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Numerical simulation of internal wave propagation in two-layer fluid under two water surface boundary conditions
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摘要: 基于FLUENT计算流体力学软件及其二次开发功能，采用VOF（Volume of Fluid）多相流模型，在
$k{\text{-}}\varepsilon$ 湍流模型下建立了模拟内波传播的分层数值水槽。设置两层稳定分层，以上下层不同密度差和水深比设置工况，利用平板拍击法造波。在刚盖和自由表面两种上边界条件下进行数值模拟并与各自的理论解进行比较，分析了两者之间的异同。研究发现密度差的改变不会明显影响理论解与数值解之间的一致程度；上下两层流体深度差值的改变会明显影响数值计算结果。上层水深很小时，在自由表面假定下水气交界面处出现了较为明显的垂向速度；在两种假定下，数值模拟的水平速度都体现了非线性的影响。而当下层水深很小时，非线性的影响微弱。鉴于在实际海洋中上层水深远小于下层水深，尤其是当计算运动幅值更大的内孤立波时，采用更为真实的自由表面假定更为合理。Abstract: Based on the FLUENT computational fluid dynamics software and its secondary development function, as well as the VOF (Volume of Fluid) multiphase flow model, a stratified numerical water flume that simulates the propagation of oceanic internal waves is established with the standard$k{\text{-}}\varepsilon$ turbulence model. In the numerical water flume, two-layer density-stratified fluid is set up, and the flapping plate method is used as the wave maker. Different combinations of density differences and water depth ratio between the upper and lower fluid are numerically simulated under two boundary assumptions of rigid lid and free surface at the still water level, and their numerical results are compared to their theoretical ones respectively. It is found that the differences of the densities of upper and lower fluid do not significantly influence the consistency between the numerical results and theoretical ones, and that the differences of the depths of upper and lower fluid significantly influence the numerical results. When the depth of the upper fluid is low, the notable vertical velocity appears at the interface between water and air under the assumption of free surface. Under the two assumptions at the still water level, the calculated horizontal velocities both reflect the nonlinear effect when the depth of the upper fluid is low, but hardly reflect the nonlinear effect when the depth of the lower fluid is low. Considering the fact that the depth of the upper fluid is much lower than that of the lower fluid in the actual ocean circumstance, it is more reasonable to adopt the assumption of free surface at the still water level, especially when the internal solitary waves with a larger amplitude are numerically simulated.-
Key words:
- internal wave /
- numerical simulation /
- free surface assumption /
- rigid lid assumption
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图 1 稳定层化两层流体中内波传播示意
Figure 1. Sketch of propagation of internal waves in density stratified two-layer fluid
图 3 (1 200, 0)处垂向速度随时间的变化过程线
Figure 3. Time series of vertical velocity at point (1 200, 0)
图 4 Case 1(750, 50)位置水平速度历时曲线
Figure 4. Time series of horizontal velocity at point (750, 50) for Case 1
图 5 Case 4(1 750, 50)位置水平速度历时曲线
Figure 5. Time series of horizontal velocity at point (1 750, 50) for Case 4
图 6 自由表面假定下波峰时水平速度沿水深的分布
Figure 6. Distribution of horizontal velocity along water depth at the moment for wave crest under the assumption of free surface
