Lumley 



root of the kinematic Reynolds stress, and noting that mean velocity differences 

 in the layer also must scale with m*, we have y^' //i* = l/K a universal constant, 

 which gives immediately the familiar logarithmic law /I/m* = lA In y/y^ , where 

 y^ is a constant yet to be determined. 



3. The viscous sublayer— that there is a layer of fluid next to the wall in 

 which dissipation is dominant, in the sense that no disturbance can be in equi- 

 librium there without energy transfer into the layer. The profile of mean ve- 

 locity there is nearly linear, since the stress is constant, and production of tur- 

 bulent energy is not important. Phenomena seem superposable in this region 

 (Sternberg (1961)) since the nonlinear convective -production terms are not sig- 

 nificant. The thickness of the layer is fixed by the Reynolds number based on 

 thickness. If we set R = y* ij.*/v as the Reynolds number based on thickness 

 where the sublayer profile, Ji/ij* - yy.*/v, meets the logarithmic profile (roughly 

 12.6 in a normal boimdary layer) then we can write 



m//^' 



1 



R- — In R 

 K 



+ — In 



K V 



which fixes the value of the constant. 



4. "Law of the Wake"— in the outer part of the layer, it is assumed that the 

 profile is similar when referred to local length and velocity scales— [m " U]//x* = 

 f(y/S) which of course also involves Reynolds number similarity. If it is as- 

 sumed that there is a region of overlap with the law of the wall, then we obtain 

 the familiar drag law 



This is the relationship which must be changed if the effect on drag of the tur- 

 bulent boundary layer is to be changed. 



Change of Viscosity 



Let us now consider ways in which the familiar drag relationship can be 

 changed (see Fig. 4). The simplest which comes to mind is a change of viscosity. 

 This will not change the structure outside the viscous sublayer, since that was 

 dominated by inertia. Therefore it does not matter whether the change in vis- 

 cosity extends to the fluid outside the sublayer. A change in viscosity will not 

 change the value of R so long as the change is imiform in the sublayer. Hence, 

 any mechanism which changes the viscosity in the viscous sublayer will produce 

 a turbulent boundary layer indistinguishable from a normal turbulent boundary 

 layer at a different length Reynolds number. Since drag is only a weak fimction 

 of length Reynolds number, this is not a particularly effective way to change 

 drag. The viscosity in the viscous sublayer might be reduced by heating the 

 wall (in a liquid). 



922 



