Eigenfunctions of the Orr-Sommerfeld/Squire Operator for Channel Flow

Knowledge of the eigenvalues and eigenfunctions of the Orr-Sommerfeld/Squire operator can be used in a variety of applications. They are relevant to the prediction of stability and transition in shear flows and could be used to construct the resolvent for turbulent wall-bounded flows. Rather than approximating the differential operators by a spectral expansion,
here we determine these eigenfunctions and eigenvalues semi-analytically for plane channel flow using the WKB approximation. The eigenfunctions are given in terms of the Bessel functions $J_{1/3}, J_{-1/3}, J_1,$ and $Y_1$. For the inviscid case, we find that for all $c$ in the real interval $[0, U_{max}]$ there is an infinite sequence of eigenfunctions
$v_j(y,c)$ with eigenvalue $k_{\perp j}^2(c)$ . We then show how these solutions may be used to solve the initial value problem. In the viscous case, we find that the eigenfunctions are composed of two types of solutions. One type is essentially an inviscid solution with a phase speed $c$ shifted by a complex factor proportional to $1/\sqrt{k_x Re}$ and is a relatively slowly varying function of $y$. The other is highly oscillatory, varying as $[(k_x Re)^{1/3}(y - y_c)]^{3/2}$ away from the critical layer at $y = y_c$.
Applications are made to plane Poiseuille flow.