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Unformatted text preview: T URBULENCE AND S IGNAL A NALYSIS Pierre Borgnat (email Pierre.Borgnat@enslyon.fr) Laboratoire de Physique (UMRCNRS 5672) ENS Lyon 46 allee dItalie 69364 Lyon Cedex 07 Abstract Turbulence deals with the complex motions in fluid at high velocity and/or involving a large range of lengthscales. 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 nonstationarity. A second part will be devoted to the study of one signal processing framework, the decomposition of selfsimilar 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; is its density. It obeys the fundamental relation of dynamics: D t u = f , where f are the forces: friction, pressure forces, gravity ( f = g ),... This equation is linear but nonlocal 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 NavierStokes (NS) equation, reads then as: D t v = t v  {z } local derivative + ( v ) v  {z } convective derivative = 1 p + v  {z } viscuous friction + X f v . (1) Friction in the fluid (supposed newtonian) is explicitely written here and it is proportional to the vicosity . For an uncompressible flow, the continuity equation v = 0 completes the problem. Remark that the pressure term is nonlocal because of a Poisson equation that relates p to v : p = 2 ( v i v j ) /x i x j . 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....
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
 Spring '11
 Staff
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