Approximate Solutions to the Linearized Navier-Stokes Equations for Incompressible Channel Flow

摘要

The linearized Navier-Stokes equations for incompressiblechannel flow are considered in which the flow is homogeneousin two directions. We study the initial-value problem for u andω_y, where y is the coordinate normal to the wall. After a Laplacetransform in time and a double Fourier transform in space weuse the WKB approximation on the resulting system of ODE’sin y to construct analytically the Green’s function for such solutions for the inviscid case in terms of the Bessel functions J_1 and Y_1, their modified counterparts, and the Airy functions Aiand Bi.