Velocity distribution variations

The a value depends on the flow velocity distribution. Variations in flow velocity will broaden the Doppler frequency spectrum and result in a large a value. Thus, the Doppler variance image can be an indicator of flow variations and can be used to study flow turbulences. In addition, standard deviation imaging can also be used to determine the transverse flow velocity [8]. 

The a value depends on the flow velocity distribution. Variations in flow velocity will broaden the Doppler frequency spectrum and result in a large a value. Thus, the... 

Figure 12. Variation of velocity distribution in a mixing tank on insertion of full side wall baffles.

The lack of a method to determine the spatial distributions of permeability has severely limited our ability to understand and mathematically describe complex processes within permeable media. Even the degree of variation of intrinsic permeability that might be encountered in naturally occurring permeable media is unknown. Samples with permeability variations will exhibit spatial variations in fluid velocity. Such variations may significantly affect associated physical phenomena, such as biological activity, dispersion and colloidal transport. Spatial variations in the porosity and permeability, if not taken into account, can adversely affect the determination of any associated properties, including multiphase flow functions [16]. 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

The hydrodynamic theory of the penetration of targets by lined cavity jets was developed, according to Cook (Ref 7, p 252), independently by Pugh (Birkhoff s et al Ref 2) and by Hill et al (Ref 1). Pack Evans (Refs 3 4) discussed the steady-state theory of penetration in which the jet-velocity. distribution was ignored and the penetration velocity was assumed constant. Pugh, 8t Eichelberger (Refs 5 6) discussed the nonsteady-state of jet penetration in which, the actual velocity distribution in the jet was taken into account as well as the variation of the velocity of penetration with.depth These theories are discussed by Cook (Ref 7) Refs 1) R. Hill, N.F. Mott D.C. Pack, Unpublished "Ministry of Supply Report, January 1944 2) G. Birkhoff, D.P. Mac-... 

A numerical solution procedure is reasonably flexible in accommodating variations of problems. For example, the Graetz problem could be solved easily for velocity profiles other than the parabolic one. Also variable properties can be incorporated easily. Either of these alternatives could easily frustrate a purely analytical approach. The Graetz problem can also be worked for noncircular duct cross sections, as long as the velocity distribution can be determined as outlined in Section 4.4. 

The pressure variation is established using the relationship between the mass-flow rate and the velocity distribution,... 

Thus, the velocity distribution and the function, e, for similar solutions must be such that d e2u )ldx is a constant equal to (2a—/3)C/. Now one possibility is that (2a p) be equal to zero. However, the variation of jq with x that provides this situation seems to have little practical significance and it will not be considered here. Therefore, (2a - P) will be assumed to be nonzero. 

There are many situations in which similarity-type solutions to the boundary layer equations cannot be obtained. Numerical solutions to these equations can be obtained in such cases. In general, such solutions first involve numerically solving for the surface Velocity distribution and then using the energy equation to obtain the temperature distribution. Here, in order to illustrate how the energy equation can be numerically solved, it will be assumed that the variation of the surface velocity with... 

A flow field is best cliaraclerizcd by the velocity distribution, and thus a flow is said to be one-, two-, or three-dimensional if Ihe flow velocity varies in one, two, or three primary dimensions, respectively. A typical fluid flow involves a three-dimensional geometry, and the velocity may vary in all three dimensions, rendering the flow three-dimensional [V (.r. y, z) in rectangular or V (r, 0, z) in cylindrical coordinates]. However, the variation of velocity in certain directions can be small relative to the variation in oUicr directions and can be ignored with negligible error, In such cases, the flow can be modeled conveniently as being one- or two-dimensional, which is easier to analyze. 

For the start up flow case, we obtain the following set of curves of the dimensionless velocity profiles evolution shown in Figure 1, where the increase in the wall velocity with time can be clearly observed. The three dimensional plot for the velocity distribution is given in Figure 2 for the periodic case, and we can observe the quasi-steady-state (periodic state) establishment, and the time variation of the dimensionless slip velocity. 

Figure 2.9 Typical mean velocity/turbulence distribution. Variation of ut / U for neutral conditions with non-dimensional heights (a) zJZq and (b) zs/zH where zs is sensor height above ground zs = Zs - Zd, Zd is zero-plane displacement height. The line in (a) is based on the log-profile, and the line in (b) is an empirical fit. (From Roth, 2000 [550]).

Structuring at the larger than film-thickness scale can include surface grooves and channels and possible incorporation of surface meshes. These act to influence the velocity distribution across the disk allowing the liquid in some cases to detach and reimpact on the surface many times as it crosses the disk. This can be used to introduce a greater variation in the transport rates for the liquid flowing over the disk compared to the more steady conditions on the smooth disk surface. 

The upper ocean wind-driven current was described realistically for the first time by Walfried Ekman s landmark theory of 1905. The velocity distribution in the near surface layer of the ocean cannot be determined without additional information about the variation of the Reynolds stress vector with depth. Ekman (1905) assumed the Reynolds stress vector to be equal to the vertical shear of the mean current vector times a constant vertical eddy viscosity. The resulting current profile below the sea surface is the well known Ekman spiral with current speed decreasing exponentially with depth and current direction turning clockwise linear with depth from 45° right-handed to the wind stress vector at the sea surface. 

To emphasize again, the preceding arguments are useful only in identifying systems in which flow will occur as a consequence of variations in the hydrostatic pressure that cannot be balanced by a body force. To analyze any details of the motion, we would have to solve the equations of motion to determine the velocity distribution, u(x, t), and the pressure p(x, t), which would be modified from the hydrostatic form because of the motion of the fluid. 

For a fixed-orifice conventional hydraulic pressure nozzle operating with a given spray liquid, the only way in which flow rate through the nozzle can be changed is by varying the pressure at the nozzle. However, variations in operating pressure result in changes to both the spray volume distribution pattern (pattemation), the droplet size (spray quality) and velocity distributions (see Table 4.2). 

A curious circumstance, however, is that calcium appears to be about half as abundant as other alpha-process elements in galaxies. The causes are not clear, but observations indicate that calcium abundance in stars is directly correlated with the mass of the star and the velocity variation within the star at the time of formation of the calcium nuclei. Further studies of supernovae, with their complex velocity distributions, should inform theories of calcium nucleosynthesis. 

The latter cases were selected because they exhibited the more approximately axisymmetric TARS outlet features. The simulations are capable of capturing the main features of the flow, including the initial shape of the recirculation zone, its lateral extent, and the development of the annular jet. More detailed comparisons between LES and LDV for Case I are shown in Fig. 11.96, in terms of radial profiles of the mean value and rms axial velocity at selected cross-stream locations. Disagreements between LDV and LES velocity data are more noticeable in terms of the rms axial velocity distributions, largely reflecting on the neglected inlet turbulent intensities in the simulations, as well as on the needed improved emulation of the laboratory inlet conditions with regards to azimuthal mean velocity variations, turbulence statistics, and spectral content. 

If the denominator in Eq. (8.47) is set equal to 1, F(0) is given by a sinB, with a = F(0) / sin(0), and an equation for Xb is obtained which only differs from Eq. (8.35) by a numerical factor. Hence, if we introduce an addition condition for a slight variation of the adsorption along the surface, both the sinusoidal velocity distribution and the relation for the retardation coefficient proposed by Levich (1962) can be verified. To compare this additional condition with the condition of strong retardation of the surface T) Xb, the following conditions result (cf Eqs (8.40), (8.47) and (8.48)) ... 

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