B,22 • MIXING LENGTH AND EDDY VISCOSITY 



B,22. Mixing Length and Eddy Viscosity in Boundary Layer 

 Flows. As mentioned in Art. 3 and 10, the transport of momentum by 

 turbulent motions may be regarded as involving an eddy viscosity. We 

 shall briefly reexamine the associated concepts in the light of certain 

 known facts about the flow in various parts of the boundary layer. 



In turbulent boundary layers three fairly distinct regions are easily 

 recognized. First there is the laminar sublayer which is typically 0.01 to 

 0.001 of the total thickness of the layer. Beyond this is a turbulent region 

 which extends to 0.1 to 0.2 of 5 and comprises the inner part of the layer 

 where the logarithmic law is valid and the mean flow is virtually un- 

 affected by pressure gradient. A short time response and rapid adjust- 

 ment to local conditions are also characteristic of this region (see dis- 

 cussion by Clauser [72]). Finally, there is the outer 0.8 to 0.9 of the layer 

 where the eddies are limited in lateral extent only by the confines of the 

 layer and mixing is relatively free. In the laminar sublayer molecular 

 diffusion predominates, being exclusively this at the wall. Turbulent dif- 

 fusion progressively increases as we enter the logarithmic region from the 

 wall side and soon predominates over molecular diffusion. For virtually 

 everything except the laminar sublayer the transfer processes should be 

 governed by a property of the motion. We wish to see whether this 

 property may be legitimately and usefully expressed in terms of an eddy 

 viscosity, e^. 



Dimensionally, e^, is a product of density, velocity, and length. Ac- 

 cording to the mixing length theory 



e^ = pvl (22-1) 



where v is the y component of turbulent velocity and I is the reach of a 

 turbulent motion while it has the velocity v and is called the mixing 

 length. Prandtl's assumption is that v = IdU/dy and I = C2y (see Art. 10). 

 It is implied in this assumption that the correlation between v and I is 

 absorbed into the value of I. 



Using these assumptions and assuming further that r is independent of 

 y and equal to r^, the value at the wall, we find the well-known expression 



cly^ 



or using Ul = r^/p 



\dy/ 



Ur = c,y^ (22-2) 



This expression may be integrated to give the velocity distribution if we 

 know the lower limits of y and U. These are their values at the edge of 

 the sublayer, which may be found from Eq. 16-2 and written in terms of 



< 143 ) 



