Abstract The breaking of gravity water waves induces both strong turbulence near the free surface and air–water mixing, which are not captured in currently-available single-phase, non-hydrostatic Reynolds Averaged Navier Stokes (RANS) models. In order to account for such dynamics, the boundary conditions proposed by Brocchini (2002) have been implemented in a two-dimensional vertical (2DV) non-hydrostatic RANS numerical model. For the theoretical boundary conditions of Brocchini (2002) to be applicable in the numerical model, appropriate closures have been adopted and mathematical equations have been proposed to calculate the coefficients introduced in the mentioned boundary conditions. Navier Stokes equations along with different turbulence closure models have been solved using finite volume method and pressure correction technique. According to the new boundary conditions, the normal-to-mean surface gradient of the turbulent kinetic energy (TKE) differs from zero and is computed as a balance between production and dissipation of TKE within the air–water mixing layer. Also, the kinematic and dynamic boundary conditions have been modified accordingly, to account for the effects of the thin two-phase layer formed at the free surface. The modified kinematic boundary condition allows for the mass exchange between the two-phase layer and the main body of the water. For the first time it is demonstrated that the proposed simple analytical model of Brocchini (2002) leads to improvements in the prediction of incipient breaking and the wave characteristics in the surf zone. This opens the way to a new, accurate but numerically low cost, approach for the computation of the air–water mixing that characterizes breaking waves. All the many benchmarking tests run to verify the ability of the new model show that significant improvements are achieved. Improvements are specifically observed in the prediction of: (I) breaking point and the breaking wave height, (II) the consequent dissipation of the wave energy observed in the form of the crest level distribution throughout the surf zone, (III) the magnitude of the horizontal velocity near the free surface, and (IV) the TKE distribution in depth. Notably, the novel numerical model does not use any parametric criteria for detecting the inception of breaking, thus it does not need calibration for different flow circumstances. Also, since no special treatments, such as hydrostatic pressure assumptions at the front face of the breaking waves are considered, the new model well captures the breaking-induced dissipation while giving a more accurate estimation of the dynamic pressure.

中文翻译：

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溢出碎波的新自由表面边界条件

摘要 重力水波的破碎会引起自由表面附近的强烈湍流和气水混合，而目前可用的单相、非静水力雷诺平均纳维斯托克斯 (RANS) 模型没有捕捉到这种情况。为了解释这种动力学，Brocchini (2002) 提出的边界条件已经在二维垂直 (2DV) 非静水力 RANS 数值模型中实现。为了使 Brocchini (2002) 的理论边界条件适用于数值模型，采用了适当的闭包，并提出了数学方程来计算上述边界条件中引入的系数。Navier Stokes 方程以及不同的湍流闭合模型已使用有限体积法和压力校正技术求解。根据新的边界条件，湍流动能 (TKE) 的法向平均表面梯度不为零，并且被计算为气水混合层内 TKE 的产生和耗散之间的平衡。此外，运动学和动态边界条件已相应修改，以考虑在自由表面形成的薄两相层的影响。修改后的运动学边界条件允许两相层和水主体之间的质量交换。首次证明 Brocchini (2002) 提出的简单分析模型改进了对初始破浪和冲浪区波浪特征的预测。这为新的、准确但数值较低的成本开辟了道路，用于计算表征破碎波的气水混合的方法。为验证新模型的能力而运行的许多基准测试表明，实现了重大改进。在预测中特别观察到改进：(I) 断裂点和断裂波高，(II) 以整个冲浪区的波峰水平分布的形式观察到的波浪能量的后续耗散，(III)自由表面附近的水平速度，以及 (IV) 深度的 TKE 分布。值得注意的是，新的数值模型不使用任何参数标准来检测破裂的开始，因此不需要针对不同的流动情况进行校准。此外，由于没有特殊处理，