Why Turbulence Burns Energy Differently ✛ ✎

Original: "Mach-number-dependent dissipative anomaly in isothermal compressible turbulence"
arXiv:2609.03945v1 · 2026-09-03 · arXiv non-exclusive · independent analysis · 1 min
When a flow moves faster than sound, energy is lost differently than in a slow flow, even if viscosity is very low.

A churning cauldron of turbulence burns energy in two distinct ways. Vortices, like funnels in a whirlpool, swirl the fluid and rub layers against each other. Shock waves, like a sudden clap, sharply compress the gas and turn motion into heat. For a long time, it was thought that at very low viscosity, the rate of these losses stops depending on it. But what happens when the flow moves faster than sound and compression becomes the main effect? It turns out that energy losses proceed differently.

Numerical simulations showed that if the flow is slower than sound, energy losses approach one constant value. If the flow is much faster than sound, losses approach another constant. It is like two roads to the same finish line: one winding, one straight. The reason is that at high speeds, shock waves become dominant rather than vortices.

At very high speeds, the regions where energy is lost most intensely become nearly flat, like sheets of paper.

Previously, it was unclear whether this independence from viscosity persists in compressible turbulence and how it depends on flow speed. Now it is clear: the independence exists, but it is not a single one. For slow flows, it is one; for fast flows, another. This result could help build turbulence models for astrophysics and engineering calculations of supersonic flows.

At Mach number 10, the regions of maximum dissipation have a dimension of about 2, meaning they are almost perfectly flat, like sheets of paper, even though the flow itself is three-dimensional.

\frac{\langle \epsilon \rangle}{u_{\mathrm{rms}}^3/L} = \mathrm{const}
The ratio of the mean dissipation to the cube of the root-mean-square velocity divided by the integral scale tends to a constant at high Reynolds numbers.
M_{\mathrm{rms}} = \frac{u_{\mathrm{rms}}}{c_s}
The ratio of the root-mean-square velocity to the speed of sound.
Re = \frac{u_{\mathrm{rms}} L_f}{\nu}
The ratio of inertial forces to viscous forces, where L_f is the characteristic forcing scale and ν is the kinematic viscosity.
Concepts
mach number reynolds number turbulence