Turbulence and Signal Analysis

Turbulence deals with the complex motions in fluid at high velocity and/or involving a large range of length-scales. Turbulence asks then many questions from modeling this complexity to measuring it. In a first part, the description of signals measured in fluid turbulence experiments will be made along with a survey of modern signal processing tools that are adapted to their properties of scaling laws, multifractalty and non-stationarity. A second part will be devoted to the study of one signal processing framework, the decomposition of self-similar signals on the Mellin oscillating functions, that is a new way to probe jointly scale invariance and local organization of a signal. 1 Turbulence: experimental signals and signal processing tools 1.1 Preliminary analysis of fluid turbulence Formalization of the problem. Turbulence is first a problem of fluid mechanics [Bat67]. Let u(r(0); t) be the Lagrangian velocity of a fluid element that is in r(0) at initial time; ρ is its density. It obeys the fundamental relation of dynamics: ρDtu = ∑ ρf , where f are the forces: friction, pressure forces, gravity (f = ρg),... This equation is linear but non-local because of the pressure term. If p is the pressure, the corresponding force is −∇p which is linked to the whole velocity field. Added to that, it is experimentally hard to track the movement of one fluid element in a fluid. So, instead of the Lagrangian velocity, the problem is often studied with the corresponding Eulerian velocity v(r, t) at the fixed position r. Both velocities are related via the change of variable u(r(0); t) = v(r(t); t). The equation for the Eulerian velocity, called the Navier-Stokes (NS) equation, reads then as: Dtv = ∂tv } {{ } local derivative + (v · ∇)v } {{ } convective derivative = − ρ ∇p+ ν∆v } {{ } viscuous friction + ∑ fv. (1) Friction in the fluid (supposed newtonian) is explicitely written here and it is proportional to the vicosity ν. For an uncompressible flow, the continuity equation ∇ · v = 0 completes the problem. Remark that the pressure term is non-local because of a Poisson equation that relates p to v: ∆p = −∂(vivj)/∂xi∂xj . One should also specify the boundary conditions: the velocity of the fluid is zero at the boundaries. One simple approach adopted by physicists is to study turbulence in open systems far from the boundaries in order to find a possible generic behaviour of a turbulent fluid, disregarding the specific geometry of the boundaries. Signals of Eulerian velocity. Experiments of turbulence consists in studying high speed motions in a fluid where the flow is disturbed by means of a grid or by creating a jet. The flow becomes turbulent, and the velocity of the fluid is recorded along time at some position. Common apparatus are hot-wire probes that measure one component v(t) of the velocity at one point only (we will discuss only single probe measurements here). A sample velocity signal v(t) is shown in figure 1, obtained during the experiment GReC [PPB+03] in a jet at high Reynolds number (up to 107) in helium at 4.5 K (so that its viscosity is very low). The erratic fluctuations are typical of such signals and one can see numerous points where the signal appears almost singular. The singular fluctuations are clearer for the dissipative signal, that is measured as the time derivative of its energy v(t)2/2; it seems made of numerous peaks of variable amplitudes, separated by periods of almost