C,22 • TURBULENT MOTION IN A COMPRESSIBLE FLUID 



Because of the difficulties encountered in the development of the 

 statistical theory of shear flow, several authors studied the more special- 

 ized problem of homogeneous turbulence in a field of uniform velocity 

 gradient. Application of the concepts of Art. 17 was made by Reis [75] 

 and later by Burgers and Mitchner [76] with almost identical assump- 

 tions and results, although the work appears to have been done inde- 

 pendently. Application of the concepts of Art. 16 to this case has recently 

 been carried out by Craya [77]. 



An entirely different approach to the problem of turbulent shear flow 

 has been proposed by Malkus [78]. The reader is referred to his original 

 paper for the details. 



CHAPTER 5. OTHER ASPECTS OF THE PROBLEM 

 OF TURBULENCE 



C,22. Turbulent Motion in a Compressible Fluid. When a com- 

 pressible gas is in turbulent motion, there are density and temperature 

 fluctuations as well as velocity fluctuations. At any instant, the velocity 

 fluctuation may be decomposed into two parts, 



Ui = w^i> + uf^ (22-1) 



such that 



div u(i) = 0, curl u^^) = (22-2) 



The rotation of the fluid is given by the first part u'-^^ and the com- 

 pression is given by the second part u'-p. 



In general, there is a continuous conversion between the rotation 

 component and the compression component of the velocity fluctuations. 

 This additional degree of freedom in the compressible case naturally 

 makes the theory of turbulence much more difficult. In the case of small 

 disturbances from a homogeneous state these modes are separable from 

 each other. 20 The study of small disturbances superimposed on a shear 

 flow is treated in connection with the instability of the boundary layer 

 at high speeds. 



Attempts have been made to extend directly to the compressible case 

 the various approaches to the theory of turbulence in the incompressible 

 case: e.g. the study of isotropic turbulence by Chandrasekhar [80], and 

 the consideration of von Karmdn's similarity theory by Lin and Shen [81] 

 for shear flow. The method discussed in Art. 21 can also be extended to 

 a compressible gas. Obviously, such approaches cannot go beyond the 

 limitations in the incompressible case. It is therefore natural that the 

 more fruitful theoretical results on turbulent motion in a compressible 

 fluid are obtained in connection with the study of the influence of com- 



*" See [79] for a detailed discussion of this case. 



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