Batchelor scale

The ability to resolve the dissipation structures allows a more detailed understanding of the interactions between turbulent flows and flame chemistry. This information on spectra, length scales, and the structure of small-scale turbulence in flames is also relevant to computational combustion models. For example, information on the locally measured values of the Batchelor scale and the dissipation-layer thickness can be used to design grids for large-eddy simulation (LES) or evaluate the relative resolution of LES resulfs. There is also the potential to use high-resolution dissipation measurements to evaluate subgrid-scale models for LES. 

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... 

Like the Kolmogorov scale in a turbulent flow, the Batchelor scale characterizes the smallest scalar eddies wherein molecular diffusion is balanced by turbulent mixing.3 In gas-phase flows, Sc 1, so that the smallest scales are of the same order of magnitude as the Kolmogorov scale, as illustrated in Fig. 3.1. In liquid-phase flows, Sc 1 so that the scalar field contains much more fine-scale structure than the velocity field, as... 

Figure 3.1. Sketch of Batchelor-scale scalar field in a gas-phase flow. 

We have set the proportionality constant in (2.68) equal to unity. Applying the resultant formula at the Batchelor scale suggests that it may be closer to 0.5 (Batchelor 1959). 

This process continues until 1 reaches the Batchelor scale, where diffusion takes over and quickly destroys all scalar gradients. The scalar field is then completely mixed, i.e.,... 

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... 

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. 

Figure 4.8. Sketch of wavenumber bands in the spectral relaxation (SR) model. The scalar-dissipation wavenumber kd lies one decade below the Batchelor-scale wavenumber kb. All scalar dissipation is assumed to occur in wavenumber band [/cd, oo). Wavenumber band [0, k ) denotes the energy-containing scales. The inertial-convective sub-range falls in wavenumber bands [k, k3 ), while wavenumber bands [/c3, /cD) contain the viscous-convective sub-range.

Further reduction in size down to the Batchelor scale with negligible change in concentration at a rate that is proportional to (e/v)1/2. 

Molecular diffusion and reaction in Batchelor-scale lamella. 

The length scales for the turbulent concentration field range from the plume width to the scale at which molecular diffusion acts to homogenize the distribution (or dissipate the variance of the scalar fluctuations). The smallest length scale is referred to as the Batchelor scale and is estimated as... 

Note that the inertial range and high Re are not necessary for the existence of this type of Batchelor scaling and it can be also produced by chaotic advection in spatially smooth unsteady large scale flows with relatively small or moderate Reynolds number, or in two-dimensional flows (Jullien et ah, 2000 Pierrehumbert, 2000). 

In liquids with Schmidt nmnbers So 10 the Batchelor scale Ag is very small in comparison to the Kolmogoroff scale A (Ag < 0.03 Aj ). 

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