Eddy viscosity, calculation

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

The use of local theories, incorporating parameters such as the eddy viscosity Km and eddy thermal conductivity Ke, has given reasonable descriptions of numerous important flow phenomena, notably large scale atmospheric circulations with small variations in topography and slowly varying surface temperatures. The main reason for this success is that the system dynamics are dominated primarily by inertial effects. In these circumstances it is not necessary that the model precisely describe the role of turbulent momentum and heat transport. By comparison, problems concerned with urban meso-meteorology will be much more sensitive to the assumed mode of the turbulent transport mechanism. The main features of interest for mesoscale calculations involve abrupt... 

The unknown Reynolds stress term can be calculated using the eddy viscosity hypothesis (1.380), or, alternatively, the term in the bracket on the RHS of this relation can be recognized as being approximately equal to the Reynolds stress ... 

A non-isotropic version of the eddy viscosity hypothesis is sometimes used in situations where such effects are essential and the particular flow in question is expected to be far from isotropic [93]. That is, when the turbulent kinetic energy is calculated, the Reynolds stresses can be approximated by ... 

From a measured average velocity profile versus y or r) in a flow and information on the viscosity and density of the fluid, we can calculate the eddy viscosity for any point in the flow. A typical plot of vjelocity versus position is shown in Fig. 11.7. Figure 16.8 shows the eddy viscosity divided by the kinematic viscosity for flow in smooth pipes, calculated from a figure like Fig. 11.7. 

From Fig. 16.8 we see that in ordinary pipe flow for regions away from the wall the eddy viscosity is typically about 100 times the molecular viscosity (i.e., the Reynolds stresses are about 100 times the stresses due to molecular viscosity), that the eddy viscosity is a strong function of position and Reynolds number, and that it is difficult to calculate values of the eddy viscosity near the center of the pipe. From Eq. 16.15 we see that the sum of the eddy and molecular viscosities is equal to Tl dVJdy) at the center of the pipe both quantities are zero. To obtain the correct limit in this ratio as both numerator and denomnator approach zero requires more precise experimental measurements of and y than are currently available. We may infer from Fig. 16.8 that in this type of pipe flow the heat transfer and mixing will be of the order of 100 times the heat transfer and mixing due to molecular thermal conductivity and molecular diffusion. 

The family of two-equation A -f models is the most widely used of the eddy viscosity models. Ak-e model consists of two transport equations, one for the turbulent kinetic energy k and one for the energy dissipation rate e. The turbulent eddy viscosity is calculated from ... 

The legitimacy of employing Blasius type models for the shear stresses in stratified flows was checked in several studies. Kowalski made direct measurements of the Reynolds shear stress in the gas for horizontal stratified flow in pipes and found that the gas-wall friction factors are well approximated by the Blasius equation provided that the hydraulic diameter is utilized [64]. For the liquid phase, Andritsos and Hanratty [28] found that the use of the Blasius equation to calculate introduces some error. However, improvements achieved by using a more complicated model for which is based on velocity profile and eddy viscosity concepts, were found to be of mild effect on the integral flow characteristics. 

A more rigorous viscous turbulent model of single-phase flow, based on a Prandtl mixing length theory was published by Bloor and Ingham. Like Rietema, these authors obtained theoretical velocity profiles, but they used variable radial velocity profiles calculated from a simple mathematical theory. The turbulent viscosity was then related to the rate of strain in the main flow and the distribution of eddy viscosity with radial distance at various levels in the cyclone was derived. 

Several models for the calculation of the eddy viscosity are available, here we apply a two-equation turbulence model, the so called k-e model. This model has been used in several lake studies (e.g. Svensson 1978 and Sahlberg, 1988, Omstedt, 1984, Elo, 1994) and provides us with a model that, in a realistical way, simulates the seasonal variation of lake temperatures. 

Since flow in a wide channel is approximately parallel, a convenient turbulence parameterization to use is the eddy viscosity equation of Prandtl [6]. The melt-velocity, calculated to leading order from the temperature solution, is then... 

In the buffer zone the value of d +/dy+ is twice this value. Obtain an expression for the eddy kinematic viscosity E in terms of the kinematic viscosity (pt/p) and y+. On the assumption that the eddy thermal diffusivity Eh and the eddy kinematic viscosity E are equal, calculate the value of the temperature gradient in a liquid flowing over the surface at y =15 (which lies within the buffer layer) for a surface heat flux of 1000 W/m The liquid has a Prandtl number of 7 and a thermal conductivity of 0.62 W/m K. 

Calculation of eddy kinematic viscosity for turbulent flow 62... 

Simonin and Viollet [122] calculated the dynamic viscosity for the gas phase from a modified k — e model. The time scale of the large eddies of the gas phase flow was given by ... 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

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