Fully turbulent flow

Mixing rate constants for fully turbulent flow regimes (Npg > 10,000)... 

Experimental evidence regarding the power law is somewhat contradictory. A constant value of 3 = 0..5 is considered to give a good fit to experimental data by many authors. According to Awbi, p depends on the flow regime and has a value of 0..5 for fully turbulent flow and 1,0 for laminar flow.- fn practice the value of P tends to be between 0.6 and 0.7. 

White (14) proposed a graphical method of determining the critical Reynolds number at which the fully turbulent flow... 

Many results with model systems and also biological particle systems indicate that the stress in technical bioreactors, in which turbulent flow conditions exist, could not be simulated by model studies in small bioreactors, where no fully turbulent flow exists, and especially with laminar flow devices such as viscosimeters, tubes or channels. 

However, several flow transition regimes have been identified between laminar and fully turbulent flow. The cessation of laminar Couette flow is marked by the appearance of Taylor vortices in the gap between the two cylinders. For the case of stationary outer cylinder, the critical angular velocity, C0crit> of inner cylinder at which these flow instabilities first appear can be estimated by using the following equations [102] ... 

For the Bingham plastic, there is no abrupt transition from laminar to turbulent flow as is observed for Newtonian fluids. Instead, there is a gradual deviation from purely laminar flow to fully turbulent flow. For turbulent flow, the friction factor can be represented by the empirical expression of Darby and Melson (1981) [as modified by Darby et al. (1992)] ... 

Although Eq. (9-17) appears to be explicit for G, it is actually implicit because the friction factor depends on the Reynolds number, which depends on G. However, the Reynolds number under choked flow conditions is often high enough that fully turbulent flow prevails, in which case the friction factor depends only on the relative pipe roughness ... 

In the current state of the art, almost all multiphase CFD models available in commercial codes use some type of turbulence model based on extending models originally developed for single-phase flows. Such CFD models are thus meant to describe fully turbulent flows (as opposed to laminar or transitional flows). Nevertheless, many of these models have not been validated... 

In the fully turbulent flow region DE, the curve becomes horizontal and the power function is independent of the Reynolds number for mixing ReM. For the region ReM > 10000... 

The data were plotted, as shown in Fig. 11, using the effective diameter of Eq. (50) as the characteristic length. For fully turbulent flow, the liquid and gas data join, although the two types of systems differ at lower Reynolds numbers. Rough estimates of radial dispersion coefficients from a random-walk theory to be discussed later also agree with the experimental data. There is not as much scatter in the data as there was with the axial data. This is probably partly due to the fact that a steady flow of tracer is quite easy to obtain experimentally, and so there were no gross injection difficulties as were present with the inputs used for axial dispersion coefficient measurement. In addition, end-effect errors are much smaller for radial measurements (B14). Thus, more experimentation needs to be done mainly in the range of low flow rates. 

Comparison of this with the radial dispersion data of Fig. 11 for an average packed bed (using de/dp = 0.38) shows good agreement. Notice that Eq. (84) predicts that D/udp should be independent of flow rate. From Fig. 11, this is true for the larger Reynolds numbers of fully turbulent flow. At lower Reynolds numbers, the mixing would become less random, and so it would be expected that the theory would break down. 

In conventional rocket engines, propellant distribution tends to be non-uniform across the injector face. Furthermore, relatively large changes in flow velocity may be associated with small fluctuations in supply pressures (4). Improved distribution of fuel and oxidizer across the injector face may be achieved by using orifice designs in which fully turbulent flow is attained reproducibly (4, 18). 

For fully turbulent flow (generally Rew > 10,000), fM is roughly constant. For laminar flow (RtH < 1000), the product fM Re is a constant. Between these values (transitional flow), fM is a function of Re. 

This form allows extension to low Reynolds number and near wall flows, unlike the standard k-e model, which is valid only for fully turbulent flows. Despite such... 

In the immediate vicinity of the solid wall, the turbulent fluctuations will be damped even in fully turbulent flow. In this thin layer adjacent to the wall, also known as the viscous sublayer, the viscous effect of the fluid outweighs that of its turbulent viscosity. 

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