Turbulent isotropic turbulence

In many types of contactors, such as stirred tanks, rotary agitated columns, and pulsed columns, mechanical energy is appHed externally in order to reduce the drop si2e far below the values estimated from equations 36 and 37 and thereby increase the rate of mass transfer. The theory of local isotropic turbulence can be appHed to the breakup of a large drop into smaller ones (66), resulting in an expression of the form... 

Reynolds Stress Models. Eddy viscosity is a useful concept from a computational perspective, but it has questionable physical basis. Models employing eddy viscosity assume that the turbulence is isotropic, ie, u u = u u = and u[ u = u u = u[ = 0. Another limitation is that the... 

In isotropic turbulence, statistical measures of fluctuations are equal in all directions. 

Davies (Turbulence Phenomena, Academic, New York, 1972) presents a good discussion of the spectrum of eddy lengths for well-developed isotropic turbulence. The smallest eddies, usually called Kolmogorov eddies (Kolmogorov, Compt. Rend. Acad. Sci. URSS, 30, 301 32, 16 [1941]), have a characteristic velocity fluctuation given by... 

Since chemical reactions are on a scale much below 1 Im, and it appears that the Komolgoroff scale of isotropic turbulence turns out to be somewhere between 10 and 30 Im, other mechanisms must play a role in getting materials in and out of reaction zones and reactants in and out of those zones. One cannot really assign a shear rate magnitude to the area around a micro-scale zone, ana it is primarily an environment that particles and reactants witness in this area. 

In these model equations it is assumed that turbulence is isotropic, i.e. it has no favoured direction. The k-e model frequently offers a good compromise between computational economy and accuracy of the solution. It has been used successfully to model stirred tanks under turbulent conditions (Ranade, 1997). Manninen and Syrjanen (1998) modelled turbulent flow in stirred tanks and tested and compared different turbulence models. They found that the standard k-e model predicted the experimentally measured flow pattern best. 

This case can also be approached using Kolmogoroff s (K9, H15) theory of local isotropic turbulence to predict the velocity of suspended particles relative to a homogeneous and isotropic turbulent flow. By examining this situation for spherical particles moving with a constant relative velocity, varying randomly in direction, Levich, (L3) has demonstrated that... 

For isotropic turbulence, from equation 12.15, this becomes ... 

Eddy kinematic viscosity, isotropic turbulence 702 penetration of laminar sub-layer 701... 

S.S. Shy, S.I. Yang, W.J. Lin, and R.C. Su 2005, Turbulent burning velocities of premixed CH4/diluent/air flames in intense isotropic turbulence with consideration of radiation losses. Combust. Flame 143 106-118. 

Fig. 1. Dimensionless stress in fully developed turbulent flow given by the theory of isotropic turbulence...

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

Stresses acting on micro-organisms in (a) to (c) are derived on the premise that the flow forces originate from the turbulent motion of the carrier medium. In almost all cases, turbulence is assumed to be locally isotropic and homogeneous which greatly simplifies the analysis and allows the application of the Kolmogoroff s theory of turbulence to the problem [81]. The Kolomogoroff micro-scale of turbulence,... 

Brunk, B. K., Koch, D. L., and Lion, L. W., Hydrodynamic pair diffusion in isotropic random velocity fields with application to turbulent coagulation. Phys. Fluids 9,2670-2691 (1997). 

As a result, the turbulent-flow field in a stirred vessel may be far from isotropic and homogeneous. Some of the cornerstones of turbulence theory, however, start from the assumption that production and dissipation of turbulent kinetic energy balance locally. In many chemical engineering flows, this... 

In dealing with the SGS terms, Revstedt et al. (1998, 2000) and Revstedt and Fuchs (2002) did not use any model rather, they assumed these terms were just as small as the truncation errors in the numerical computations. This heuristic approach lacks physics and does not deserve copying. A most welcome aspect of LES is that the SGS stresses may be conceived as being isotropic, i.e., insensitive to effects of the larger scales, to the way the turbulence is induced and to the complex and varying boundary conditions of the flow domain. Exactly this... 

While the theoretical value (based on homogeneous, isotropic turbulence) of the Smagorinsky coefficient cs amounts to 0.165 (Mason and Callen, 1986), in many simulation studies lower values for cs proved to result in a better reproduction of experimental data. This may have to do with the abundant presence of shear flows in process equipment. Derksen (2003) reported that varying cs values in the range 0.08-0.14 does not have a large impact on the simulation results. A value of 0.12 is recommended. 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

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