A METHOD FOR THE CALCULATION OF THE TURBULENT BOWDARY LAYER 

 IN A PRESSURE GRADIENT 



by 

 Paul S. Granville 



ABSTRACT 



A method is described for the calculation of the development of incompress- 

 ible turbulent boundary layers in pressure gradients, so that drag and separation of 

 flow may be predicted more accurately. The principal contribution is the formulation 

 of new relations for the variation with pressure gradient of 1, the shearing stress at 

 the wall (local skin friction) and of 2, the integral of the shearing stresses across 

 the boundary layer for use in a moment-of-momentum equation recently derived by 

 Tetervin and Lin. 



With the assumption of the one-parameter characterization of velocity profiles 

 by a shape parameter H, the moment-of-momentum equation and the von Karman mo- 

 mentum equation constitute a working method involving the solution of a pair of simul- 

 taneous first-order differential equations. 



As an introduction to the subject of turbulent boundary layers in pressure 

 gradients, a review of previous methods is included in this report. 



INTRODUCTION 



An accurate calculation of the growth of turbulent boundary layers 

 in a pressure gradient is required for many problems in hydrodynamics, aero- 

 dynamics and hydraulics where high Reynolds numbers are involved. Some of 

 these problems are the calculation of the drag of airfoils and of bodies of 

 revolution, the prediction of separation of flow on airfoils, the prediction 

 of flow conditions in the test sections and diffusers of water tunnels and 

 wind tunnels, and the prediction of the flow (frictional wake) around the 

 tails of bodies of revolution for use in designing propellers and control 

 surfaces. 



Unlike laminar boundary layers, turbulent boundary layers are not 

 very amenable to mathematical analysis owing to the present incomplete state 

 of knowledge of the mechanics of turbulent flow. Nevertheless, considerable 

 success in analyzing turbulent boundary layers on flat plates with zero pres- 



