Pseudo-turbulent fluctuations

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. 

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 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 ... 

The role of subgrid scale (pseudo) turbulent fluctuations on the collision probabilities and coUision outcomes of deformable particles should be investigated. This is especiaUy important for applications of stochastic models such as DSMC. 

Another variant devised by Martem Yanov et al. [16] ensures strong electrolyte stirring for generating turbulent fluctuations in solution. Assuming a pseudo hydrodynamical white noise, the responding current can be analyzed in the frequency domain to provide the same information as that obtained from any of the techniques mentioned above. 

We see that in addition to the convective, pressure, and viscous terms, we have an additional term, which is the gradient of the nonlinear term pu v, which represents the average transverse transport of longitudinal momentum due to the turbulent fluctuations. It appears as a pseudo-stress along with the viscous stress pdUldy, and is called the Reynolds stress. This term is usually large in most turbulent shear flows (Lieber and Giddens, 1988). 

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]. 

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 ... 

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... 

Mehrabadi M, Tenneti S, Garg R, Subramaniam S Pseudo-turbulent gas-phase velocity fluctuations in homogeneous gas-solid flow fixed particle assemblies and fireely evolving suspensions, J Fluid Mech 770 210-246, 2015. 

---