Eddy diffusivity for momentum

If we define AT as the eddy diffusivity for momentum, the vertictil eddy diffusion coefficient under unstable conditions can be expressed as... 

Let us define an eddy viscosity or eddy diffusivity for momentum eM such that... 

In which et is the turbulent viscosity or eddy diffusivity for momentum transfer (SI units m2/s). This then allows the total shear stress to be expressed by... 

On page 55 an eddy diffusivity for momentum transfer was defined. A corresponding eddy diffusivity for heat transfer can be defined by... 

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... 

The quantity is called the eddy diffusivity for momentum, and ew is the eddy diffusivity for heat. They are related through the turbulent Prandtl number Pr, = eM/eH. 

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... 

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... 

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. 

The eddy diffusitives for momentum and heat, and Ejj, respectively, are not properties of the fluid but depend on the conditions of flow, especially on all factors that affect turbulence. For simple analogies, it is sometimes assumed that and jf are both constants and equal, but when determined by actual velocity and temperature measurements, both are found to be functions of the Reynolds number, the Prandtl number, and position in the tube cross section. Precise measurement of the eddy diffusivities is diflScult, and not all reported measurements agree. Results are given in standard treatises. The ratio Sh/sm also varies but is more nearly constant than the individual quantities. The ratio is denoted by i/f. For ordinary liquids, where Np > 0.6, is close to 1 at the tube wall and in boundary layers generally and approaches 2 in turbulent wakes. For liquid metals is low near the wall, passes through a maximum of about unity at j/r X 0.2, and decreases toward the center of the pipe. ... 

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. 

Momentum eddy diffusivity For turbulent flow in circular pipes. Re = 50 000 to... 

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 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... 

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. 

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. 

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. 

At present analytical solutions of the equations describing the microscopic aspects of material transport in turbulent flow are not available. Nearly all the equations representing component balances are nonlinear in character even after many simplifications as to the form of the equation of state and the effect of the momentum transport upon the eddy diffusivity are made. For this reason it is not to be expected that, except by assumption of the Reynolds analogy or some simple consequence of this relationship, it will be possible to obtain analytical expressions to describe the spatial variation in concentration of a component under conditions of nonuniform material transport. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

Because an air packet and the molecules within it move as a unit, the eddy diffusion coefficients for different gaseous species are equal. In fact, Kj is often assumed to be the same for the transfer of gases, heat, and momentum (expressed in the same units), a relation that is referred to as the similarity principle. Therefore Kj is generally measured for the most... 

The meteorological input required in the Unified EMEP model are the 3D horizontal and vertical wind fields, specific humidity, potential temperature cloud cover, and precipitation. The transferred surface 2D fields for use in the chemical transport model are surface pressure, 2 m temperature, surface flux of momentum, sensible and latent heat, and surface stress. All variables are given in 3-h interval. Table 13.1 lists the variables and their main purposes in the EMEP model. Inside the model different boundary layer parameters like the stability, eddy diffusion, and mixing height are calculated based on MOST. 

A schematic representation of the boundary layers for momentum, heat and mass near the air—water interface. The velocity of the water and the size of eddies in the water decrease as the air—water interface is approached. The larger eddies have greater velocity, which is indicated here by the length of the arrow in the eddy. Because random molecular motions of momentum, heat and mass are characterized by molecular diffusion coefficients of different magnitude (0.01 cm s for momentum, 0.001 cm s for heat and lO cm s for mass), there are three different distances from the wall where molecular motions become as important as eddy motions for transport. The scales are called the viscous (momentum), thermal (heat) and diffusive (molecular) boundary layers near the interface. 

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