Evaluation of Spectral Behavior for Large Ensembles of Exact Solutions to Burgers’ Equation for Thomas Initial Conditions

Abstract

Spectral and cumulative spectral transfer, as well as energy and dissipation spectra, have been computed by FFT from exact solutions to Burgers’ equation for Thomas [1] random initial conditions at initial turbulence Reynolds’ numbers of Re ₀ = 400 and 1000. Accurate, efficient numerical evaluation of the exact solution on a fine spatial mesh enables spectral computations to be carried out at high wave numbers for large ensembles, thereby confirming the exponential viscous cutoff. The low wave-number white narrow band of the energy spectrum gives way to the well-known k ⁻²-subrange, independently of Rep in the high Rep range. The k ⁻²-subrange, in turn, gives way with increasing wave-number k to an exp(−αk) high wave-number viscous cutoff at a wave number depending on Re ₀. Spectral transfer and cumulative spectral transfer functions look very much like their counterparts for three-dimensional isotropic turbulence.