B • TURBULENT FLOW 



Nothing has yet been said about the character of the motions them- 

 selves. Naturally we should like to know whether the sustaining mecha- 

 nism dictates some particular form of motion. It is a known fact that 

 even though the energizing of turbulence is expressible in terms of shear 

 stresses, turbulent pressure gradients are required, and they must arise 

 from interactions within the flow itself. These interactions can be im- 

 agined to take the form of collisions between fluid elements; but since 

 all streams are connected, the interaction paths are curved and continu- 

 ous. The resulting motions may best be described as a superposition of 

 eddies with various orientations. The shearing action stretches the eddies 

 with axes lying along directions in which the fluid is being strained and 

 intensifies their vorticity. Some concentrated vortex motions can there- 

 fore be expected to exist in the complex jumble of motions. 



Before we can proceed further we must consider the various scales of 

 motion encountered in turbulent flow and examine their role. It is gener- 

 ally assumed that the largest scale is that characteristic of the size of the 

 mean flow field, such as the thickness of a boundary layer. Next come the 

 turbulent motions where the superimposed jumble of eddies have various 

 sizes ranging from near that of the mean flow down to the so-called micro- 

 scales. All turbulent motions are agents responsible for shearing stress in 

 the presence of a mean shear, and therefore all extract energy from the 

 mean flow to sustain themselves. However, this action decreases with 

 decreasing scale, and from an over-all point of view it is generally assumed 

 that the energy enters the turbulence by way of the larger eddies. Corre- 

 spondingly, the damping action of viscosity is assumed to be negligible 

 in the mean flow and among the larger eddies but to increase progressively 

 with decreasing size until it finally becomes dominant among the smallest 

 eddies. The effect of viscosity is the more removed from the larger eddies 

 as the Reynolds number becomes higher. 



It is obvious that if energy enters the turbulence more by way of the 

 large eddies than by the small ones and leaves more by way of the small 

 eddies than by the large ones, there must be a transfer of energy from 

 larger scales to smaller scales. The succession of transfer is generally re- 

 garded as taking place from size to size down the scale, with the number 

 of stages increasing with the Reynolds number. 



Except for the laminar sublayer next to a wall and its immediate 

 vicinity, it is an observed fact that if the Reynolds number is sufficiently 

 high for transition to have occurred, the succession of transfer is already 

 long. In the usual terms, the turbulent energy spectrum is broad. This 

 signifies that turbulent flows as a class show comparatively minor effects 

 of Reynolds number in their over-all character. The mean velocity dis- 

 tribution, for example, changes little with Reynolds number, and the 

 mean flow field shows a tendency to remain similar in form as it grows to 



{ 78) 



