2 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


Introduction
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 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 4 th and the 6 th 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 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.

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): 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 2 / 1.047 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 , () , as a fraction of its initial value (0) 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).

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 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=10L from the vortex centre which is offset from the centre of the square computational box domain of linear size 40L.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 =2.5L is considered first.The solution of the acoustic wave scattered by the vortex is sought in the form of the scattered wave component where p is the full pressure field obtained as the solution of the sound wave interaction with the vortex, () a p is the solution that corresponds to the acoustic wave propagating in the free www.intechopen.comspace without any hydrodynamic perturbation, () v 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 () t p pp      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 10 5 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 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 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.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 ~ 70 0 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.

Dynamics of counter-rotating vortices
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, Re=5000-10000 with the conservative Navier-Stokes CABARET method.The vortex system undergoes a jittering motion with one vortex pair sleeping through the other and taking turns.The centre of mass travels at a positive subsonic speed, which corresponds to the right horizontal direction in Fig. 8.After each sleep-through, the distance between the vortices decreases until they finally coalesce and continue the movement as a single core.By using a point-vortex approximation with neglecting viscous effects, the dynamics of the vortex system before coalescence can be described by the classical analytical solution of Hicks (1922).This solution gives the following expressions for the centre mass velocity U and the slip through period p T of the vortex system: .
where k=1/[1+( y /  x ) 2 ]=0.5 and () ,() Kk Ek are the complete elliptic integrals of the first and second kind.For the specified parameters of the model, the centre mass velocity and slipthrough period are 2 54.46 0.1437 12.21, 0.1282 The velocity of the centre mass corresponds to the Mach number 0.1083 M   .Because of the problem symmetry with regard to axis y=0, one-half of the computational domain is considered with the symmetry boundary condition.The problem is solved in the inertial frame of reference which velocity with regard to the absolute frame equals the velocity of the centre of mass, U .The latter is available from the analytical point-vortex solution.Because of this choice the centre of mass is approximately stationary in the reference coordinate system.The latter is helpful for minimising the size of the computational domain.The computational domain of size 440 x 220 (axially times vertically) is covered by a Cartesian grid that has a uniform grid spacing in the central block (60 x 30).Exponential grid stretching is applied near the outer boundaries to reduce numerical reflections (Fig. 9).Three grid resolutions are considered: 6, 9 and 12 grid cells per vortex core radius.Initial conditions in conservation variables are computed in the following way.By combining the initial vorticity distribution (5) with the solenoidal velocity field condition, the velocity field is computed from solving the Laplace equation for velocity potential.The Fig. 10 shows the time evolution of vorticty field of the system as the vortex pairs sleep through each other.The grid density is 6 grid cells per vortex radius and the Reynolds number of the CABARET simulation is 9400.This particular Reynolds number is chosen as the best match for the reference Lagrangian particle solution of Eldridge ( 2007) that corresponded Re=10000.The 5% difference in the Reynolds numbers may be attributed to the differences in numerical approximation of viscous terms in the momentum and energy equations in the conservative CABARET and the non-conservative vortex particle method.For Re=9400, the vortex pairs merge after 4 sleep-through events.The sleep-though events correspond to Fig10(c),(d),(e),(f). For Re=5000, the coalescence happens earlier in comparison with the Re=9400 case: for the lower Re-number the vortices coalesce already after 3 sleepthroughs.This change can be compared with results for the fully inviscid follow case calculation that was conducted with the same CABARET Euler method.In the latter case the vortices manage to undergo 6 sleep-through events before coalescence.The capability of the CABARET model to capture the qualitative differences between the Re=9400 and the fully inviscid solution is noted as a good indication of how low-dissipative the method is.
To zoom into the flow details,    is the time period before the second and the third sleep-through.The agreement for the meanflow velocity between the point-vortex theory value (0.1282) and the numerical values on 3 different grids is within 1%.For the sleep-through period, the values obtained on different grids are converged within 0.5%.The numerically predicted time period, however, is 10-15% shorter in comparison with the point-vortex theory (12.21).This discrepancy is within the order of accuracy the point-vortex model, (r 0 /) 2 =0.16 and thus characterises how non-compact the viscous vortex core is in comparison with the distance between the adjacent vortex centres.
In addition to the near-filed, the far-field pressure field has been computed on a circle control surface at distance of 20 from the vortex centre of vortices.The control surface is located in the same reference system as the centre of mass that moves at a small subsonic speed, 0.1083 M   with respect to the absolute frame.Fig. 13 shows the pressure signals obtained at the control points corresponding to 30 0 and 90 0 angle to the flow direction for Re=5000 and 9400 on the grids of different resolution.
The pressure fluctuations are defined with the reference to the pressure field value at infinity, 1 p   .The peaks of the pressure signatures correspond to the vortex sleepthrough events and the number of the peaks corresponds to the total number of vortex sleep-throughs, respectively.The phase of intense vortex interaction during the vortex pairing is followed by a "calming" period that corresponds to the vortex roll-up after the coalescence.In comparison with the pre-coalescence time history that is dominated by largetime scales the post-coalescence signal is dominated by small-time-scale events.For the higher Re-number case, the amplitude of the last acoustic "burst" that corresponds to /1 1 0 at   has some 20% higher amplitude in comparison with other peaks.This loud acoustic event corresponds to the last vortex sleep-through, which takes place at /9 0 at   and which is well-captured on the grids of different resolution.After the vortex coalescence, the increase of Reynolds number from 5000 to 9400 also leads to a notable prolongation of small-scale acoustic fluctuations in the post-coalescence phase.These effects may be associated with the small spatial structures that are generated shortly before the vortices coalesce (e.g., fig.10f,g) and which are more sensitive to viscous dissipation.For the pre-coalescence period of vortex evolution, the numerical solutions that correspond to the grids of different resolution are converged within 1-2% for both Reynolds numbers.For the post-coalescence time history, the grid convergence for the high Reynolds-number case, Re=9400, slows down in comparison with the Re=5000 case.For both Re-number cases, however, the CABARET solution on the grid resolution 12 cells per vortex radius appears adequate to capture the fine pressure field fluctuations well.

