Nonlinear subcritical free-surface flow past topography: exponential-asymptotics solution

A theoretical study is made of steady, subcritical (Froude number F < 1 ) two-dimensional free-surface flow due to a uniform stream flowing over smooth, locally confined bottom topography of large horizontal extent ( L 1 ) and finite peak height ( ε = O(1) ). In earlier work, this flow was analysed based on the nonlinear shallow-water equations which neglect the effects of dispersion altogether. This so-called hydraulic theory predicts a steady disturbance confined in the vicinity of the topography if ε is below a critical value ε crit(F) . The present asymptotic analysis of the full potential flow equations focuses on how dispersive effects (controlled by μ = 1/L 1 ) influence this steady state, particularly in regard to a steady short-scale radiating wave downstream that is ignored by hydraulic theory. Utilizing exponential asymptotics, it is shown that as ε is increased this dispersive wave, whose amplitude is formally exponentially small with respect to μ , grows sharply and ultimately it becomes comparable with the hydraulic wave disturbance when ε approaches ε crit . Thus, the nonlinear shallow-water equations break down in the vicinity of ε crit regardless of μ 1 . The asymptotic results are supported by numerical solutions of the full potential flow theory, which also reveal a limiting ε , ε ₗᵢₘ ≈ ε _ crit , above which steady wave responses cannot be computed. For ε just below ε ₗᵢₘ , the downstream wave resembles a steep steady Stokes periodic wave, while for ε slightly above ε ₗᵢₘ unsteady computations suggest that the downstream disturbance steepens and breaks.