N-S方程

作者: 越狱_29c6 | 来源:发表于2022-08-19 15:32 被阅读0次

变量说明

$\begin{array}{c} X=[x_1,x_2,x_3]^T=x_i=x_j^T\\ V=[v_1,v_2,v_3]^T=v_i=v_j^T \end{array}$

$V(r,\theta,z)=\left[ \begin{array} {c} v_r\\ v_\theta\\ v_z \end{array} \right]$

圆柱坐标系下相关变量及梯度

$\nabla p = \left[ \begin{array}{c} \frac{\partial p}{\partial r}\\ \frac{1}{r}\frac{\partial p}{\partial \theta}\\ \frac{\partial p}{\partial z}\\ \end{array} \right]$

N-S方程

$\rho\frac{dV}{dt}=\rho g-\nabla p + \mu\nabla^2V$

$\rho\big( \frac{\partial V}{\partial t} + (V\cdot \nabla)V \big)=\rho g-\nabla p + \mu\nabla^2V$

$\frac{\partial V}{\partial t} + (V\cdot \nabla)V = g-\frac{1}{\rho}\nabla p + \frac{\mu}{\rho}\nabla^2V$

$\frac{dV}{dt} = \frac{\partial v_i}{\partial t}+(V\cdot \nabla)V = \frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j}$

笛卡尔坐标系下的N-S方程

$\frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j}=g-\frac{1}{\rho}\frac{\partial p}{\partial x_i}+\frac{\mu}{\rho}\frac{\partial^2v_i}{\partial x^2_j}$

对称

有关 $\theta$ 的变量全都为0

V的定义

$V(r,\theta,z) =\left[ \begin{array} {c} v_r\\ v_\theta\\ v_z \end{array} \right] =\left[ \begin{array} {c} v_r\\ 0\\ v_z \end{array} \right]$

$\frac{\partial V}{\partial t}$ 的求解

$\frac{\partial V}{\partial t} =\left[ \begin{array} {c} \frac{\partial v_r}{\partial t}\\ \frac{\partial v_\theta}{\partial t}\\ \frac{\partial v_z}{\partial t} \end{array} \right] =\left[ \begin{array} {c} \frac{\partial v_r}{\partial t}\\ 0\\ \frac{\partial v_z}{\partial t} \end{array} \right] =\left[ \begin{array} {c} 0\\ 0\\ 0 \end{array} \right]$

$\nabla p$ 的求解

$\nabla p =\left[ \begin{array}{c} \frac{\partial p}{\partial r}\\ \frac{1}{r}\frac{\partial p}{\partial \theta}\\ \frac{\partial p}{\partial z}\\ \end{array} \right] =\left[ \begin{array}{c} \frac{\partial p}{\partial r}\\ 0\\ \frac{\partial p}{\partial z}\\ \end{array} \right]$

$\nabla V$ 的求解

$\nabla V =\left[ \begin{array} {c} \frac{\partial v_r}{\partial r} & \frac{1}{r}\big(\frac{\partial v_r}{\partial \theta}-v_\theta \big) & \frac{\partial v_r}{\partial z} \\ \frac{\partial v_\theta}{\partial r} & \frac{1}{r}\big(\frac{\partial v_\theta}{\partial \theta}+v_\theta \big) & \frac{\partial v_\theta}{\partial z} \\ \frac{\partial v_z}{\partial r} & \frac{1}{r} \frac{\partial v_z}{\partial \theta} & \frac{\partial v_z}{\partial z} \\ \end{array} \right] =\left[ \begin{array} {c} \frac{\partial v_r}{\partial r} & 0 & \frac{\partial v_r}{\partial z} \\ 0 & 0 & 0 \\ \frac{\partial v_z}{\partial r} & 0 & \frac{\partial v_z}{\partial z} \\ \end{array} \right]$

