C • STATISTICAL THEORIES OF TURBULENCE 



pressibility on turbulent motion, principally for small Mach numbers of 

 turbulence. It is then plausible that the chief influence of compressibility 

 is that acoustic energy is constantly being radiated, causing the turbulent 

 motion to dissipate faster than in the incompressible case. 



Lighthill [82] has shown that, in the absence of solid boundaries, 

 turbulent motion acts as quadrupole sources of sound. He also showed 

 that the amount of energy radiated per unit volume of turbulence is 

 proportional to pV^/aH, where p is the density, F is a typical velocity 

 of the turbulent motion, a is the acoustic speed, and I is a typical linear 

 scale. Since the rate of energy conversion in turbulent motion is propor- 

 tional to pV^/l, the acoustic efficiency is proportional to the fifth power 

 of the root mean square Mach number. At low Mach numbers, this 

 would be a very small amount indeed, if it were not for a numerical factor 

 of proportionality of the order of 40, as shown by Proudman [83]. In the 

 cases where the theory is applicable, the experimental results bear out 

 the general theoretical conclusions. 



If solid boundaries are present, such as in the problem of the noise 

 from the boundary layer of a flat plate, Phillips [84] found that dipole 

 sources are present if the plate is semi-infinite. Acoustic sources are again 

 of the quadrupole type if the plate is infinite and the motion is statisti- 

 cally the same along the plate. 



The scattering of energy due to the interaction of turbulence with 

 sound or shock waves has been considered by Lighthill [85] and others. 



All of the above results are for low Mach numbers of turbulent mo- 

 tion. At the present time, only speculation can be made for the cases of 

 higher Mach numbers where shock waves may appear. 



C,23. Magneto-Hydrodynamic Turbulence. In astrophysics, one 

 important problem is the turbulent motion of an electrically conducting 

 gas in the presence of magnetic fields. One is then dealing with the con- 

 version of energy from the mechanical form to the electro-magnetic form. 

 There is an extensive and rapidly growing literature on this subject, and 

 it is perhaps inappropriate to try to survey it at the present time in a 

 volume on high speed aerodynamics. 



One of the central problems at issue is the partition of energy between 

 the two modes. Batchelor [86] noted that the equation for the magnetic 

 field is exactly the same as that for vorticity, and suggested that the 

 energy spectrum of magnetic energy is proportional to kW{k). This would 

 mean that there is little magnetic energy in the large scales. Other authors, 

 however, contended that there should be equipartition of energy of the 

 two modes. Recently, Chandrasekhar [87] undertook a systematic de- 

 velopment of the theory of turbulent motion for magneto-hydrodynamics 

 along the lines of Art. 16 and 17, and found solutions which are in agree- 

 ment with the latter opinion. However, since his assumption limited him 



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