Scaling of Equilibrium Boundary Layers Under Adverse Pressure Gradient Using Turbulence Models

Abstract

The large Reynolds number behavior of four commonly used turbulence models was investigated by numerically solving the boundary-layer equations up toReo = 108 for turbulent boundary layers under an adverse pressure gradient with an equilibrium parameter (3 = (^*/rw)(dp/djt:) in the range between 0 and 200. All models reproduce the same classical scalings in the inner and outer layer. The solution in the outer layer asymptotes to the defect law described by the defect-layer equation as derived by Tennekes and Lumley (Tennekes, H., and Lumley, J. L., A First Course in Turbulence, MIT Press, Cambridge, MA, 1972, pp. 186-188) and not to the one derived by Wilcox (Wilcox, D. C., Turbulence Modeling, DCW Industries, Inc., La Canada, CA, 1993, pp. 110-121). The asymptotic state is obtained at a higher Reynolds number for increasing f3 value. The turbulence models show that the same outer-edge velocity can generate two different equilibrium boundary layers. Comparison with existing experiments shows that the differential Reynolds stress model is very accurate; the algebraic model and the k-ui model are also reasonably good. The k-e model is not accurate for these adverse-pressure gradient boundary layers, and it gives a far too large wall-shear stress.