VI. TURBULENCE 



K. F. BOWDEN 



1. General Properties of Turbulence 



A . Reynolds Stresses and Turbulent Transports 



In the turbulent flow of a fluid, an irregular fluctuating motion is super- 

 imposed, on the general pattern of movement. The velocity measured at a 

 given point in the fluid varies rapidly with time and there are comparable 

 variations from one point to another at a given time. The laws of hydro- 

 dynamics are applicable to the instantaneous distribution of velocities at all 

 points in the fluid and would, in principle, enable the subsequent motion to be 

 determined. To render the problem tractable in practice, however, it is neces- 

 sary to separate the complete flow into mean motion and turbulent motion. 

 While it is possible in this way to deal with the details of the mean motion, the 

 properties of the turbulent flow can only be treated statistically. 



Let rectangular axes OX, OY, OZ be taken, with OX, OY in a horizontal 

 plane and OZ vertically upwards. Let u', v', w' be the corresponding com- 

 ponents of the instantaneous velocity at a point {x, y, z). Then 



u' = U + u, v' = V + v, W = W + w, (1) 



where U, V, W are the components of the mean velocity and u, v, w are the 

 components of turbulent motion. U, V, W may be defined by an averaging 

 process which is carried out over a specified length of time at a given point or 

 over a specified volume at a given instant. The method of averaging and the 

 fundamental interval or volume will depend on the scale of the motion and the 

 aspect of it which is being considered. 



It was shown by Reynolds (1894), in his pioneer work on turbulence, that 

 the effect on the mean flow of the existence of the turbulent fluctuations of 

 velocity was to introduce additional components of internal stress, given by 



Pxz = -pU^, Pyy = -PV^, Pzz = - pw"^, 



_ _ (2) 



Pxy = — pUV, Pxz = — pUW, etc., 



where Pxy denotes the stress per unit area on a surface perpendicular to OX in 

 the OY direction, and the bar denotes a mean value taken over the fundamental 

 interval. These stresses, known as the Reynolds stresses, are formally analogous 

 to the viscous stresses arising from molecular viscosity which, in the case of the 

 mean motion, are given by 



i>.. = 2/x— , p,, = ^(^— + —j, etc., (3) 



[MS received July, 1960 ] 802 



