**4. Sound scattering by a steady vortex**

 We next consider an isolated viscous-core vortex from Colonius at al (1994) that corresponds to a slowly decaying velocity field of constant circulation 0

$$\begin{aligned} \upsilon\_r &= 0\\ \upsilon\_\theta &= \frac{\Gamma\_\alpha}{2\pi r} (1 - \exp(-\alpha(r/L)^2))\\ \alpha &= 1.256431, \,\mathrm{M}\_{\mathrm{max}} = \left(\upsilon\_\theta\right)\_{\mathrm{max}} / c\_\circ \end{aligned} \tag{2}$$

where *c* is acoustic far-field sound and *L* is the vortex core radius. The density and pressure field satisfy the usual isentropic relation and the steady tangential momentum equation:

$$\begin{aligned} \rho \cdot \rho^{-\gamma} &= \text{const};\\ \frac{\delta p}{\delta r} &= \rho \frac{v\_{\theta}^{2}}{r} \end{aligned} \tag{3}$$

The vortex is specified in the centre of an open square Cartesian domain, three sides of which are open boundaries and the forth one corresponds to an incident acoustic wave that is monochromatic with frequency *f*. and normal to the boundary. The incident wave boundary condition is imposed at distance *R*=10*L* from the vortex centre which is offset from the centre of the square computational box domain of linear size 40*L*.

The velocity perturbations of the incident acoustic wave are several orders of magnitude as small as the maximum velocity of the vortex, *u'*=1.e-5 max *v* . The problem has two length and time scales associated with the vortex circulation and the acoustic wave. The case of long acoustic wavelength =2.5*L* is considered first.

The solution of the acoustic wave scattered by the vortex is sought in the form of the scattered wave component

$$p' = p - p^{(a)} - \delta p, \quad \delta p = (p^{(v)} - p\_{\circ}) \tag{4}$$

where *p* is the full pressure field obtained as the solution of the sound wave interaction with the vortex, ( ) *<sup>a</sup> p* is the solution that corresponds to the acoustic wave propagating in the free 26 Advanced Fluid Dynamics

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)

We next consider an isolated viscous-core vortex from Colonius at al (1994) that

(1 exp( ( / ) )) <sup>2</sup> 1.256431, M /

where *c* is acoustic far-field sound and *L* is the vortex core radius. The density and pressure field satisfy the usual isentropic relation and the steady tangential momentum equation:

 

The vortex is specified in the centre of an open square Cartesian domain, three sides of which are open boundaries and the forth one corresponds to an incident acoustic wave that is monochromatic with frequency *f*. and normal to the boundary. The incident wave boundary condition is imposed at distance *R*=10*L* from the vortex centre which is offset

The velocity perturbations of the incident acoustic wave are several orders of magnitude as

and time scales associated with the vortex circulation and the acoustic wave. The case of

The solution of the acoustic wave scattered by the vortex is sought in the form of the

( ) ( ) ' , () *a v p pp p p p p* 

where *p* is the full pressure field obtained as the solution of the sound wave interaction with the vortex, ( ) *<sup>a</sup> p* is the solution that corresponds to the acoustic wave propagating in the free

from the centre of the square computational box domain of linear size 40*L*.

small as the maximum velocity of the vortex, *u'*=1.e-5 max *v*

long acoustic wavelength =2.5*L* is considered first.

scattered wave component

2 *p const*; *p v r r*

*<sup>v</sup> r L <sup>r</sup>*

max max

2

*v c*

(4)

. The problem has two length

(2)

(3)

(a) (b) (c)

MinMod limiter at time t=4, and (c) with the limiter deactivated at time t=5.

corresponds to a slowly decaying velocity field of constant circulation 0

0

*r v*

**4. Sound scattering by a steady vortex** 

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 form (5) instead of using 0 ( ) *<sup>t</sup> p p p* in order to account for a small systematic approximation error of the round vortex on a rectilinear Cartesian grid.

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 boundary conditions to minimise numerical reflections from the boundaries.

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

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

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.

