General Three‐Dimensional Disturbances to Inviscid Couette Flow

The general solution to the linearized equations governing three‐dimensional disturbances to inviscid Couette flow has been obtained. This result extends the Orr solution to initial conditions that do not consist of a single Fourier sine component in the cross‐stream coordinate and a plane wave in the streamwise/spanwise coordinates. The time evolution of a measure of disturbance energy for some specific pulsed initial conditions is examined, and it is concluded that, while the rapid algebraic growth to large amplitude followed by decay exemplified by the Orr solution can be of importance for individual cross‐stream Fourier components, more realistic initial conditions, which in general consist of the sum of an infinite number of components, often display uniform decay to zero amplitude. However, an interesting example is described in which one positive definite measure of disturbance amplitude remains constant, yet the streamwise/spanwise velocity components grow linearly in time if the initial disturbance is three‐dimensional.