Transition to dilatation-dominated compressible turbulence

Abstract

The kinetic energy dissipation rate is of central importance for the small-scale statistics in turbulent flows. Here, we determine the transition to the dilatation-dominated regime of 3d fully compressible, homogeneous, isotropic turbulence by moments of energy dissipation and its components up to order 4 for turbulent Mach numbers $0.1\le M_t\le 10$. Our high-resolution numerical simulations show a crossover from incompressible to $M_t$-independent, Burgers turbulence-like moment scaling with respect to Reynolds number $Re$. This confirms the statistical dominance of shocks for $M_t\gtrsim 1$.

Figures (4)

• Figure 1: Structure of the kinetic energy dissipation rate field. Isosurface snapshots of total energy dissipation $\epsilon({\bm x}, t_0)$ at the level $\epsilon = 10 \langle \epsilon \rangle_{V,t}$ (top row) and its solenoidal component $\epsilon_s({\bm x}, t_0)$ at $\epsilon_s = 10 \langle \epsilon_s \rangle_{V,t}$ (bottom row) for turbulent Mach numbers $M_t = 0.1$ (a, d), $M_t = 1$ (b, e) and $M_t = 10$ (c, f). All data are obtained at the highest Reynolds number $Re \approx 2400$; isosurfaces are displayed for the decadic logarithm of the fields.
• Figure 2: (a) Operating points of the solenoidally forced compressible turbulence simulations in the $\delta$--$M_t$ parameter plane (panel a). The incompressible limit and the $\delta$-asymptote are indicated, which is consistently not exceeded by our simulations. Panels b and c show the ratios of the mean dissipation rates versus the turbulent Mach number $M_t$: (b) $\langle \epsilon_d \rangle_{V,t}/\langle \epsilon \rangle_{V,t}$ and (c) $\langle \epsilon_d \rangle_{V,t}/\langle \epsilon_s \rangle_{V,t}$. The dotted horizontal lines indicate asymptotic behavior toward a 1:1 ratio.
• Figure 3: Reynolds number-scaling exponents of the 2nd-, 3rd- and 4th-order moments of the total dissipation rate (a), solenoidal component (b), dilatational component (c), and incompressible component (d), as functions of the turbulent Mach numbers $M_{t}$. In the incompressible limit $M_t \ll 1$, $\beta_{{\rm inc}, n} = \beta_n$ and are consistent with those reported for incompressible isotropic turbulence YS2004yakhot2006SSY07; $\beta_2 = 0.157$, $\beta_3 = 0.489$ and $\beta_4 = 0.944$ . In this limit, the solenoidal exponents (b) take the values $\beta_{s,2} = 0.242$, $\beta_{s,3} = 0.693$, and $\beta_{s,4} = 1.31$, in close agreement with Elsinga:JFM2023 after necessary adjustment, see Supplementary Material. For $M_t > 1$, the exponents are very close to the Burgers' scaling exponents. In panel (a), the data points are connected by a tanh fitting curve. The solid and dotted reference lines indicate the scaling exponents of incompressible and Burgers turbulence limits, respectively.
• Figure 4: Scaling exponents $\beta_n^{\rho}$ of the density moments $M_n^{\rho}$ versus $M_t$. The inset shows a density contour plot in a two-dimensional plane at $M_t=10$ and $Re=2400$, plotted as $\log_{10}\left(\rho/\langle \rho \rangle_{V,t}\right) \in (-3,\,1)$.