NUMERICAL SIMULATION AND THEORETICAL CALCULATION OF THE FORMATION AND EVOLUTION OF THE BACKFLOW BORE
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摘要: 当潮波进入喇叭状河口时, 由于地形的升高和河口的逐渐缩窄, 会发生剧烈变形的特殊水力现象, 即涌潮. 一旦涌潮与涉水结构物(如海塘)相互作用, 在这些结构物的阻挡下往往使涌潮往相反的方向传播, 从而形成回头潮. 回头潮因其威力大, 一定程度上危及人们生命. 过去对回头潮的研究较少, 关于回头潮产生的机理研究还不够充分. 文章通过高精度数值模拟对垂直海塘上的回头潮水动力特性进行研究, 详细分析了入射涌潮高度和潮前水深两个主要因素的影响. 同时基于一维连续性方程、动量守恒方程和经典的伯努利方程, 推导了回头潮高度、传播速度以及冲击载荷的理论计算公式. 并且应用高精度数值仿真的计算结果对理论计算公式的准确性进行了系统的验证. 结果表明: 入射涌潮高度和潮前水深对回头潮的高度、传播速度、冲击载荷及形态产生较明显影响. 提出的回头潮高度、传播速度以及冲击载荷的理论计算公式具有较高的精度, 满足回头潮分析的计算要求, 可用于相关工程实践中.Abstract: When the tidal wave enters the trumpet-shaped estuary, due to the elevation of the terrain and the gradual narrowing of the estuary, a special hydraulic phenomenon of severe deformation will occur, that is, the tidal bore. Once the tidal bore interacts with water-related structures (such as seawalls), the tidal bore often propagates in the opposite direction under the obstruction of these structures, thus forming a backflow bore. Because of its great power, the backflow bore endangers people 's lives to a certain extent. In the past, there were few studies on the backflow bore, and the research on the mechanism of the backflow bore was not sufficient. In this paper, the hydrodynamic characteristics of the backflow bore on the vertical seawall are studied by high-precision numerical simulation, and the influence of the two main factors of the incident tide height and the initial water depth is analyzed in detail. At the same time, based on the one-dimensional continuity equation, the momentum conservation equation and the classical Bernoulli equation, the theoretical calculation formulas of the height of the backflow bore, the propagation speed and the impact load are derived. The accuracy of the theoretical calculation formula is systematically verified by the calculation results of high-precision numerical simulation. The results show that the incident tide height and the initial water depth have a significant effect on the height, propagation velocity, impact load and height of the backflow bore. The theoretical calculation formulas of the height, propagation velocity and impact load of the backflow bore proposed in this paper have high accuracy and meet the calculation requirements of the backflow bore analysis, which can be used in related engineering practice.
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Keywords:
- backflow bore /
- theoretical calculation /
- numerical simulation /
- impacting load
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引 言
当潮波在喇叭形河口中传播时, 因受到地形抬高和河口束窄的影响, 会逐步形成一种特殊的水位暴涨的水流现象, 称为涌潮[1]. 其中, 最具影响力的就是我国的钱塘江涌潮. 涌潮在传播的过程中受到海塘、丁坝等涉水建筑物的影响会形成气势磅礴的回头潮, 成为受人关注的潮景. 然而, 回头潮具有排山倒海之势, 极易冲毁海塘, 危及人民群众的生命安全. 因此, 系统研究回头潮的水动力学演变过程具有重要的学术和工程应用价值.
近些年来, 围绕涌潮的生成、传播与演变过程, 国内外的学者开展了广泛的现场观测[2-4]、物理模型试验[5-7]和数值计算工作[8-10]. Pan等[11]通过分析长期观测的数据, 发现上游河流流量可以显著影响现实潮汐河口涌潮高度的空间变化. Zeng等[12]通过物理实验分析了堤线调整对涌潮水动力产生的影响. 他们的实验数据表明, 通过调整堤线与涌潮入射角的相互作用角度, 可以显著影响涌潮的传播方向和强度. 除此之外, 潮汐河口涌潮的产生和转化过程也可以通过求解浅水方程, 采用合适的数值方法来模拟, 如TVD-MUSCL格式[13]. 在恶劣天气下, 强风与水流[14]之间存在复杂的相互作用. Wang等[15]通过数值模拟研究了台风对钱塘江涌潮的水动力影响, 发现强风能显著增加涌潮高度, 显著增加海堤涌潮的冲击载荷和越浪量. 此外, Qu等[16]基于两相流方法数值研究了强风对海堤处涌潮越浪过程的影响. Wang等[17]采用大型近岸流模型FVCOM, 数值研究了钱塘江直角岸线对涌潮传播的影响. 计算结果表明, 回头潮传播速度可达到入射涌潮传播速度的80 %.
