Turbulent - Boundary - Layer Behaviour and the Auxiliary Equation

In addition to the K&m& momentum-integral equation, two further equations are required for the purposes of calculating the development of the incompressible turbulent boundary layer-a "skin-friction law" and an "auxiliary equrtion". The problem of deriving a satisfactory form of auxiliary equation is a major one. Indeed, despite the effort devoted to this question for meny years, few derived forms of auxiliary equotion can be relied upon to account for the boundary-layer development in more th'an a restricted number of cases. As a contribution to the elucidation of the problem en examination is made of several sets of experimental data covering different types of pressure distribution. Among the data certain basic types of boundary-layer behaviour ccn be distinguished. The equilibrium boundmJ-lcyer may be regarded as a datum condition and the other types of behaviour are discussed in the context of tendencies towards, or departures from, equilibrium. Simple forms of auxiliary equation are postulated and an examination is made of the extent to which they cnn be reconciled, even qualitatively, with the observed types of boundcry-layer behaviour. It is shown that several forms of the equation must be rejected as inadequate. The most economical form which appears to be capable of describing cl1 the various trends of the data is a second-order differential equation involving the shape factor. A limited number of comparisons with experiment indicate that values can be ascribed to the free constants in the equation which lead to qu'antitatively acceptable predictions.