$\nabla^2 V$ 的求解

$\nabla^2 V =\left[ \begin{array} {c} \frac{\partial^2 v_r}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 v_r}{\partial \theta^2}+ \frac{\partial^2 v_r}{\partial z^2}+ \frac{1}{r}\frac{\partial v_r}{\partial r}- \frac{v_r}{r^2}- \frac{2}{r^2}\frac{\partial v_\theta}{\partial \theta} \\ \frac{\partial^2 v_\theta}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 v_\theta}{\partial \theta^2} + \frac{1}{r}\frac{\partial v_\theta}{\partial r}+ \frac{2}{r^2}\frac{\partial v_r}{\partial \theta} - \frac{v_\theta}{r^2} \\ \frac{\partial^2 v_z}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 v_z}{\partial \theta^2} + \frac{1}{r}\frac{\partial v_z}{\partial r} + \frac{\partial^2 v_z}{\partial z^2} \\ \end{array} \right]\\ =\left[ \begin{array} {c} \frac{\partial^2 v_r}{\partial r^2} + 0 + \frac{\partial^2 v_r}{\partial z^2}+ \frac{1}{r}\frac{\partial v_r}{\partial r}- \frac{v_r}{r^2}- 0 \\ 0 + 0 + 0+ 0 - 0 \\ \frac{\partial^2 v_z}{\partial r^2} + 0 + \frac{1}{r}\frac{\partial v_z}{\partial r} + \frac{\partial^2 v_z}{\partial z^2} \\ \end{array} \right]\\ =\left[ \begin{array} {c} \frac{\partial^2 v_r}{\partial r^2} + \frac{\partial^2 v_r}{\partial z^2}+ \frac{1}{r}\frac{\partial v_r}{\partial r}- \frac{v_r}{r^2} \\ 0 \\ \frac{\partial^2 v_z}{\partial r^2} + \frac{1}{r}\frac{\partial v_z}{\partial r} + \frac{\partial^2 v_z}{\partial z^2} \\ \end{array} \right]$

代入公式

原始公式
$\frac{\partial V}{\partial t} + (V\cdot \nabla)V = g-\frac{1}{\rho}\nabla p + \frac{\mu}{\rho}\nabla^2V$
由于定长，忽略质量力，公式简化为
$(V\cdot \nabla)V = -\frac{1}{\rho}\nabla p + \frac{\mu}{\rho}\nabla^2V$
代入公式，即
$\left[ \begin{array} {c} v_r\frac{\partial v_r}{\partial r} + v_z\frac{\partial v_r}{\partial z} \\ 0 \\ v_r\frac{\partial v_z}{\partial r} + v_z\frac{\partial v_z}{\partial z} \\ \end{array} \right] =-\frac{1}{\rho} \left[ \begin{array}{c} \frac{\partial p}{\partial r}\\ 0\\ \frac{\partial p}{\partial z}\\ \end{array} \right]+ \frac{\mu}{\rho} \left[ \begin{array} {c} \frac{\partial^2 v_r}{\partial r^2} + \frac{\partial^2 v_r}{\partial z^2}+ \frac{1}{r}\frac{\partial v_r}{\partial r}- \frac{v_r}{r^2} \\ 0 \\ \frac{\partial^2 v_z}{\partial r^2} + \frac{1}{r}\frac{\partial v_z}{\partial r} + \frac{\partial^2 v_z}{\partial z^2} \\ \end{array} \right]$

$v_r = \overline {v_r}+v'_r\\ v_z = \overline {v_z}+v'_z\\ p = \overline {p}+p'$

化简

step1

step2

$v_r\frac{\partial v_z}{\partial r} + v_z\frac{\partial v_z}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial z}+ \frac{\mu}{\rho}(\frac{\partial^2 v_z}{\partial r^2} + \frac{1}{r}\frac{\partial v_z}{\partial r} + \frac{\partial^2 v_z}{\partial z^2})\\ (\overline {v_r}+v'_r)\frac{\partial (\overline {v_z}+v'_z)}{\partial r} + (\overline {v_z}+v'_z)\frac{\partial (\overline {v_z}+v'_z)}{\partial z} = -\frac{1}{\rho}\frac{\partial (\overline {p}+p')}{\partial z}+ \frac{\mu}{\rho}(\frac{\partial^2 (\overline {v_z}+v'_z)}{\partial r^2} + \frac{1}{r}\frac{\partial (\overline {v_z}+v'_z)}{\partial r} + \frac{\partial^2 (\overline {v_z}+v'_z)}{\partial z^2})$

$\overline {v_r}\frac{\partial \overline {v_z}}{\partial r}+ \overline {v_z}\frac{\partial \overline {v_z}}{\partial z}+ \overline{v_r'\frac{\partial v_z'}{\partial r}}+ \overline{v_z'\frac{\partial v_z'}{\partial z}}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial z}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_z}}{\partial r^2} + \frac{1}{r}\frac{\partial \overline {v_z}}{\partial r} + \frac{\partial^2 \overline {v_z}}{\partial z^2})$

