Published online by Cambridge University Press: 07 December 2015
We investigate how compressibility affects the turbulent statistics from a Lagrangian point of view, particularly in the parameter range where the flow transits from the incompressible type to a state dominated by shocklets. A series of three-dimensional simulations were conducted for different types of driving and several Mach numbers. For purely solenoidal driving, as the Mach number increases a new self-similar region first emerges in the Lagrangian structure functions at sub-Kolmogorov time scale and gradually extends to larger time scale. In this region the relative scaling exponent saturates and the saturated value decreases as the compressibility becomes stronger, which can be attributed to the shocklets. The scaling exponent for the inertial range is still very close to that of incompressible turbulence for small Mach number, and discrepancy becomes visible when the Mach number is high enough. When the driving force is dominated by the compressive component the shocklet-induced self-similar region occupies a much wider range of time scales than that in the purely solenoidal driving case. Regardless of the type of driving force, the probability density functions of the velocity increment collapse onto one another for the time scales in the new self-similar region after proper normalization.
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