Karman s equation

TABLE 6.2. Typical Velocities and Pressure Drops in Pipelines Karman s equation... 

For fully turbulent flow, the friction coefficient f is independent of Reynolds number and may be approximated by the Karman s equation (Equation 6-14). Note that the three walls of the channel are smooth, but the fourth one is the porous gas diffusion layer. 

Cheremisinoff and Davis (1979) relaxed these two assumptions by using a correlation developed by Cohen and Hanratty (1968) for the interfacial shear stress, using von Karman s and Deissler s eddy viscosity expressions for solving the liquid-phase momentum equations while still using the hydraulic diameter concept for the gas phase. They assumed, however, that the velocity profile is a function only of the radius, r, or the normal distance from the wall, y, and that the shear stress is constant, t = tw. ... 

For the liquid phase, Cheremisinoff and Davis (1979) solved the momentum equation using von Karman s and Deissler s eddy viscosity expressions. 

In this equation p<> is the density of the air and vi and vi the mean wind velocities in a specified direction at two corresponding heights Z and 22, and Ci and C% the corresponding concentrations as weight per weight of air. The constant ko is von Karman s turbulence constant which, as we have elsewhere noted, has a value of approximately 0.4 and is independent of the fluid. The value of Cot is simply the mass of material raised from a unit surface in a unit time. 

Two traditional approaches to the closure of the Reynolds equation are outlined below. These approaches are based on Boussinesq s model of turbulent viscosity completed by Prandtl s or von Karman s hypotheses [276, 427]. For simplicity, we confine our consideration to the case of simple shear flow, where the transverse coordinate Y = Xi is measured from the wall (the results are also applicable to turbulent boundary layers). According to Boussinesq s model, the only nonzero component of the Reynolds turbulent shear stress tensor and the divergence of this tensor are defined as... 

The simplest way to close equations (3.1.37) is to use the hypothesis that the turbulent Prandtl number for the examined process is a constant quantity. Then it readily follows from Eq. (3.1.39) that the turbulent diffusion coefficient is proportional to the turbulent viscosity >t = i /Pr,. By using the expression for vx borrowed from the corresponding hydrodynamic model, one can obtain the desired value of Dt. In particular, following Prandtl s or von Karman s model, one can use formula (1.1.21) or (1.1.22) for vx. 

The lattice component was calculated using the harmonic approximation, in which all the acoustic and low-frequency optical vibrations are included with the help of a single Debye fimction, while high-frequency crystal vibrations are taken into accoimt by Einstein s equation. According to Kelley s derivations (Gurvich et al., 1978-1984) based on the Born-von Karman d)mamic crystal lattice theory, we therefore have... 

The buckling equations developed by Von Karman and Tsien are the basis of the design equations developed by ASME. Von Kantian s equations, which are substantiated by tests, give a more accurate prediction of buckling strength of... 

If translational symmetry is taken into account and Born-Von Karman s cyclic conditions are applied the dynamical equation (5) can be rewritten in the form [24]... 

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

The formula of Zel dovich and Frank-Kamenetskii [7], [34] may be obtained from the simplest result of von Karman by performing an asymptotic expansion of the integral 1 for large values of P and retaining only the first term in the expansion. In view of equation (43), by transforming from the variable t to z s P (l - r)/(l/a + t) in equation (64), we find that... 

We now consider a three-dimensional periodic system with (2N -f 1) unit cells and m orbitals within the cell. If we again apply the Bom-von Karman periodic boundary conditions the matrices F , F, and S become cyclic hypermatrices of order m(2N -h 1). Therefore we can again apply to equations (1.97) the unitary transformation described in Section 1.1. Hence... 

Consider the simplest model of a crystal as a one-dimensional chain of identical atoms. Direct the chain along an axis JC. Denote a period along the chain by the letter a, and let / be the number of the atom counted from an arbitrary chosen atom (Figure 9.13). Value x, represents instant coordinate of atom 1. hi this model one more simplification is entered, namely, the nearest neighboring atoms are considered to be interacting among themselves only. Such a model is called the Born-Karman model. The equation of movement (Newton s second law) can be written for the atom I as... 

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