Pseudo-turbulent

The last terms in each of Eqn. (5.2-9) and Eqn. (5.2-10) represent the divergence of the deviatoric stress including viscosity and pseudo-turbulence. The quantities /n L and uG are effective viscosities of each phase including bulk and shear viscosities. Fl and Fg represent the volume-averaged forces exerted on the liquid-and gas-phase (respectively) by the other phases across the common interfaces. 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

Van t Riet K, Bruijn W, Smith JM (1976) Real and Pseudo-Turbulence in the Discharge Stream Prom a Rushton Turbine. Chem Eng Sci 31 407-412 Van t Riet K, Smith JM (1975) The trailing vortex system produced by Rushton turbine agitators. Chem Eng Sci 30 1093-1105... 

The flow regimes discussed above are often used to describe systems with 0i 1. Other flow regimes occur for 0i 1 (e.g. bubbly flows) because additional force models (such as added mass) introduce new dimensionless parameters, and because the momentum of the continuous phase becomes dominant. For example, in bubbly flow when RCp > 1 the turbulent liquid wakes behind bubbles lead to pseudo-turbulence (Mudde et al, 2009 Riboux et al, 2010 Sato Sekoguchi, 1975) that changes the nature of bubble-bubble interactions through the continuous phase. 

CONSERVATION EQUATIONS FOR THE DISPERSED PHASE, 128 CONSERVATION EQUATIONS FOR THE CONTINUOUS PHASE, 134 SIMPLIFIED MODELS OF DISPERSE FLOW, 134 MODELING PSEUDO-TURBULENCE, 136... 

On the other hand, an analysis of the extreme case of coarse dispersions is more difficult, in a sense, than an analysis of the opposite extreme of fine suspensions. This is due to the mere fact that particles in Ene suspensions interact only hydro-dynamically. Although this means that there is no need to consider direct particle collisions, the problem of formulating both the conservation and rheological equations remains difficult because hydrodynamic interactions involve many particles simultaneously in fine particle suspensions. A sophisticated statistical theory of Brownian suspensions is now being developed by Brady and his co-workers that might help to tackle this problem [11-13]. An attempt to take into account pseudo-turbulent fluctuations in finely dispersed suspensions is described in [14,15]. It is quite evident that any generalization of these models of fine collisionless suspensions to coarse collisional suspensions involves, first of all, the necessity to account for direct collisions, and this is certainly a matter of some difficulty. 

Equations 3.1-3.5 determine both the mean force of interphase interaction and the fluctuation of this force. The mean force will be used later when formulating conservation equations for mean suspension flow. Force fluctuation is sorely needed to study properties of pseudo-turbulent motion. It should be noted that the last term in Equation 3.5 had been omitted in a similar analysis of pseudo-turbulence [25]. 

This flux approximately equals the volume flux in a fictitious suspension of the same particles at the same mean concentration but without fluctuations. However, these two fluxes by no means identically coincide. Pseudo-turbulent fluctuations cause the appearance of an additional component that is added to the total flux and that usually differs from zero. 

The other constituent of the work done by fluctuations is connected with the action of the interphase interaction force fluctuation. It is described by the two terms in Equation 4,3 that contain (f ) and f. These terms give both the energy input into the pseudo-turbulent motion from the mean relative fluid flow and the dissipation of fluctuation energy by viscous forces, henceforth denoted by q and q, respectively. The sum of the two mentioned terms is equal to q - q Obviously,... 

In the first approximation, it is evidently permissible to neglect the second term on the right-hand side of Equation 5.3 as opposed to the first. We can do this in view of the fact that thermal motion velocity of fluid molecules considerably exceeds the characteristic fluid fluctuation velocity. For the same reason, the difference between p and fluid molecular pressure p may be ignored. Obviously, to improve the accuracy of this model it is necessary, on the basis of a detailed theory of pseudo-turbulent motion, to calculate tensor (v v ) and, moreover, to determine flux (( ) v ) which enters into Equation 5.1. 

As a first approximation, small pseudo-turbulent corrections to volume fluid and particle fluxes may be ignored. Similarly, the difference between interphase interaction forces that act in fluctuating suspensions and in corresponding suspensions without fluctuations may be overlooked. Ignoring these factors, we arrive at a full set of hydrodynamic equations for both suspension phases which includes ... 

Thus, the next key problem that requires our attention is the formulation of a consistent model of pseudo-turbulence which would offer an opportunity to calculate T. ... 

The general approach in dealing with pseudo-turbulent fluctuations was already outlined in great detail [9,14,23] and thereafter applied to homogeneous fluidized beds [25]. A serious deficiency in the latter analysis [25] lies in the fact that the authors... 

Equations to describe random pseudo-turbulent fluctuations have to be derived 1) from fluid mass and momentum conservation laws, and 2) from the Langevin equation for one particle. Taking the fluctuation parts of the mass and momentum conservation equations (the corresponding mean equations were formulated in Section 5) and multiplying the Langevin equation by the particle number concentration, we arrive at the following set of equations governing particle and fluid fluctuations ... 

Mean flow variables are related to each other by Equations 4.1-4.3 and 5.1, 5.2 closed with the help of rheological equations for 1) the mean interphase interaction force, 2) the stresses acting in both suspension phases, and 3) the fluctuation energy flux. It is easy to see that only momentum conservation Equations 4.2 and 5.2 remain informative for the uniform steady state under consideration. If pseudo-turbulent contributions to the interaction force are neglected in accordance with the discussion in Sections 4-6, these equations assume the form... 

Attempts to generalize the developed model to dispersions for which these assumptions are not satisfied pose a number of tempting new problems. Some of these problems can be successfully solved without much ado. For instance, it is not difficult to allow for the effeet of collision inelasticity on the properties of pseudo-turbulent motion by means of replacing simple Equation 8.1 by other equations in which collisional energy dissipation is duly taken into account, as has been previously done in reference [25]. However, the repudiation of other assumptions is by no means a simple matter and requires a great deal of work. Fortunately, this work seems to be much facilitated by the mere fact that there exists a sound tentative model which plays the role of a certain initial approximation. It is the formulation of Just such a model that should be regarded as a main achievement of the present article. 

Buyevich, Y. A. Statistical hydromechanics of disperse systems. Part 3. Pseudo-turbulent structure of homogeneous suspensions. J. Fluid Mech. 56, 313-336 (1972). 

Fan, F.G. and Ahmadi, G. (1995a). Dispersion of EUipsoidal Particles in an Isotropic Pseudo-Turbulent Flow Field. ASME J. Fluid Eng., Vol. 117, pp. 154-161. 

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