Single-Phase Flow in a Curved Pipe

It is shown in Fig. 11.9 that when a pipe is curved, the distribution of velocity in a bend is appreciably altered by the secondary flow and the highest axial velocity occurs 

Let / b be the radius of curvature of the pipe axis and R4 be the radius of the circular cross section of the pipe. Define U as the axial velocity component and (=/ d — r) as the distance normal to the wall. Denote 0 as the angle in the transverse plane with respect to the outward direction of the symmetry line and 4 as the angle measured in the plane of the curved pipe axis, as shown in Figs. 11.9(a) and (c). Assume that the changes of the flow pattern along the axis of the bend can be neglected. Thus, the momentum integral equations 

On the other hand, W increases from zero at the wall to a positive value and then decreases to zero at the edge of the shedding layer. Thus, W is assumed to have the form 

The shear stress at the wall can be given in terms of Vt and as [Schlichting, 1979] 

Since the maximum velocity in the secondary flow is lower than the axial velocity at the edge of the shedding layer, B is much smaller than unity. Thus, we have 

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