KARMAN equation

This equation is called the von Karman equation (or, sometimes, the law of the wall ), and can be written in the following dimensionless form... 

For rough tubes in turbulent flow (7VRc > 4000), the von Karman equation was modified empirically by Colebrook to include the effect of wall roughness, as follows ... 

Dodge and Metzner (1959) modified the von Karman equation to apply to power law fluids, with the following result ... 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

The most widely accepted relationship for turbulent flow in smooth pipes is the von Karman equation... 

It was learned that e is a measure of the absolute roughness of the inside of a pipe. The question naturally arises as to whether e and re may be functionally related to one another. Such a correlation may be accomplished by use of the Prandtl-Karman equation for rough pipes. Written in terms of R, recalling that for a circular pipe D = 4R,... 

Schacham equation, 94 von Karman equation, 95 Froth flotation, 636,638 equipment, 645 performance, 638 Fuel cells, 646 characteristics, 655, 656 Fugacity coefficient, 372... 

A more exact procedure is to solve the Bom-von Karman equations of motions 38) to obtain frequencies as a function of the wave vector, q, for each branch or polarization. These will depend upon unit-cell symmetry and periodicity, force constants, and masses. Thus, for a simple Bravais lattice with identical atoms per unit cell, one obtains three phase-frequency relations for the three polarizations. For crystals having two atoms per unit cell, six frequencies are obtained for each value of the phase or wave vector. When these equations have been solved for a sufficient number of wave-vectors, g hco) can, in principle, be obtained by direct count . Thus, a recent calculation (13) of g to) based upon a normal-mode calculation that included intermolecular forces gave an improved fit to the specific heat data of Wunderlich, and showed additional peaks of 140, 90 and 60 cm in the frequency distribution. Even with this procedure, care must be exercised, since it has been shown that significant features of g k(o) may be rormded out. Topological considerations have shown that significant structure in g hco) vs. ho may arise from extreme or saddle points in the phase-frequency curves (38). 

For smooth pipe for Reynolds numbers from 4000 to 10 , the von Karman equation provides an implicit relation between the friction factor and the Reynolds number ... 

By proceeding in a manner similar to that employed in deriving the von Karman equation for a developing boundary layer, the system of partial differential equations (Equations 4.21 and 4.22) can be reduced to the ordinary differential equation... 

If the magnitude of the center point deflection of the film wq increases to values on the order of film thickness h, then the potential arises for generation of significant membrane stress in the film due to transverse deflection (in addition to any residual membrane stress which may be present in the film prior to deflection). As in the case of film buckling, the von Karman plate theory provides a useful and effective framework for describing response with center point deflection wq of magnitude equal to several times the film thickness. In the present case, the von Karman equations reduce to the pair of ordinary differential equations... 

The horizontal wind near the surface of the Earth is retarded by viscous frictional effects imposed on the flow by the rough surface, so that the change in velocity with height (Figure 16.4) follows a log-linear relation in the lowest 10% of the atmospheric boundary layer, as characterized by the Prandtl-von Karman equation ... 

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