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

Direct Numerical Simulations of Compressible Vortex Flow Problems 29

For the Roe-MUSCL scheme, the comparison with the reference solution on the finest grid (400x400) is shown in Fig.5c. In comparison to the CABARET results (fig.5a), for the Roe-MUSCL scheme there is some 30% overprediction of the peak sound directivity that is associated with numerical dispersion. Clearly, the acoustic peak corresponds to the downstream vortex direction where the sound waves spend more time inside the strongest vortex-induced hydrodynamic field and which direction is more sensitive to the linear

The next case considered is the high frequency acoustic wave imposed as the inflow boundary condition. It is well known (e.g., Kinsler and Frey, 2000) that in the high-

The latter, for example, describe the effect of focusing and defocusing of acoustic rays as they pass through a non-uniform medium. In particular, the focusing of acoustic rays creates caustics which loci can be found from the solution of eikonal equation (Georges, 1972). On the other hand, caustic locations correspond to the most intense root-meansquare (r.m.s) fluctuations of the pressure field that can be obtained directly from solving

To illustrate this numerically, let's consider the incident acoustic wave at a high-frequency

this calculation, the computational grid with the resolution of 7-8 cells per acoustic wavelength that corresponds to (1000x1200) grid cell points is used. Fig.6a shows the scattered pressure r.m.s. field obtained from the Euler solution, where the loci of the caustics bifurcating into two branches, as obtained in Colonius at al (1994), are shown. The centre of the vortex corresponds to the origin of the coordinate system. The caustics branches outline the acoustic interference zone that develops behind the vortex. Fig.6b

distance R=*L* from the vortex centre. Two grid resolutions are considered, 7 and 14 grid cells per acoustic wavelength. The polar angle variation corresponds to the top half of the computational domain which intersects one of the caustics bifurcation point at ~ 700

For the solution grid sensitivity study, the scattered pressure r.m.s. solutions are computed with two grid densities, as shown in Figs 6c,d. It can be seen the main features such as the caustic point location and the peak amplitudes are well captured on both grids. The scattered acoustic pressure solution component is then further used to compute the trajectories of sound rays. The trajectories are defined as the normal to the scattered pressure r.m.s. fronts. In particular, from this vector field, the maximum angle of the acoustic ray deflected by the vortex can be compared with the ray-tracing solution. According to the ray theory, the maximum deflection angle scales linearly with the vortex

Fig. 7 shows the maximum deflection angles obtained from the Euler calculation (Euler) and the reference values obtained from the ray-tracing solutions. All solutions are in a good agreement and follow the linear trend expected. In particular, the Euler solution almost coincides with the eikonal solution of Tucker and Karabasov (2009) that corresponds to the same computational domain size. The slight disagreement with the other ray-tracing solutions is likely to be caused by the differences in the domain size, i.e., the proximity of

 *<< L* the Euler equations can be reduced to the ray-theory equations.

 *=*0.076 *L* and solve the Euler equations with the CABARET method. For

of the computed solution at

dispersion error of the numerical scheme.

frequency limit

the Euler equations.

relative to the incident wave direction.

shows the pressure r.m.s. directivity, *rms p f R* . . .( ') ( , )

inflow boundary conditions, as discussed by Colonius et al, 1994.

wavenumber

Mach number.

*rms p f R* . . .( ') ( , ) (acoustic pressure directivity) at a large distance from the vortex centre *R*=10*L* are considered. Figs.5 show the acoustic pressure directivity *rms p f R* . . .( ') ( , ) with respect to the polar angle defined anti-clock-wise from the positive *x*-direction. The comparison of the CABARET solution with the reference solution of Colonius et al (1994) is shown in Fig.5a. To monitor the grid convergence, the Euler equations are solved on a Cartesian grid whose resolution is gradually increasing: (100,100), (200х200) и (400х400) cells (2.5, 5 and 10 grid points per vortex radius, respectively). For CABARET, the r.m.s. distributions on the two finer grids virtually coincide.

Fig. 5. Sound wave scattering by a vortex: (а) grid convergence of the scattered pressure r.m.s. field of the CABARET solution, (b) comparison of the fine-grid CABARET results with the reference solution of Colonius et al (1994), (c) comparison of the fine-grid 3rd order Roe/MUSCL results with the reference solution of Colonius et al (1994).

