Turbulent mixing length scales

Prandtl mixing length, length scale of turbulence... 

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 ... 

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

Two important length scales for describing turbulent mixing of an inert scalar are the scalar integral scale L, and the Batchelor scale A.B. The latter is defined in terms of the Kolmogorov scale r] and the Schmidt number by... 

The scalar integral scale characterizes the largest structures in the scalar field, and is primarily determined by two processes (1) initial conditions - the scalar field can be initialized with a characteristic that is completely independent of the turbulence field, and (2) turbulent mixing - the energy-containing range of a turbulent flow will create scalar eddies with a characteristic length scale I.,p that is approximately equal to Lu. 

In the LEM, turbulence is modeled by a random rearrangement process that compresses the scalar field locally to simulate the reduction in length scales that results from turbulent mixing. For example, with the triplet map, defined schematically in Fig. 4.2, a random length scale / is selected at a random point in the computational domain, and the scalar field is then compressed by a factor of three.14 The PDF for /,... 

The one-point PDF contains no length-scale information so that scalar dissipation must be modeled. In particular, the mixing time scale must be related to turbulence time scales through a model for the scalar dissipation rate. 

Cremer, M. A., P. A. McMurtry, and A. R. Kerstein (1994). Effects of turbulence length-scale distribution on scalar mixing in homogeneous turbulent flow. Physics of Fluids 6, 2143-2153. 

Turbulent mixing coefficient Constant of porportionality used in Pick s First and Second Laws to predict changes in solute concentration over time or fluxes from solute gradients that arise from turbulent mixing. The constant is a function of length scales and the degree of density stratification and hence is termed a coefficient . 

The mixing length relation caiijalso be deduced by starting with the assumption that the turbulence at any point in the flow can be characterized by a single length scale, 7, and a single velocity scale, r- From this it follows that ... 

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