Kolmogorov turbulent

The phase structure function for a separation r is defined as the mean value of the square difference of phase for all points with that separation. For wave-fronts affected by Kolmogorov turbulence it is given by... 

Davies (Turbulence Phenomena, Academic, New York, 1972) presents a good discussion of the spectrum of eddy lengths for well-developed isotropic turbulence. The smallest eddies, usually called Kolmogorov eddies (Kolmogorov, Compt. Rend. Acad. Sci. URSS, 30, 301 32, 16 [1941]), have a characteristic velocity fluctuation given by... 

Typical quoted values for the Kolmogorov microscale of turbulence for agitated vessels are normally in the range 25 —50 pm. 

For turbulent fluid-indueed stresses aeting on partieles it is neeessary to eon-sider the strueture and seale of turbulenee in relation to partiele motion in the flow field. There is as yet, however, no eompletely satisfaetory theory of turbulent flow, but a great deal has been aehieved based on the theory of isotropie turbulenee (Kolmogorov, 1941). 

Abstract This is a tutorial about the main optical properties of the Earth atmosphere as it affects incoming radiation from astrophysical sources. Turbulence is a random process, of which statitical moments are described relying on the Kolmogorov model. The phase structure function and the Fried parameter ro are introduced. Analytical expressions of the degradation of the optical transfer function due to the turbulence, and the resulting Strehl ratio and anisoplanatism are derived. 

Keywords atmosphere, turbulence, Kolmogorov model, phase stmcture function, transfer... 

By its random nature, turbulence does not lend itself easily to modelling starting from the differential equations for fluid flow (Navier-Stokes). However, a remarkably successful statistical model due to Kolmogorov has proven very useful for modelling the optical effects of the atmosphere. 

The Kolmogorov velocity field mixes packets of air with different passive scalars a passive scalar being one which does not exchange energy with the turbulent velocity flow. (Potential) temperature is such a passive scalar and the temperature fluctuations also follow the Kolmogorov law with a different proportionality constant... 

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

Although vortices of small scale, such as Kolmogorov scale or Taylor microscale, are significant in modeling turbulent combustion [4,6-9], vortices of large scale, in fhe order of millimeters, have been used in various experiments to determine the flame speed along a vorfex axis. 

In reactor design, it is very important to know how and where turbulence is generated and dissipated. In a liquid phase, it is also important that the smallest eddies are sufficiently small. The ratio between the reactor scale (I) and the smallest turbulent scale, the Kolmogorof scale rj), usually scales as L/x]aR . The Kolmogorov scale can also be estimated from the viscosity and the power dissipation T] = (v 30 xm in water with a power input of 1W kg and from the Bachelor scale 3 pm in liquids. For a liquid, the estimation of the time... 

Rate of turbulence production (m s ) Velocity of a turbulent eddy of size X Rate of turbulence dissipation (m s ) Kolmogorov length scale (m)... 

The validity of Eqs. (3-5) are bond on the condition of fully developed turbulent flow which only exists if the macro turbulence is not influenced by the viscosity. This is the case if the macro turbulence is clearly separated from the dissipation range by the inertial range. This is given if the macro scale A is large in comparison to Kolmogorov s micro scale qp Liepe [1] and Mockel [24] found out by measurement of turbulence spectra s the following condition ... 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

An alternative approach (e.g., Patterson, 1985 Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent flow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still fluctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS fluctuations. 

Frisch, U., Turbulence, the Legacy of A. N. Kolmogorov . Cambridge University Press, Cambridge, UK (1995). 

In this definition, ps and pt are the solid and fluid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the flow in the drag law). The kinematic viscosity of the fluid is vf and y is a characteristic strain rate for the flow. In a turbulent flow, y can be approximated by l/r when ds is smaller than the Kolmogorov length scale r. (Unless the turbulence is extremely intense, this will usually be the case for fine particles.) Based on the Stokes... 

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