这些研究工作在一定程度上理清了涌潮演变的生成机理[6], 分析了上游流量[18]、强风[16]等因素对涌潮传播演变特性的影响, 量化了典型涉水结构物在涌潮冲击过程中的压强[19]和载荷特性[20], 以及涉水结构物建设对涌潮传播演变过程的影响[21]. 然而, 目前采用理论方法分析涌潮传播演变特性的研究工作还不够充分. 潘存鸿等[22]基于一维连续性方程和动量方程推导了涌潮传播速度的计算公式, 并且建立了潮后水位、涌潮流速与涌潮高度、潮前水位和退潮流速的关系. Li等[23]分析了涌潮在缓坡地形上的传播过程, 提出了潮前和潮后水深比的计算公式. 目前, 对于涉水建筑物, 尤其是直立式海塘处回流涌潮的水动力生成问题, 目前还缺乏足够的研究, 采用理论方法系统分析回头潮演变特性的研究还未见报道.
本文基于一维连续性方程、动量守恒方程和经典的伯努利方程, 推导了回头潮高度、回头潮传播速度以及冲击载荷的理论计算公式, 开展了回头潮生成演变的高精度数值仿真, 并且利用数值计算结果对回头潮高度、回头潮传播速度以及涌潮冲击载荷的理论计算公式进行了系统的验证, 为研究回头潮的水动力特性及相关工程实践提供参照.
1. 涌潮的定义
涌潮的传播过程可视为一种运动的水跃现象(图1). 其中, $ {H_0} $代表入射涌潮的高度, $ {u_0} $代表落潮速度, $ {u}_{ 1} $代表入射涌潮处的水流速度, $ {c_1} $代表涌潮传播速度, 入射涌潮后的水深为$ {h}_{ 1} $, 并且$ {h}_{ 1} = {h}_{ 0} + {H}_{0} $.
选取图1中的断面1和断面2之间的水体作为控制体, 应用一维连续性方程和动量守恒方程, 可以计算出入射涌潮的传播速度($ {c_1} $)
$$ {c}_{ 1} = {u}_{0} + \sqrt{\frac{g{h}_{ 1}({h}_{1} + {h}_{0})}{2{h}_{0}}} $$ (1) $$ {c}_{ 1} = {u}_{1} + \sqrt{\frac{g{h}_{0}({h}_{1} + {h}_{0})}{2{h}_{ 1}}} $$ (2) 基于式(1)和式(2)可以计算出入射涌潮处的流速($ {u}_{ 1} $)为
$$ {u}_{1} = {u}_{0} + \sqrt{\frac{g{h}_{ 1}({h}_{1} + {h}_{0})}{2{h}_{ 0}}}-\sqrt{\frac{g{h}_{0}({h}_{1} + {h}_{0})}{2{h}_{1}}} $$ (3) 在本研究中, 方程(3)计算的速度用作入射涌潮的生成, 给定落潮流速($ {u}_{ 0} $)、入射涌潮的高度($ {H_0} $)和潮前水深($ {h}_{ 0} $). 当考虑垂直海塘处的回头潮时, 本研究中的落潮水速度为0. Froude数$ \left({F_r} = \dfrac{{c + {u_0}}}{{\sqrt {g{h_0}} }}\right) $用于涌潮类型区分. 如果Fr < 1.3, 则为波状的涌潮; 如果为1.3 < Fr < 1.5, 则为弱漩滚状的涌潮; 如果Fr > 1.5, 则为强漩滚状的涌潮. 强漩滚是最猛烈、最强大的涌潮, 本文将主要讨论.
