**3. Steady vortex solution in a finite domain**

Let's first consider a steady problem of isolated compressible Gaussian vortex in a square periodic domain. The vortex is specified in the centre of the box domain, as a perturbation to a uniform background flow with zero mean velocity (*ρ∞,***u∞**,*p∞*)=(1,0,1):

$$\begin{split} \rho^\* = \rho\_\* \left[ \left( 1 - \frac{(\gamma - 1)}{4 \alpha \gamma} a^2 \exp\{2\alpha(1 - \tau^2)\} \right)^{\frac{1}{\gamma - 1}} - 1 \right], u^\* = \sigma \tau \exp\{\alpha(1 - \tau^2)\} \sin \theta, \\\ v^\* = -\sigma \tau \exp\{\alpha(1 - \tau^2)\} \cos \theta, p^\* = p\_\alpha \left[ \left( 1 - \frac{(\gamma - 1)}{4 \alpha \gamma} a^2 \exp\{2\alpha(1 - \tau^2)\} \right)^{\frac{\gamma}{\gamma - 1}} - 1 \right], \\\ \tau = r \,/\ L; r = \sqrt{(\mathbf{x} - \mathbf{x}\_o)^2 + (\mathbf{y} - \mathbf{y}\_o)^2}; \theta = \tan^{-1}((\mathbf{y} - \mathbf{y}\_o) \, (\mathbf{x} - \mathbf{x}\_o)), \sigma = 0.3, L = 0.05, \alpha = 0.204. \end{split} \tag{1}$$

To simplify the treatment of external boundary conditions, the box size is set 20 times as large as the vortex radius, *L* so that the vortex induced velocity vanishes at the boundaries. The vortex field corresponds to a steady rotation that is a stable solution of the governing compressible Euler equations (e.g., Colonius at al, 1994). The characteristic space scale of the problem is the vortex core radius *L*. It is also useful to introduce the time scale based on the vortex circulation time*T L* 2 / 1.047 .

24 Advanced Fluid Dynamics

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.

> -4 -3 -2 -1 0 1 2 3 4

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

Let's first consider a steady problem of isolated compressible Gaussian vortex in a square periodic domain. The vortex is specified in the centre of the box domain, as a perturbation to

<sup>1</sup> <sup>2</sup> 2 2

0.3, 0.05, 0.204. *L*

To simplify the treatment of external boundary conditions, the box size is set 20 times as large as the vortex radius, *L* so that the vortex induced velocity vanishes at the boundaries. The vortex field corresponds to a steady rotation that is a stable solution of the governing compressible Euler equations (e.g., Colonius at al, 1994). The characteristic space scale of the problem is the vortex core radius *L*. It is also useful to introduce the time scale based on the

 *u*

 

 

 

1 <sup>1</sup> 2 2 <sup>2</sup>

( 1) ' 1 exp{2 (1 )} 1 , ' exp{ (1 )}sin , <sup>4</sup>

0 0 0 0

( 1) ' exp{ (1 )}cos , ' 1 exp{2 (1 )} 1 , <sup>4</sup>

**cg**

0.E+00 2.E-01 4.E-01 6.E-01 8.E-01 1.E+00 **kh/**

 

(1)

DRP E6 E4 E2 LUI CABARET0.1 CABARET0.4 CABARET0.6 CABARET0.9

DRP E6 E4 E2 LUI CABARET0.1 CABARET0.4 CABARET0.6 CABARET0.9 CABARET0.51

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

1.E+00 1.E+01 1.E+02 **N**

**3. Steady vortex solution in a finite domain** 

*v p p*

 

 

vortex circulation time*T L* 2 / 1.047

a uniform background flow with zero mean velocity (*ρ∞,***u∞**,*p∞*)=(1,0,1):

 

 

> 

 .

2 21

*r Lr x x y y y y x x*

/ ; ( ) ( ) ; tan (( ) /( )),

and (b) normalised group speeds.

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

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 energy integral has been computed , ( ) *i i x y K t uu* , as a fraction of its initial value *K*(0) . The

relative error ( ) 1 ( ) / (0) *t Kt K* of this nonlinear problem at *t* =100 shows approximately a 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 until they completely contaminate the vortex solution (fig3c).

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).

Direct Numerical Simulations of Compressible Vortex Flow Problems 27

space without any hydrodynamic perturbation, ( ) *<sup>v</sup> p* is the steady solution vortex without any incoming acoustic wave, and *p* is the pressure at the far field. Note, that from the numerical implementation viewpoint it is preferable to compute the scattered solution in

Colonius et al (1994) obtain the benchmark solution to this problem by using the 6-th order Pade-type compact finite-difference scheme in space and 6-th order Runge-Kutta integration in time with the grid density of 7-8 grid points per vortex radius. The reference solution corresponds to the Navier-Stokes equations at Reynolds number 105 integrated over four acoustic wave time periods in the open computational domain with well-tailored numerical

It is interesting to compare the reference solution with the results obtained with the CABARET scheme and the third-order Roe-MUSCL-Runge-Kutta method from the previous section. To reduce the numerical dissipation error of the latter, the MinMod limiter has been deactivated. For CABARET, the complete formulation including the nonlinear flux correction is used. For the sake of comparison, the vortex with core Mach number max *M* 0.25 is considered. Characteristic-type nonreflecting boundary conditions and grid

Fig.4 shows the computational problem configuration and the distribution of the rootmeans-square (r.m.s.) of the scattered pressure fluctuations for the CABARET solution,

(a) (b)

The main emphasis of this subsection is the effect of non-uniform hydrodynamic flow on sound scattering, hence, the numerical solutions for the scattered pressure field intensity

Fig. 4. Sound wave scattering by a non-zero circulation vortex of M=0.25: (a) problem configuration, (b) computed r.m.s of the scattered pressure field of the CABARET solution

on coarse grid of 2.5 cells per vortex core radius.

*p p p* in order to account for a small systematic

form (5) instead of using 0 ( ) *<sup>t</sup>*

approximation error of the round vortex on a rectilinear Cartesian grid.

boundary conditions to minimise numerical reflections from the boundaries.

stretching close to the open boundaries are used to minimise artificial reflections.

where the vortex centre corresponds to the origin of the system of coordinates.

Fig. 3. Steady compressible Gaussian vortex in a periodic box domain: vorticity levels of the Roe-MUSCL solution on grid (240x240) cells with (a) MinMod limiter at time t=100, (b) MinMod limiter at time t=4, and (c) with the limiter deactivated at time t=5.
