Lumley 



edge of the sublayer as being essentially the free stream value. Then the shear 

 at the outer edge will be nearly that without heating at the same wall stress. If 

 we take the thickness Reynolds number as being determined largely by the shear 

 at the outer edge, then this will be essentially the same (although it may increase 

 somewhat due to the favorable curvature of the profile) so that the thickness will 

 be essentially the same. Thus the whole effect will appear from outside the sub- 

 layer as a slip at the wall of value (to first order) 



_ _ RyLi* Av 



Assuming self-preservation, negligible laminar length, and large Reynolds num- 

 ber, we may obtain the effect on ^* of a change in u by this mechanism: 



which is of the order of 1/4 at moderate Reynolds number. The sensitivity of 

 viscosity to temperature in liquids suggests that (at ordinary pressures) changes 

 in M* by 20% may be possible before boiling occurs. 



It should be mentioned in passing that surface heating in a gravitational field 

 may produce secondary motions which will only increase the momentum trans- 

 port and the drag. 



Change in the Wall Layer 



A slightly more sophisticated way in which the principles outlined above 

 could be violated is by a change in the "law of the wall." This could be done by 

 the introduction either of a length scale or of a velocity scale. These are essen- 

 tially equivalent, since a height can be defined at which the mean velocity equals 

 the velocity scale selected. Thus a new parameter is introduced, say the ratio 

 of /a* to the new velocity scale. A simple way in which this can be done is by 

 coating the wall with a nonrigid material having a Rayleigh wave speed below the 

 free-stream speed. Then convected fluctuating pressure fields can exchange 

 energy with the wall in the same manner as described by Phillips (1955) for the 

 generation of ocean surface waves by turbulent wind. It is not obvious a priori 

 why such an interchange should necessarily result in a reduction of drag. The 

 random wave motion of the surface would necessarily be associated with dissi- 

 pation of energy in the surface so that the simple existence of such an interaction 

 would only increase the total dissipation, if it did not drastically alter the struc- 

 ture of the boundary layer so as to reduce the dissipation in the fluid. Again, we 

 have, as before for the laminar layer, that damping in the wall material is prob- 

 ably detrimental, and it seems likely that we will not achieve favorable effects 

 unless the damping in the surface material is considerably smaller than that in 

 the fluid. This is the case for an air boundary layer over water, and P. A. 

 Shepphard (private communication) has observed drag reduction in such bound- 

 ary layers. Unfortiinately, it is more difficult to find wall materials of viscosity 

 lower than water. 



924 



