Energy cascade and scaling in supersonic isothermal turbulence
  Supersonic turbulence plays an important role in a number of extreme
astrophysical and terrestrial environments, yet its understanding remains
rudimentary. We use data from a three-dimensional simulation of supersonic
isothermal turbulence to reconstruct an exact fourth-order relation derived
analytically from the Navier-Stokes equations (Galtier and Banerjee, Phys. Rev.
Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like
inertial energy cascade in supersonic turbulence previously discussed on a
phenomenological level. We show that two compressible analogues of the
four-fifths law exist describing fifth- and fourth-order correlations, but only
the fourth-order relation remains `universal' in a wide range of Mach numbers
from incompressible to highly compressible regimes. A new approximate relation
valid in the strongly supersonic regime is derived and verified. We also
briefly discuss the origin of bottleneck bumps in simulations of compressible
turbulence.
