Universality and scaling in homogeneous compressible turbulence

Universality in compressible turbulence has proven to be elusive as no unifying set of parameters was found to yield universal scaling laws. This severely limits our understanding of these flows and the successful development of theoretically sound models. Using results in specific asymptotic limits of the governing equations in the absence of a mean flow, we show that universal scaling is indeed observed when the set of governing parameters is expanded to include internally generated dilatational scales regardless of driving mechanisms that produce the turbulence. The analysis, though restricted to homogeneous flows, demonstrates why previous scaling laws fail, and it suggests new venues to identify physical processes of interest and aid in the development of more general turbulence models. We support our results with a new massive database of highly resolved direct numerical simulations along with data from the literature comprising isotropic flows with different forcing mechanisms as well as homogeneous shear flows. We also include flows with considerable bulk viscosity. In search of universal features, we postulate the existence of classes that bundle the evolution of flows in the new parameter space. An ultimate asymptotic regime predicted by renormalization-group theories and statistical mechanics is also assessed with available data.

Figure 1

Scaling of variance of pressure with $M_t$ (a), $M_d$ (b), and $\mathcal{D}$ (c). In (a), dashed lines are ${\gamma }^2{M}_t^4/9$ [33]. In (b), dashed lines are ${\gamma }^2{M}_d^2/F_T$ (high $M_d$) and $\propto M_d$ for reference (low $M_d$.) In (c), dashed lines are ${\mathcal{D}}^{-2}$ (low $\mathcal{D}$) for different values of $A$ and the asymptotic DDE horizontal lines (high $\mathcal{D}$).

Figure 2

Scaling of $w_{d,\mathrm{rms}}/w_{s,\mathrm{rms}}$ vs $\delta$ where ${\mathit{w}}_{\mathit{d}}=\sqrt{\rho }{\mathit{u}}_{\mathit{d}}$ and ${\mathit{w}}_{\mathit{s}}=\sqrt{\rho }{\mathit{u}}_{\mathit{s}}$.

Figure 3

Contours of $\left|\mathbf{\nabla }p\right|/{\left|\mathbf{\nabla }p\right|}_{\text{rms}}$ for $(\mathcal{D},R_{\lambda },M_t,\delta )=(0.04,58,0.09,0.0035)$ (a), (0.27,102,0.36,0.097) (b), (12,50,0.20,1.4) (c), and (52,51,0.06,1.65) (d).

Figure 4

Scaling of the ratio of dilatational to solenoidal dissipation with (a) $M_t$ and (b) $\delta$. The dashed line in (b) with a slope of 2 is for reference.

Figure 5

Skewness of the longitudinal velocity gradient.

Figure 6

Regions in the $\delta \text{−}{M}_t$ diagram: Dilatationally dominated $p$-equipartition (DDE, $\mathcal{D}>{\mathcal{D}}_{\text{cr}}$), $S$-divergence (${\delta }^2{M}_t>2\times {10}^{-2}$), and $\langle {\epsilon }_d\rangle /\langle {\epsilon }_s\rangle >1$. (a) Entire database in Table 1. (b) Selected trajectories, which include isotropic cases with two types of solenoidal forcing, cases with dilatational forcing, and homogeneous shear flows.