Logarithmic velocity profile

It is interesting to note that this logarithmic velocity profile is followed over most of the cross section of the pipe, not just where f tw. 

Then, eliminating w u from equations (5.35) and (5.37) results in the well-known logarithmic velocity profile ... 

For a fully developed tubular flow, assuming a logarithmic velocity profile, Taylor derived the equation... 

Turbulent flow in a pipe, assuming logarithmic velocity profile (Taylor, 1954) 10.1 Ru, ... 

In RANS models, the solid wall boundary conditions have traditionally been modeled using wall functions. Wall functions use empirical profiles to replace the actual boundary conditions, such as no-slip (zero velocity) condition at solid surfaces. An example of an empirical law is the logarithmic velocity profile ... 

Hence, both models will give the proper logarithmic velocity profile [Eq. (12)], provided the coefficients in brackets are unity in each case. 

Wai and Fumeaux [1990] applied CFD to crossflow membrane filtration to provide an array of data such as local pressures and fluid velocities on both sides of a membrane, shear stresses on the membrane surface and local concentrations of retentate species. This type of information is useful for designing the membrane unit as a separator or a reactor. With a commercial CFD code, the authors simulated the fluid flow, on both the feed and permeate sides, along the membrane channel and through the membrane. Frictional effects between the fluid and membrane surfaces depend on the nature of the fluid flow. For flow parallel to the essentially flat membrane surface, standard wall friction expressions based on logarithmic velocity profiles adjacent to the wall arc used. 

Jackson, P.S. (1981) On the displacement height in the logarithmic velocity profile, Journal of Fluid Mechanics 111, 15-25. 

There are several factors that contribute to the difficulty of predicting the vertical transport of aerosols. Most obviously, the flow field is not homogeneous in the vertical direction. For neutral atmospheric stability conditions, the mean wind velocity follows an approximately logarithmic velocity profile, given by (Wieringa 1980)... 

FIGURE 3.1 (a) Logarithmic velocity profile for a neutral atmospheric surface layer for a... 

Application of the Governing Eqnations to Turbulent Flow 127 Finally, integrating with respect to y+, we obtain a logarithmic velocity profile ... 

The last parameter, CLi, is determined investigating inhomogeneous high Reynolds number, fully developed channel flows (i.e., these flows are sometimes referred to as 2D Couette flows). Actually, the turbulence model is applied describing the flow in regions near walls, where the logarithmic velocity profile applies. 

The velocity gradient near a wall can be estimated from the logarithmic velocity profile (1.373) ... 

The velocity components parallel to the wall are estimated from the logarithmic velocity profile relation (1.373). For example, using Cartesian coordinates, the x-component of the velocity vector is given as ... 

B model parameter in logarithmic velocity profile (—) b constant of integration in laminar boundary layer theory (—)... 

The basic logarithmic velocity profile (16.66) is applicable only to adiabatic conditions. However, the atmosphere is seldom adiabatic, and the velocity profiles for stable and unstable conditions deviate from this logarithmic law. For the more frequently encountered nonadiabatic atmosphere (also called stratified), the Monin-Obukhov similarity theory is usually employed (Monin and Obukhov 1954). 

The dynamics of turbulent plumes relevant to most crustaceans are complicated by the fact that many are produced in boundary layer flows. A crustacean moving across the substratum does so in a velocity gradient characterized by no motion of fluid in contact with the substratum and a nominal or ffee-stream velocity at some distance away. The region in between is characterized by a roughly logarithmic velocity profile that comprises approximately 30% of the water depth (Schlichting 1987). 

The well-known logarithmic velocity profile law follows from Prandtl mixing length theory. It applies well to the constant flux surface layer and has been verified numerous times from measurements taken both in the laboratory and in the field for neutral and near-neutral stability atmospheric surface layers. The result is the following relationship for the skin friction ... 

From a theoretical perspective, our understanding of the flow field above this aquatic surface tracks that presented above for the atmospheric boundary layer. For the neutral-stability class of turbulent flows the logarithmic velocity profile, the constant flux layer assumption and so on, apply as well. Although Equation 2.21 is valid for use in estimating Cf less measurement on yo, the bottom roughness parameters are available in aquatic environments for producing summary results as shown in Table 2.1. In the absence of these site-specific y values, an alternative approach is used to estimate Cf for hydraulic flows it is presented next. 

The unstable boundary layer is a commonly occurring regime in environmental flows, and reverts to the stable boundary layer if the stability terms are zero. Assume a logarithmic velocity profile in the air boundary layer, altered by a stability function (Brutsaert, 1982), and assign = pw to be the flux of momentum at height z. Then, the momentum/unit volume difference between heights z and z = 0 is... 

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