Karman constant

Turbulent layer Since Tj = 0, and shearing velocity u can be considered proportional to the gradient and the distance from the wall y, i.e., u = Ky, where k is a constant (Karman constant), we have at constant... 

W. Tubes, turbulent, smooth tubes. Constant surface concentration. Von Karman analogy... 

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. ... 

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 steady flow field created by an infinite disk rotating at a constant angular velocity in a fluid with constant physical properties was first studied by Von Karman [31]. 

While fishing in Transylvania, Theodore von Karman noticed that downstream of the rocks the distance between the shed vortices was constant, regardless of flow velocity (Figure 3.104). From that observation evolved the three types of vortex meters the vortex shedding, the vortex precession, and the fluidic oscillation (Coanda) versions. All three types detect fluid oscillation. They have no moving components and can measure the flow of gas, steam, or liquid. 

Figure 6.5 Distinct differences in transport behavior between pools of surface and bottom floes over several tidal cycles, as determined by R0 values, in the ACE Basin (USA). Hatched areas are times of maximum current speed. ws = sediment settling velocity, 0 = proportionality coefficient between eddy viscosity and diffusivity, k = von Karman s constant, and n = frictional velocity. (From Milligan et al., 2001, with permission.)...

The constant k0 is called von Karman s turbulence constant. Using this value of X... 

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. 

This is the momentum equation of the laminar boundary layer with constant properties. The equation may be solved exactly for many boundary conditions, and the reader is referred to the treatise by Schlichting ll] for details of the various methods employed in the solutions. In Appendix B we have included the classical method for obtaining an exact solution to Eq. (5-13) for laminar flow over a flat plate. For the development in this chapter we shall be satisfied with an approximate analysis which furnishes an easier solution without a loss in physical understanding of the processes involved. The approximate method is due to von Karman [2],... 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

Up to this point, the analysis is rigorously correct and general it is simply a restatement of the Born-Von Karman expansion of the energy in terms of relative displacements—.see Eq. (8-17). However, we shall now make a major approximation in taking the force constants from the very simple valence force field that we described in Chapter 8. This will give us a clear and correct qualitative description of the vibration spectra and will even give semiquantitativc estimates of the frequencies. Afterward, we shall consider the influence of the many terms that are omitted in this simple model. 

Cartesian coordinate vector (x, y, z) Molecular thermal diffusivity Turbulent thermal diffusivity Molecular kinematic viscosity Turbulent kinematic viscosity Karman constant Mass density See Eq. (26)... 

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). 

In this Hamiltonian k is the wave vector of the phonons, y is the phonon mode branch, go and Vmk are the electron-strain and the electron-phonon interaction constants. It is important to remind that as it was noted for the first time by Kanamori [3], the electron interaction with the homogeneous strain U should be considered separately from the electron-phonon interaction as that type of strain can not be represented by phonons. The introduction of the last ones depends upon the Born-Karman conditions that are changing at the structural phase transition. 

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