Scalar length scales

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. 

The total mixing time can be computed by inserting the mixing rate y(Jp,) into the following simple phenomenological model for the scalar length scale ... 

As shown below, the rate of change of / (t) is determined from the length-scale distribution of the scalar field as characterized by the scalar energy spectrum. 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

As mentioned above, the one-point PDF description does not provide the length-scale information needed to predict the decay rate of the scalar variance. For this purpose, a... 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

Thus, E k, t) Ak represents the amount of scalar variance located at wavenumber k. For isotropic turbulence, the scalar integral length scale is related to the scalar energy spectrum by... 

Figure 3.15. The scalar-to-velocity length-scale ratio for a fully developed scalar energy spectrum as a function of the Schmidt number at various Reynolds numbers Rk = 50,100, 200, 400, and 800. The arrow indicates the direction of increasing Reynolds number.

Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. For the simplest case of two inert scalars, this implies F / and y 2 > 1 (see (3.140)). In homogeneous turbulence, one effect of differential diffusion is to de-correlate the scalars. This occurs first at the diffusive scales, and then backscatters to larger scales until the energy-containing scales de-correlate. Thus, one of the principal difficulties of modeling differential diffusion is the need to account for this length-scale dependence. 

When the Schmidt number is greater than unity, addition of a scalar transport equation places a new requirement on the maximum wavenumber K. For Sc > 1, the smallest characteristic length scale of the scalar field is the Batchelor scale, 7b- Thus, the maximum wavenumber will scale with Reynolds and Schmidt number as... 

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 last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

The SR model introduced in Section 4.6 describes length-scale effects and contains an explicit dependence on Sc. In this section, we extend the SR model to describe differential diffusion (Fox 1999). The key extension is the inclusion of a model for the scalar covariance (4> aft p) and the joint scalar dissipation rate. In homogeneous turbulence, the covari-... 

Reduction in size down to the Kolmogorov scale with no change in concentration at a rate that depends on the initial scalar length scale relative to the Kolmogorov scale. 

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. 

Couple it with a model for the joint scalar dissipation rate that predicts the correct scalar covariance matrix, including the effect of the initial scalar length-scale distribution. 

Most molecular mixing models concentrate on step (1). However, for chemical-reactor applications, step (2) can be very important since the integral length scales of the scalar and velocity fields are often unequal (L / Lu) due to the feed-stream configuration. In the FP model (discussed below), step (1) is handled by the shape matrix H, while step (2) requires an appropriate model for e. 

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. 

Mell, W. E., G. Kosaly, and J. J. Riley (1991). The length-scale dependence of scalar mixing. Physics of Fluids A Fluid Dynamics 3, 2474—2476. 

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