2. 数值模拟
2.1 控制方程
涌潮冲击海塘的水动力过程可以用非定常两相流来描述, 本文基于开源程序OpenFOAM建立涌潮数值水槽, 其控制方程, 连续性方程和动量守恒方程 如下
$$ \frac{{\partial {u_i}}}{{\partial {x_i}}} = 0 $$ (4) $$\frac{{\partial \rho {u_i}}}{{\partial t}} + \rho {u_j}\frac{{\partial {u_i}}}{{\partial {x_j}}} = - \frac{{\partial p}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left[{\mu _{{\mathrm{eff}}}}\left(\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}\right)\right] + \rho gi$$ (5) 式中, $ t $为时间, $ {u}_{i} $为流速, $ p $为压力, $ \rho $为气-水混合物的密度, $ {\mu }_{{\mathrm{eff}}} $为有效黏性系数, 并且$ {\mu }_{{\mathrm{eff}}} = {\mu }_{l} + {\mu }_{t} $. 其中, $ {\mu }_{l} $为层流黏性系数, $ {\mu }_{t} $ 为湍流黏性系数. 本文采用方程$ k-\omega $湍流模型[24]计算湍流黏性系数, 其湍动能($ k $)和湍流耗散率($ \omega $)的控制方程可表示为
$$ \begin{split} & \frac{{\partial \rho k}}{{\partial t}} + \rho {u_j}\frac{{\partial k}}{{\partial {x_j}}} = \frac{\partial }{{\partial {x_j}}}\left[\left({\mu _l} + \frac{{{\mu _t}}}{{{\sigma _k}}}\right)\frac{{\partial k}}{{\partial {x_j}}}\right] + \\ &\qquad 2{\mu _t}|S{|^2} - \rho k\omega \end{split} $$ (6) $$ \begin{split} & \frac{{\partial \rho \omega }}{{\partial t}} + \rho {u_j}\frac{{\partial \omega }}{{\partial {x_j}}} = \frac{\partial }{{\partial {x_j}}}\left[\left({\mu _l} + \frac{{{\mu _t}}}{{{\sigma _\omega }}}\right)\frac{{\partial \omega }}{{\partial {x_j}}}\right] + \\ &\qquad 2{c_\mu }{c_{\omega 1}}\rho |S{|^2} - {c_{\omega 2}}\rho {\omega ^2}\end{split} $$ (7) 式中, $ {\sigma _\omega } = {\sigma _k} = 2 $, $ {c_\mu } = 0.09,\;{c_{\omega 1}} = {5}/{9} $和$ {c_{\omega 2}} = {5}/{6} $. $ S $为应变张量. 湍流黏性系数$ {\mu _t} = \rho {k}/{\omega } $.
水-气界面采用流体体积分数(VOF)方法进行计算, 其控制方程如下
$$ \frac{{\partial \alpha }}{{\partial t}} + \frac{{\partial \alpha {u_i}}}{{\partial {x_i}}} + \frac{{\partial \alpha (1 - \alpha ){u_i}}}{{\partial {x_i}}} = 0 $$ (8) 式中, $ \dfrac{{\partial \alpha (1 - \alpha ){u_i}}}{{\partial {x_i}}} $为人工压缩项, α代表计算网格中水相的体积分数, 其含义如下所示
$$ \left.\begin{aligned} &\alpha = 0,\quad {\mathrm{air}}\\ &0 < \alpha < 1,\quad \text{air-water interface}\\ &\alpha = 1,\quad \text{water}\end{aligned}\right\} $$ (9) 空气-水混合物的密度和层流黏性系数可定义为$ \mathrm{\alpha } $的函数, 即
$$ \rho = {\rho }_{{\mathrm{air}}} + \alpha \cdot ({\rho }_{{\mathrm{w}}}-{\rho }_{{\mathrm{air}}}) $$ (10) $$ {\mu }_{l} = {\mu }_{{\mathrm{air}}} + \alpha \cdot ({\mu }_{{\mathrm{w}}}-{\mu }_{{\mathrm{air}}}) $$ (11) 本文采用延迟修正的数值格式离散控制方程中的对流项[25], 采用经典的中心差分格式离散耗散项和压力梯度项, 把重力项作为源项, 并且采用PIMPLE方法求解压力泊松方程[26-27]. 为保证数值求解的稳定性, 动量平衡插值法用来实现从计算网格中心到计算网格面中心的速度传递.
