Solutions to the linearized Navier-Stokes equations for channel flow via the WKB approximation

Progress on determining semi-analytical solutions to the linearized Navier-Stokes equations for incompressible channel flow, laminar and turbulent, is reported. Use of the WKB approximation yields, e.g., solutions to initial-value problem for the inviscid Orr-Sommerfeld equation in terms of the Bessel functions $J_{+1/3}, J_{-1/3}, J_1,$ and $Y_1$ and their modified counterparts for any given wave speed $c = \omega/k_x$ and $k_{\perp}, (k_{\perp}^2 = k_x^2 + k_z^2)$. Of particular note to be discussed is a sequence $i = 1, 2,. . .$ of homogeneous inviscid solutions with complex $k_{\perp i}$ for each speed $c$, ($0 < c \leq U_{max}$), in the downstream direction. These solutions for the velocity component normal to the wall $v$ are localized in the plane parallel to the wall. In addition, for limited range of negative $c $, $( - c* \leq c \leq 0)$, we have found upstream-traveling homogeneous solutions with real $k_{\perp}(c)$. In both cases the solutions for $v$ serve as a source for corresponding solutions to the inviscid Squire equation for the vorticity component normal to the wall $\omega_y$.