抛物线型地形上的干溃坝问题
Dry dam-break over parabolic bathymetry
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中文总结 AI 辅助
研究抛物线型地形上的干溃坝问题，通过微扰框架求解一维浅水方程和无粘伯格斯方程，得到广义稀疏波解，其与海岸波浪爬高相关，分析结果与数值模拟一致，并探讨了与相关方程的联系。
中文摘要 AI 辅助
受干溃坝水流的启发，开发了一个微扰框架，用于求解抛物线型地形上的一维浅水方程（SWE）以及无粘伯格斯方程。用黎曼变量表示，解被展开为地形曲率\(\omega^2\)的解析幂级数，并以三角函数的形式封闭求和。这些解被称为广义稀疏波（GRWs），具有有限时间奇点，并描述了流体在干（真空）点附近的渐近行为。GRWs与海岸波浪爬高直接相关。还考虑了一个排斥性抛物线形山丘地形，得到了双曲函数形式的解。讨论了与模拟限制在谐波势中的玻色 - 爱因斯坦凝聚体的非线性薛定谔/格罗斯 - 皮塔耶夫斯基方程的联系。分析结果与不同场景下伯格斯方程和SWE的直接数值模拟结果一致。
英文摘要
Motivated by dry dambreak flows, a perturbative framework for solving the 1D shallow water equations (SWE) over parabolic bathymetry, together with the inviscid Burgers' equation, is developed. Expressed in Riemann variables, the solutions are expanded as analytic power series in the bathymetric curvature $ω^2$ and summed in closed form in terms of trigonometric functions. The solutions, termed generalised rarefaction waves (GRWs), exhibit finite-time singularities and are shown to describe the asymptotic behaviour of the fluid near dry (vacuum) points. The GRWs bear direct relevance to wave run-up at a shore. A repulsive parabolic hill bathymetry is also considered, yielding solutions in terms of hyperbolic functions. The connection to the nonlinear Schrödinger/Gross-Pitaevskii equation that models Bose-Einstein condensates confined in a harmonic potential is discussed. The analytical results agree with direct numerical simulations of the Burgers' equation and of the SWE in different scenarios.
