The Nonlinear Critical‐Wall Layer in a Parallel Shear Flow

Abstract

The nonlinear critical layer theory is developed for the case where the critical point is close enough to a solid boundary so that the critical layer and viscous wall layers merge. It is found that the flow structure differs considerably from the symmetric "cat's eye" pattern obtained by Benney and Bergeron [1] and Haberman [2]. One of the new features is that higher harmonics generated by the critical layer are in some cases induced in the outer flow at the same order as the basic disturbance. As a consequence, the lowest-order critical layer problem must be solved numerically. In the inviscid limit, on the other hand, a closedform solution is obtained. It has continuous vorticity and is compared with the solutions found by Bergeron [3], which contain discontinuities in vorticity across closed streamlines.

Long Nonlinear Surface Waves in the Presence of a Variable Current

“…The remaining question concerns the magnitude of the constant multiplying the regular Frobenius solution. For this case, |A/B| = 4.9705 which more or less confirms the scaling in [13]. To support the basic idea, the ratio of |A| to |B| is larger by a factor of 25.6 compared with the reference point, G = 1.0.…”

“…The only condition that we have yet to impose is (7), the kinematic condition at the free surface. At lowest order, we use (14) to replace η (0,0) and the wave Equation (13) can then be employed to achieve separation of variables. The result can be written…”

“…The procedure described earlier is appropriate when there is no critical layer because the integral in (16) is determined uniquely. However, for singular modes, it is preferable to deal with the zeroth order vorticity equation which, after using (13) to separate variables, leads to the Rayleigh equation with wavenumber zero, namely,…”

“…The primary difference however, compared with the case when y c is in the interior of the fluid, is that A can be much greater than B. The nonlinear critical layer theory for that case was formulated by Finlay, Liu and Maslowe [13]; as in [5], B was assumed to be O(1). However, A, the constant multiplying the regular Frobenius solution, was taken to be O(ε −1/2 ) so that A(y − y c ) is O(1) near the critical layer.…”

“…where (θ, Y ) = (εū c ) −1 ψ(θ, y), Y = (y − y c )/ε 1/2 , θ = kx, and λ 1/3 is the ratio of the viscous to the nonlinear critical layer thickness. Solutions of (24) were obtained numerically in [13] for various values of λ and streamline patterns were computed showing the influence of the wall. Here, we are interested primarily in large Reynolds number flows, so it is worth recalling [13] that a solution of (24) in the inviscid limit, λ = 0, is given by…”