The prediction of separation of the turbulent boundary layer

Abstract

A rapid method for the prediction of flow separation results from an approximate solution of the equations of motion; a single empirical factor is required. The equations are integrated by a modified ‘inner and outer solutions’ technique developed recently for laminar boundary layers, the criterion for separation being obtained as a simple formula applying directly to the separation position. At Reynolds numbers of the order of 10⁶, the criterion is $C_p(xdC_p|dx)^{\frac {1}{2}} = 0 \cdot 39 (10^{-6}R)^{\frac {1}{10}},$ when d²p/dx² [ges ] 0 and C_p [les ] 4/7; the coefficient 0·39 is replaced by 0·35 when d²p/dx² < 0.

The prediction of the pressure rise to separation is likely to be from 0 to 10% too low, which puts it second in accuracy to those methods, such as Maskell's (1951), which utilize the Ludweig-Tillmann skin friction law. However, the convenience of the method makes the present error acceptable for many applications, while a greater accuracy should be attainable from an improved allowance for the quantity d²p/dx².

The main derivation is for arbitrary pressure distributions, while an extension leads to the pressure distribution which just maintains zero skin friction throughout the region of pressure rise.

The concept of a turbulent inner layer with zero wall stress is put forward, and it is deduced that in the neighbourhood of the wall the velocity is proportional to the square root of the distance from the wall.