Regime Transitions

Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically statistical methods are used to quantify turbulence. 

The Intermediate Regime (Best known as the Transition Regime Law, for 2 [Pg.273]

Those particles with sizes d > d" at a given set of conditions (v, p, Pp, and a ) will settle only in the turbulent flow regime. For particles with sizes d < d, d" will settle only when the flow around the object is in the transitional regime. Recall that the transitional zone occurs in the Reynolds number range of 0.2 to 500. The sedimentation numbers corresponding to this zone are 3.6 < S, < 82,500 and 0.0022 < S2 < 1,515. 

For the same case of n = 1200 rpm and r = 0.5, we obtain u,/Ug = 800, whereas for the turbulent regime the ratio was only 28. This example demonstrates that the centrifugal process is more effective in the separation of small particles than of large ones. Note that after the radial velocity u, is determined, it is necessary to check whether the laminar condition. Re < 2, is fulfilled. For the transition regime, 2 < Re < 500, the sedimentation velocity in the gravity field is ... 

This ratio represents an average between similar ratios for the laminar and turbulent regimes. In the most general case, u, = f(D, Pp, p, /r, r, w), and hence we may ignore whether the particle displacement is laminar, turbulent or within the transition regime. This enables us to apply the dimensionless Archimedes number (recall the derivation back in Chapter 5) ... 

Transition Regime the kinks become unstable and are able to propagate but remain localized global intermittent pattern cannot yet be formed. The patterns arc reniiniscent of class c4 behavior. 

However, for the lubricants with lower viscosity, e.g.. Polyglycol oil 1 and 2 with the kinetic viscosity of 47 mm /s to 145 mm /s in Table 1, the transition from EHL to TFL can be seen at the speed of 8 mm/sand23 mm/s, i.e., the relationship between film thickness and speed becomes much weaker than that in EHL. The transition regime can be explained when the film reduces to several times the thickness of the molecular size, the effect of solid surface forces on the action of molecules becomes so strong that the lubricant molecules become more ordered or solid like. The thickness of such a film is related to the lubricant viscosity or molecular size. 

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] ... 

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

Ramilison, J. M., and J. H. Lienhard, 1987, Transition Boiling Heat Transfer and the Film Transition Regime, Trans. ASME. J. Heat Transfer 109 146. (2)... 

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

Recently, des Cloizeaux has conceived a rubber -like model for the transition regime to local reptation [56]. He considered infinite chains with spatially fixed entanglement points at intermediate times. In between these... 

Fig. 12. Dynamic moduli master curves of PBD 44 precursor (p = 0) and PBD 44 critical gel [60]. The entanglement and glass transition regime is hardly affected by the crosslinking. Open symbols correspond to G, filled ones to G"...

Formulas for the conductance in this transition regime (see Fig. 1.3) are quite complicated and can be found in ref. [9], It is worth noting that the flow in the transition region is usually greater than that calculated assuming a molecular flow. 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

A further application of the coplanar cell configuration showed in Fig. 3.1(c) concerns the study of the time dependence of the photocurrent following carrier excitation by means of a short pnlse of illnmination. This transient photodecay technique enables the examination of the interaction of initially free carriers with varions localized states. In principle, the decay of photocnrrent measured in this manner should (in the absence of recombination effects and phenomena associated with drift close to the surface of a thin film) correspond to the behavior in the initial pre-transit regime of a TOF pnlse. Becanse it allows measurements to be performed on very thin films under conditions appropriate to their nse in many device applications, and because the photocurrent may be examined over several decades of time withont the complications associated with carrier extraction, the techniqne has become rather popular over recent years. 

The addition of solids has been found to increase or even decrease the gas holdup, depending on the experimental conditions. When the solids addition promotes bubble break up, the average bubble size is smaller, the bubble rising velocity is reduced, and the gas holdup increases (DOE, 1985). The following correlation of Koide et al. (1963) is used in slurry bubble columns and for the heterogeneous and transition regimes (Koide et al., 1984 Koide, 1996) ... 

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