2.2 数值验证
为验证本文数值模型的可靠性, 开展了涌潮生成、传播的数值计算, 并且将计算得到的水位和流速与Zhang等[28]所采集的水槽试验数据进行了对比. 计算布置与物理模型试验相同. 数值验证的计算模型中初始条件底面均采用壁面边界, 顶端采用压力出口边界, 出流端采用自由出流边界, 入口水边界条件基于潘存鸿等[22]推导的涌潮速度边界, 具体设置参照式(3), 出口为消波边界. 水深方向的最小网格分辨率$ {\mathrm{d}}z $ = 0.001 m, 涌潮传播方向的最小网格分辨率$ {\mathrm{d}}x $ = 0.01 m. 给定潮前水深$ {h}_{0} $ = 0.05 m, 选取两种试验入射涌潮高度: $ {H}_{0} $ = 0.03, 0.07 m.
图2对比了距离涌潮入口10 m处水位和流速随时间的变化. 可以看出, 数值计算的结果同模型试验测量结果吻合较好, 满足计算要求.
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图 2 水位与流速的时程曲线对比Figure 2. Comparisons of time series of water elevation and flow velocity2.3 计算设置
本文涌潮数值计算比尺根据弗劳德数相似准则设计, 模型几何比尺为1:10, 对应钱塘江天然涌潮的水流条件: 潮前水深h0的范围为1.5 ~ 6.0 m、涌潮高度$ {H}_{0} $的范围为0.5 ~ 3.9 m的涌潮[29-31]. 入射涌潮类型根据$ {F_r} $的大小, 分为波状涌潮、弱漩滚涌潮和强漩滚涌潮3种类型[32]. 考虑到强漩滚涌潮的复杂性及其显著的破坏性, 本文入射涌潮均为强漩滚涌潮($ {F}_{r} $ > 1.5). 具体工况设置如表1所示, 包括潮前水深$ {h}_{0} $、入射涌潮高度$ {H}_{0} $、入射涌潮处的流速$ {u}_{1} $及弗劳德数$ {F}_{r} $. 数值计算布置如图3所示. 计算区域的总长度17 m, 高度1.2 m. 计算网格分辨率与2.2节设置相同.
表 1 工况设置Table 1. Numerical setupsCondition $ {h}_{0} $/m $ {H}_{0} $/m $ {u}_{1} $/(m·$ {{\mathrm{s}}}^{-1} $) $ {F}_{r} $ 1 0.15 0.24 1.61 2.16 2 0.15 0.25 1.68 2.21 3 0.15 0.26 1.74 2.26 4 0.15 0.27 1.80 2.31 5 0.15 0.28 1.86 2.35 6 0.21 0.24 1.40 1.84 7 0.21 0.25 1.46 1.87 8 0.21 0.26 1.51 1.90 9 0.21 0.27 1.56 1.94 10 0.21 0.28 1.62 1.97 11 0.27 0.24 1.27 1.65 12 0.27 0.25 1.31 1.68 13 0.27 0.26 1.36 1.71 14 0.27 0.27 1.41 1.73 15 0.27 0.28 1.46 1.76 16 0.33 0.24 1.16 1.53 17 0.33 0.25 1.21 1.56 18 0.33 0.26 1.25 1.58 19 0.33 0.27 1.30 1.60 20 0.33 0.28 1.34 1.62 3. 数值计算结果
3.1 回头潮的生成及复杂水动力现象
本节分析了入射涌潮冲击海塘生成回头潮的水动力学过程. 计算选取潮前水深($ {h}_{0} $)为0.21和0.33 m, 入射涌潮高度($ {H}_{0} $)均为0.26 m. 如图4所示: 在涌潮向海塘传播过程中, 涌潮前部会出现连续的水体漩滚(图4(a)和图5(a)), 存在明显的自由表面破碎、水滴飞溅和水气掺混等现象. 当入射涌潮冲击海塘后, 水体沿着海塘向上流动(图4(b)和图5(b)), 最终反射回来形成回头潮(图4(c)和图5(c)). 对比不同潮前水深情况下的计算结果可以看出: 给定相同入射涌潮高度, 回头潮的形态分别呈现出漩滚涌潮和波状涌潮特征. 在给定入射涌潮高度的情况下, 存在一个临界潮前水深. 高于这个临界潮前水深, 回头潮会呈现出波状涌潮的特征. 低于这个临界潮前水深, 回头潮仍然会呈现出漩滚涌潮的特征, 且漩滚程度弱于入射情况.
