Burgers equation with a passive scalar: Dissipation anomaly and Colombeau calculus

A connection between dissipation anomaly in fluid dynamics and Colombeau’s theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the well-known viscosity-independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate ϵθϵθ of the passive scalar on the Prandtl number PrPr is given by a simple analysis: we show, in particular, ϵθ∝1/Pr−−√ϵθ∝1/Pr for large PrPr. The passive scalar profile is shown to have a form of a sum of tanh2n+1 xtanh2n+1 x with suitably scaled xx, thereby implying the necessity to distinguish HH from HnHn when PrPr is large, where HH is the Heaviside function and nn is a positive integer. An incorrect result of ϵθ∝1/Prϵθ∝1/Pr would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole–Hopf-type transform is also given for the case of unit Prandtl number.