B • TURBULENT FLOW 



Two contributions following the general method proposed by Tetervin 

 and Lin are those of Granville [104\ and Rubert and Persh [l^t-O]. Gran- 

 ville's work suggested that the difficulty in using the moment of momen- 

 tum equation for H might be overcome. By examining a limited amount 

 of experimental data he showed that the integral of the shearing stress 

 across the layer in terms oi y/h* was the same in adverse pressure gradi- 

 ents as in constant pressure flow. Rubert and Persh chose the kinetic 

 energy equation for the determination of H and hence had to evaluate 

 the integral of the dissipation across the layer. This they did empirically 

 using experimental data for a variety of conditions. They also included 

 the Reynolds normal stress in the momentum equation. Values of d and 

 H calculated by Rubert and Persh showed reasonably close agreement 

 with experiment for two-dimensional boundary layers and flow in dif- 

 fusers. Both of these methods draw on the work of Ludwieg and Tillmann 

 {89\ for the shearing stress at the wall and the existence of the law of the 

 wall in an adverse pressure gradient. 



Two methods based on dividing the treatment between the inner 

 part of the boundary layer and the outer part are those of Ross [I4I] 

 and Spence [142]. Each uses a separate similarity for the inner and outer 

 parts. Both use the law of the wall for the region next to the wall. Ross 

 adopts a f-power velocity-deficiency expression for the outer region with 

 a new parameter D, thus avoiding the use of the shape parameter H. 

 Spence retains the i7-parameter for the outer region, but evaluates it 

 by means of an expression for the velocity at the distance d from the wall, 

 obtained from the equation of motion formulated for the distance y = 6. 



The several methods here mentioned show that progress is being 

 made on this difficult problem. In some cases more tests are needed to 

 judge the amount of progress. There is general agreement that more 

 information is needed on the behavior patterns of turbulent flow before 

 a universally valid method can come within reach. 



B,25. Three-Dimensional Effects. It may seem that undue atten- 

 tion is given to two-dimensional mean flows when in their totality all 

 flows are three-dimensional. The justification for the convenience of 

 avoiding the complications introduced by a third dimension is that 

 motion in the third dimension is in many cases locally absent or so 

 insignificant that two-dimensionality is an acceptable assumption. This 

 fortunate circumstance comes about because boundary layers are usually 

 thin compared to the expanse and radius of curvature of a wall. 



Obviously there are many cases where the edges are too close to the 

 region in question or the boundary layers are too thick for three-dimen- 

 sional effects to be ignored even under local inspection. Common exam- 

 ples are flow in noncircular pipes, flow near wing tips, and flow near the 



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