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图 4 不同时刻的涌潮流速云图($ {h}_{0} $ = 0.21 m)Figure 4. Tidal bore flow velocity contours at different moments ($ {h}_{0} $ = 0.21 m)![]()
图 5 不同时刻的涌潮流速云图($ {h}_{0} $ = 0.33 m)Figure 5. Tidal bore flow velocity contours at different moments ($ {h}_{0} $ = 0.33 m)图6展示了不同空间位置的水面高程随时间的变化. 当水深相对较低, 为0.21 m时, 回头潮形态展现为较稳定的破碎涌潮向上游传播. 但当水深相对较大, 为0.33 m时, 回头潮最初的形态为波状起伏(图6(e)和图6(f)). 随着波状的回头潮进一步向上游传播, 回头潮潮头趋于破碎(图6(d)、图6(c)和图6(b)), 因为随着回头潮向上游传播, 能量耗损, 回头潮越来越弱. 由图7可知, 在相同的入射涌潮高度下, 海塘上的涌潮上升高度随着潮前水深的增加逐渐增加. 且在相同的入射涌潮高度下, 涌潮对海塘上的冲击载荷也随潮前水深的增加而增大(图8), 水动力载荷的时间变化与入射涌潮的水位高程变化一致.
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图 7 垂直海塘处的涌潮上升高度Figure 7. The rising height of the tidal bore at the vertical seawall![]()
图 8 涌潮在垂直海塘上冲击载荷的时程变化Figure 8. Time series of the impacting load at the vertical seawall exerted by incident tidal bore3.2 入射涌潮高度的影响
本节分析入射涌潮高度($ {H}_{0} $)变化对回头潮水动力影响. 数值模拟中取潮前水深$ {h}_{0} $ = 0.33 m, 设置5个入射涌潮高度, $ {H}_{0} $ = 0.24, 0.25, 0.26, 0.27和0.28 m. 图9为不同入射涌潮高度下的涌潮流速云图, 分别展示涌潮入射传播过程及受垂直海塘阻碍回头潮传播过程. 同一初始潮前水深下, 涌潮流速随涌潮高度的增加显著增大, 同时湍流强度也随涌潮高度的增加而增大, 这极大影响回头潮的水动力形成过程. 当$ {H}_{0} $ = 0.24 ~ 0.28 m, 回头潮的高度$ {H}_{1} $提高了31.1%. 说明随着入射涌潮的能量增加, 回头潮的能量也增加. 图10(a)可看出回头潮的传播速度$ {c}_{2} $随$ {H}_{0} $的变化呈现非线性趋势, 而涌潮在垂直海塘上水平冲击载荷$ {F}_{H} $及回头潮高度$ {H}_{1} $均为线性增加(图10(b)和图10(c)).
