The initial development of a jet caused by fluid, body and free surface interaction. Part 4. the large-time structure

The free surface and flow field structure generated by the uniform acceleration of a rigid plate, inclined at an angle α ∈ (0, π) to the exterior horizontal, as it advances into an initially stationary and horizontal strip of inviscid incompressible fluid, is studied in the large-time limit via the method of matched asymptotic expansions. The small-time limit of the solution to this problem has been studied in detail in Needham et al. (2008, The initial development of a jet caused by fluid, body and free-surface interaction. Part 3. An inclined accelerating plate. Q. Jl. Mech. Appl. Math., 64, 581-615). We assume that after an initial constant acceleration, the velocity of the plate reaches a terminal value in finite time and remains at that velocity thereafter. The structure of the large-time solution includes a steady bore propagating away from the plate with constant speed, together with an algebraically decaying wave field behind the bore, up to the plate, which demonstrates that there are no localized trapped waves excited near to the plate, with all energy radiating away from the plate as t → ∞.