The Statistical Theory of One-Dimensional Turbulence in a Compressible Field

Abstract

The statistical properties of one-dimensional compressible turbulence are investigated by means of the reductive perturbation method and the numerical integration of the governing equations. The mechanical energy of turbulence (Remark: Graphics omitted.), which is the sum of the kinetic energy and the adiabatic internal energy, is shown to decay in time t as (Remark: Graphics omitted.), and the numerical result (Remark: Graphics omitted.) confirms well this prediction. The kinetic and the adiabatic internal energies are found to oscillate in time with large amplitude around the common average (Remark: Graphics omitted.), and thus the equipartition of both energies in the average is concluded. The numerically calculated mechanical energy spectrum is expressed as E(k)∝k⁻² at low wave-numbers and E(k)∝e^−αk, α being a constant, at higher wave-numbers. This behaviour of the energy spectrum reflects the structure of compressible turbulence composed of random trains of shocks and expansion waves.