Spectral properties of exact random solutions to Burgers' equation for modified Thomas initial conditions

Abstract

Random sawtooth Thomas [1] initial conditions can be specified on a fixed spatial mesh d or on a random one d n (with average mesh spacing 〈 d n 〉) termed modified Thomas initial conditions. Beyond the white-noise band common to both at small wave number k, the energy spectrum K(k, 0) for the modified Thomas has a smooth transition from a k −4 to a k −2 asymptote, whereas Thomas has pronounced ringing. For modified Thomas conditions, there is (i) a narrow band (small k) white-noise spectrum E(k, t) = E 0 like that for Thomas conditions, (ii) a k −2 subrange of increasing bandwidth with increasing initial turbulence Reynolds' number Re 0, and (iii) an exponential viscous cutoff at high wave numbers which decays more rapidly with decreasing Re 0. The small time spectral transfer, T(k, t), for modified Thomas conditions does not have a spike at k = 2π/〈d n 〉 , whereas it does for Thomas ones at k = 2π/d. The evolution of the energy and dissipation spectra and transfer and cumulative transfer spectra for modified Thomas conditions is otherwise similar in form to that for Thomas conditions for t > 1.