Number turbulent Schmidt

Miller, P. L. (1991). Mixing in High Schmidt Number Turbulent Jets. Ph. D. thesis, California Institute of Technology. 

Miller, P. L. and P. E. Dimotakis (1991). Reynolds number dependence of scalar fluctuations in a high Schmidt number turbulent jet. Physics ofFluids A FluidDynamics 3,1156-1163. (1996). Measurements of scalar power spectra in high Schmidt number turbulent jets. 

Morton number Reynolds number of a bubble Schmidt number turbulent Schmidt number Weber number... 

Matrix of inverted mass transfer coefficients, m s Ratio of fluemating velocity dissipation time and fluctuating concentration dissipation time Source term Schmidt number Turbulent Schmidt number Time, s... 

It is not possible to translate the above reasoning to turbulent flow, as turbulent flow equations are not reliable. However, in practice it is typical to assume that the same analogy is also valid for turbulent flow. Because of this hypothesis level, it is quite futile to use the diffusion factor D g in the Schmidt number instead we will directly use the number D g as in the Sherwood number. Hence in practical calculations Sc = v/D b-... 

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

EjEh is termed the Turbulent Prandtl Number and E/Ep the Turbulent Schmidt Number. 

Deissler, R. G., 1955, Analysis of Turbulent Heat Transfer, Mass Transfer, and Friction in Smooth Tubes at High Prandtl and Schmidt Numbers, NACA Rep. 1210, Lewis Res. Ctr., Cleveland, OH. (5)... 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

Hubbard and Lightfoot (HI la) earlier reported a Sc,/3 dependence on the basis of measurements in which the Schmidt number was varied over a very large range. The data did not exclude a lower Reynolds number exponent than 0.88, and reaffirmed the value of the classical Chilton-Colburn equation for practical purposes. Recent measurements on smooth transfer surfaces in turbulent channel flow by Dawson and Trass (D8) also firmly suggest a Sc13 dependence and no explicit dependence of k+ on the friction coefficient, with Sh thus depending on Re0,875. The extensive data of Landau... 

To summarize, a comprehensive understanding of turbulent transport is not yet achieved, and information will be needed from optical as well as from further mass-transfer measurements. The latter will have to be made at high Reynolds numbers (> 50,000 in channel flow) and at very high Schmidt numbers (> 10,000) to yield critical information about the transfer process. 

The turbulent diffusivity is defined by introducing a so-called turbulent Schmidt number ScT ... 

In Fig. 9, the distribution of reactant C is shown in each environment. As cc is a linear combination of and Y2 (Eq. 78), we can distinguish features of both Fig. 7 and Fig. 8 in the plots in Fig. 9. In particular, because C is injected in the right-hand inlet stream, cC2 and 2 appear to be quite similar. Finally, as shown in Liu and Fox (2006), the CFD predictions for the outlet conversion X are in excellent agreement with the experimental data of Johnson and Prud homme (2003a). For this reactor, the local turbulent Reynolds number ReL is relatively small. The good agreement with experiment is thus only possible if the effects of the Reynolds and Schmidt numbers are accounted for using the correlation for R shown in Fig. 4. Further details on the simulations and analysis of the CFD results can be found in Liu and Fox (2006). 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

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... 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

The sub-grid-scale turbulent Schmidt number has a value of Scsgs 0.4 (Pitsch and Steiner 2000), and controls the magnitude of the SGS turbulent diffusion. Note that due to the filtering process, the filtered scalar field will be considerably smoother than the original field. For high-Schmidt-number scalars, the molecular diffusion coefficient (T) will be much smaller than the SGS diffusivity, and can thus usually be neglected. 

Note that the right-hand sides of these expressions can be extracted from DNS data for homogeneous turbulence in order to explore the dependence of the rate constants on Rei and Sc. Results from a preliminary investigation (Fox and Yeung 1999) for Rx = 90 have revealed that the backscatter rate constant from the dissipative range has a Schmidt-number dependence like/Son Sc1/2 for Schmidt numbers in the range [1/8, 1], On the other hand, for cut-off wavenumbers in the inertial-convective sub-range, one would expect a 1) and... 

Calmet, I. and J. Magnaudet (1997). Large-eddy simulation of high-Schmidt number mass transfer in a turbulent channel flow. Physics of Fluids 9,438 155. 

The isotropic turbulent mixer Part II. Arbitrary Schmidt number. AIChE Journal 10, 870-877. 

Dahm, W. J. A. and P. E. Dimotakis (1990). Mixing at large Schmidt number in the self-similar far field of turbulent jets. Journal of Fluid Mechanics 217, 299-330. 

Yeung, P. K., S. Xu, and K. R. Sreenivasan (2002). Schmidt number effects on turbulent transport with uniform mean scalar gradient. Physics of Fluids 14, 4178 -191. Zwietering, T. N. (1959). The degree of mixing in continuous flow systems. Chemical Engineering Science 11, 1-15. 

---