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Unformatted text preview: ME 563 - Intermediate Fluid Dynamics - Su Lecture 36 - Turbulence: more on scaling, and the Reynolds stress In the last class we looked at the Kolmogorov similarity hypotheses, which expressed the idea that the smaller scales of turbulence approach a universal state for high enough Reynolds number. (The Reynolds number we’re interested in is Re = u l /ν , where u is a characteristic velocity of the flow large scale, and l is a characteristic large-scale flow dimension.) That is, while the large scales will obviously be different, being set by the different flow boundary conditions, etc., the statistics of the small scales of the flow will be the same between flows if the Reynolds numbers are high enough. Also, if you look at the smallest turbulent scales, the turbulence at those scales will be isotropic, meaning that all of the spatial directions will look the same. Based on Kolmogorov’s hypotheses, we can also extimate the size of the smallest turbulence length, time and velocity scales. From Kolmogorov’s first similarity hypothesis, the universal form of the smallest turbulence scales is dependent only on the viscosity, ν and the rate of energy dissipation, ε . The units of these terms are ν = length 2 time , ε = velocity 2 time = length 2 time 3 (1) where we’ll note that ε is really the specific rate of energy dissipation, i.e. the rate per unit mass. From these, we can form a unique length scale, η , time scale, τ η , and velocity scale, u η , as: η = ν 3 ε 1 / 4 τ = ν ε 1 / 2 u η = ( εν ) 1...
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