Turbulence profile

Figure 8. Isoplanatic angle at 2.2pm as a function of conjugate altitude of the deformable mirror for different turbulence profiles obtained at the Observatorio del Roque de los Muchachos.

The relative shapes for the velocity profiles in laminar and turbulent flow are indicated in Fig. 5-1. The laminar profile is approximately parabolic, while the turbulent profile has a portion near the wall which is very nearly linear. This linear portion is said to be due to a laminar sublayer which hugs the surface very closely. Outside this sublayer the velocity profile is relatively flat in comparison with the laminar profile. 

The Reynolds stress distributions in Fig. 8.5c indicate that turbulent momentum transport is also modified at high levels of counterflow. Since the mean velocity profiles (shown in Fig. 8.4) display independence of i, but the Reynolds stress experiences enhanced transport, the overall turbulent production of the layer is considerably increased above 13%) counterflow. In fact, a comparison of the self-similar stress profiles in Fig. 8.5 indicates that a common state is achieved for < 0.13(/i, and a second common self-similar state is achieved for U2 > 0.24(7i. From 0% to 13% counterflow, the turbulent profiles collapse, indicating a mechanism for generating the turbulence that scales with the growth rate parameter o- and velocity difference AU. Above 13% counterflow, there is an increase in turbulence level across the entire cross-stream extent of the layer. This increase seems to be dependent on velocity ratio, but not on the parameter c. Since the mean profiles display similar shape, there is likely an additional mechanism for turbulence production when Ibol is greater than approximately 0.13f7i. 

Gasljevic, K., Aguilar, G., and Matthys, E. R, On two distinct types of drag-reducing fluids, diameter scaling, and turbulent profiles, /. Non-Newtonian Fluid Mech.,96,405-425 (2001). 

It, however, has to be noticed that this statement is valid provided that the fluctuations of the gas flow do not contribute to droplet coalescence, which can occur when the gas stream reaches a fuUy developed turbulent profile around the liquid jet breakup region. Above the critical We value of 40, Gafidn-Calvo pointed out that the sinuous non axisymmetric disturbances become apparent, coupled to the axisymmetric ones. It is also mentioned that increasing again the We number will lead to a nonlinear growth rate of the sinous disturbances, which will overcome the axissymmetric ones and produce polydisperse drops. ... 

Turbulence profiles across the tube were measured by laser doppler anemometry in all three directions separately at 5 m from the tube entrance. A short thin wall section of plastic tube... 

Although these results for the velocity fluctuations are in agreement with the mixing length experiment, it is not directly possible to relate the mixing performance to the turbulence profiles found. 

The analysis of concentration profiles showed that measured turbulent profiles could be successfully approximated by the Rouse-Schmidt turbulent dif sion model with the implemented settling velocity effect [2]. For both the fine sand (0.10-0.15 mm s uid) and the medium sand (0.2-0. 5 imn sand) (Fig. 1), the solids dispersion coefficient s,meaii (the mean value obtained by integrating local s values over the flow core) seems to be virtually independent of solids concentration in a pipehne. 

Many authors refer to the critical Reynolds number (Recrit) as 2300. The transition from laminar to turbulent profiles can actually occur over a relatively broad range and is a function of the channel geometry and surface roughness. In this text, Recrit = 3000 is used as a general guideline. 

Figure 5.20 Laminar and turbulent profile boundaries (a) fully developed laminar profile (b) fully developed turbulent profile. Notice the turbulent du/dy profile is much steeper, leading to a higher pressure drop per unit length via Eq. (5.71).

The time-to-distance transfonnation requires fast mixing and a known flow profile, ideally a turbulent flow with a well-defined homogeneous composition perpendicular to the direction of flow ( plug-flow ), as indicated by tire shaded area in figure B2.5.1. More complicated profiles may require numerical transfomiations. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Although molecular diffusion itself is very slow, its effect is nearly always enhanced by turbulent eddies and convection currents. These provide almost perfect mixing in the bulk of each Hquid phase, but the effect is damped out in the vicinity of the interface. Thus the concentration profiles at each... 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Fig. 1. Flow profiles, where N is velocity (a) laminar, and (b) turbulent for fluids having Reynolds numbers of A, 2 x 10, and B, 2 x 10 .

Most flow meters are designed and caHbrated for use on turbulent flow, by far the more common fluid condition. Measurements of laminar flow rates may be seriously in error unless the meter selected is insensitive to velocity profile or is specifically caHbrated for the condition of use. 

Fig. 7. Axial density profiles in the (—) bubbling, (------) turbulent, and (----) fast and ( ) riser circulating fluidization regimes. Typical gas velocities for...

In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

The overall benefits of this high efficiency combustor over a conventional bubbling- or turbulent-bed regenerator are enhanced and controlled carbon-bum kinetics (carbon on regenerated catalyst at less than 0.05 wt %) ease of start-up and routiae operabiUty uniform radial carbon and temperature profiles limited afterbum ia the upper regenerator section and uniform cyclone temperatures and reduced catalyst iaventory and air-blower horsepower. By 1990, this design was well estabUshed. More than 30 units are ia commercial operation. 

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. 

In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between. 

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. 

Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker Ind. Eng. Chem., 38, 622-624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile ... 

The universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the eddy viscosity,... 

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