The A^, B^, C^ in /A denote 5x5 matrices. The 

 solution of (55) is obtained by the procedure 

 described in Cebeci and Bradshaw (1977) . 



Ue 



+ (1 - Ys) 



197 

 (66) 



Here yg is an interraittency factor defined by the 

 following empirical formulas: 



< T- f 0.05 Y = 1 



«0 



0.05 < ^ < 0.3 



^0 



0.3 < 



0.7 



Yg = 1 - 2.64214 



4.40531 ^ 0.5 



\O0 



0.05 

 3 



1.85021 i 0.5 



\°0 



0.5 



RESULTS 



0.7 < ^ < 0.95 

 00 



,„ - 2.64214 ( i 0.05 



Turbulent Flow Calculations for a Curved Duct and 

 Comparison with Experiment 



The turbulence model described in Section 2 has been 

 used with considerable success to compute a wide 

 range of two-dimensional turbulent boundary layers 

 [see for example Cebeci and Smith (1974) ] . The 

 model has also been used to compute three-dimensional 

 flows and again is found to yield accurate results 

 [see for example Cebeci (1974, 1975) and Cebeci, 

 Kaups , and Moser (1976)]. To further test the model 

 for three-dimensional flows, we have considered the 

 experimental data taken in a 60° curved duct of rect- 

 angular cross section. Figure 6 shows a sketch of 

 the flow geometry. The experimental data are due 

 to Vermeulen (1971) . Here z denotes the distance 

 from the outer wall, measured along normals to the 

 wall; X denotes the arc length along the outer wall; 

 and y denotes distance normal to the plane x,z. 



To test the computed results with the data, it 

 is necessary to specify the initial profiles given 

 by experiment. This can be done in a number of ways. 

 In the study reported by Cebeci , Kaups , and Moser 

 (1975) the profiles were generated by using Coles' 

 velocity profile formula. That formula, which repre- 

 sents the experimental data rather well for two- 

 dimensional flows, was not very satisfactory for 

 three-dimensional flows. Here we abandon the use 

 of Coles' formula in favor of Thompson's two- 

 parameter velocity profiles as described and im- 

 proved by Galbraith and Head (1975) . According 

 to this formula, the dimensionless u/Ug velocity 

 profile is given by 



> 0.95 



0.0 



^0 



The dimensionless velocity profile for the inner 

 layer, that is, (u/ug) inner' is given by 



y+ < 4 



u+ = y+ 



4 < y"'' < 30 



cj + C2ln y + C3 (In y"*") ^ 



y"^ > 30 



+ CLt (In y"*") 3 

 u"*" = 5.50 In y"^ + 5.45 



Here ci = 4.187, C2 = -5.745, 03 = 5.110, C14 = 

 -0.767, y+ = yu.^/v, u^ = (T^^/p)>5, u+ = u/u^ , and 6^ 

 is a parameter which is a function of 9 , c^, and H. 



To find the functional relationship between 6 

 C£, 6, and H, we use the definitions of displacement 

 thickness, &* , and momentum thickness, 6. Substitut- 

 ing (66) into the definition of &* , after some alge- 

 bra, we get 



Alt In 



6* 



A3 



A2lniR^.j^ 



(67) 



where 



Ai = 50.679, A2 = 1.1942, A3 = 0.7943, Ai^ = 1.195. 



CONDITIONS 



INITIAL CONDITIONS 



MEASURING LOCATIONS 



FIGURE 6. Coordinate system and notation for the 

 . curved duct . 



An expression similar to that given by (67) can also 

 be obtained if we substitute (66) into the defini- 

 tion of 9. However, the resulting expression is 

 quite complicated. For this reason, the expression 

 for 9 is obtained nimierically , and for a given value 

 of 6 and H, the corresponding values of Cj and 6^ 

 are computed from that equation and from (67) . 



Equation (66) is recommended for two-dimensional 

 flows. Here we assume that it also applies to the 

 streamwise velocity profile by replacing u/u^ by 

 Ug/ug with C£ now representing the streamwise skin- 

 friction coefficient. 



In order to generate the crossflow velocity com- 

 ponent (u^/Ug ) , we use Mager ' s expression and 

 define Un/Ug by 



