Batchelor length scales

In stirred chemical reactors, unlike in combustion and with other gas-phase reactions, these closure terms should take into account that for liquids the Schmidt number (Sc = v/D) is in the order 100-1,000, and, hence, the role of species diffusion at scales within the Kolmogorov eddies should explicitly be taken into account (Kresta and Brodkey, 2004). Essential is that diffusion of chemical species is governed by the Batchelor length scale rjB which obeys to... 

An alternative approach (e.g., Patterson, 1985 Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent flow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still fluctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS fluctuations. 

The initial length L(0) is usually set equal to the Batchelor length scale >.B, in which case the initial conditions are... 

For the species concentration field, the scalar integral scale and the Batchelor scale characterize respectively the largest and the smallest eddies. In most cases, the scalar integral scale is approximately equal to the turbulence integral scale. The Batchelor length scale, on the other hand, is related to the Kolmogorov length scale via the Schmidt number, Sc=nlpD ... 

Batchelor length scale for mass transfer where Sc is large... 

So the use of N = 60 rpm (or higher) and exact geometric similarity will ensure that the Batchelor length scale for scalar mixing is preserved on scale-up. 

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

Batchelor (1959) developed an expression for the smallest concentration (or temperature) striation based on the argument that for diffusion time scales longer than the Kolmogorov scale, turbulence would continue to deform and stretch the blobs to smaller and smaller lamellae. Only once the lamellae could diffuse at the same rate as the viscous dissipation scale would the concentration striations disappear. The Batchelor length scale is the size of the smallest blob that can diffuse by molecular diffusion in one Kolmogorov time scale. Using the lamellar diffusion time from eq. (13-6) gives... 

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... 

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. 

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... 

Vector fields whose divergence vanishes are sometimes referred to as solenoidal. A more comprehensive discussion of the conditions for approximating the velocity field as solenoidal has been given by Batchelor.8 These imply that, in cases in which the fluid is subjected to an oscillating pressure, the characteristic velocity in the Mach number condition should be interpreted as the product of the frequency times the linear dimension of the fluid domain, and that the difference in static pressures over the length scale of the domain must be small compared with the absolute pressure. Because our subject matter will frequently deal with incompressible, isothermal fluids, we shall often make use of (2-20) in lieu of the... 

This is the Batchelor spectrum, that is valid in the viscous-convective range that extends from the Kolmogorov scale down to the diffusive scale, where the scalar variance is finally dissipated by molecular diffusion. The diffusive scale in this case is the length scale at which the diffusion time l2/D is comparable to the timescale of advection corresponding to the Kolmogorov scale eddies, that gives... 

This analysis is limited to cases where the molecular diffusivity is slow relative to the momentum diffusivity (kinematic viscosity), so that Sc > 1. In the same way that the Kolmogorov scale provides a limit where turbulent stresses are balanced by viscous stresses, the Batchelor scale provides a limiting length scale where the rate of molecular diffusion is equal to the rate of dissipation of turbulent kinetic energy. Below this scale, distinct packets of dye will quickly be absorbed into the bulk fluid by molecular diffusion, where our meaning of quickly is now consistent between the energy dissipation and molecular diffusion. 

Figure 2-13 shows the gross characteristics of the velocity and concentration spectra. For a low viscosity liquid. Sc can be on the order of 1000, so the Batchelor scale can be 30 times smaller than the Kolmogorov scale. The ultimate scale of mixing needed for reaction is the size of a molecule, so in liquid-phase reactions, molecular diffusion is critically important for the final reduction in scale. For a gas, Sc is closer to 1, so the ratio is closer to 1, and the competition between the turbulent reduction in scale and molecular diffusion occurs at the same range of wavenumbers. The various length scales shown in Figure 2-13 are also summarized in Table 2-3... 

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