Inviscid criterion for decomposing scales

The proper scale decomposition in flows with significant density variations is not as straightforward as in incompressible flows, with many possible ways to define a “length scale.” A choice can be made according to the so-called inviscid criterion [Aluie, Physica D 24, 54 (2013)]. It is a kinematic requirement that a scale decomposition yield negligible viscous effects at large enough length scales. It has been proved [Aluie, Physica D 24, 54 (2013)] recently that a Favre decomposition satisfies the inviscid criterion, which is necessary to unravel inertial-range dynamics and the cascade. Here we present numerical demonstrations of those results. We also show that two other commonly used decompositions can violate the inviscid criterion and, therefore, are not suitable to study inertial-range dynamics in variable-density and compressible turbulence. Our results have practical modeling implication in showing that viscous terms in Large Eddy Simulations do not need to be modeled and can be neglected.

Figure 1

Shock profiles at Mach 1.2 (M1) and 3 (M3) in the reference frame of the shock.

Figure 2

Mach 1.2 (top row) and 3 (bottom row). The viscous contributions are calculated at length scale $L/8$. The data are normalized by the unfiltered dissipation. The Favre decomposition yields a smaller viscous contribution at large length scales compared to the other two decompositions. Moreover, the disparity between decompositions increases with Mach number.

Figure 3

Mach 1.2 (top row) and 3 (bottom row). The figure shows both the average and the maximum dissipation as a function of $k=L/\ell$, and the vertical line in each figure marks the viscous cutoff wave number $k_d=k_{\mu }=L/{\ell }_{\mu }$. Data are normalized by the unfiltered dissipation. Left two panels show how the non-Favre decompositions yield significant viscous contamination at large length scales. Right two panels show that ${\mathrm{\Sigma }}^{F,\text{diss}}$ decays as ${\ell }^{-2}$ as proven mathematically, whereas the non-Favre definitions decay at a much slower rate of ${\ell }^{-1}$ due to the dilution of the shock's effect in one dimension.

Figure 4

Density field of 2D buoyancy-driven flows R2–R4 carried out with successively higher initial density ratios (see Table 2). Flows R4vv and R4vc in the bottom two panels test the sensitivity of our results to the temperature dependence of viscosity, $\mu$, and thermal conductivity coefficient, $\kappa$.

Figure 5

Comparing the viscous contribution from the decompositions as a function of scale using three flows R2–R4 with increasing density ratio from left to right (see Table 2). Horizontal axes are $k=L_z/\ell$, and the vertical line in each figure marks the viscous cutoff wave number $k_d=L_z/\eta$. Top row shows the full viscous contribution $\langle {\mathrm{\Sigma }}_{\ell }\rangle$. Bottom row shows the $L^1$ norm scaling of the viscous dissipation, $\parallel {\mathrm{\Sigma }}_{\ell }^{\mathrm{diss}}{\parallel }_1$. Plots are normalized by the unfiltered dissipation. In all cases, the Favre dissipation decays at least as ${\ell }^{-2}$ at large scales. The other two definitions do not show a clear decay trend and even become negative on average at the largest scales.

Figure 6

Similar to Fig. 5. Testing the sensitivity of results to a spatially varying viscosity and thermal conductivity (runs R4vv and R4vc in Table 2). Plots are normalized by the unfiltered dissipation. The plots are consistent with those in Fig. 5, showing that our conclusions also hold when $\mu \left(\mathbf{x}\right)$ and $\kappa \left(\mathbf{x}\right)$ are spatially varying, as proved in Ref. [5].

Figure 7

Visualization of density in the RTI flow at the instant of time we use in our analysis.

Figure 8

$L^1$ norm scaling of the viscous dissipation, $\parallel {\mathrm{\Sigma }}_{\ell }^{\mathrm{diss}}{\parallel }_1$ using the original 3D RT field in Fig. 7. Plot is normalized by the unfiltered dissipation. In the horizontal axis $k=L_z/\ell$, and the vertical line marks the viscous cutoff wave number $k_d=L_z/\eta$. The Favre dissipation decays the fastest, at least as ${\ell }^{-2}$ at large scales.

Figure 9

The spectra and PDF of density fields from the snapshot shown in Fig. 7. Spectra are calculated by a 2D Fourier transform in the horizontal directions and then averaging along the $z$ direction, horizontal axis $k=L_z/\ell$. Density spectra in cases D1–D8 have similar scaling but with different fluctuation intensity.

Figure 10

Sensitivity to increasing density variations: log-log plot of the $L^1$ norm of dissipation for each decomposition. Horizontal axis are $k\equiv L_z/\ell$, and the vertical line in each figure marks the viscous cutoff wave number $k_d=L_z/\eta$. This shows that the Favre dissipation term decays at least as fast as ${\ell }^{-2}$ regardless of the intensity of density variability, unlike the other two decompositions. Note that plots of case D1 are the same as in Fig. 8.

Figure 11

Ma 3 constant spatial viscosity results. Left: average dissipation, right: $L^{\infty }$ of dissipation. Data are normalized by the unfiltered dissipation, and the vertical line in each figure marks the viscous cutoff wave number $k_d=L/{\ell }_{\mu }$.

Figure 12

Visualization of density in the R3 case of 2D RT flow at later time.

Figure 13

The viscous contribution and viscous dissipation of R3 at a later time than that used in Fig. 5 above. Left: average viscous contribution, right: $L^1$ norm of dissipation. Data are normalized by the unfiltered dissipation, and the vertical line in each figure marks the viscous cutoff wave number $k_d=L_z/\eta$.