B,4 • EQUATIONS OF CONTINUITY AND MOMENTUM 



derivatives, such as was the case for incompressible flow, but also con- 

 taining the product of velocity by density. The latter must moreover 

 satisfy the conditions imposed by the equation of continuity and the 

 equation of heat conduction. Second, in an incompressible flow the 

 Reynolds equations have a form similar to the original Navier-Stokes 

 equations, provided additional fictitious forces, called Reynolds stresses, 

 are introduced. These stresses also characterize the turbulent friction 

 and give the rate of production of turbulence when multiplied by mean 

 velocity gradients. However, in a compressible flow, such fictitious 

 stresses are more complicated and involve other roles in addition to the 

 production of turbulence. 



It is well to mention briefly why averages are used and what they 

 mean. Turbulent motions of fluid elements are so complex that they can- 

 not be treated individually. By averaging we can obtain mean motions 

 which include turbulent properties statistically. The average can be 

 taken at a given point over a certain interval of time, or over a certain 

 region at a particular instant of time, or finally over a great number of 

 realizations represented by identical fields at corresponding points and 

 instants. These are the three kinds of Eulerian mean values, termed 

 respectively, temporal, spatial, and statistical mean values. Finally we 

 can follow the motion of an individual particle as a function of time and 

 find the temporal and statistical mean value of any physical property 

 associated with the particle. This would be the Lagrangian mean value. 

 It is beyond our scope to discuss the different mean values. We shall 

 adhere to the Eulerian description in which any one of the three averages 

 may be used as far as the formalism is concerned. Commonly used meth- 

 ods of measurement and observation require the use of the temporal mean 

 value, and this mean value will subsequently be inferred. The time inter- 

 val does not need to be considered for present purposes, especially when 

 we are concerned with steady motion. 



The motion of the fluid is decomposed into a mean motion with ve- 

 locity components Ui parallel to the Xi axis, with the running indices 

 z = 1, 2, 3, and the superimposed turbulent motions or fluctuations, with 

 velocity components w^.^ The velocity components of the total motion 

 will be Ui + Ui. Likewise the scalar quantities, pressure, density, and 

 temperature are also decomposed into their mean parts and fluctuating 

 parts, and are respectively 



where p denotes the mean pressure and p' its fluctuations, and p and T 



^ In this section as well as in Sec. C a departure from the usual notation of the 

 Series wherein u is used to denote the mean velocity in the x direction and u' the 

 fluctuation about its mean value has been necessary in order to eliminate the confusion 

 that would result in referring to the fluctuations that might exist simultaneously at 

 two points. 



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