On the correlation function in Burgers’ model of turbulence

Abstract

SurnrnaryThe equation for the correlation function in Burgers' recent approach (Istanbul, 1952) is investigated. It is shown that the parameter fJ must have the value zero. Both from qualitative arguments and numerical integration, it appears that the solution of the equation is consistent with experiments: thus Burgers' approach might be profitably pursued.In a recent paper 1) Bur g e r s has derived from the central equation in his well-known model of turbulence (see 2) and references given there) ov ov EJ2v -+v-=v-at oy ay2 (y distance, t time, v velocity, v kinematic viscosity) an equation for the velocity correlation function R(7j, t) = v(y, t) v(y + 7j, t).(2)The indicated average is an ensemble average; however, the ensemble is assumed "homogeneous" and hence the correlation function is, as indicated by the notation, dependent only on the separation 7j and the time t. The equation given by Bur g e r s is Here R = R (0, t), and f3 is an unspecified pure number (we shall show below that one must set f3 = 0). R(t) is at time t twice the average energy per unit mass of the "pseudo-turbulence" governed -361-