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Where is the integral time scale, L is the integral length scale, and () and () are the autocorrelation with respect to time and space respectively. In isotropic homogeneous turbulence, the integral length scale ℓ {\displaystyle \ell } is defined as the weighted average of the inverse wavenumber , i.e.,
The Taylor microscale is the intermediate length scale at which fluid viscosity significantly affects the dynamics of turbulent eddies in the flow. This length scale is traditionally applied to turbulent flow which can be characterized by a Kolmogorov spectrum of velocity fluctuations. In such a flow, length scales which are larger than the ...
where ε is the average rate of dissipation of turbulence kinetic energy per unit mass, and; ν is the kinematic viscosity of the fluid.; Typical values of the Kolmogorov length scale, for atmospheric motion in which the large eddies have length scales on the order of kilometers, range from 0.1 to 10 millimeters; for smaller flows such as in laboratory systems, η may be much smaller.
The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as
Here l is the turbulence or eddy length scale, given below, and c μ is a k – ε model parameter whose value is typically given as 0.09; =. The turbulent length scale can be estimated as =, with L a characteristic length. For internal flows this may take the value of the inlet duct (or pipe) width (or diameter) or the hydraulic diameter.
It is a two-equation model which gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows. k–ω (k–omega)
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All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov microscales), up to the integral scale, associated with the motions containing most of the kinetic energy.