ON ASYMPTOTIC SOLUTION OF J' - (ikU/v_k)J = 0 FOR LARGE k

摘要

Asymptotic solutions of the differential equation J" - (ikU/v_k)J = 0 derived for the complex amplitude of turbulent shear stress for turbulent flow over linear (Miles [2]) and nonlinear (Sajjadi [6]) waves near the surface of the wave (kη 1) is investigated. The above equation for logarithmic mean velocity profile is first converted to a more general form (d~2J)/(dz~2)=z~(-2) {k~2(η_0 - z)p_1(z) + q_1(z)}J for large positive values of the wavenumber k, where p_1 (z) and q_1 (z) are regular functions of the complex variable z in a domain in which p_1 (z) does not vanish. The point z = 0 is a regular singularity of the equation and a branch-cut extending from z = 0 is taken through the point z = η_0 which is assumed to lie on the positive real z axis. Asymptotic expansions for the solutions of the equation, valid uniformly with respect to z in domains including z = 0, z = η_0 ± i0 are derived in terms of modified Bessel functions of large order.