ROUGH DRAFT authorea.com/109637
Main Data History
Export
Show Index Toggle 0 comments
  •  Quick Edit
  • Вывод уравнения вихря

    Уравнения движения и уравнение неразрывности: \[\begin{aligned} \overset{(1)}{ \frac{\partial u}{\partial t}} + \overset{(2)}{u\frac{\partial u}{\partial x}} + \overset{(3)}{v\frac{\partial u}{\partial y}} - \overset{(4)}{K\left(\frac{{\partial }^2u}{\partial x^2}+\frac{{\partial }^2u}{\partial y^2}\right)} &= 0 \\ %newline \frac{\partial v}{\partial t }+ u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}-K\left(\frac{{\partial }^2v}{\partial x^2}+\frac{{\partial }^2v}{\partial y^2}\right)&=0 \\ %newline \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &= 0\end{aligned}\]

    На основании уравнения неразрывности можно ввести потенциал \(\psi\), так что \[u=-\frac{\partial \psi}{\partial y}, v=\frac{\partial \psi}{\partial x}\].

    Применим к \(x\) уравнению операцию \(\partial / \partial y\), а к \(y\) уравнению - \(\partial / \partial y\), затем результаты вычтем один из другого. Это будет аналогично взятию \(z\)-компоненты ротора. Потому есть подозрение, что данная операция эквивалентна применению приближения мелкой воды и гидростатики к уравнению движения в форме Громеки-Лэмба

    \[\begin{aligned} (1)\; \frac{\partial}{\partial y}: \; \frac{\partial u}{\partial t} \to \frac{\partial}{\partial y} \left( \frac{\partial u}{\partial t}\right) &= \frac{\partial}{\partial y}\left[ \frac{\partial}{\partial t} \left( -\frac{\partial \psi}{\partial y} \right) \right] = - \frac{\partial^2 \psi}{\partial y^2} \\ %newline \frac{\partial}{\partial x}: \; \frac{\partial v}{\partial t} \to \frac{\partial}{\partial x} \left( \frac{\partial v}{\partial t}\right) &= \frac{\partial}{\partial x}\left[ \frac{\partial}{\partial t} \left( \frac{\partial \psi}{\partial x} \right) \right] = \frac{\partial^2 \psi}{\partial x^2}\end{aligned}\]

    \[\begin{aligned} (2)\; % d/dy \frac{\partial}{\partial y}: \; % u*u_x u \frac{\partial u}{\partial x} \to \frac{\partial}{\partial y} \left( u \frac{\partial u}{\partial x}\right) = \frac{\partial u}{\partial y} \cdot \frac{\partial u}{\partial x} + u \frac{\partial^2 u}{\partial x \partial y} &= \frac{\partial}{\partial y}\left[ \left( - \frac{\partial \psi}{\partial y} \right) \cdot \frac{\partial}{\partial x} \left( -\frac{\partial \psi}{\partial y} \right) \right] = \\ %newline &= \frac{\partial^2 \psi}{\partial y^2} \cdot \frac{\partial^2 \psi}{\partial x \partial y } + \frac{\partial \psi}{\partial y } \cdot \frac{\partial^3 \psi}{\partial x \partial y^2}\\ %newline % d/dx \frac{\partial}{\partial x}: \; % u*v_x u \frac{\partial v}{\partial x} \to \frac{\partial}{\partial x} \left( u \frac{\partial v}{\partial x}\right) = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial x} + u \frac{\partial^2 v}{\partial x^2} &= \frac{\partial}{\partial x}\left[ \left(- \frac{\partial \psi}{\partial y} \right) \cdot \frac{\partial}{\partial x} \left( \frac{\partial \psi}{\partial x} \right) \right] = \\ %newline &= - \frac{\partial^2 \psi}{\partial x \partial y} \cdot \frac{\partial^2 \psi}{\partial x^2 } - \frac{\partial \psi}{\partial y } \cdot \frac{\partial^3 \psi}{\partial x^3 }\\\end{aligned}\]

    \[\begin{aligned} (3)\; % d/dy \frac{\partial}{\partial y}: \; % v*u_y v \frac{\partial u}{\partial y} \to \frac{\partial}{\partial y} \left( v \frac{\partial u}{\partial y}\right) = \frac{\partial v}{\partial y} \cdot \frac{\partial u}{\partial y} + v \frac{\partial^2 u}{\partial y^2} &= \frac{\partial}{\partial y} \left[ \frac{\partial \psi}{\partial x} \cdot \frac{\partial}{\partial y} \left( -\frac{\partial \psi}{\partial y} \right) \right] = \\ %newline &=-\frac{\partial^2 \psi}{\partial x \partial y } \cdot \frac{\partial^2 \psi}{\partial y^2 } + -\frac{\partial \psi}{\partial x } \cdot \frac{\partial^3 \psi}{\partial y^3 }\\ %newline % d/dx \frac{\partial}{\partial x}: \; % v*v_y v \frac{\partial v}{\partial y} \to \frac{\partial}{\partial x} \left( v \frac{\partial v}{\partial y}\right) = \frac{\partial v}{\partial x} \cdot \frac{\partial v}{\partial y} + v \frac{\partial^2 v}{\partial x \partial y} &= \frac{\partial}{\partial x}\left[ \frac{\partial \psi}{\partial x} \cdot \frac{\partial}{\partial y} \left( \frac{\partial \psi}{\partial x} \right) \right] = \\ %newline &= \frac{\partial^2 \psi}{\partial x^2 } \cdot \frac{\partial^2 \psi}{\partial x \partial y } - \frac{\partial \psi}{\partial x } \cdot \frac{\partial^3 \psi}{\partial x^2 \partial y }\\\end{aligned}\]

    [Someone else is editing this]

    You are editing this file