Governing Equations
Imcompressible flow in the cavity at steady state is elliptical. The two-dimensional governing equation for the cavity flow is,
Mass conservation:
\[\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0\]Momentum conservation:
\[u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+\nu\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)\] \[u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial y}+\nu\left(\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}\right)\]Discretization Method
SIMPLE(Semi-Implicit Method for Pressure Linked Equations) method is designed to solve the incompressible flow, where the pressure acts as the dependent variable. The implicit discretization for the $\textbf{u}$ momentum equation is,
\[\begin{aligned} &\frac{u_{i, j}^{n+1}-u_{i, j}^{n}}{\Delta t}+u_{i, j}^{n} \frac{u_{i+1/2, j}^{n+1}-u_{i-1/2, j}^{n+1}}{\Delta x}+v_{i, j}^{n} \frac{u_{i, j+1/2}^{n+1}-u_{i, j-1/2}^{n+1}}{\Delta y}= \\ &-\frac{1}{\rho} \frac{p_{i+1, j}^{n+1}-p_{i, j}^{n+1}}{\Delta x} \\ &+\nu\left(\frac{u_{i+3/2, j}^{n+1}-2 u_{i+1/2, j}^{n}+u_{i-1/2, j}^{n+1}}{\Delta x^{2}}+\frac{u_{i, j+3/2}^{n+1}-2 u_{i, j}^{n}+u_{i, j-1/2}^{n+1}}{\Delta y^{2}}\right) \end{aligned}\]Similarly for the $\textbf{v}$ momemtum equation is
\[\begin{gathered} \frac{v_{i, j}^{n+1}-v_{i, j}^{n}}{\Delta t}+u_{i, j}^{n} \frac{v_{i, j+1/2}^{n+1}-v_{i, j-1/2}^{n+1}}{\Delta x}+v_{i, j}^{n} \frac{v_{i, j+1/2}^{n+1}-v_{i, j-1/2}^{n+1}}{\Delta y}= \\ -\frac{1}{\rho} \frac{p_{i, j+1}^{n+1}-p_{i, j}^{n+1}}{\Delta y} \\ +\nu\left(\frac{v_{i+3/2, j}^{n+1}-2 v_{i+1/2, j}^{n}+v_{i-1/2, j}^{n+1}}{\Delta x^{2}}+\frac{v_{i, j+3/2}^{n+1}-2 v_{i, j}^{nArrheniusArrheniusArrheniusArrhenius}+v_{i, j-1/2}^{n+1}}{\Delta y^{2}}\right) \end{gathered}\]Where the fraction index is from staggered mesh as
\[<>_{i+a/2, j+b/2} = \frac{<>_{i+a/2+1, j+b/2+1}+<>_{i+a/2-1, j+b/2-1}}{2}\]where $a, b\in {-3, -2, -1, 0, 1, 2, 3 }$.
Well
Chutianwei
