B • TURBULENT FLOW 



dimensional flow where pressure gradients in the x and y directions are 

 encountered. 



Boundary layers are usually so thin compared to the relatively large 

 distances over which pressure changes occur that the changes across the 

 layer are so small that they have insignificant effects. The pressure may 

 change even more gradually in the x direction, but here the boundary 

 layer extends over the full range of the pressure changes, and cumulative 

 effects become important. The thickness of the layer is always affected, 

 and the mean velocity profile will change form as the flow progresses 

 unless conditions are so arranged that it is held in equihbrium by the 

 balance between inertial, pressure, and friction forces. Pipe and channel 

 flows are examples of equilibrium flows in which the pressure drop is 

 exactly balanced by wall friction. As shown by Clauser [83], a balance is 

 possible in boundary layer flows under certain conditions, and his con- 

 tributions to this subject will be taken up in Art. 20. In general, mean 

 velocity distributions undergo progressive changes when subjected to 

 pressure gradients — the less so when the flow proceeds toward lower 

 pressures, and the more so when the flow proceeds toward higher pres- 

 sures. The latter therefore deserves, and usually receives, the greater 

 attention. 



The importance of flow to higher pressures is emphasized by the 

 possibility, and often the occurrence, of flow separation. Separation is 

 the result of flow reversal and an accumulation of stagnant fluid over 

 which the moving fluid passes without having to follow the contour of a 

 body. An adverse pressure gradient opposes motion in the direction of 

 the main flow and can set up motion in the reverse direction when the 

 fluid has lost sufficient momentum through friction with a wall. Since 

 the momentum approaches zero at a wall, only the shear stresses between 

 the faster- and slower-moving fluids can prevent flow reversal. Whether 

 or not reversal will occur depends on an interplay between the shear 

 stresses and the pressure gradient. In any case the fluid movement is 

 retarded, and shear stresses are expended against internal forces on the 

 fluid arising from the pressure gradient. The maximum shear stress is no 

 longer at the wall, as it is for constant pressure, but now occurs some 

 fraction of the boundary layer thickness away from the wall depending 

 on the state of retardation of the layer. These effects reduce skin friction 

 and the momentum losses from this source, but only in exchange for even 

 greater internal momentum losses resulting from shear stresses applied to 

 pressure-retarded flow. 



A classic example of the typical evolution of velocity profiles occurring 

 when a boundary layer is subjected to a monotonically increasing pres- 

 sure sufficient to bring about eventual flow separation is the set of curves 

 compiled by von Doenhoff and Tetervin [84\ shown in Fig. B,19a. Here 

 I7e is the local free stream velocity just outside the boundary layer, and 



< 130) 



