Mean square velocity

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

The parameter used to design rapid mix and flocculation systems is the root mean square velocity gradient G, which is defined by equation... 

It is thus seen that the kinematic viscosity, the thermal diffusivity, and the diffusivity for mass transfer are all proportional to the product of the mean free path and the root mean square velocity of the molecules, and that the expressions for the transfer of momentum, heat, and mass are of the same form. 

Turbulent mass burning rate versus the turbulent root-mean-square velocity by Karpov and Severin [18]. Here, nis the air excess coefficient that is the inverse of the equivalence ratio. (Reprinted from Abdel-Gayed, R., Bradley, D., and Lung, F.K.-K., Combustion regimes and the straining of turbulent premixed flames. Combust. Flame, 76, 213, 1989. With permission. Figure 2, p. 215, copyright Elsevier editions.)... 

Since the masses of the molecules are proportional to their molecular weights and the average velocity of the molecules is a measure of the rate of effusion or diffusion, all we have to do to this equation to get Graham s law is to take its square root. (The square root of v2 is not quite equal to the average velocity, but is a quantity called the root mean square velocity. See Problem 12.18.)... 

The square root of the average of the square of the velocity, vrms, is not the average velocity, but a quantity called the "root mean square velocity. (b) The pressure of the gas does not matter, (c) The identity of the gas is important, because the mass of the molecule is included in the calculation. (Contrast this conclusion with that of the previous problem.)... 

Even though the Reynolds number gives some measure of turbulent phenomena, flow quantities characteristic of turbulence itself are of more direct relevance to modeling turbulent reacting systems. The turbulent kinetic energy q may be assigned a representative value <7o at a suitable reference point. The relative intensity of the turbulence is then characterized by either q()KH2 U2) or (77(7, where (/ = (2q0)m is a representative root-mean-square velocity fluctuation. Weak turbulence corresponds to U /U < 1 and intense turbulence has (77(7 of the order unity. 

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined the integral scale l0, which characterizes... 

Since measurements of u v ), u w ), etc., usually indicate that the magnitudes of these covariances are much smaller than the mean square velocity fluctuations (m ), etc., it is generally assumed that the off-diagonal elements of P are negligible. Also, we use the notation, al = ((,1), cr, =... 

Figures The mean-square velocity for two different cases of the double-well problem are displayed above. In the top panel, C, - 0 corresponds to a stationary environment. In the bottom panel, = 1 corresponds to selfconsistent heterogeneous nonstationary environments of degree w = 16 vis-a-vis the WiGLE model...

The velocity relaxation time is again f/rn and the mean square velocity (up = k T/m. Schell et al. [272] have used the Langevin equation to model recombination of reactants in solutions. Finally, from the properties of the fluctuating force (see above)... 

The flow field is symmetric over the period, with velocities in both directions at different times. Because of the symmetry there is no net flow through the duct, and thus the mean velocity profile is exactly zero. The average root-mean-square velocity, however, does have a radial dependence as shown in Fig. 4.11. The root-mean-square velocity is defined as... 

This average root-mean-square velocity has a peak value away from the centerline (i.e., the overshoot). The magnitude of the velocities depend on fi and w. The root-mean-square profile has a relatively weak dependence on [Pg.176]

Fig. 4.11 Instantaneous nondimensional velocity profiles in a circular duct with a purely oscillating pressure gradient. The average root-mean-square velocity, averaged over one full period, shows a region of high velocity away from the centerline. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

Determine the total translational energy of 0.472 kg of CH4 at 700 K. For these conditions, calculate methane s root-mean-squared velocity. 

Note that Eq. 10.30 could have been deduced directly from Eq. 10.28 directly. The root-mean-squared velocity is thus... 

The hitherto unknown r can now be identified by employing the fact that fort -> oo the mean square velocity must have the known thermal value,... 

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