B,22 • MIXING LENGTH AND EDDY VISCOSITY 



He noted that the principal difference in appearance between constant 

 pressure laminar profiles and turbulent profiles was that the turbulent 

 profiles dropped so abruptly at the wall as to appear to extrapolate to a 

 nonzero velocity at the wall, whereas laminar profiles went to zero much 

 more gradually and did not give this impression. The characteristic shape 

 of the turbulent profile arises from the circumstance that the laminar 

 sublayer next to the wall and the flow adjacent to it has a lower viscosity 

 than the eddy viscosity prevailing in the main body of the turbulent flow. 

 Consequently a large part of the velocity change from the wall to the 

 free stream occurs in this low viscosity region. If the same situation were 

 made to prevail in a laminar layer, say by placing a layer of fluid of 

 lower viscosity next to the wall, a laminar profile could be made to re- 

 semble a turbulent profile. Clauser therefore proceeded to simulate this 

 condition in a family of laminar profiles obtained by solving the Blasius 

 equation for slip velocities Uy, at the wall, U^/Ue amounting to 0, 0.2, 

 0.4, 0.5, 0.6, 0.7, and 0.8. He then attempted to collapse the family to 

 a single curve by dividing {U — Ue)/Ue and y/d by suitable factors. 

 Leaving details to the original paper [72], we merely point out the sig- 

 nificant fact that exact coincidence proved to be impossible, but that two 

 procedures each resulted in a narrow band of curves. Clauser concluded 

 that the same basic dissimilarity would prevent turbulent profiles, which 

 pertain to different values of Ur, from collapsing to a single curve on the 

 basis of the velocity-defect law. Accordingly there is an almost-but-not- 

 quite universal curve. 



The next step was to relate the laminar profiles to turbulent profiles 

 on a velocity-defect-law basis by an appropriate eddy viscosity, e^. The 

 appropriate velocity and length were chosen by the same reasoning proc- 

 ess that leads to a reference velocity Ur and a reference length 5 in the 

 velocity-defect law, and e^ was expressed by 



^ = aUA 

 P 



where a is a constant of proportionality to be determined. Since A is 

 equal to Ue8*/Ur (see Art. 20) 



-" = aUeS* (22-4) 



P 



which is an expression for e^ in readily available quantities. 



The original article must be consulted for the details of the fitting 

 process and the curves showing comparisons with data of Fig. B,16. Best 

 agreement was obtained with a = 0.018. Considering that a narrow band 

 of laminar curves is obtained rather than a single curve and that experi- 

 mental data are expected to show a similar dispersion, the agreement is 

 excellent for the outer 80 to 90 per cent of the layer. The method pro- 



< 145 ) 