方程
公式1
$\overline {v_r}\frac{\partial \overline {v_r}}{\partial r}+ \overline {v_z}\frac{\partial \overline {v_r}}{\partial z}+ \overline{v_r'\frac{\partial v_r'}{\partial r}}+ \overline{v_z'\frac{\partial v_r'}{\partial z}}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_r}}{\partial r^2} + \frac{\partial^2 \overline {v_r}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {v_r}}{\partial r}- \frac{\overline {v_r}}{r^2})\\ \frac{\partial \overline {v_r^2}}{\partial r}+ \overline {v_r}(\frac{\partial \overline{v_z}}{\partial z}+\frac{\overline{v_r}}{r})+ \overline {v_z}\frac{\partial \overline {v_r}}{\partial z}+ \overline{v_r'\frac{\partial v_r'}{\partial r}}+ \overline{\frac{\partial v_r'v_z'}{\partial z}}- \overline{v_r'\frac{\partial v_z'}{\partial z}} = -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_r}}{\partial r^2} + \frac{\partial^2 \overline {v_r}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {v_r}}{\partial r}- \frac{\overline {v_r}}{r^2})$

$\frac{\partial \overline {v_r^2}}{\partial r}+ \frac{\overline{v_r^2}}{r}+ \frac{\partial \overline {v_rv_z}}{\partial z}+ \overline{v_r'\frac{\partial v_r'}{\partial r}}+ \overline{\frac{\partial v_r'v_z'}{\partial z}}- \overline{v_r'\frac{\partial v_z'}{\partial z}} = -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_r}}{\partial r^2} + \frac{\partial^2 \overline {v_r}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {v_r}}{\partial r}- \frac{\overline {v_r}}{r^2})$

隔断

公式2
$\overline {v_r}\frac{\partial \overline {v_z}}{\partial r}+ \overline {v_z}\frac{\partial \overline {v_z}}{\partial z}+ \overline{v_r'\frac{\partial v_z'}{\partial r}}+ \overline{v_z'\frac{\partial v_z'}{\partial z}}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial z}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_z}}{\partial r^2} + \frac{1}{r}\frac{\partial \overline {v_z}}{\partial r} + \frac{\partial^2 \overline {v_z}}{\partial z^2})$

公式1+公式2

$\overline {v_r}\frac{\partial \overline {v_r}}{\partial r}+ \overline {v_z}\frac{\partial \overline {v_r}}{\partial z}+ \overline{v_r'\frac{\partial v_r'}{\partial r}}+ \overline {v_r}\frac{\partial \overline {v_z}}{\partial r}+ \overline {v_z}\frac{\partial \overline {v_z}}{\partial z}+ \overline{v_r'\frac{\partial v_z'}{\partial r}}+ \overline{v_z'\frac{\partial v_z'}{\partial z}}+ \overline{v_z'\frac{\partial v_r'}{\partial z}}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_r}}{\partial r^2} + \frac{\partial^2 \overline {v_r}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {v_r}}{\partial r}- \frac{\overline {v_r}}{r^2})+ -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial z}+ \frac{\mu}{\rho}(\frac{\partial^2 \overline {v_z}}{\partial r^2} + \frac{1}{r}\frac{\partial \overline {v_z}}{\partial r} + \frac{\partial^2 \overline {v_z}}{\partial z^2})$

$\overline {v_r}\frac{\partial \overline {v_r}}{\partial r}+ \overline {v_r}\frac{\partial \overline {v_z}}{\partial r}+ \overline {v_z}\frac{\partial \overline {v_r}}{\partial z}+ \overline {v_z}\frac{\partial \overline {v_z}}{\partial z}+ \overline{v_r'\frac{\partial v_r'}{\partial r}}+ \overline{v_r'\frac{\partial v_z'}{\partial r}}+ \overline{v_z'\frac{\partial v_z'}{\partial z}}+ \overline{v_z'\frac{\partial v_r'}{\partial z}}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r} -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial z}+ \frac{\mu}{\rho}\big( \frac{\partial^2 \overline {v_r}}{\partial r^2}+ \frac{\partial^2 \overline {v_z}}{\partial r^2}+ \frac{\partial^2 \overline {v_r}}{\partial z^2}+ \frac{\partial^2 \overline {v_z}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {v_r}}{\partial r} + \frac{1}{r}\frac{\partial \overline {v_z}}{\partial r} + \frac{\overline {v_r}}{r^2} \big)$