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图 9 不同涌潮高度下的流速云图Figure 9. Contours of the flow velocity under different tidal bore heights![]()
图 10 $ {c}_{2} $, $ {F}_{H} $和$ {H}_{1} $随涌潮高度的变化 (h0 = 0.33 m)Figure 10. The variation of $ {c}_{2} $, $ {F}_{H} $ and $ {H}_{1} $ with the height of tidal bore (h0 = 0.33 m)3.3 潮前水深的影响
潮前水深的变化, 同样对回头潮产生影响. 本节通过相同入射涌潮高度$ {H}_{0} $ = 0.26 m, 对$ {h}_{0} $ = 0.15, 0.21, 0.27, 0.33 m 四个不同潮前水深的涌潮工况分析. 图11展现了不同潮前水深下的涌潮流速云图. 当$ {h}_{0} $相对较小时, 涌潮传播表现为复杂的流动且涌潮流速很大(图11(a)). 在涌潮撞击垂直海塘开始形成回头潮时, 回头潮后方的涌潮流速几乎为零. 高速的水体区仅在回头潮的前端可观察到(图11(b)). 随着$ {h}_{0} $逐渐增加, 涌潮流速逐渐降低(图11(c)). 然而回头潮的破碎强度和流速增加(图11(d)). 此外, 当潮前水深$ {h}_{0} $大于0.21 m, 回头潮的形态显示为波状(图11(f)和图11(h)). 这与上文所提到, 存在临界水深使回头潮形态产生变化对应. 即当$ {h}_{0} $小于此临界水深, 则回头潮形态仍为漩滚状, 而$ {h}_{0} $大于此临界水深, 回头潮形态转变为波状. 图12(c)描绘了回头潮高度$ {H}_{1} $随潮前水深的变化. $ {H}_{1} $随$ {h}_{0} $增加单调减小, 且从$ {h}_{0} $ = 0.15 m变化到0.33 m, 减少为16.7%. 然而回头潮的传播速度$ {c}_{2} $随之增加(图12(a)). 这一定上可归因为质量守恒的原因, 尤其是回头潮高度随潮前水深增加而减少. 与此同时, 海塘上冲击载荷$ {F}_{H} $显然随潮前水深的增加单调递增, 从$ {h}_{0} $ = 0.15 ~ 0.33, 增加为30.4% (图12(b)).
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图 11 不同潮前水深下的流速云图Figure 11. Contours of the flow velocity under different initial water depths![]()
图 12 $ {c}_{2} $, $ {F}_{H} $和$ {H}_{1} $随涌潮高度的变化 (H0 = 0.26 m)Figure 12. The variation of $ {c}_{2} $, $ {F}_{H} $ and $ {H}_{1} $ with initial water depths (H0 = 0.26 m)4. 理论分析
4.1 理论解的推导
给定潮前水深($ {h}_{0} $), 落潮速度($ {u}_{0} $)和入射涌潮高度($ {H}_{0} $), 就可以确定出入射涌潮处的流速($ {u}_{1} $). 因此, 式(3)可以作为生成涌潮的边界条件, 其合理性已经在前人的研究中得到了广泛的验证[2,6,16,22].
受到海塘的阻碍作用, 涌潮冲击海塘以后会形成回头潮(如图13所示). 其中, $ {h}_{2} $代表回头潮潮后水深, $ {H}_{1} $代表回头潮高度, 并且$ {H_1} = {h_2} - {h_1} $. $ {c}_{2} $代表回头潮传播速度. 海塘存在阻碍了水流的运动, 海塘处的水流速度$ {u}_{2} $ = 0 m/s.
当回头潮向左传播时, 所到之处, 将引起水体的运动, 形成非恒定流. 采用运动坐标, 将坐标固定在回头潮潮头处, 坐标系随着回头潮的传播以波速$ {c}_{2} $自左向右移动, 因此相对于此运动坐标系, 回头潮的水流运动可视为恒定流动. 对于上述相对于动坐标的恒定流动. 在图13中, 选取距离回头潮潮头很近的两个断面1和2, 忽略水流摩阻力的作用, 应用经典的伯努利方程, 可得
$$ {h}_{ 1} + \frac{{({u}_{ 1} + {c}_{2})}^{2}}{2g} = {h}_{2} + \frac{{c}_{2}^{2}}{2g} $$ (12) 入射涌潮的流动受到海塘的阻碍作用, 速度降低, 水位抬高(图13). 当回头潮潮头以传播速度$ {c}_{2} $自右向左移动时, 回头潮潮后的水深增长为$ {h}_{2} $, 应用一维连续性方程, 可得
$$ {u}_{ 1}{h}_{1} = {c}_{2}{H}_{1} $$ (13) 将回头潮高度($ {H}_{1} $)和回头潮传播速度($ {c}_{2} $)作为未知量, 求解式(12)和式(13), 可得
$$ {H_1} = \frac{{{h_1}}}{{\sqrt {\dfrac{1}{{16}} + \dfrac{{g{h_1}}}{{u_1^2}} - \dfrac{1}{4}} }} $$ (14) $$ {c_2} = {u_1}\left(\sqrt {\frac{1}{{16}} + \frac{{g{h_1}}}{{u_1^2}}} - \frac{1}{4}\right) $$ (15) 选取回头潮潮头和海塘之间的水体作为控制体, 应用动量守恒方程, 可以计算出入射涌潮作用在海塘上的水平载荷($ {F}_{H} $)为
$$ {F_H} = \rho \left(c_2^2{H_1} + u_1^2{h_1} + \frac{1}{2}gh_1^2\right) $$ (16) 通过上面的推导过程可知, 给定入射涌潮高度($ {H}_{0} $)和潮前水深($ {h}_{0} $), 通过式(14) ~ 式(16)就可以计算出回头潮高度($ {H}_{1} $)、回头潮传播速度($ {c}_{2} $)以及涌潮作用在海塘上的水平载荷($ {F}_{H} $).
