B • TURBULENT FLOW 



a relatively short distance. The layer in which this occurs is called the 

 boundary layer. A knowledge of the flow behavior within this layer is of 

 prime importance, especially when effects associated with compressibility 

 and aerodynamic heating come into play. 



The turbulent boundary layer occurs more generally than the laminar 

 boundary layer, but is less well understood theoretically. The exceedingly 

 complex character of turbulent flow and the inadequacy of theories of 

 turbulence make an exact mathematical treatment of the flow impossible 

 at present. Therefore a great number of approximations are necessary, 

 and it is to be expected that the various proposed theories may turn out 

 different results which are not always reconcilable. In order to clarify 

 many obscure points in the theories, and to display in a simple manner 

 the essential physical features governing boundary layer flow, it seems 

 worthwhile to outline the main approaches of the analytical treatments, 

 and especially to elucidate the bases and assumptions underlying the 

 theories. Where possible the theoretical results will be compared with 

 existing experimental results. 



First the fundamental hydrodynamic equations, as developed earlier, 

 will be simplified in Art. 8 under the special conditions of the boundary 

 layer. Consequently some simple relations between pressure, temperature, 

 and velocity can be derived in Art. 9. These will at once show some 

 features of heat transfer in the boundary layer, and especially of the 

 recovery factor, without going into the turbulent transport processes. 

 For a deeper understanding of the problems, some statistical methods of 

 transport phenomena become necessary. Existing theories make extensive 

 use of the concept of mixing length as a parameter of the turbulent ex- 

 change of properties. Since several fundamental questions arise in con- 

 nection with the application of mixing length to various types of transport 

 (mass, momentum, and heat) governing the boundary layer, and in the 

 analogy theories between heat transfer and skin friction (the so-called 

 Reynolds analogy), the statistical foundation of the transport processes 

 will be studied in Art. 10. As an immediate application, the Reynolds 

 analogy can be better understood and will be treated in Art. 11. 



Theories relating to velocity profiles in a compressible turbulent 

 boundary layer do not seem to differ much from the corresponding 

 theories for the incompressible boundary layer, especially concerning 

 their basis and method of attack. Therefore we shall reserve these for 

 Chap. 4 where incompressibility is assumed, and be content here to give 

 only some experimental data on the velocity distribution. 



The skin friction in a compressible boundary layer deserves special 

 attention, because of its important compressibility effect and its practi- 

 cal significance. The basis of the theories will be described in Art. 12; the 

 empirical formulas illustrating the essential behavior of skin friction will 

 be given in Art. 13; and finally the comparison between theories and 



< 88 ) 



