Kinematic viscosity mixing length

Whereas the kinematic viscosity fx/p, the thermal diffusivity k/Cpp, and the diffusivity D are physical properties of the system and can therefore be taken as constant provided that physical conditions do not vary appreciably, the eddy coefficients E, Eh, and ED will be affected by the flow pattern and will vary throughout the fluid. Each of the eddy coefficients is proportional to the square of the mixing length. The mixing length will ... 

In the universal velocity profile a dimensionless velocity + is plotted against lnyf, where y+ is a dimensionless distance from the surface. For the region where eddy transport dominates (eddy kinematic viscosity kinematic viscosity), the ratio of the mixing length (Ag) to the distance (>>) from the surface may be taken as approximately constant and equal to 0.4. Obtain an expression for d +/dy+ in terms of y+. 

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

Comparing equations 2.59 and 2.62, it follows that the eddy kinematic viscosity can be expressed in terms of the mixing length ... 

Transfer by molecular diffusion is discussed in Section 12.2 and the concept of the mixing length in Section 12.3.2. By analogy with kinetic theory, the eddy kinematic viscosity, E, is given by ... 

The importance of the universal velocity profile is discussed in Section 12.4. From equation 12.18, for isotropic turbulence, the eddy kinematic viscosity, EaXEuE where XE is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. The momentum transfer rate per unit area in a direction perpendicular to the surface at position y is then ... 

The target quantity of this stirring operation, the mixing time 6 or 0m - in the following for the sake of simplicity shortened to 0 - depends, for a material system without density and viscosity differences and for a particular stirrer type under given geometric conditions, only upon the characteristic measurement of length (stirrer diameter d), both material parameters (density p and kinematic viscosity v) and the stirrer speed n ... 

Instead, the Kolmogorov micro-scales (1.328) to (1.330) characterize the dissipation scales of turbulence and might indicate the effectiveness of micromixing in the flow provided that the turbulent energy dissipation rate and the kinematic viscosity of the fluid is known. If the Kolmogorov micro length scale is much larger than the molecular scales, the molecules are not efficiently mixed by turbulent diffusion. 

The Smagorinsky Model (cf. Ref. [51]) is an algebraic model in the same spirit as the Prandtl mixing length model discussed in section RANS Turbulence Modeling. In the Smagorinsky model, the SGS stresses are assumed to be proportional to the rate of strain, that is, = VtS, and the kinematic eddy viscosity is determined from the expression... 

Batchelor length scale, (vD /e) " (m) absolute viscosity (kg/m s) kinematic viscosity (m /s) density (kg/m ) mixing time constant (s)... 

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