28 Advanced Fluid Dynamics

respect to the polar angle defined anti-clock-wise from the positive *x*-direction. The comparison of the CABARET solution with the reference solution of Colonius et al (1994) is shown in Fig.5a. To monitor the grid convergence, the Euler equations are solved on a Cartesian grid whose resolution is gradually increasing: (100,100), (200х200) и (400х400) cells (2.5, 5 and 10 grid points per vortex radius, respectively). For CABARET, the r.m.s.

(a) (b)

0

0.25

0.5

0.75

**P***rms***/P***I*

1

1.25

**M=0.25**

MUSCL3-RK3, 400x400 Colonius et al, JFM 1994

**M=0.25**



(c)

Fig. 5. Sound wave scattering by a vortex: (а) grid convergence of the scattered pressure r.m.s. field of the CABARET solution, (b) comparison of the fine-grid CABARET results with the reference solution of Colonius et al (1994), (c) comparison of the fine-grid 3rd order

Roe/MUSCL results with the reference solution of Colonius et al (1994).

*R*=10*L* are considered. Figs.5 show the acoustic pressure directivity *rms p f R* . . .( ') ( , )

grid (100x100) grid (200x200) grid (400x400)

(acoustic pressure directivity) at a large distance from the vortex centre

Current prediction, 400x400 Colonius et al, JFM 1994

with

*rms p f R* . . .( ') ( , )

0

0.25

0.5

0.75

**P***rms***/P***I*

1

1.25

distributions on the two finer grids virtually coincide.

**M=0.25**


> 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

**P***rms***/P***I*

For the Roe-MUSCL scheme, the comparison with the reference solution on the finest grid (400x400) is shown in Fig.5c. In comparison to the CABARET results (fig.5a), for the Roe-MUSCL scheme there is some 30% overprediction of the peak sound directivity that is associated with numerical dispersion. Clearly, the acoustic peak corresponds to the downstream vortex direction where the sound waves spend more time inside the strongest vortex-induced hydrodynamic field and which direction is more sensitive to the linear dispersion error of the numerical scheme.

The next case considered is the high frequency acoustic wave imposed as the inflow boundary condition. It is well known (e.g., Kinsler and Frey, 2000) that in the highfrequency limit  *<< L* the Euler equations can be reduced to the ray-theory equations. The latter, for example, describe the effect of focusing and defocusing of acoustic rays as they pass through a non-uniform medium. In particular, the focusing of acoustic rays creates caustics which loci can be found from the solution of eikonal equation (Georges, 1972). On the other hand, caustic locations correspond to the most intense root-meansquare (r.m.s) fluctuations of the pressure field that can be obtained directly from solving the Euler equations.

To illustrate this numerically, let's consider the incident acoustic wave at a high-frequency wavenumber  *=*0.076 *L* and solve the Euler equations with the CABARET method. For this calculation, the computational grid with the resolution of 7-8 cells per acoustic wavelength that corresponds to (1000x1200) grid cell points is used. Fig.6a shows the scattered pressure r.m.s. field obtained from the Euler solution, where the loci of the caustics bifurcating into two branches, as obtained in Colonius at al (1994), are shown. The centre of the vortex corresponds to the origin of the coordinate system. The caustics branches outline the acoustic interference zone that develops behind the vortex. Fig.6b shows the pressure r.m.s. directivity, *rms p f R* . . .( ') ( , ) of the computed solution at distance R=*L* from the vortex centre. Two grid resolutions are considered, 7 and 14 grid cells per acoustic wavelength. The polar angle variation corresponds to the top half of the computational domain which intersects one of the caustics bifurcation point at ~ 700 relative to the incident wave direction.

For the solution grid sensitivity study, the scattered pressure r.m.s. solutions are computed with two grid densities, as shown in Figs 6c,d. It can be seen the main features such as the caustic point location and the peak amplitudes are well captured on both grids. The scattered acoustic pressure solution component is then further used to compute the trajectories of sound rays. The trajectories are defined as the normal to the scattered pressure r.m.s. fronts. In particular, from this vector field, the maximum angle of the acoustic ray deflected by the vortex can be compared with the ray-tracing solution. According to the ray theory, the maximum deflection angle scales linearly with the vortex Mach number.

