Renormalization-group theory for the propagation of a turbulent burst

Abstract

We consider the propagation of a plane front separating a turbulent region of fluid from a quiescent region. Initially, the turbulent-energy distribution as a function of z, the displacement normal to the front, is assumed to be localized, and after a time t, general renormalization-group arguments show that there is a similarity solution of the form q(z,t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}(2/3+2\mathrm{\ensuremath{\alpha}}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}})}$f (${\mathit{zt}}^{\mathrm{\ensuremath{-}}(2/3+\mathrm{\ensuremath{\beta}})}$, \ensuremath{\epsilon}), where \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{} and \ensuremath{\beta} are \ensuremath{\epsilon}-dependent anomalous dimensions, satisfying the scaling law \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}+\ensuremath{\beta}=0 and \ensuremath{\epsilon} is a measure of the dissipation. Using perturbation theory, we calculate values of \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{} and \ensuremath{\beta} to O(\ensuremath{\epsilon}), which are in good agreement with numerical calculations, and we explicitly verify the above scaling law and find the form of the scaling function f.