09-kolm

# 09-kolm - Lecture 9 Kolmogorovs Theory Applied...

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1 Lecture 9 - Kolmogorov’s Theory Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org © André Bakker (2002-2006)

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2 Eddy size Kolmogorov’s theory describes how energy is transferred from larger to smaller eddies; how much energy is contained by eddies of a given size; and how much energy is dissipated by eddies of each size. We will derive three main turbulent length scales: the integral scale, the Taylor scale, and the Kolmogorov scale; and corresponding Reynolds numbers. We will also discuss the concept of energy and dissipation spectra.
3 Jets at two different Reynolds numbers Relatively low Reynolds number Relatively high Reynolds number Source: Tennekes & Lumley. Page 22.

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4 Turbulent eddies Consider fully turbulent flow at high Reynolds number Re= UL / ν . Turbulence can be considered to consist of eddies of different sizes. An ‘eddy’ preludes precise definition, but it is conceived to be a turbulent motion, localized over a region of size l , that is at least moderately coherent over this region. The region occupied by a larger eddy can also contain smaller eddies. Eddies of size l have a characteristic velocity u(l) and timescale τ (l) l/u(l). Eddies in the largest size range are characterized by the lengthscale l 0 which is comparable to the flow length scale L . Their characteristic velocity u 0 u(l 0 ) is on the order of the r.m.s. turbulence intensity u’ (2 k /3) 1/2 which is comparable to U . Here the turbulent kinetic energy is defined as: The Reynolds number of these eddies Re 0 u 0 l 0 / ν is therefore large (comparable to Re) and the direct effects of viscosity on these eddies are negligibly small. ) ' ' ' ( 2 2 2 2 1 2 1 w v u u u k i i + + = < =
5 Integral scale We can derive an estimate of the lengthscale l 0 of the larger eddies based on the following: Eddies of size l 0 have a characteristic velocity u 0 and timescale τ 0 l 0 /u 0 Their characteristic velocity u 0 u(l 0 ) is on the order of the r.m.s. turbulence intensity u’ (2 k /3) 1/2 Assume that energy of eddy with velocity scale u 0 is dissipated in time τ 0 We can then derive the following equation for this length scale: Here, ε (m 2 /s 3 ) is the energy dissipation rate. The proportionality constant is of the order one. This lengthscale is usually referred to as the integral scale of turbulence. The Reynolds number associated with these large eddies is referred to as the turbulence Reynolds number Re L , which is defined as: 3/ 2 0 k l ε 1/ 2 2 0 Re L k l k ν εν = =

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6 Energy transfer The large eddies are unstable and break up, transferring their energy to somewhat smaller eddies. These smaller eddies undergo a similar break-up process and transfer their energy to yet smaller eddies.
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