Eddy diffusivity of momentum

It has been assumed that the density is constant in writing these equations, which are therefore strictly valid only for incompressible flow. ed is called the eddy diffusivity and eh the eddy thermal diffusivity. Although s can be interpreted as the eddy diffusivity of momentum, it is usually called the eddy viscosity and sometimes by the better name eddy kinematic viscosity. 

B Eddy diffusivity 8 for eddy diffusivity of momentum 8 for eddy diffusivity of heat mVs fft/h... 

According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady... 

During food engineering operations, many fluids deviate from laminar flow when subjected to high shear rates. The resulting turbulent flow gives rise to an apparent increase in viscosity as the shear rate increases in laminar flow, i.e., shear stress = viscosity x shear rate. In turbulent flow, it would appear that total shear stress = (laminar stress + turbulent stress) x shear rate. The most important part of turbulent stress is related to the eddies diffusivity of momentum. This can be recognized as the atomic-scale mechanism of energy conversion and its redistribution to the dynamics of mass transport processes, responsible for the spatial and temporal evolution of the food system. 

S = ntul/f V can also be interpreted as a dimensionless relaxation time r, where tn/f is a characteristic time for particle motion and v/u] h a characteristic time for the turbulent fluctuations. Hence S" " = r". The viscous sublayer is the region near a smooth wall where momentum transport is dominated by the viscous forces, which are large compared with eddy diffusion of momentum. Fol lowing the usual practice and taking the sublayer thickness to extend to y = 5, particles with a slop distance < 5 would not reach the wall if the sublayer were truly stagnant. 

Quantity is analogous to fi, the absolute viscosity. Also, in analogy with the kinematic viscosity v the quantity called the eddy diffusivity of momentum, is defined as e,j = EJp. 

To further verify the above conclusion on the failure of the analogy between momentum and heat transfer in the case of viscoelastic fluids, the approximate values of the eddy diffu-sivities of momentum and heat transfer corresponding to the minimum asymptotic cases will be compared. The eddy diffusivity of momentum corresponding to the minimum asymptotic case was calculated by Kale [84] directly from Deissler s continuous eddy diffusivity model ... 

Least squares curves in the wall, core and surface regions calculated from the velocity measurement results. The three regions have overlapping zones to prevent discontinuities. The results were substituted in Eqs. (16) and (17) to calculate the eddy diffusivity of momentum or the normalized eddy... 

Until now, open channel viscoelastic eddy diffusivity measurements have not been available. Only few measurements were reported with water by Jobson, et al. C21) and Ueda, et al, C22) who measured the eddy diffusivity of momentum in wide open channels as shown in Fig, 3. 

In Fig. 3, these experimental results agree fairly well with Eq. (15). The eddy diffusivity of momentum results show their maximum at 0.5 - 0.55 for both slopes (s = 0.0045 and s = 0.01). The nximerical calculation results are also shown in Fig. 3 at two different Reynolds numbers. These numberical results underestimate the eddy diffusivity. 

The three eddy diffusivities of momentum, heat, and mass can be computed from measured velocity, temperature, and concentration gradients, respectively, in that order of increasing difficulty. 

In the case of turbulent flow, the differential equations will contain time-averaged velocities and in addition the eddy diffusivities of momentum, mass, and heat transfer. The resulting equations cannot be solved for lack of information about the eddy diffusivities, but one might expect results of the form... 

A simple conceptual model for turbulent flow deals with eddies, small portions of fluid in the boundary layer that move about for a short time before losing their identity [8], The transport coefficient, which is defined as eddy diffusivity for momentum transfer eM, has the form... 

Eddy Diffusivity Models. The mean velocity data described in the previous section provide the bases for evaluating the eddy diffusivity for momentum (eddy viscosity) in heat transfer analyses of turbulent boundary layers. These analyses also require values of the turbulent Prandtl number for use with the eddy viscosity to define the eddy diffusivity of heat. The turbulent Prandtl number is usually treated as a constant that is determined from comparisons of predicted results with experimental heat transfer data. 

Mizushina, T., and H. Usui, "Reduction of Eddy Diffusivity for Momentum and Heat in Viscoelastic Fluid Flow in a Circular Tube", Phys. Fluids Suppl., S 100 (1977)... 

Mizushina, T. Usui, H. Reduction of eddy diffusion for momentum and heat in viscoelastic fluid flow in a circular tube. Phys. Fluids 20 (1977) S100-S108. 

The first shear stress xl is due to laminar flow, X( is due to turbulence, and v is the velocity component in y direction. The turbulent sheer stress is given in terms of eddy viscosity or eddy diffusivity for momentum... 

Heat and mass eddy diffusivity The evidence is that with Prandtl and Schmidt numbers close to unity, as for most gases, the eddy diffusivities of heat and mass are equal to the momentum eddy diffusivity for all regions of turbulence [15], For turbulent fluids where Prandtl and Schmidt numbers exceed unity, the ratios E jand E /E will vary with location relative to the wall and in the turbulent core will lie generally in the range 1,2 to 1.3, with E and essentially equal [44, 62], For = 0 to 45, with Pr and Sc > 1, a critical analysis of the theoretical and experimental evidence [44] led to... 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

If there is a temperature gradient within the fluid, the eddies will be responsible for heat transfer and an eddy thermal diffusivity Ep may be defined in a similar way. It is suggested that, since the mechanism of transfer of heat by eddies is essentially the same as that for transfer of momentum, Eh is related to mixing length and velocity gradient in a similar manner. 

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

There are shown in Fig. 19 values of the eddy diffusivity calculated from the measurements by Sherwood (SI6). These data show the same trends as were found in thermal transport, indicating that the values of eddy diffusivity are determined primarily from the transport of momentum for situations where the molecular Schmidt numbers of the components do not differ markedly from each other. 

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