Circular cylinder separation, boundary-layer

Fig. 4. Boundary layer development around a circular cylinder where A represents the point of separation.

There are some cases where this approach fails. One such case is that in which significant regions of separated flow exist. In this case, although the boundary layer equations are adequate to describe the flow upstream of the separation point, the presence of the separated region alters the effective body shape for the outer inviscid flow and the velocity outside the boundary layer will be different from that given by the inviscid flow solution over the solid surface involved. For example, consider flow over a circular cylinder as shown in Fig. 2.16. Potential theory gives the velocity, ui, on the surface of the cylinder as ... 

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

The cause of large drag in the case of a body like a circular cylinder is the asymmetry in the velocity and pressure distributions at the cylinder surface that results from separation. All bodies in laminar streaming flow at large Reynolds number are subjected to viscous stresses that boundary-layer analysis shows must be... 

Figure 10-9. The dimensionless shear stress as a function of position on the surface of a circular cylinder as calculated with the approximate Blasius series solution. Note that x is measured in radians from the front-stagnation point. The predicted point of boundary-layer separation corresponds to the second zero of du/dY 0. and is predicted to occur just beyond the minimum pressure point atx = jt/2.

Experimental observations of the flow past a circular cylinder show that separation does indeed occur, with a separation point at 0S — 110 . It should be noted, however, that steady recirculating wakes can be achieved, even with artificial stabilization,24 only up to Re 200, and it is not clear that the separation angle has yet achieved an asymptotic (Re —> oo) value at this large, but finite, Reynolds number. In any case, we should not expect the separation point to be predicted too accurately because it is based on the pressure distribution for an unseparated potential flow, and this becomes increasingly inaccurate as the separation point is approached. The important fact is that the boundary-layer analysis does provide a method to predict whether separation should be expected for a body of specified shape. This is a major accomplishment, as has already been pointed out. 

The problem of start-up flow for a circular cylinder has received a great deal of attention over the years because of its role in understanding the inception and development of boundary-layer separation. An insightful paper with a comprehensive reference list of both analytical and numerical studies is S. I. Cowley, Computer extension and analytic continuation of Blasius expansion for impulsive flow past a circular cylinder, J. Fluid Mech. 135, 389-405 (1983). 

The position of the separation point depends on the Reynolds number. At low Reynolds numbers, we have a flow without separation (see Figure 1.7). In a cross flow around a circular cylinder, the separation occurs if the Reynolds number (the cylinder diameter is taken as the characteristic length) is greater than 5 [486]. For 5 < Re < 40, a separation region with steady-state symmetric adjacent vortices is formed (there is no boundary layer yet). 

In the range 103 < Re < 105 (here the radius a of the circular cylinder is taken as the characteristic length, Re = aUi/u), the laminar boundary layer approximation is valid, and the separation point do moves from 109° to 85° [427], In this case, retaining only the first three terms in the expansion (1.8.13) provides fairly good accuracy. 

A particularly interesting phenomenon connected with transition in the boundary layer occurs with blunt bodies, e.g., spheres or circular cylinders. In the region of adverse pressure gradient (i.e., dP/dx > 0 in Fig. 1.9) the boundary layer separates from the surface. At this location the shear stress goes to zero, and beyond this point there is a reversal of flow in the vicinity of the wall, as shown in Fig. 1.9. In this... 

In the range of Reynolds number Re = 103 to 107 (based on cyhnder diameter and free stream velocity), the flow aronnd a solid circular cylinder is periodic and transitional in character. The range of interest of the present work is located in a sub-critical flow regime (103 < Re < 105, corresponding to air velocities of - 0.1-10 m/s around a typically sized 0.1m diameter limb), in which, dne to the vortex shedding at the cylinder surface, the flow is highly unstable. The boundary layer remains fidly laminar up to the separation point and transition to turbulence... 

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