Time scale Kolmogorov

Finally, Bakker and Van den Akker calculated local values for the specific mass transfer rate kfl, by estimating local Ay-values from local values of the Kolmogorov time scale /(v/e) and by deriving local values of the specific interfacial area a from local values for bubble size and bubble hold-up. 

In the CRE literature, turbulence-based micromixing models have been proposed that set the micromixing time proportional to the Kolmogorov time scale ... 

Note that the turbulence integral time scale is related to the Kolmogorov time scale r = (-) (2-39)... 

By definition, the dissipation range is dominated by viscous dissipation of Kolmogorov-scale vortices. The characteristic time scale rst in (2.74) can thus be taken as proportional to the Kolmogorov time scale rn, and taken out of the integral. This leads to the final form for (2.70),... 

This expression was derived originally by Batchelor (1959) under the assumption that the correlation time of the Kolmogorov-scale strain rate is large compared with the Kolmogorov time scale. Alternatively, Kraichnan (1968) derived a model spectrum of the form... 

Again, the use of the Kolmogorov time scale in the second term implies that 1 < Sc. 

The standard example is A + B - P, where the reaction time scale is much shorter than the Kolmogorov time scale. 

Surface tension, Pa m Stress tensor. Pa Particle response time, s Kolmogorov time scale, s Chemical conversion... 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

Performing a dimensional analysis and expressing the characteristic time as a function of e and v, one obtains the Kolmogorov time scale given by... 

The latter of these numbers corresponds to the inverse of the number of Damkbhler, defined with the Kolmogorov time scale. With these definitions, the following relations are given for nV-SL and ///f in terms of the dimensionless numbers Re, Da, and Ka... 

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... 

A proper analysis is needed to assess which of the above time scales are relevant in the case of interest. The nondimensional Stokes number is widely used to denote the ratio of the particle relaxation time to a fluid relaxation time (see also Ghatage et al, 2013). Depending on the apphcation and the pertinent fluid-particle interaction model, the fluid relaxation time is either the Kolmogorov time scale (Balachandar, 2009 Collins and Keswani, 2004 Olivieri et al, 2014 Pai and Subramaniam, 2012) or the integral time scale (Derksen, 2003 Lane et al, 2005 Derksen et al, 2008 Van Wageningen et al, 2004) or the eddy lifetime (Eaton, 2009 Moraga et al, 2003). 

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