Secondary boundary layer and wall shear for fully developed flow in curved pipes

The secondary motion of incompressible fluid for a fully developed turbulent flow in horizontally laid curved pipes is theoretically analysed with the aid of the turbulent boundary-layer approach, integrating the Navier-Stokes equations and satisfying the continuity equation, having mean-time-average parameters. Outside the boundary layer that is developed due to the secondary flow, the inviscid motion of fluid is treated using the Euler and continuity equations. Inside the thin boundary layer, the Pohlhausen method is applied considering a one-seventh power distribution of velocity for solving the integrals of the Navier-Stokes equations. The expression of shear given by Blasius for the turbulent flow in pipes is incorporated into the integrals of the Navier-Stokes equations as an apparent shear to derive an implicit equation that is numerically solved to determine the boundary-layer thickness and subsequently the wall shear stress. The computational results of the boundary-layer thickness and the wall shear stresses are presented in non-dimensional (graphical) form for different Reynolds numbers and curvature ratios. The model is applicable over a wide range of Dean numbers (D) 57 less than or equal to D less than or equal to 5 x 10(5).