Solenoidal Scaling Laws for Compressible Mixing

Mixing of passive scalars in compressible turbulence does not obey the same classical Reynolds number scaling as its incompressible counterpart. We first show from a large database of direct numerical simulations that even the solenoidal part of the velocity field fails to follow the classical incompressible scaling when the forcing includes a substantial dilatational component. Though the dilatational effects on the flow remain significant, our main results are that both the solenoidal energy spectrum and the passive scalar spectrum assume incompressible forms, and that the scalar gradient essentially aligns with the most compressive eigenvalue of the solenoidal part, provided that only the solenoidal components are consistently used for scaling. A slight refinement of this statement is also pointed out.

Figure 1

Parameter space of simulations for the scalar field. (a) $M_t-R_{\lambda }$ plane: red, $\delta <{10}^{-3}$; blue, ${10}^{-3}<\delta <{10}^{-2}$; green, ${10}^{-2}<\delta <{10}^{-1}$; orange, ${10}^{-1}<\delta <1$; brown, $1<\delta <10$. (b) $\delta -R_{\lambda }$ plane: blue, $M_t<0.2$; green, $0.2<M_t<0.4$; brown, $0.4<\delta <0.7$. Symbols in both figures correspond to different percentages of dilatational forcing, $\sigma$: diamonds, incompressible simulations; circles, $\sigma =0$; triangles, $\sigma =10–30$; squares, $\sigma =30–65$; stars, $\sigma =65–100$.

Figure 2

(a) Kolmogorov-compensated total energy spectra using total energy dissipation and Kolmogorov length scale [$\eta ={({\mu }^3/⟨\rho {⟩}^2⟨\epsilon ⟩)}^{1/4}$] [5]. (b) Kolmogorov-compensated solenoidal energy spectra using total dissipation and the Kolmogorov length scale based on it. Here and in all figures to follow except Fig. 7, different colors correspond to different Reynolds numbers. Red, $R_{\lambda }<40$; blue, $40<R_{\lambda }<75$; green, $75<R_{\lambda }<115$; and orange, $115<R_{\lambda }<170$. The velocity data of this figure and in Fig. 5 include a larger set of conditions than shown for the scalar field in Fig. 1.

Figure 3

The Obukhov-Corrsin compensated scalar spectra using total dissipation and Kolmogorov length scale based on the total energy dissipation. No scaling is observed. The dashed line is for the incompressible case.

Figure 4

Alignment of scalar gradient ($\nabla \varphi$) with the eigendirections of the $S_{ij}$, i.e., $e_{\gamma }$, $e_{\beta }$, $e_{\alpha }$, which correspond to the eigenvalues $(\gamma ,\beta ,\alpha )$ with $\gamma <\beta <\alpha$.

Figure 5

Kolmogorov-compensated solenoidal energy spectra (a) and Obukhov-Corrsin compensated scalar spectra (b) using solenoidal dissipation $⟨{\epsilon }_s⟩$ and solenoidal Kolmogorov length scale ${\eta }_s$. The dashed line in the bottom figure is for the incompressible case.

Figure 6

(a) Normalized eigenvalues of the solenoidal symmetric velocity gradient tensor $S_{ij}^s$: (i) ${\widehat{\beta }}_s=\sqrt{6}{\beta }_s/\sqrt{{\alpha }_s^2+{\beta }_s^2+{\gamma }_s^2}$; (ii) ${\beta }_s/{\gamma }_s$. (b) Alignment of scalar gradient ($\nabla \varphi$) with $e_{\gamma }$, $e_{\beta }$, $e_{\alpha }$, the eigenvectors of $S_{ij}^s$.

Figure 7

(a) Probability that the scalar gradient ($\nabla \varphi$) aligns perfectly with the eigendirection $e_{\gamma }$ corresponding to the most compressive eigenvalue. Symbols in the figure correspond to different percentages of dilatational forcing. $\sigma$: incompressible simulations (diamonds), $\sigma =0$ (circles), 10–30 (triangles), 30–65 (squares), and 65–100 (stars).