All Reynolds Numbers

The friction factor for a Bingham plastic can be calculated for any Reynolds number, from laminar through turbulent, from the equation 

---

Radial-flow impellers include the flat-blade disc turbine, Fig. 18-4, which is labeled an RlOO. This generates a radial flow pattern at all Reynolds numbers. Figure 18-17 is the diagram of Reynolds num-ber/power number curve, which allows one to calculate the power knowing the speed and diameter of the impeller. The impeller shown in Fig. 18-4 typically gives high shear rates and relatively low pumping capacity. 

Figure 8 shows that the group Di /wde is roughly constant for all Reynolds numbers. Jacques and Vermeulen (Jl) found, however, that their data with regular arrangements of the particles showed a definite break in the values. This seems to be caused by a transition from laminar to turbulent flow. 

Experimental data are available for large particles at Re greater than that required for wake shedding. Turbulence increases the rate of transfer at all Reynolds numbers. Early experimental work on cylinders (VI) disclosed an effect of turbulence scale with a particular scale being optimal, i.e., for a given turbulence intensity the Nusselt number achieved a maximum value for a certain ratio of scale to diameter. This led to speculation on the existence of a similar effect for spheres. However, more recent work (Rl, R2) has failed to support the existence of an optimal scale for either cylinders or spheres. A weak scale effect has been found for spheres (R2) amounting to less than a 2% increase in Nusselt number as the ratio of sphere diameter to turbulence macroscale increased from zero to five. There has also been some indication (M15, S21) that the spectral distribution of the turbulence affects the transfer rate, but additional data are required to confirm this. The major variable is the intensity of turbulence. Early experimental work has been reviewed by several authors (G3, G4, K3). 

P 62] A Lagrangian particle tracking technique, i.e. the computation of trajectories of massless tracer particles, which allows the computation of interfacial stretching factors, was coupled to CFD simulation [47]. Some calculations concerning the residence time distribution were also performed. A constant, uniform velocity and pressure were applied at the inlet and outlet, respectively. The existence of a fully developed flow without any noticeable effect of the inlet and outlet boundaries was assured by inspection of the computed flow fields obtained in the third mixer segment for all Reynolds numbers under study. 

The Reynolds number range between 2100 and 4000 is commonly designated as the critical region. In this range, there is considerable doubt as to whether the flow is viscous or turbulent. For design purposes, the safest practice requires the assumption that turbulent flow exists at all Reynolds numbers grater than 2100. 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

An important problem is to analyze the stability of fluid flows. With the exception of the Taylor-Couette and Saffman Taylor problems, this chapter has focused on stability questions when the base state of the system was one with no motion (or rigid-body motion), so that instability addresses the conditions for spontaneous onset of flow. An equally valid question is whether a particular flow, such as Poiseuille flow in a pipe (or any of the other flows that we have analyzed in previous chapters of this book), is stable, especially to infinitesimal perturbations as linear instability determines whether the particular flow is actually realizable in experiments. This question was first mentioned back in Chapter 3 when we analyzed simple unidirectional flow problems and noted that solutions such as Poiseuille s solution for flow through a tube was a valid solution of the Navier-Stokes equations for all Reynolds numbers, even though common experience tells us that beyond some critical Reynolds number there is a transition to turbulent flow in the tube. 

A combined effect of all five corrections can reduce the ideal heat transfer coefficient by up to 60 percent. A comparison with a large number of proprietary experimental data indicates the shellside h predicted using all correction factors is from 50 percent too low to 200 percent too high with a mean error of 15 percent low (conservative) at all Reynolds numbers. 

Crespo and Manuel [36] and, independently, Balasubramaniam and Chai [37] recognized that the Stokes flow solution for small Marangoni numbers gave aggregate predictions at all Reynolds numbers provided that Ma 1 (i.e., that the convective transport of energy is negligible). Balasubramaniam and Chai [37] also showed that, at finite Reynolds numbers, a bubble or drop deforms into an oblate spheroid and obtained an asymptotic result for the shape for small inertial corrections. Their analysis is based on the assumption that both the capillary number and the Weber number are small compared to unity ... 

Estimates of (kf,ka) and (ka,hw) are virtually uncorrelated except possibly at low Reynolds number those for (kr.hw) a strongly correlated at all Reynolds numbers. While ky. is not a conductivity in the true sense, it nevertheless has a sound theoretical basis, as proposed by Argo and Smith (9 ) h on the other hand is perhaps no more than an empirical parameter needed... 

In fully turbulent flow, viscous forces become negligible relative to turbulent stresses and can be neglected (except for their action at the dissipative scales of motion). This has an important implication above a certain Reynolds number, all velocities will scale with the tip speed of the impeller, and the flow characteristics can be reduced to a single set of dimensionless information, regardless of the fluid viscosity. One experiment in the fiiUy turbulent regime can be applied for all tanks that are exactly geometrically similar to the model, at all Reynolds numbers... 

---