$\overline {v_r}(\frac{\partial \overline {v_r}}{\partial r}+\frac{\partial \overline {v_z}}{\partial r})+ \overline {v_z}(\frac{\partial \overline {v_r}}{\partial z}+ \frac{\partial \overline {v_z}}{\partial z})+ \overline{v_r'(\frac{\partial v_r'}{\partial r}+\frac{\partial v_z'}{\partial r})}+ \overline{v_z'(\frac{\partial v_z'}{\partial z}+\frac{\partial v_r'}{\partial z})}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r} -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial z}+ \frac{\mu}{\rho}\big( \frac{\partial^2 \overline {v_r}}{\partial r^2}+ \frac{\partial^2 \overline {v_z}}{\partial r^2}+ \frac{\partial^2 \overline {v_r}}{\partial z^2}+ \frac{\partial^2 \overline {v_z}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {v_r}}{\partial r} + \frac{1}{r}\frac{\partial \overline {v_z}}{\partial r} + \frac{\overline {v_r}}{r^2} \big)$

$\overline {v_r}\frac{\partial \overline {(v_r+v_z)}}{\partial r}+ \overline {v_z}\frac{\partial \overline {(v_r+v_z)}}{\partial z}+ \overline{v_r'\frac{\partial (v_r'+v_z')}{\partial r}}+ \overline{v_z'\frac{\partial (v_r'+v_z')}{\partial z}}= -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r} -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial z}+ \frac{\mu}{\rho}\big( \frac{\partial^2 \overline {(v_r+v_z)}}{\partial r^2}+ \frac{\partial^2 \overline {(v_r+v_z)}}{\partial z^2}+ \frac{1}{r}\frac{\partial \overline {(v_r+v_z)}}{\partial r} + \frac{\overline {v_r}}{r^2} \big)$

方程

$\frac{\partial v_r}{\partial r}+\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+\frac{\partial v_z}{\partial z}+\frac{v_r}{r}=0\\ \frac{\partial \overline{v_r}}{\partial r}+ \frac{\partial v_r'}{\partial r}+ \frac{\partial \overline{v_z}}{\partial z}+ \frac{\partial v_z'}{\partial z}+ \frac{\overline{v_r}}{r}+ \frac{v_r'}{r}=0\\ \frac{\partial \overline{v_r}}{\partial r}+ \frac{\partial \overline{v_z}}{\partial z}+ \frac{\overline{v_r}}{r}=0$

delete

step1

$v_r\frac{\partial v_r}{\partial r} + v_z\frac{\partial v_r}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial r}+ \frac{\mu}{\rho}(\frac{\partial^2 v_r}{\partial r^2} + \frac{\partial^2 v_r}{\partial z^2}+ \frac{1}{r}\frac{\partial v_r}{\partial r}- \frac{v_r}{r^2})\\ (\overline {v_r}+v'_r)\frac{\partial (\overline {v_r}+v'_r)}{\partial r} + (\overline {v_z}+v'_z)\frac{\partial (\overline {v_r}+v'_r)}{\partial z} = -\frac{1}{\rho}\frac{\partial (\overline {p}+p')}{\partial r}+ \frac{\mu}{\rho}(\frac{\partial^2 (\overline {v_r}+v'_r)}{\partial r^2} + \frac{\partial^2 (\overline {v_r}+v'_r)}{\partial z^2}+ \frac{1}{r}\frac{\partial (\overline {v_r}+v'_r)}{\partial r}- \frac{(\overline {v_r}+v'_r)}{r^2})\\ \overline {v_r}\frac{\partial \overline {v_r}}{\partial r} + \overline {v_r}\frac{\partial v'_r}{\partial r} + v'_r\frac{\partial \overline {v_r}}{\partial r} + v'_r\frac{\partial v'_r}{\partial r} + \overline {v_z}\frac{\partial \overline {v_r}}{\partial z} + \overline {v_z}\frac{\partial v'_r}{\partial z} + v'_z\frac{\partial \overline {v_r}}{\partial z} + v'_z\frac{\partial v'_r}{\partial z} \\ = -\frac{1}{\rho}\frac{\partial \overline {p}}{\partial r} -\frac{1}{\rho}\frac{\partial p'}{\partial r}+ \frac{\mu}{\rho}\big( \frac{\partial^2 (\overline {v_r}+v'_r)}{\partial r^2} + \frac{\partial^2 (\overline {v_r}+v'_r)}{\partial z^2}+ \frac{1}{r}\frac{\partial (\overline {v_r}+v'_r)}{\partial r}- \frac{(\overline {v_r}+v'_r)}{r^2} \big)$

N-S方程

变量说明

圆柱坐标系下相关变量及梯度

N-S方程

笛卡尔坐标系下的N-S方程

对称

V的定义

$\frac{\partial V}{\partial t}$ 的求解

$\nabla p$ 的求解

$\nabla V$ 的求解

$\nabla^2 V$ 的求解

代入公式

化简

step1

step2

隔断

delete

step1

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