Evolution of Vortex Dipole Disturbances in Channel Flow

The initial-value problem for the linearized Navier-Stokes equations for channel flow with mean velocity and mean vorticity given by $\mathbf{U} = (U(y),0,0)$ and $\boldmath{\Omega} = (0,0, -dU(y)/dy)$ is investigated for initial disturbances consisting of vortex dipoles. Approximate analytical solutions to the resulting Orr-Sommerfeld (OS)/ Squire equations are found in terms of special functions, thus avoiding the need of large-scale computation. Under the assumption of linearity, the initial vortex dipole merely travels downstream with the local mean velocity $U(y) $ as a diffusive scalar. However, additional disturbances are produced continually because the quantity $ U''(y) \partial v/\partial x $ appears in the OS equation as a source for transported quantity $ \nabla^2 v$. In turn, $ - U'(y) \partial v/\partial z $ is the source term in the Squire equation for the transported quantity $\omega_y$. We find it particularly effective to represent this additional $v$ field as an expansion in terms of eigenfunctions of the OS equation with complex values of $k_\perp$ ($k_\perp^2 = k_x^2 + k_z^2$). For the inviscid case, e.g., these solutions satisfy \begin{equation} \frac{d^2 v_j}{dy^2} - \frac{U^{\prime \prime}(y) v_j}{U(y) - c} = k^2_ {\perp j}v_j \end{equation}