4.2 理论解与数值解的对比验证
图14对比了数值计算回头潮高度($ {H}_{1} $)与理论回头潮高度(式(15)). 由图14可知, 理论回头潮高度和计算回头潮高度吻合较好. 同时, 随着潮前水深逐渐减小时, 相对误差有所增加, 但最大误差不超过7.61%. 除此之外, 回头潮高度随着入射涌潮高度的增大和涌潮潮前水深的减小而单调增大. 图15对比了数值计算回头潮传播速度($ {c}_{2} $)、理论回头潮传播速度(式(15))和Wang等[17]中回头潮传播速度理论公式结果. 由图15结果可知, 理论回头潮传播速度和计算回头潮传播速度同样吻合较好, 最大误差不超过5.6%, 对比Wang等[17]中关于回头潮传播速度理论公式
$$ {U_2} = - \sqrt {g\frac{{{d_3}}}{{{d_4}}}\left(\frac{{{d_3} + {d_4}}}{2}\right)} $$ (17) 式中, $ {U}_{2} $为回头潮的传播速度, 对应文中的$ {c}_{2} $, $ {d}_{3} $为入射涌潮传播时的水深, 对应文中$ {h}_{1} $, $ {d}_{4} $为回头潮传播时的水深, 对应文中$ {h_2} = {H_1} + {h_1} $.
结果显示Wang等[17]中的公式结果和数值计算的结果误差最大为4.97%. 而相比于本文式(15)的结果要略大一些, 但整体上误差较小, 最大误差为7.21%. 同时根据Wang等[17]中回头潮强度和入射涌潮强度的关系式
$$ \begin{split} & \xi _1^3{({\xi _0} + 1)^2} + 2\xi _1^2{({\xi _0} + 1)^2} - \\ &\qquad {\xi _1}(\xi _0^3 + 2\xi _0^2) - \xi _0^3 - 2\xi _0^2 = 0 \end{split}$$ (18) 式中, $ {\xi _0} = {d_2}/{d_1} - 1,{d_2} = {d_3} $, 为入射涌潮传播时的水深, 对应文中$ {h}_{1} $, $ {d}_{1} $为潮前水深, 对应文中$ {h}_{0} $; $ {\xi _1} = {d_4}/{d_3} - 1 $.
将文中公式计算的相关参数代入关系式(18), 发现部分工况有所差异, 但大体上吻合Wang等[17]中的关系式. 结合数值与Wang等[17]中关系式计算结果对比, 验证了理论回头潮计算公式(14)和式(15)的可靠性. 除此之外, 入射涌潮高度对回头潮传播速度的影响较小. 入射涌潮潮前水深对回头潮传播速度影响显著. 随着入射涌潮潮前水深的增大, 回头潮传播速度随之增大. 图16对比了计算得到的涌潮水平载荷($ {F}_{H} $)与理论水平载荷(式(16)). 由图16可知, 理论公式(16)计算的涌潮冲击载荷与数值模拟计算的涌潮冲击载荷吻合良好, 最大误差仅为1.78%. 涌潮冲击海塘的水动力载荷随着入射涌潮高度的增大以及涌潮潮前水深的增大而单调增大.
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图 16 涌潮作用在海塘上的水平载荷对比Figure 16. Comparison of horizontal load of tidal bore on seawall5. 结论
本研究针对垂直海塘处回头潮的水动力生成展开一系列高分辨率模拟, 深入分析了潮前水深与入射涌潮高度所产生的水动力影响. 主要研究结果如下.
