The Reduction of Skin Friction Drag 



thickness) no matter how delayed by a change in shape. A boundary layer whose 

 growth is prevented may never reach its critical Reynolds number. 



The ways in which the profile can be altered can be placed in two categories, 

 depending on whether constant or variable fluid properties are necessary. Under 

 the heading of methods which work with constant fluid properties we can include 

 pressure gradients and suction. The suction may be either distributed, or it may 

 be through discrete slots (see particularly the work of Pfenninger et. al.). Dis- 

 crete slots are satisfactory so long as the boundary layer is caught by the next 

 downstream slot before disturbances have time to grow to a significant extent. 

 Among the methods that involve variable fluid properties, most are dependent on 

 a variation of the ordinary viscosity m- An increase of m with distance from the 

 wall increases the curvature. The viscosity h- can be varied in several ways: in 

 water it can be changed by heating the wall, a film of a different fluid can be 

 placed next to the wall, such as a gas film or a liquid with a lower ^ —such a film 

 being produced by injection, film boiling, cavitation, sublimation or chemical 

 reaction. Finally, an additive could be placed in the boundary layer so that the 

 fluid becomes non-Newtonian, in particular "shear -thinning"; then the high shear 

 near the wall will mean a lower m there and the i-l will increase with distance 

 from the wall. It should be mentioned that there is some disagreement as to 

 whether the low m fluid film should be considered primarily as a stabilization 

 technique; this seems to be largely a matter of taste, and I have taken the posi- 

 tion that if it did not stabilize, it would not work, since the low m fluid would be 

 mixed with the high. 



Flexible Boundary 



To the best of my knowledge there are only two methods that do not depend 

 on changing the profile; the first of these is the stabilization of the laminar 

 boundary layer by a compliant boundary. This does not damp the distiirbances; 

 as a matter of fact, it is a result of the theory (Betchov (1959), Benjamin (1960) 

 Boggs & Tokita (1960), Landahl(1962)) that damping in the wall is in general 

 destabilizing. Rather, the compliant boundary acts to change the phase rela- 

 tions between the pressure and the velocity in the neighborhood of the wall, re- 

 sulting in an alteration of the Reynolds stresses there, and changing the energy 

 budget of a disturbance. While a passive wall in general changes the lower 

 critical Reynolds number, Betchov (1958) has shown that an active wall may be 

 expected to eliminate it entirely. In a tenuously related investigation Wu (1959), 

 has shown that a suitable active wall can propel. 



In Fig. 3 are shown the phase relations induced at the siurface by a visco- 

 elastic material, together with the phase relations corresponding to a small dis- 

 turbance in the region between the inner viscous layer and the critical layer of 

 the laminar boiindary layer over a rigid surface. If the former are added to the 

 latter as a first order approximation, the influence on the disturbance Reynolds 

 stress may be seen. 



919 



