The Reduction of Skin Friction Drag 

 Reynolds Number Similarity 



Another way in which the boundary layer may be attacked is through the 

 principle of "Reynolds number similarity." Violating this principle is not a 

 straight-forward matter. For instance, if the fluid viscosity is increased, there 

 will be no important change (other than the slow increase in drag associated with 

 the weak dependence on length Reynolds number) until the dissipative and energy- 

 containing scales are nearly equal, at which point the turbulence can no longer 

 extract energy from the mean motion at a sufficient rate to maintain itself, and 

 the flow will become laminar. This would, of course, result in a drag reduction, 

 but falls more properly in the realm of stabilization. One might suggest using 

 a non-Newtonian medium which is shear-thinning. If indeed it behaved as though 

 it had a simple shear -dependent viscosity (Lumley (1964)) it would change noth- 

 ing. In the high-shear viscous sublayer, its viscosity might be expected to be 

 nearly the value of the solvent; in any event, R would remain unchanged. If the 

 flow outside the sublayer were turbulent, then it would be inertia dominated, and 

 nothing would be changed. Only by increasing the effective viscosity outside the 

 sublayer until the layer became laminar could a change be made, and again this 

 falls more properly under the heading of stabilization. Evidently, in order to in- 

 fluence Reynolds number similarity, it is necessary to have a material whose 

 constitutive equation is such that terms in the energy equation, arising from that 

 part of the stress which is not a pressure, are appreciable in the energy contain- 

 ing range of wave numbers, without being dissipative in character, so as not to 

 turn the turbulence off. That is, they must be non-negligible in the energy con- 

 taining range of wave numbers without being viscous in character. There is both 

 theoretical (Lumley (1964)) and experimental (see particularly Fabula (1963)) 

 support for the conclusion that only a material having viscoelastic properties 

 can behave in this manner, although the exact mechanism is not understood. 



Particles and Fibers 



There has been reliable observation of drag reduction in flows containing 

 particles and fibers. Although this effect is described as "damping" the turbu- 

 lence, the intensities are observed to increase (Elata, Ippen (1961)). From the 

 principle of Reynolds number similarity, we know that a simple change in the 

 mechanism of dissipation, so long as the flow remained turbulent, would be un- 

 likely to change the turbulent structure, since this is determined by inertia. 

 There is a known interaction of suspended particles with the viscous sublayer, 

 which will be described below, but if the observed drag reduction does not arise 

 from this source, then it seems likely that it is due to a violation of Reynolds 

 number similarity by the introduction of other length and time scales. Depend- 

 ing on the ratios of these scales to others in the flow, this may also be regarded 

 as a violation of the law of the wall, of course, since particles having relatively 

 small length or time scales may leave the outer part of the flow unaffected, be- 

 ginning to exert an influence only as the scales of the energy containing eddies 

 shrink to corresponding size as the wall is approached. Length scales may be 

 introduced in a very direct way by long fibers, while velocity scales may be in- 

 troduced by the settling velocity (in a gravitational field), and time scales by the 

 characteristic time of the particles (the response time to a step function in rela- 

 tive velocity). The mechanism associated with this latter may be similar to 



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