图 7 刚盖假定下波谷时水平速度沿水深的分布
Figure 7. Distribution of horizontal velocity along water depth at the moment for wave trough under the assumption of rigid lid
图 8 Case 5(1 200, 0)位置垂向速度历时曲线
Figure 8. Time series of vertical velocity at point (1 200, 0) for Case 5
图 9 Case 5(1 200, 10)位置垂向速度数值解历时曲线
Figure 9. Time series of numerical vertical velocity at point (1 200, 10) for Case 5
图 10 Case 5(1 200, 10)位置水平速度数值解历时曲线
Figure 10. Time series of numerical horizontal velocity at point (1 200, 10) for Case 5
图 11 Case 8(1 200, 0)位置垂向速度历时曲线
Figure 11. Time series of vertical velocity at point (1 200, 0) for Case 8
图 12 Case 8(1 200, 90)位置水平速度数值解历时曲线
Figure 12. Time series of numerical horizontal velocity at point (1 200, 90) for Case 8
图 13 波峰时垂向速度沿水深的分布图
Figure 13. Distribution of vertical velocity along water depth at the moment for wave crest
图 14 波峰时水平速度沿水深的分布图
Figure 14. Distribution of horizontal velocity along water depth at the moment for wave crest
表 1 上下层不同密度工况设置
Table 1. Cases with different densities for upper and lower layers
工况 $\;{\rho }_{1}\text{/}（\text{kg}\cdot {\text{m} }^{-3}）$ $\;{\rho }_{2}\text{/}（\text{kg}\cdot {\text{m} }^{-3}）$ $ {b_1}{\text{/m}} $ $ {b_2}{\text{/m}} $ $ k $ 测点坐标/m Case 1 1 020 1 022 50 50 0.028 7 (750, 0), (750, 50) Case 2 1 020 1 024 50 50 0.018 0 (1 200, 0), (1 200, 50) Case 3 1 020 1 026 50 50 0.014 1 (1 500, 0), (1 500, 50) Case 4 1 020 1 028 50 50 0.012 0 (1 750, 0), (1 750, 50) 
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表 2 上下层不同水深工况设置
Table 2. Cases with different water depths of upper and lower layers
工况 $\; {\rho }_{1}\text{/}（\text{kg}\cdot {\text{m} }^{-3}）$ $\; {\rho }_{2}\text{/}（\text{kg}\cdot {\text{m} }^{-3}）$ $ {b_1}{\text{/m}} $ $ {b_2}{\text{/m}} $ $ k $ 测点坐标 Case 5 1 020 1 028 10 90 0.019 8 (1 200, 0), (1 200, 10) Case 6 1 020 1 028 20 80 0.014 9 (1 500, 0), (1 500, 20) Case 7 1 020 1 028 80 20 0.014 9 (1 500, 0), (1 500, 80) Case 8 1 020 1 028 90 10 0.019 8 (1 200, 0), (1 200, 90)
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表 3 各工况水平和垂向速度的数值解和理论解一致性分析
Table 3. Consistency analysis between the numerical and analytical results for the horizontal and vertical velocities of different cases
d Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 $ d{\left(w峰\right)}_{\text{R}} $ 0.999 8 0.999 8 0.998 7 0.997 3 0.938 9 0.988 4 0.993 2 0.994 2 $ d{\left(w峰\right)}_{\text{F}} $ 0.996 4 0.996 0 0.998 9 0.998 2 0.927 5 0.984 2 0.997 0 0.996 6 $ d{\left(u峰\right)}_{\text{R}} $ 0.992 2 0.994 4 0.997 1 0.997 6 0.971 1 0.995 4 0.992 5 0.968 3 $ d{\left(u峰\right)}_{\text{F}} $ 0.994 1 0.996 0 0.997 1 0.997 0 0.962 8 0.996 2 0.994 4 0.979 8 $ d{\left(w谷\right)}_{\text{R}} $ 0.999 8 0.999 8 0.998 5 0.999 8 0.993 4 0.995 2 0.994 2 0.977 7 $ d{\left(w谷\right)}_{\text{F}} $ 0.994 7 0.995 9 0.997 3 0.998 9 0.979 0 0.997 4 0.992 5 0.978 0 $ d{\left(u谷\right)}_{\text{R}} $ 0.993 8 0.995 5 0.997 8 0.998 2 0.977 4 0.996 0 0.996 8 0.972 6 $ d{\left(u谷\right)}_{\text{F}} $ 0.995 1 0.997 1 0.998 0 0.998 3 0.971 3 0.995 8 0.996 8 0.987 5 注：R代表刚盖，F代表自由表面。
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