limiter leads to a notable solution smearing, which builds up with time, and which affects even the solution on the fine grid (120x120) (fig.3a). It is, therefore, tempting to deactivate the TVD limiter since in the case considered there are no shocks involved. Without the limiter, the Roe-MUSCL scheme initially preserves the vortex shape well (as in fig.3b). However, after a few vortex circulation times, spurious oscillations that correspond to the nonphysical propagation direction of the short scales (cf. fig.1b) grow

( ) 1 ( ) / (0) *t Kt K* of this nonlinear problem at *t* =100 shows approximately a

, as a fraction of its initial value *K*(0) . The

tailored to the initial vortex shape tends to be a challenge.

until they completely contaminate the vortex solution (fig3c).

(a) (b) (c)

Fig. 2. Steady compressible Gaussian vortex in a periodic box domain: vorticity levels of the CABARET solution at time t=100 on (a) grid (30x30), (b) grid (60x60) , and (c) grid (120x120).

energy integral has been computed

relative error

CFL=0.1), the CABARET dispersion error is similar to that of the conventional fourth-order scheme. Fig1b shows that the numerical group speed of central finite-difference schemes on coarse grids is negative that leads to spurious wave reflection and sets the limit to the minimum grid resolution if numerical backscatter is to be avoided (Colonius and Lele, 2004). In comparison with the central schemes, the CABARET group speed remains in the physically correct direction for all wavenumbers, i.e., the non-physical backscatter is always absent.

Fig. 1. Linear wave properties of several spatial finite-difference schemes: (a) phase errors and (b) normalised group speeds.
