The simulation of shallow flows over obstacles is an important problem in environmental fluid dynamics, including exchange flows over seabed sills, atmospheric flows past steep mountains and water flows over river bedforms. A common mathematical treatment consists in using vertically-averaged models instead of vertically-resolved ones by introducing a suitable shallow water approximation. The dispersionless Saint Venant equations are a useful tool, albeit accuracy is not enough in many circumstances. The next approach consists in resorting to the Serre–Green–Naghdi theory, which is well known to produce good solutions for long non-breaking waves. However, a common feature of flows over obstacles is the generation of breaking waves at its lee side, which are important to model, given their role in the mixing and transport of passive scalars downstream of the terrain barrier. The Serre–Green–Naghdi theory fails to model these flows, producing unrealistic trains of undular waves. A widely used practice consist in resorting to a patching approach in a numerical setting where the solutions of Serre–Green–Naghdi and Saint Venant equations are assembled once wave breaking is detected by case-dependent empirical parameters. In this work an alternative method to dealt with wave breaking over obstacles within the Boussinesq-type approximation is proposed. The exact depth-averaged equations for flows over uneven beds are developed and presented as function of the vertical acceleration and non-uniformity of velocity with elevation. By introducing a suitable kinematic field, a new high-order phase resolving system of non-hydrostatic equations is presented, containing the usual dispersive corrections of Serre–Green–Naghdi theory plus high-order corrections for velocity profile modeling. It is found that the new theory allows the simulation of both breaking and non-breaking waves in shallow flows over obstacles without introducing any case-dependent calibration parameter. The new shallow water approximation is thus an alternative method to deal with wave breaking in Boussinesq type models.

浅水流越过障碍物的模拟是环境流体动力学中的一个重要问题，包括海床基台上的交换流、陡峭山脉上的大气流和河床上的水流。一种常见的数学处理包括通过引入合适的浅水近似来使用垂直平均模型而不是垂直解析模型。无色散圣维南方程是一个有用的工具，尽管在许多情况下精度不够。下一种方法是求助于 Serre-Green-Naghdi 理论，众所周知，该理论可以为长的非破波产生良好的解决方案。然而，流过障碍物的一个共同特征是在其背风面产生破碎波，这对建模很重要，考虑到它们在地形屏障下游的被动标量的混合和传输中的作用。Serre-Green-Naghdi 理论未能模拟这些流动，产生不切实际的波状波列。一种广泛使用的做法是在数值环境中采用修补方法，一旦通过与案例相关的经验参数检测到波破，就会组装 Serre-Green-Naghdi 和 Saint Venant 方程的解。在这项工作中，提出了一种在 Boussinesq 型近似内处理波突破障碍的替代方法。不均匀床层上流动的精确深度平均方程被开发并呈现为垂直加速度和速度随高度不均匀性的函数。通过引入合适的运动场，提出了一种新的非流体静力学方程的高阶相位解析系统，其中包含 Serre-Green-Naghdi 理论的通常色散校正以及速度剖面建模的高阶校正。发现新理论允许在不引入任何与情况相关的校准参数的情况下模拟跨越障碍物的浅水流中的破碎波和非破碎波。因此，新的浅水近似是处理 Boussinesq 类型模型中波浪破坏的替代方法。发现新理论允许在不引入任何与情况相关的校准参数的情况下模拟跨越障碍物的浅水流中的破碎波和非破碎波。因此，新的浅水近似是处理 Boussinesq 类型模型中波浪破坏的替代方法。发现新理论允许在不引入任何与情况相关的校准参数的情况下模拟跨越障碍物的浅水流中的破碎波和非破碎波。因此，新的浅水近似是处理 Boussinesq 类型模型中波浪破坏的替代方法。