Isotropic turbulence assumption

As already noted, Eq. (14.13) applies for drops suspended in a homogeneous and isotropic turbulent flow. The movement of a gas-liquid mixture may be considered as homogeneous and isotropic in the bulk flow, but near a wall this assumption does not hold. Besides, drops can be blown away from a liquid film at the wall, but other drops may be deposited at the wall. These phenomena result in a change of Eq. (14.13). Measurements of the average steady-state size of drops of gas-liquid mixtures in a turbulent flow in a pipe, as described in ref [3], have shown that the average drop size is described by the following relationship ... 

The assumption of isotropic turbulence arises when one assumes fliat flic various mean turbulent velocity fluctuations are equal, and thereby obtains a mean dissipation for a given mass (knowing the energy input) ... 

With the assumption of isotropic turbulence (approximately valid in stirred vessels) the mean values of the fluctuating velocities are not a function of the direction. The definition of the effective value is... 

Recently, Pozorski and Apte [39] performed a systematic study of the direct effect of subgrid scale velocity on particle motion for particle-laden forced isotropic turbulence using a stochastic model based on filtered particle tracking (FPT). The FPT approach statistically reconstructs the imresolved carrier-phase velocity along particle trajectories. A reasonable assumption for LES is to consider the residual turbulent motion as locally homogeneous and isotropic. Then, the fluid velocity... 

Turbulent mixing of a chemical species in the liquid in a bubble column is difficult to describe based on first principles, unless the assumption of isotropic turbulence is invoked. Then, the dispersion coefficient at any location a in the column can be estimated by (Brodkey, 1967) ... 

Mixing of newtonian fluids in conventional stirred tanks is now reasonably well understood, although most models rely on the assumption of homogeneous isotropic turbulence. This is only a rough approximation. For instance, the fluctuating components u - and u 0 of the radial and tangential velocity were recently measured by Laser-Doppler Anemometry in a 6.3 dm reactor,. Values of the Reynolds Tension... 

The terms on the diagonal are the normal stresses or variances, and these squared terms will always be positive. In an idealized flow with no directional preferences, they will all be equal. The off-diagonal elements are symmetric (uv = vu), so only three of them are unique. If the turbulence has no directional preference and there are no velocity gradients in the flow, the individual fluctuations will be completely random and the covariances will be equal to zero. This assumption of no directional preference or isotropic turbulence is an important concept for understanding the different classes of time-averaged turbulence models. With these two conditions, the six unknowns can be reduced to a single unknown ... 

A statistical theory of turbulence which is applicable to continuous movements and which satisfies the equations of motion was introduced by Taylor [160-162] and [163, 164], and further developed by von K m n [177, 178]. Most of the fundamental ideas and concepts of the statistical turbulence theory were presented in the series of papers published by Taylor in 1935. The two-point correlation function is a central mathematical tool in this theory. Considering the statistics of continuous random functions the complexity of the probability density functions needed in a generalized flow situations was found not tractable in practice. An idealized flow based on the assumption of statistical homogeneity greatly simpUfled the analysis. Taylor [160-162] went further still and considered isotropic turbulence. 

Other authors (e.g., Dhotre et al, 2008, 2009 HiU et al, 1995 Lain et al, 2000 Masood and Delgado, 2014 Moraga et al, 2003 Mudde and Simonin, 1999 Niceno et al, 2008) have incorporated such a term into the phase interaction force in the momentum balances, its basis being in the above concept of a drifting velocity (Dehbi, 2008 VioUet and Simonin, 1994). In aU cases, assumptions have to be made about the (isotropic) turbulent dif-fusivity or the turbulent diffusivity tensor in this term. Also this topic is covered by the recent review by Pourtousi et al (2014). 

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... 

Concept (b) is less useful, except in rare cases where the energy spectrum has been measured. It is common to assume that the turbulence is homogeneous and isotropic and that the eddies in question are in the inertial ( — 5/3 power) subrange. This assumption is unlikely to be valid in an overall sense though it may be reasonable locally (GIO) or for the high wavenumber (small) eddies which are of primary interest. For an example of the application of the theory, see Middleman (Ml3). 

Mellor and Herring also examine MRS closures, and show how the MTEN closure results from the MRS equations with the additional assumption of small departures from isotropy. While this approach is academically interesting, even the most weakly strained flows are far from isotropic (C4), and hence the main selling point for MTE methods is that they work very well for predicting a wide class of turbulent shear flows. Examples are given in the following section. 

Hinze interpreted data from Clay (31) in order to determine a value of 0.725 for C which allows the diameter in Couette flow to be deduced [see Eq. (4)]. Karabelas (32) questioned the assumptions made in Eq. (3), as turbulent flows are sometimes neither isotropic nor homogeneous. However, a number of workers have found the expression to be satisfactory. The dg diameter may also be expressed as a function of the Weber number ... 

While the fully isotropic assumption is not a good match to physical reality, the implications of isotropy are profound for turbulence modeling and measurements. Isotropy allows the entire turbulent spectrum to be defined from one component of fluctuating velocity, because the flow is perfectly without directional preference. It allows simplification of the equations to include only the normal stresses. It also allows one to make spectral arguments to simplify the measurement of the dissipation. This assumption is so powerful that it is often invoked in the hope that it will be good enough for a flrst approximation, despite the fact that it is a poor match for the full physical reality. 

---