sure gradient has been achieved by such investigators as Prandtl and von 

 Karman. No comparable progress has been attained to date, however, in the 

 theory of turbulent boundary layers in a pressure gradient, especially in an 

 adverse (positive) pressure gradient . 



Briefly, the current objective for analyzing turbulent boundary 

 layers in a pressure gradient is to develop an equation for the variation of 

 the shape of the velocity profile within the boundary layer. This formulation 

 is required as an auxiliary equation for complementing the well-known 

 von Karman momentum equation. Auxiliary equations have been obtained empir- 

 ically by a number of investigators. Because of significant differences ex- 

 isting among these empirical formulations, Tetervin and Lin attempted a the- 

 oretical approach to the problem. One of the equations obtained by them is a 

 moment -of -momentum equation derived from the basic Prandtl boundary-layer 

 equations and from a one-parameter characterization for the shape of the ve- 

 locity profiles. Although the moment -of -momentum equation of Tetervin and Lin 

 has the required form for an auxiliary equation, it lacks essential relations 

 needed for numerical computations — namely, the variation with pressure gra- 

 dient of the shearing stress at the surface (local skin friction) and of the 

 integral of the distribution of the shearing stresses across the boundary 

 layer. The principal aim of this paper is to supply the necessary shearing- 

 stress relations to this moment -of -momentum equation in order to develop a 

 suitable auxiliary equation with a theoretical basis, to be used in conjunc- 

 tion with the von Karman momentum equation. 



The relation for the local skin friction in a pressure gradient is 

 derived on the basis of the recent work of Ludwieg and Tillmann who demon- 

 strated the validity of applying the so-called "law of the wall" to the inner 

 flow in the boundary layer. The integral of the transverse distribution of 

 the shearing stresses in a pressure gradient is found empirically in this 

 paper to be a function of similar flat-plate data for zero pressure gradient. 



The moment -of -momentum equation, after being modified by the inclu- 

 sion of the shearing-stress relations, is demonstrated to agree with the aver- 

 age of other existing auxiliary equations whose formulation is based on purely 

 empirical grounds. 



FUNDAMENTAL RELATIONS FOR TURBULENT BOUNDARY LAYERS 



Some of the fundamentals of turbulent boundary layers are briefly 

 reviewed here as an introduction to the subject. The comprehensive summaries 

 by Prandtl, Goldstein, and Schllchting, References 1, 2, and 3 respectively, 

 should be referred to for fuller details.* 



♦References are listed on page 



