水深平均的二维浅水方程推导

最新推荐文章于 2024-09-10 10:09:41 发布

SCU张德帅

最新推荐文章于 2024-09-10 10:09:41 发布

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分类专栏： CFD 文章标签：流体力学

本文链接：https://blog.youkuaiyun.com/qq_40567160/article/details/118066416

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浅水方程推导

将三维的基本方程沿水深积分平均，即可得到沿水深平均的平面二维流动基本方程。

定义水深为 $H=\zeta-Z_{0}$ ， $\zeta$ 、 $Z_{0}$ 为基准面下液面水位和河床高程：

定义沿水深平均流速 $U_{i}$ 为：

$H=\zeta-Z_{0}U_{i}=\frac{1}{H} \int_{z_{0}}^{\zeta} \bar{u}_{i} d z$

引用莱布尼兹公式

$\frac{\partial}{\partial x_{i}} \int_{b}^{a} f d z=\int_{a}^{b} \frac{\partial f}{\partial x_{i}} d z+\left.f\right|_{b} \frac{\partial b}{\partial x_{i}}-\left.f\right|_{a} \frac{\partial a}{\partial x_{i}}$

自由表面及底部运动学条件

$\begin{array}{l} \left.\bar{u}_{z}\right|_{z=\zeta}=\frac{d \bar{\zeta}}{d t}=\frac{\partial \bar{\zeta}}{\partial t}+\left.\frac{\partial \bar{\zeta}}{\partial x} \bar{u}_{x}\right|_{z=\zeta}+\left.\frac{\partial \bar{\zeta}}{\partial y} \bar{u}_{y}\right|_{z=\zeta} \\ \left.\bar{u}_{z}\right|_{z=z_{0}}=\frac{d \overline{z_{0}}}{d t}=\frac{\partial \overline{z_{0}}}{\partial t}+\left.\frac{\partial \overline{z_{0}}}{\partial x} \overline{u_{x}}\right|_{z=z_{0}}+\left.\frac{\partial \overline{z_{0}}}{\partial y} \overline{u_{y}}\right|_{z=z_{0}} \end{array}$

以x方向为例三维流动的运动方程沿水深平均为

$\large \int_{z_{0}}^{\zeta}\left[\frac{\partial \overline{u_{x}}}{\partial t}+\frac{\partial}{\partial x}\left(\overline{u_{x}} \overline{u_{x}}\right)+\frac{\partial}{\partial y}\left(\bar{u}_{x} \overline{u_{y}}\right)+\frac{\partial}{\partial z}\left(\overline{u_{x}} \overline{u_{z}}\right)+\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x}-v_{t}\left(\frac{\partial^{2} \overline{u_{x}}}{\partial x^{2}}+\frac{\partial^{2} \overline{u_{y}}}{\partial y^{2}}+\frac{\partial^{2} \overline{u_{z}}}{\partial z^{2}}\right)\right] d z=0$

非恒定项积分

$\large \begin{array}{l} \int_{z_{0}}^{\zeta} \frac{\partial u_{x}}{\partial t} d z=\frac{\partial}{\partial t} \int_{z_{0}}^{\zeta} \overline{u_{x}} d z-\left.\frac{\partial \zeta}{\partial t} \overline{u_{x}}\right|_{z=\zeta}+\left.\frac{\partial z_{0}}{\partial t} u_{x}\right|_{z=z_{0}} \\ =\frac{\partial H U_{x}}{\partial t}-\left.\frac{\partial \bar{\zeta}}{\partial t} \overline{u_{x}}\right|_{z=\zeta}+\left.\frac{\partial \overline{z_{0}}}{\partial t} \overline{u_{x}}\right|_{z=z_{0}} \end{array}$

对流项积分

首先将时均流速分解为 $\large \bar{u}_{i}=U_{i}+\Delta \bar{u}_{i}$ ，式中 $\large U_{i}$ 为垂线平均流速， $\large \Delta \bar{u}_{i}$ 为时均流速 $\large \bar{u}_{i}$ 与垂线平均流速 $\large U_{i}$ 的差值。

$\large \begin{array}{l} \int_{z_{0}}^{\zeta} \overline{u_{x}} \bar{u}_{x} d z=\int_{z_{0}}^{\zeta}\left(U_{x}+\Delta \overline{u_{x}}\right)\left(U_{x}+\Delta \overline{u_{x}}\right) d z \\ =\int_{z_{0}}^{\zeta}\left(U_{x} U_{x}+\Delta \overline{u_{x}} \Delta \overline{u_{x}}+2 U_{x} \Delta \overline{u_{x}}\right) d z \\ =H U_{x} U_{x}+\int_{z_{0}}^{\zeta} \Delta \overline{u_{x}} \Delta \overline{u_{x}} d z=\beta_{x x} H U_{x} U_{x} \end{array}$

