Boundary layer similarity flow driven by power-law shear

SummarySimilarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βyα 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 −1−2/3 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 −1−1/2 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.