(1)涌潮撞击海塘时, 水体沿海塘向上流动并产生回头潮, 回头潮向上游传播过程中, 其后水体近乎静止, 回头潮的水动力现象主要由潮前水深主导. 在给定入射涌潮高度的情况下, 存在临界潮前水深, 当潮前水深小于该临界值时, 回头潮仍呈典型漩滚状; 潮前水深大于临界值时, 为典型波状, 且随着回头潮继续向上游传播, 回头潮越来越弱.
(2)回头潮高度随潮前水深增加而减小, 随入射涌潮高度增加而增加; 垂直海塘的最大水动力载荷随入射涌潮高度和潮前水深增加而增加; 入射涌潮高度变化对回头潮传播速度影响甚微, 而回头潮传播速度随潮前水深增加呈单调递增.
(3)此外, 本文基于一维连续性方程、动量守恒方程和经典的伯努利方程推导了回头潮高度、回头潮传播速度以及冲击载荷的理论计算公式, 并与高精度数值仿真的结果及现有相关理论公式计算结果进行了系统对比, 验证了相应理论计算公式的准确性, 满足回头潮分析的计算要求.
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图 2 水位与流速的时程曲线对比
Figure 2. Comparisons of time series of water elevation and flow velocity
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图 4 不同时刻的涌潮流速云图($ {h}_{0} $ = 0.21 m)
Figure 4. Tidal bore flow velocity contours at different moments ($ {h}_{0} $ = 0.21 m)
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图 5 不同时刻的涌潮流速云图($ {h}_{0} $ = 0.33 m)
Figure 5. Tidal bore flow velocity contours at different moments ($ {h}_{0} $ = 0.33 m)
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图 7 垂直海塘处的涌潮上升高度
Figure 7. The rising height of the tidal bore at the vertical seawall
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图 8 涌潮在垂直海塘上冲击载荷的时程变化
Figure 8. Time series of the impacting load at the vertical seawall exerted by incident tidal bore
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图 9 不同涌潮高度下的流速云图
Figure 9. Contours of the flow velocity under different tidal bore heights
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图 10 $ {c}_{2} $, $ {F}_{H} $和$ {H}_{1} $随涌潮高度的变化 (h0 = 0.33 m)
Figure 10. The variation of $ {c}_{2} $, $ {F}_{H} $ and $ {H}_{1} $ with the height of tidal bore (h0 = 0.33 m)
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图 11 不同潮前水深下的流速云图
Figure 11. Contours of the flow velocity under different initial water depths
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图 12 $ {c}_{2} $, $ {F}_{H} $和$ {H}_{1} $随涌潮高度的变化 (H0 = 0.26 m)
Figure 12. The variation of $ {c}_{2} $, $ {F}_{H} $ and $ {H}_{1} $ with initial water depths (H0 = 0.26 m)
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图 16 涌潮作用在海塘上的水平载荷对比
Figure 16. Comparison of horizontal load of tidal bore on seawall
表 1 工况设置
Table 1 Numerical setups
Condition $ {h}_{0} $/m $ {H}_{0} $/m $ {u}_{1} $/(m·$ {{\mathrm{s}}}^{-1} $) $ {F}_{r} $ 1 0.15 0.24 1.61 2.16 2 0.15 0.25 1.68 2.21 3 0.15 0.26 1.74 2.26 4 0.15 0.27 1.80 2.31 5 0.15 0.28 1.86 2.35 6 0.21 0.24 1.40 1.84 7 0.21 0.25 1.46 1.87 8 0.21 0.26 1.51 1.90 9 0.21 0.27 1.56 1.94 10 0.21 0.28 1.62 1.97 11 0.27 0.24 1.27 1.65 12 0.27 0.25 1.31 1.68 13 0.27 0.26 1.36 1.71 14 0.27 0.27 1.41 1.73 15 0.27 0.28 1.46 1.76 16 0.33 0.24 1.16 1.53 17 0.33 0.25 1.21 1.56 18 0.33 0.26 1.25 1.58 19 0.33 0.27 1.30 1.60 20 0.33 0.28 1.34 1.62 
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