Transient Numerical Solutions of an Extended Korteweg-De Vries Equation Describing Solitary Waves in Open-Channel Flow

Richard Jurisits

doi:10.3311/ppme.9334

Abstract

A solitary wave in two-dimensional, incompressible, turbulent free-surface flow over a plane bottom with small, constant slope is considered. The flow is assumed to be slightly supercritical with Froude numbers close to 1. If the flow far upstream and far downstream is fully developed, a simple argument based on the law of momentum shows that for a solitary wave to exist, the bottom friction cannot be constant all along the channel bed. In [1] the situation was considered where the bottom roughness of the channel is constant over some distance and slightly higher than in the rest of the channel bed, giving rise to a higher bottom friction coefficient. In an asymptotic analysis in [1] an extended Korteweg-de Vries (KdV) equation was derived to describe the surface elevation of the fluid. Adopting this equation, we solved it numerically by posing a coupled boundary-value eigenvalue problem and obtained results for stationary and transient wave solutions as well as for the eigenvalue, which corresponds to distinct values of the bottom friction coefficient. While the numerical solutions as compared to the asymptotic solutions agree qualitatively in the stationary case, there were major differences found in case of the transient solutions. Preliminary studies to this work were reported in [2].

Highlights

A solitary wave in two-dimensional, incompressible, turbulent free-surface flow over a plane bottom with small, constant slope α is considered, see Fig. 1. g ur hr u(−∞, y) α y h(x, t) xIf the flow far upstream and far downstream is fully developed, a simple argument based on the law of momentum shows that for a solitary wave to exist, the bottom friction cannot be constant all along the channel bed
While the unstable asymptotic solution for the solitary wave initially shifted to the right by the value + β decays in the limit of large times, the corresponding numerical solution converges to a different stationary solution, with its values for the crest height and position seen as the lower terminating point of the black dashed line in Figure 4 shows the shapes of the numerical solutions of
4 Conclusion Solitary waves in turbulent open-cannel flow are governed by an extended Korteweg-de Vries (KdV) equation which was derived in [1] without any recourse to turbulence modelling

Summary

Introduction

A solitary wave in two-dimensional, incompressible, turbulent free-surface flow over a plane bottom with small, constant slope α is considered, see Fig. 1. g ur hr u(−∞, y) α y h(x, t) xIf the flow far upstream and far downstream is fully developed, a simple argument based on the law of momentum shows that for a solitary wave to exist, the bottom friction cannot be constant all along the channel bed. We solved it numerically by posing a coupled boundary-value eigenvalue problem and obtained results for stationary and transient wave solutions as well as for the eigenvalue, which corresponds to distinct values of the bottom friction coefficient. For the special case of V ≡ 0 , Eq (10) gives just an algebraic condition with the asymptotic values for the stationary crest positions and as solutions, which were obtained earlier in the asymptotic analysis.

Results

Conclusion