On the production of dissipation through the interaction of forced oscillating waves in fluid dynamics

Christophe Cheverry (1) , Aurélien Klak (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

This article is devoted to the study of some bidimensional fluid model with a source term implying oscillations. In this context, we can construct a family of exact solutions u(epsilon) indexed by epsilon is an element of ]0,1], defined on a strip [0, T] x R-2 with T is an element of R+* fixed, and involving three types of scales: epsilon(-2)t for a boundary layer at time t = 0, epsilon(-2)x(1) for oscillations in the direction x(1), and epsilon(-1)x(2) for a concentration near the position x(2) = 0. We show that the propagation and the interaction of these singularities produce a turbulent effect that manifests itself through the addition of some (unusual) diffusion when describing the underlying modulation equations.

Mots clés: nonlinear geometric optics; WKB calculus; oscillations; turbulent flows; instabilities in fluid dynamics; equations of mixed hyperbolic and parabolic type

Domaines: Equations aux dérivées partielles [math.AP]

Christophe Cheverry, Aurélien Klak. On the production of dissipation through the interaction of forced oscillating waves in fluid dynamics. Analysis and Applications, 2014, 12 (1), pp.1-61. ⟨10.1142/S0219530513500231⟩. ⟨hal-00951252⟩