Velocity profile similar

Figures 10.9 and 10.11 show the burner exit axial velocities with the baffle solution implemented. More uniform velocity profiles similar to that of a plug flow can be seen.

For external flow such as flow over a flat stationary plate whose surface temperature is different from the bulk fluid temperature, both hydrodynamic and thermal boundary layers develop along the direction of the flow. Inside the hydrodynamic boundary layer viscous forces are dominant resulting in velocity profile. Similarly as thermal boundary layer develops along the flow direction and heat is being transferred to or from the surface results in a temperature profile. In Figure 22.8 the hydrod5mamic and thermal boundary layers are shown for flow over a heated flat plate. Both velocity and temperature inside the boundary layer reach 99% of the free stream velocity (Vf) and temperature (T ), respectively, at the edge of the boundary layer. 

Enough space must be available to properly service the flow meter and to install any straight lengths of upstream and downstream pipe recommended by the manufacturer for use with the meter. Close-coupled fittings such as elbows or reducers tend to distort the velocity profile and can cause errors in a manner similar to those introduced by laminar flow. The amount of straight pipe required depends on the flow meter type. For the typical case of an orifice plate, piping requirements are normally Hsted in terms of the P or orifice/pipe bore ratio as shown in Table 1 (1) (see Piping systems). 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

Similar to the case with compact air jet supply, theoretical values of characteristic depend upon the type of velocity profile equation and supply conditions. According to Shepelev, = 2.62. The Gortler profile results in Kj = 2.43 and the Tollmein profile in iCj = 2.51. Becher " reported the A, characteristic for a linear jet to be equal to 2.55. Experimental results by Heskestad, Miller and Comings,van der Hegge Zijnen, 238 Gutmark and Wygnanski, and Kotsovinos and List appear to satisfy A, = 2.43,... 

These relations, which describe the velocity profile sketched in Fig. 10.63, have a similarity property, behind the form of the equations. Guttmark and Wignanski indicate that the similarity profile could be found up to a length ot 120 times the outlet opening width. 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

In any liquid flowing down a surface, a velocity profile is established with the velocity increasing from zero at the surface itself to a maximum where it is in contact with the surrounding atmosphere. The velocity distribution may be obtained in a manner similar to that used in connection with pipe flow, but noting that the driving force is that due to gravity rather than a pressure gradient. 

Similarity solutions of the velocity profile functions for the Von Karman problem. (From Von Karman, Th, Z., Angew. Math. Mech., 1,231,1921.)... 

The density profile for the micropore fluid was determined as In the equilibrium simulations. In a similar way the flow velocity profile for both systems was determined by dividing the liquid slab Into ten slices and calculating the average velocity of the particles In each slice. The velocity profile for the bulk system must be linear as macroscopic fluid mechanics predict. 

In Figure 10, we present flow velocity predictions of the high density approximation, Equations 32 - 33, 38 and 39, of Davis extension of Enskog s theory to flow In strongly Inhomogeneous fluids (1 L). The velocity profile predicted In this way Is also plotted In Figure 10. The predicted profile, the simulated profile, and the profile predicted from the LADM are quite similar. 

At large distances from the nozzle, the axial velocity exhibits self-preserving similarity in that, when plotted in the above dimensionless form, velocity profiles at all cross-sections downstream collapse onto a single curve. It is probable that strict similarity is preserved only at axial distances in excess of 30-40 diameters. However, the above formulation is frequently used to describe the velocity profile at all points downstream of the flow development region. For work performed with M. citrifolia, the numerical values recommended by Panchapakesan and Lumley [131] for the constants K and C in Eqs. (12) and (13), i.e. 75.2 and 6.06, respectively, are employed. 

Although NMRI is a very well-suited experimental technique for quantifying emulsion properties such as velocity profiles, droplet concentration distributions and microstructural information, several alternative techniques can provide similar or complementary information to that obtained by NMRI. Two such techniques, ultrasonic spectroscopy and diffusing wave spectroscopy, can be employed in the characterization of concentrated emulsions in situ and without dilution [45],... 

For a number of flow situations, the mass-transfer rate can be derived directly from the equation of convective diffusion (see Table VII, Part A). The velocity profile near the electrode is known, and the equation is reduced to a simpler form by appropriate similarity transformations (N6). These well-defined flows, therefore, are being exploited increasingly by electrochemists as tools for the kinetic characterization of electrode reactions. Current distributions at, or below, the limiting current, transient mass transfer, and other aspects of these flows are amenable to analysis. Especially noteworthy are the systematic investigations conducted by Newman (review until 1973 in N7 also N9b, N9c, H6b and references in Table VII), by Daguenet and other French workers (references in Table VII), and by Matsuda (M4a-d). Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and mass-transfer experiment. 

The resulting velocity profiles and the flow pattern inside and around the jet are shown in Figs. 29 and 30 for a jet velocity of 32.6 m/s and with two different aeration flows. The jet boundary at Vz = 8 m/s shown in Figs. 29 and 30 was calculated from Tollmien similarity. The boundary where the tracer gas concentration becomes zero, C = 0, was determined from the normalized experimental tracer gas concentration profiles shown in Figs. 

The axial velocity profiles, calculated on the basis of Tollmien similarity and experimental measurement in Yang and Kcaims (1980) were integrated across the jet cross-section at different elevations to obtain the total jet flow across the respective jet cross-sections. The total jet flows at different jet cross-sections are compared with the original jet nozzle flow, as shown in Fig. 31. Up to about 50% of the original jet flow can be entrained from the emulsion phase at the lower part of the jet close to the jet nozzle. This distance can extend up to about 4 times the nozzle diameter. The gas is then expelled from the jet along the jet height. 

Qualitatively similar velocity profiles are set up within gas cyclones, discussed in Section 1.6.2, which are extensively used for the removal of suspended solids from gases. In this case, the velocities are generally considerably higher. 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

Upstream and downstream disturbances in the flow field are caused by valves, elbows, and other types of fittings. Two upstream elbows in two perpendicular planes will impart swirl in the fluid downstream. Swirl, similar to atypical velocity profiles, can lead to erroneous flow measurements. Although the effect is not as great as in upstream flow disturbances, downstream flow disturbances can also lead to erroneous flow measurements. 

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