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Shock-turbulence interaction

The scientific understanding of shock/turbulence interactions remains limited, despite decades of efforts. The most fundamental problem is arguably that of isotropic turbulence passing through a nominally normal shock wave, termed `canonical shock/turbulence interaction' in this report. This problem has received considerable attention in past theoretical, experimental, and computational studies. The linear theory by Ribner (NACA report, 1953 predicts that the velocity and vorticity variances are amplified during the shock interaction, and that there is an inviscid adjustment region behind the shock where turbulence adjusts to its post-shock state. This theory is formally valid in the limits of infinite Reynolds number and zero turbulent Mach number, but has thus far (Lee et al., J. Fluid Mech. 1993; 1997) only been tested at Reynolds numbers of Re_l=20. At these low Reynolds numbers, the viscous decay behind the shock is comparable to or larger than the inviscid adjustment, which makes direct comparisons to the linear theory difficult. Furthermore, the spectrum of turbulence is not truly broadband at such Reynolds numbers. And, finally, there is reason to believe that previous studies have not truly resolved the dissipative motions of the turbulence.

The Hybrid code (Larsson et al., CTR Annual Research Briefs 2007) solves the compressible Navier-Stokes equations for a perfect gas using solution-adaptive finite-difference schemes on Cartesian (but stretched) grids. Near shock waves or other discontinuities, a 5th order accurate weighted essentially non-oscillatory (WENO) scheme is used. In these regions, the equations are discretized in divergence (or conservative) form, thereby ensuring convergence to the correct weak solution. Away from shock waves, a 6th-order accurate central difference scheme is used. This scheme has nominally zero numerical dissipation, which is an important attribute for the prediction of broadband turbulence.  The coupling of the different discretizations globally conserves mass, momentum, and energy, and was shown to be linearly stable. The semi-discrete system is integrated in time using classic 4th-order, fully explicit Runge-Kutta. The method is implemented using C++, and parallelized through domain decomposition with communication handled by MPI.  The code has been run on several machines.

Figure 1. Snapshot of shock/turbulence interaction at M=2, Mt=0.15, Re_l = 40. The flow is from left to right, with the shock visualized by transparent isosurfaces of compression. Vortex cores are visualized by isosurfaces of the second invariant of the velocity gradient tensor, colored by the vorticity magnitude.

The essence of shock/turbulence interaction is shown in Figure 1. The incoming turbulence is isotropic, as evidenced by the random orientation of the vortex cores. The shock compresses the turbulence in the x direction, increasing the vorticity and making the post-shock turbulence axisymmetric with vortex cores predominantly oriented in the y-z plane.

Figure 2. Streamwise (thick) and transverse (thin) Reynolds stress for canonical shock/turbulence interaction. Variance of Reynolds stress and vorticity for (M,M_t)=(2.0,0.15) (solid), (M,M_t)=(2.0,0.30) (dashed), and (M,M_t)=(1.1,0.15) (dash-dotted).

Figure 3. Streamwise (thick) and transverse (thin) vorticityfor canonical shock/turbulence interaction. Variance of Reynolds stress and vorticity for (M,M_t)=(2.0,0.15) (solid), (M,M_t)=(2.0,0.30) (dashed), and (M,M_t)=(1.1,0.15) (dash-dotted).

The variances of velocity and vorticity fluctuations are shown in figures 2-3 for a sequence of cases. The transverse vorticity is directly amplified at the shock due to the compression, while the streamwise vorticity is initially unchanged. Behind the shock, the out-of-equilibrium turbulence adjusts towards an isotropic state, although the Reynolds stresses never reach isotropy in these runs. One outstanding question in shock/turbulence interaction is whether the turbulence truly returns to isotropy. The relatively low Reynolds number of the present (and all previous) runs implies a quick viscous decay behind the shock, which overwhelms any return-to-isotropy. Large-scale calculations at higher Reynolds numbers are needed to illuminate this issue.

Figure 4. Compression for canonical shock/turbulence interaction. Instantaneous (thin) and average (thick) profiles of compression (negative dilatation) and density for (M,M_t)=(1.1,0.15)..

Figure 5. Density for Canonical shock/turbulence interaction. Instantaneous (thin) and average (thick) profiles of compression (negative dilatation) and density for (M,M_t)=(1.1,0.15)..

At high enough turbulent Mach number, M_t, the turbulent pressure fluctuations become comparable to the pressure jump associated with the shock, which significantly alters the instantaneous shock profile. Some instantaneous profiles of the compression (negative dilatation) and density along the x axis are shown in figures 4-5 for such a case. The instantaneous structure of the shock varies wildly, from being twice as strong as on average, to being replaced by a smooth compression wave, to being replaced by two weaker shocks. This intermittency is largely absent at lower values of M_t, and represents a second outstanding fundamental question in shock/turbulence interaction. In fact, the regime of high-M_t shock/turbulence interaction remains largely unexplored in the literature.




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