Turbulent/laminar transition

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

Re(MR)c Critical value of Reyn at laminar-turbulent transition ... 

Putting the constant equal to zero, implies that <5 = 0 when x = 0, that is that the turbulent boundary layer extends to the leading edge of the surface. An error is introduced by this assumption, but it is found to be small except where the surface is only slightly longer than the critical distance xc for the laminar-turbulent transition. 

An experimental study of the laminar-turbulent transition in water flow in long circular micro-tubes, with diameter and length in the range of 16.6-32.2 pm and 1-30 mm, respectively, was carried out by Rands et al. (2006). The measurements allowed to estimate the effect of heat released by energy dissipation on fluid viscosity under conditions of laminar and turbulent flow in long micro-tubes. 

The conflicting predictions of the various equations for the transition point have led us to experimentally determine the laminar-turbulent transition for the particular configuration employed in this work. This is reported in the section on results. 

A summary of the nine batch reactor emulsion polymerizations and fifteen tubular reactor emulsion polymerizations are presented in Tables III IV. Also, many tubular reactor pressure drop measurements were performed at different Reynolds numbers using distilled water to determined the laminar-turbulent transitional flow regime. 

Figure 3. Log of friction factor ratio vs, log D used to determine the laminar-turbulent transition in the reactor...

It was found that the maximum rate of polymerization occurred at (NRe)e 5000. This shift in (NRe) corresponds to the shift of the laminar turbulent transition in a helically coiled tube as reported by White ( ). Further, no plugging of this reactor, under any conditions of operation, was noticed. The reaction mechanism appears to be very close to the Smith-Ewart model, although conversions were not always a3 complete as expected. 

Under normal circumstances, the laminar-turbulent transition occurs at a Reynolds number of about 2100 for Newtonian fluids flowing in pipes. 

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

The dimensionless Reynolds number (Re) is used to characterize the laminar-turbulent transition and is commonly described as the ratio of momentum forces to viscous forces in a moving fluid. It can be written in the form... 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

Based on these data, particle-liquid Reynolds numbers were calculated to range from Re = 25 (50 rpm) to Re = 90 (150 rpm) for coarse grade particles with a median diameter of 236 pm. In contrast, Reynolds numbers for a batch of micronized powder of the same chemical entity with a median diameter of 3 pm were calculated to be significantly lower (Re < 1), indicating less sensitivity towards convective hydrodynamics [(10), Chapter 12.3.8]. Based on the aforementioned considerations for spheres, bulk Reynolds numbers of about Re > 50 appear to be sufficient to produce the laminar-turbulent transition around a rough drug particle of coarse grade dimensions. 

The Reynolds number characterizing laminar-turbulent transition for bulk flow in a pipe is about Re 2300 provided that the fluid moves unidirectionally, the pipe walls are even and behave in a hydraulically smooth manner, and the internal diameter remains constant. However, intestinal walls do not fulfill these hydraulic criteria due to the presence of curvatures, villi, and folds of mucous membrane, which are up to 8 mm in the duodenum, for instance (Fig. 18). Furthermore, the internal diameter of the small intestine is estimated to... 

Fig. 5.11 Position of boundary layer separation and laminar/turbulent transition in the critical region and beyond. Experimental results of Achenbach (A3) and Raithby and Eckert (R3).

Ogawa K, Kuroda C (1986) Experimental study on the effect of elasticity on drag reduction and turbulent fluctuations in the laminar-turbulent transition region in pipe flow of dilute polymer solutions Can J Chem Eng 64 497... 

It is clear that the flow regime is a complicated but predictable function of the physical properties of the liquid, the flow rate, and the slope of the channel. It has been shown that, for water films, gravity waves first appear in the region NrT = 1-2, capillary surface effects become important in the neighborhood of JVw = I, and the laminar-turbulent transition occurs in the zone ArRe = 250-500 (F7). 

The parameter R is the laminar-turbulent transition value of R and has the numerical value 183.3 for Newtonian fluids. For non-Newtonian fluids it would have to be computed from the various results presented above. 

Arnal, D. (1986). Three-dimensional boundary layer Laminar-turbulent transition. AGARD eport No. 7fl 1-34. 

Arnal, D., Casalis, G. and Jullien, J.C. (1989). Experimental and theoretical analysis of natural transition on infinite swept wing. In the lU-TAM S3mip. Proc. on Laminar- Turbulent Transition (Eds. D. Arnal, R. Michel), Springer-Verlag, 311-326. 

Leehey, P. and Shapiro, P. (1979). Leading edge effect in laminar boundary layer excitation by sound. In Laminar Turbulent Transition (eds. R. Eppler and H. Fasel), 321-331, Springer Verlag. 

Michel, R., Arnal. D., Coustols, E. and Jullien, J.C. (1984). Experimental and theoretical studies of boundary layer transition on a swept infinite wing. In Proc. of lUTAM S3unp. On Laminar-Turbulent Transition Novosibirsk, USSR, Springer Verlag. 

Hanks, R. W. and Ricks, B. L. 1974. Laminar-turbulent transition in flow of pseudoplastic fluids with yield stress. 7. Hydronautics 8 163-166. 

The evolution of the Poiseuille number f Re) as a function of the Reynolds number is shown on figure 12. It is observed that the classical value for the laminar regime is obtained if the Reynolds number is less than 2000. The laminar turbulent transition occurs for the conventional value. The authors [22] investigated the entrance effects. They conclude that the friction factor is insensitive to the channel height and that there was no sign of a faster transition to turbulence compared to conventional channel flows. 

To verify the flow regime and the laminar-turbulent transition, a bronze block was replaced by an transparent altuglas plate. Visualisation with dye revealed a very stable flow for Reynolds numbers up to 2500 for both smooth and rough channels. On the contrary, large eddies were visualised for Reynolds numbers over 3800. Between these two values a stable flow region following turbulent structures were observed. 

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