Boundary layer similarity flow driven by power-law shear

Summary

Similarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βy ^α are determined for a range of exponents α and amplitudes β. Asymptotic analysis of the equations shows that for α<−1 no similarity solutions with proper algebraic decay exist. For wall-bounded flow, exact solutions found at α=−1/2 and α=1 correspond to an Airy function wall jet and uniform planar Couette flow. Numerical integration of the governing similarity equation reveals singular behaviour for wall-bounded flows as α→α₀ = −2/3, and no solutions are found in the range −1<α≦−2/3. For α>−2/3 the shear stressf″(0) parameter is determined as a function of α and β. Symmetric free-shear flow solutions become singular as α→α₀ = −1/2 and no solutions are found in the range −1<α≦−1/2. For α>−1/2 the centerline velocityf′(0) is determined as a function of α and β. An asymptotic analysis of the singular behavior of these two problems as α→α₀, given in a separate Appendix, shows excellent comparison with the numerical results. Similarity solutions at the critical values α₀ have exponential decay in the far field and correspond to the Glauert wall jet for wall-bounded flow and to the Schlichting/Bickley planar jet for symmetric free-shear flow.