式中， $\large \beta_{x x}=1+\frac{\int_{z_{0}}^{\zeta} \Delta \overline{u_{x}} \Delta \overline{u_{x}} d z}{H U_{x} U_{x}}$ ，是由于流速沿垂线分布不均匀而引入的修正系数，类似于水力学中的动量修正系数，其数值一般在1.02—1.05，可以近似取1.0，因此

$\large \int_{z_{0}}^{\zeta} \frac{\partial \overline{u_{x}} \overline{u_{x}}}{\partial x} d z=\frac{\partial H U_{x} U_{x}}{\partial x}-\left.\frac{\partial \bar{\zeta}}{\partial x} \overline{u_{x}} \overline{u_{x}}\right|_{z=\zeta}+\left.\frac{\partial \overline{z_{0}}}{\partial x} \overline{u_{x}} \overline{u_{x}}\right|_{z=z_{0}}$

类似，可以得到

$\large \int_{z_{0}}^{\zeta} \frac{\partial \overline{u_{x}} \overline{u_{y}}}{\partial y} d z=\frac{\partial H U_{x} U_{y}}{\partial y}-\left.\frac{\partial \bar{\zeta}}{\partial y} \overline{u_{x}} \overline{u_{y}}\right|_{z=\zeta}+\left.\frac{\partial \overline{z_{0}}}{\partial y} \overline{u_{x}} \overline{u_{y}}\right|_{z=z_{0}}$

$\large \int_{z_{0}}^{\zeta} \frac{\partial \overline{u_{x}} \overline{u_{z}}}{\partial_{z}} d z=\left.\overline{u_{x}} \overline{u_{z}}\right|_{z=\zeta}-\left.\overline{u_{x}} \overline{u_{z}}\right|_{z=z_{0}}$

上几式相加，并利用底部及自由表面运动学条件可得

$\large \begin{array}{l} \int_{z_{0}}^{\zeta}\left[\frac{\partial \bar{u}_{x}}{\partial t}+\frac{\partial}{\partial x}\left(\bar{u}_{x} \bar{u}_{x}\right)+\frac{\partial}{\partial y}\left(\bar{u}_{x} \bar{u}_{y}\right)+\frac{\partial}{\partial z}\left(\bar{u}_{x} \bar{u}_{z}\right)\right] d z \\ =\frac{\partial H U_{x}}{\partial t}+\frac{\partial H U_{x} U_{x}}{\partial x}+\frac{\partial H U_{x} U_{y}}{\partial y} \end{array}$

压力项积分

$\large \int_{z_{0}}^{\zeta} \frac{\partial \bar{p}}{\partial x} d z=\frac{\partial}{\partial x} \int_{z_{0}}^{\zeta} \bar{p} d z-\left.\frac{\partial \bar{\zeta}}{\partial x} \bar{p}\right|_{z=\zeta}+\left.\frac{\partial \overline{z_{0}}}{\partial x} \bar{p}\right|_{z=z_{0}}$ （莱布尼茨公式）

将 $\large \bar{p}=\rho g(\bar{\zeta}-z)$ 代入上式后化简得：

$\large \int_{z_{0}}^{\zeta} \frac{\partial \bar{p}}{\partial x} d z=\rho g H \frac{\partial H}{\partial x}+\rho g H \frac{\partial \overline{z_{0}}}{\partial x}=\rho g H \frac{\partial \bar{\zeta}}{\partial x}$

扩散项积分

$\large \int_{z_{0}}^{\zeta}\left[v_{t}\left(\frac{\partial^{2} \bar{u}_{x}}{\partial x^{2}}+\frac{\partial^{2} \bar{u}_{y}}{\partial y^{2}}+\frac{\partial^{2} \bar{u}_{z}}{\partial z^{2}}\right)\right] d z =v_{t}\left(\frac{\partial^{2} H U_{x}}{\partial x^{2}}+\frac{\partial^{2} H U_{x}}{\partial y^{2}}\right)-g \frac{n^{2} U_{x} \sqrt{U_{x}^{2}+U_{y}^{2}}}{H^{1 / 3}}+C_{w} \frac{\rho_{a}}{\rho} \omega^{2} \cos \beta$

上式右边后两项分别为由底部创面阻力和表面风阻力引起的阻力项。式中， $\large C_{w}$ 为无因次风应力系数； $\large \rho_{a}$ 为空气密度； $\large \omega$ 为风速； $\large \beta$ 为风向与x方向的夹角。

最后运动方程写成张量形式为

$\large \frac{\partial H U_{i}}{\partial t}+\frac{\partial H U_{i} U_{j}}{\partial x_{j}}+g H \frac{\partial \varsigma}{\partial x_{i}}+g \frac{n^{2} U_{i} \sqrt{U_{j}^{2}}}{H^{1 / 3}}=v_{t} \frac{\partial^{2} H U_{i}}{\partial x_{j}^{2}}$