Laminar wakes

Ret < 300 Unsteady laminar inertial flow in which laminar wake oscillations appear in the pores and vortices form at around Ret — 250 ... 

Fig. 7.11 Wake configurations for drops in water (highly purified systems), reproduced from Winnikow and Chao (W8) with permission, (a) nonoscillating nitrobenzene drop = 0.280 cm, Re = 515 steady thread-like laminar wake (b) nonoscillating m-nitrotoluene drop 4 = 0.380 cm. Re = 688 steady thread accompanied by attached toroidal vortex wake (c) oscillating nitrobenzene drop 4 = 0.380 cm. Re = 686 central thread plus axisymmetric outer vortex sheet rolled inward to give inverted bottle shape of wake (d) oscillating nitrobenzene drop = 0.454 cm. Re = 775 vortex sheet in c has broken down to form vortex rings (e) oscillating nitrobenzene drop d = 0.490 cm. Re = 804 vortex rings in d now shed asymmetrically and the drop exhibits a rocking motion.

The effect of an external straining flow on a laminar wake caused by other bodies is illustrated with linearised calculations (see Fig. 7.2b). Consider a point source of momentum in a planar linear straining flow,... 

In order to better characterize the laminar wakes, Campos and Guedes de Carvalho [4] performed a visual study inspired on the work of Maxworthy [II], The liquid in the lower half of the column, where the slug was injected, was made dark by means of a soluble dye. As the slug rose into the clear liquid, in the upper half of the column, the dark wake could be seen clearly in photographs like those shown in Figure 2, obtained with a still camera. 

As the Reynolds number rises above about 40, the wake begins to display periodic instabiUties, and the standing eddies themselves begin to oscillate laterally and to shed some rotating fluid every half cycle. These still laminar vortices are convected downstream as a vortex street. The frequency at which they are shed is normally expressed as a dimensionless Strouhal number which, for Reynolds numbers in excess of 300, is roughly constant ... 

For 40 < Re < 200 the vortices are laminar and the Strouhal number has a nearly constant value of 0.2 for flow past a cylinder. Between Re = 200 and 400 the Strouhal number is no longer constant and the wake becomes irregular. Above about Re = 400 the vortices become turbulent, the wake is once again stable, and the Strouhal number remains constant at about 0.2 up to a Reynolds number of about 10. ... 

The Reynolds number, which is directly proportional to the air velocity and the size of the obstacle, is a critical quantity. According to photographs presented elsewhere, a regular Karman vortex street in the wake ot a cylinder is observed only in the range of Reynolds numbers from about 60 to 5000. At lower Reynolds numbers, the wake is laminar, and at higher Reynolds numbers, there is a complete turbulent mixing. 

You want to perform an experiment that illustrates the wake behind a sphere falling in water at the point where the boundary layer undergoes transition from laminar to turbulent. (See Fig. 11-4.) If the sphere is made of steel with a density of 500 lbm/ft3, what should the diameter be ... 

On increasing the Reynolds number further, a point is reached when the boundary layer becomes turbulent and the point of separation moves further back on the surface of the sphere. This is the case illustrated in the lower half of Figure 9.1 with separation occurring at point C. Although there is still a low pressure wake, it covers a smaller fraction of the sphere s surface and the drag force is lower than it would be if the boundary layer were laminar at the same value of Rep. 

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

At Re = 130, a weak long-period oscillation appears in the tip of the wake (T2). Its amplitude increases with Re, but the flow behind the attached wake remains laminar to Re above 200. The amplitude of oscillation at the tip reaches 10% of the sphere diameter at Re = 270 (GIO). At about this Re, large vortices, associated with pulsations of the fluid circulating in the wake, periodically form and move downstream (S6). Vortex shedding appears to result from flow instability, originating in the free surface layer and moving downstream to affect the position of the wake tip (Rll, R12, S6). 

For Re < 110 the wake is closed and laminar as discussed above. Transfer over the front portion of the cap is again described by Eq. (8-20). Transfer from the base occurs by diffusion into the wake fluid as it moves along the bubble base, producing a concentration boundary layer. The solute in this... 

The previous intent has been to use kinetics simply as a tool to describe qualitatively the particular aspect of combustion under study. Numerical values of the kinetic constants were thus assumed for illustrative purposes or approximated from other types of data by making admittedly questionable major assumptions. Approximations include, for example, the extrapolation of low temperature hydrocarbon oxidation rates to high temperature hydrocarbon combustion rates. Other schemes involve application of semiempirical laminar flame speed theories or of flow patterns in the wake of a bluff body immersed in an air stream (43). 

Figure 4. The longitudinal vortex. A longitudinal vortex showing laminar flow about the central axis. The coldest water filaments are always closest to the central axis of flow. Thermal stratification occurs even with minimal differences in water temperature. The central core water is subjected to the least turbulence and acclerates ahead, drawing the rest of the water body in its wake.

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

FIGURE 716 Laminar boundary layer separation witli a turbulent wake flow over a circular cylinder at Re = 7.000. 

---