wavelet transforms and their applications to turbulence - Wavelets ...

Annual Reviewswww.annualreviews.org/aronlineWAVELET TRANSFORMS 433each location has the same energy spectrum, which then corresponds tothe Fourier energy spectrum. I(l, x0) -- 10 means that the point x0 contributes10 times more than the average over x to the Fourier energyspectrum at scale I.Frick & Mikishev (1990) had previously defined a similar intermittencymeasure using the Zimin hierarchical model (Zimin 1981).SPACE-SCALE REYNOLDS NUMBER AND GLOBAL INTERMITTENCY MEASUREFarge et al (1990b) have introduced a space-scale Reynolds number~(l, x)Re (l, x) , (60)where v is the kinematic viscosity of the fluid and g the characteristic rmsvelocity at scale l and location x such as:f(l,x) = (3C0) -I ~ [~7,(l,x)[ 2 . (61)At large scales 1 ~ L,.Re (L) = Re (L, x)cocoincides with the usual large-scale Reynolds number Re = v’L/v, wherev" is the rms turbulent velocity and L the integral scale of the flow. At theKolmogorov scale l ~ r/, where dissipation effects equilibrate nonlineareffects, Re(r/, x)= 1. In plotting the iso-surface Ref/,x)= 1 we can thencheck whether it is fiat or not. If it is flat, then 1 = r/ everywhere, asassumed by Kolmogorov’s theory. If it is not flat, then the turbulent flow isintermittent and we can no longer define a unique Kolmogorov scalebut only a range of scales from r/rain to r/~-~x" The ratio /(Re) = r/max/r/rain,for Re = 1, is a global measure of the flow intermittency in the dissipativerange. The ratio/(Re) lmax(Re)/lmin(Re), for Re>> 1, measures the globalflow intermittency in the inertial range.For direct numerical simulations of turbulent flow, this iso-surfaceRe(l, x) = 1 may be useful in detecting numerical errors and verifyingthe space resolution Ax is sufficient to resolve the dissipative scales everywherein the flow. We have to verify that r/rain > AX everywhere in the flow,otherwise for points where this inequality is not verified some numericalinstability may develop.For numerical simulations we can define a better space-scale Reynoldsnumber. For this we must compute the nonlinear term N = II v o Vv II and