Conclusion
The computational of compressible vortical flows is challenging because of the multi-scale phenomena involved.Computational approaches and numerical methods for the solution of compressible vortical flow problems have been discussed.In particular, the key elements of a successful computational method have been outlined that include low numerical dissipation and low dispersion, as well as the good vortex preservation property.For the sake of illustration, several two-dimensional problems are considered that typically present a challenge for conventional Eulerian numerical schemes.The problems include the preservation of steady vortex in a box domain, acoustic wave scattering by a vortex field and the dynamics and acoustics of counter-rotating vortices pairs.For these problems, several computational solutions are presented and discussed, including those obtained with the CABARET scheme developed by the authors.Analytical and reference solutions are provided where applicable.All test problems considered are promoted as the benchmark problems for new Computational Fluid Dynamics codes that are to be used in application for hydrodynamics and acoustics of vortex resolving simulations.

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

Fig. 4 .
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.

Fig. 5 .
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 ofColonius et al (1994), (c) comparison of the fine-grid 3 rd order Roe/MUSCL results with the reference solution ofColonius et al (1994).

Fig. 6 .
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 M max =0.295 with the grid density of (c) 7 cells per acoustic wavelength and (d) 14 cells per acoustic wavelength.

Fig. 7 .
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.

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    and a Gaussian distribution of the vorticity with the core radius 0 r distance between the adjacent vortex centres.In non-dimensional variables, the flow parameters at infinity are taken to be 1location of the centre of mass of the system corresponds to x=0.

Fig. 9 .
Fig. 9. Computational grid: (a) full domain view and (b) zoom in the beginning of the grid stretching zone.

Fig. 10 .
Fig.10shows the time evolution of vorticty field of the system as the vortex pairs sleep through each other.The grid density is 6 grid cells per vortex radius and the Reynolds number of the CABARET simulation is 9400.This particular Reynolds number is chosen as the best match for the reference Lagrangian particle solution ofEldridge (2007) that corresponded Re=10000.The 5% difference in the Reynolds numbers may be attributed to the differences in numerical approximation of viscous terms in the momentum and energy equations in the conservative CABARET and the non-conservative vortex particle method.For Re=9400, the vortex pairs merge after 4 sleep-through events.The sleep-though events correspond to Fig10(c),(d),(e),(f). For Re=5000, the coalescence happens earlier in comparison with the Re=9400 case: for the lower Re-number the vortices coalesce already after 3 sleepthroughs.This change can be compared with results for the fully inviscid follow case calculation that was conducted with the same CABARET Euler method.In the latter case the vortices manage to undergo 6 sleep-through events before coalescence.The capability of the CABARET model to capture the qualitative differences between the Re=9400 and the fully inviscid solution is noted as a good indication of how low-dissipative the method is.To zoom into the flow details, Fig.11compares instantaneous vorticity contours obtained from the CABARET solution at two grid resolutions, 6 and 12 cells per core radius, with the reference Lagrangian particle solution fromEldridge (2007) at one time moment.This time moment corresponds to the last vortex sleep-through before the coalescence.At this time the vortices are very close and strongly interact with each other through fine vorticity filaments.For all three solutions, the same contour levels are plotted that show a good agreement down to a small detail.It is also interesting to compare the 2D solution computed with the results of the vortical structure visualization obtained experimentally by Bricteux et al 2011 for a 3D high-Reynolds number jet.In this experimental work a moving window technique is used in the framework of Particle Image Velocimetry (PIV) method to visualise vortex paring in the jet shear layer.Fig.12shows the results of the visualisation in the jet symmetry plane that appear qualitatively very similar to the results of the 2D simulation (cf.fig.10a,b,d,g).For a quantitative comparison, the centre mass velocity of the vortex system and the values of first few vortex slip-through periods are compared with the reference analytical solution for point vortices in inviscid flow.
Fig. 11.Vorticity distribution of the leapfrogging vortex pairs at /9 0 at  

Fig. 12 .
Fig. 12. PIV of vorticity distribution of leapfrogging vortex rings in the symmetry plane of a high Reynolds number jet obtained with a moving window technique by Bricteux et al 2011.

Table 1 .
Integral characteristics of the vortex system as obtained from the numerical solution Here (T p ) 1 is the time period between the first and the second vortex sleep-through and (T p ) 2