Mixing length

These models are usually categorized according to the number of supplementary partial differential transport equations which must be solved to supply the modeling parameters. The so-called zero-equation models do not use any differential equation to describe the turbulent quantities. The best known example is the Prandtl (19) mixing length hypothesis ... 

Prandtl mixing length, length scale of turbulence... 

The universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the eddy viscosity,... 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

Zero equation model—mixing length model. 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

TABLE 12.9 Required Mixing Lengths (L/D) to Keep the Error due to Incomplete Mixing Smaller Than 5% or 10% ... 

A proper representation of the effective viscosity is often problematic. Based on the Prandtl mixing length model for turbulence, Bloor and Ingham-suggest that the variation in p, should be of the form... 

Prandtl has suggested that uE is likely to increase as both the mixing length kE and the modulus of the velocity gradient d t/d v increase. The simplest form of relation between the three quantities is that ... 

This is tantamount to saying that the velocity change over a distance equal to the mixing length approximates to the eddy velocity. This cannot be established theoretically but is probably a reasonable assumption. 

If the distance between the two locations is approximately equal to the mixing length XE, and if the velocity gradient is nearly constant over that distance ... 

It is assumed throughout that no mixing takes place with the intervening fluid when an eddy transports fluid elements over a distance equal to the mixing length. 

Inserting K = 0.4 in equation 12.27 gives the relation between mixing length (Ag) and distance (y) from the surface ... 

If ur is expressed as the product of the mixing length and if the proportionality constant is equal to unity ... 

If there is a temperature gradient within the fluid, the eddies will be responsible for heat transfer and an eddy thermal diffusivity Ep may be defined in a similar way. It is suggested that, since the mechanism of transfer of heat by eddies is essentially the same as that for transfer of momentum, Eh is related to mixing length and velocity gradient in a similar manner. 

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 ... 

K Ratio of mixing length to distance from surface — —... 

Explain Prandtl s concept of a mixing length . What parallels can you draw between the mixing length and the mean free path of the molecules in a gas ... 

The ratio of Ihe mixing length to the distance from the pipe wall has a constant value of 0.4 for the turbulent flow of a fluid in a pipe. What is the value of the pipe friction factor if the ratio of the mean velocity to the... 

Explain the importance of the universal velocity profile and derive the relation between the dimensionless derivative of velocity trf, and the dimensionless derivative of distance from the surface y+, using the concept of Prandtl s mixing length kE. 

It may be assumed that the fully turbulent portion of the boundary layer starts at y+ — 30, that the ratio of the mixing length to the distance y from the surface, Af/v = 0.4, and that for a smooth surface u+ = I4 at v 30. 

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+. 

Vandenberg JA, Ryan MC, Nuell DD, Chu A (2005) Field evaluation of mixing length and attenuation of nutrients and fecal coliform in a wastewater effluent plume. Environ Monit Assess 107 45-57... 

In addihon to the CFD results, estimates of the mixing length based on the Fourier number ... 

With injection mixers (Figures 10.52b,c), in which the one fluid is introduced into the flowing stream of the other through a concentric pipe or an annular array of jets, mixing will take place by entrainment and turbulent diffusion. Such devices should be used where one flow is much lower than the other, and will give a satisfactory blend in about 80 pipe diameters. The inclusion of baffles or other flow restrictions will reduce the mixing length required. 

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