Fig. 7 shows the maximum deflection angles obtained from the Euler calculation (Euler) and the reference values obtained from the ray-tracing solutions. All solutions are in a good agreement and follow the linear trend expected. In particular, the Euler solution almost coincides with the eikonal solution of Tucker and Karabasov (2009) that corresponds to the same computational domain size. The slight disagreement with the other ray-tracing solutions is likely to be caused by the differences in the domain size, i.e., the proximity of inflow boundary conditions, as discussed by Colonius et al, 1994.

Direct Numerical Simulations of Compressible Vortex Flow Problems 31

(a) (b) Fig. 7. Extracting sound ray trajectories from the Euler solution (a) for vortex Mach number 0.55 and (b) comparing the extracted maximum ray deflection angle as a function of various

As the final example, we consider the test problem of interacting counter-rotating vortices that involves both their nonlinear dynamics and, as a by-product, sound generation. For small viscosity, the direct simulation of vortex dynamics and acoustics by solving the compressible Navier-Stokes equations on a Eulerian grid is a challenging problem because of the thin vorticity filaments that are generated as the process evolves in time. These are difficult to capture because of numerical dissipation-dispersion problems mentioned in the introduction. In the literature, examples of flow simulations have Reynolds number, as defined based on the velocity circulation, in the range of 1000-4000 (e.g., Inoue, 2002). Eldridge (2007) manages to accurately compute the problem of dynamics and acoustics of counter-rotating vortex pairs at a high Reynolds number, Re=10000 with the use of a Lagrangian vortex particle method. In the latter, the governing fluid flow equations are solved in a non-conservative form and the advected vortex solution is regularly reinitialised on a Eulerian grid to reduce the complexity of thin vorticity filaments and stabilise the solution. In the present subsection, the problem of counter-rotating vortices is solved on a fixed Eulerian grid for the range of Reynolds numbers,

Fig. 8 shows the problem setup. Four viscous-core counter-rotating vortices are initiated in an open domain. Each of the vortices has a constant velocity circulation at infinity , 0

> 1.25 *<sup>r</sup> <sup>r</sup> e*

*a* that corresponds to the vortex core Mach number 0 *M* 0.3 and

*r* is half-distance between the adjacent vortex centres. In non-dimensional

2 0

*r*

2 0 1.25

(5)

 and 0.2 

. The

vortex Mach numbers with several ray-tracing solutions.

Re=5000-10000 with the conservative Navier-Stokes CABARET method.

variables, the flow parameters at infinity are taken to be *p* 1, 1

initial location of the centre of mass of the system corresponds to **x**=0.

and a Gaussian distribution of the vorticity with the core radius 0*r*

where 0.24

*x y* 3 0 

 

10

**5. Dynamics of counter-rotating vortices** 

Fig. 6. Euler solution of the sound scattering by a vortex at high frequency: (а) pressure r.m.s. field where the loci of caustic branches are shown with the open symbols, (b) pressure r.m.s. directivity in the top half of the domain for the grid resolutions 7 and 14 cell per acoustic wavelength (7 ppw and 14 ppw); the scattered pressure r.m.s. for vortex core Mach number Mmax=0.295 with the grid density of (c) 7 cells per acoustic wavelength and (d) 14 cells per acoustic wavelength.

30 Advanced Fluid Dynamics

(a) (b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

**p***rms* **/p***I*

> 0 90 180

7ppw 14ppw

(c) (d)

Fig. 6. Euler solution of the sound scattering by a vortex at high frequency: (а) pressure r.m.s. field where the loci of caustic branches are shown with the open symbols, (b) pressure r.m.s. directivity in the top half of the domain for the grid resolutions 7 and 14 cell per acoustic wavelength (7 ppw and 14 ppw); the scattered pressure r.m.s. for vortex core Mach number Mmax=0.295 with the grid density of (c) 7 cells per acoustic wavelength and (d) 14

cells per acoustic wavelength.

Fig. 7. Extracting sound ray trajectories from the Euler solution (a) for vortex Mach number 0.55 and (b) comparing the extracted maximum ray deflection angle as a function of various vortex Mach numbers with several ray-tracing solutions.
