對于二維不可壓流體，寫出速度壓力形式得連續性方程和運動方程；在如圖所示的交錯網格下，寫出 p(1,j) 滿足的差分格式

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重寫為 u_{i+1/2,j}^{n+1}=u_{i+1/2,j}^n-\frac{\Delta t}{4\Delta x}\left[u_{i+\frac{3}{2},j}^n\left(u_{i+\frac{3}{2},j}^n+2u_{i+\frac{1}{2},j}^n\right)-u_{i-\frac{1}{2},j}^n\left(u_{i-\frac{1}{2},j}^n+2u_{i+\frac{1}{2},j}^n\right)\right]-\frac{\Delta t}{4\Delta y}\left[\left(u_{i+\frac{1}{2},j}^n+u_{i+\frac{1}{2},j+1}^n\right)\left(v_{i,j+\frac{1}{2}}^n+v_{i+1,j+\frac{1}{2}}^n\right)-\left(u_{i+\frac{1}{2},j-1}^n+u_{i+\frac{1}{2},j}^n\right)\left(v_{i,j-\frac{1}{2}}^n+v_{i+1,j-\frac{1}{2}}^n\right)\right]-\frac{1}{\rho}\frac{\Delta t}{\Delta x}\left(p_{I+1,J}^n-p_{I,J}^n\right)+\frac{\mu\Delta t}{\left(\Delta x\right)^2}\left(u_{i+\frac{3}{2},j}^n-2u_{i+\frac{1}{2},j}^n+u_{i-\frac{1}{2},j}^n\right)+\frac{\mu\Delta t}{\left(\Delta y\right)^2}\left(u_{i+\frac{1}{2},j+1}^n-2u_{i+\frac{1}{2},j}^n+u_{i+\frac{1}{2},j-1}^n\right) v_{i,j+1/2}^{n+1}=v_{i,j+1/2}^n-\frac{\Delta t}{4\Delta x}\left[\left(u_{i+\frac{1}{2},j+1}^n+u_{i+\frac{1}{2},j}^n\right)\left(v_{i+1,j+\frac{1}{2}}^n+v_{i,j+\frac{1}{2}}^n\right)-\left(u_{i-\frac{1}{2},j+1}^n+u_{i-\frac{1}{2},j}^n\right)\left(v_{i,j+\frac{1}{2}}^n+v_{i-1,j-\frac{1}{2}}^n\right)\right]-\frac{\Delta t}{4\Delta y}\left[v_{i,j+\frac{3}{2}}^n\left(v_{i,j+\frac{3}{2}}^n+2v_{i,j+\frac{1}{2}}^n\right)-v_{i,j-\frac{1}{2}}^n\left(v_{i,j-\frac{1}{2}}^n+2v_{i,j+\frac{1}{2}}^n\right)\right]-\frac{1}{\rho}\frac{\Delta t}{\Delta y}\left(p_{I,J+1}^n-p_{I,J}^n\right)+\frac{\mu\Delta t}{\left(\Delta x\right)^2}\left(v_{i+1,j+\frac{1}{2}}^n-2v_{i,j+\frac{1}{2}}^n+v_{i-1,j+\frac{1}{2}}^n\right)+\frac{\mu\Delta t}{\left(\Delta y\right)^2}\left(v_{i,j+\frac{3}{2}}^n-2v_{i,j+\frac{1}{2}}^n+v_{i,j-\frac{1}{2}}^n\right) \frac{1}{\left(\Delta x\right)^2}\left(p_{i+1,j}^n-2p_{i,j}^n+p_{i-1,j}^n\right)+\frac{1}{\left(\Delta y\right)^2}\left(p_{i,j+1}^n-2p_{i,j}^n+p_{i,j-1}^n\right)=\left(S_p\right)_{i,j}^n \left(S_p\right)_{i,j}^n=\rho\left\{\frac{D_{i,j}^n}{\Delta t}-\frac{1}{{4\left(\Delta x\right)}^2}\left[\left(u_{i+\frac{3}{2},j}^n\right)^2-\left(u_{i+\frac{1}{2},j}^n\right)^2-\left(u_{i-\frac{1}{2},j}^n\right)^2+\left(u_{i-\frac{3}{2},j}^n\right)^2+2u_{i+\frac{1}{2},j}^n\left(u_{i+\frac{3}{2},j}^n-u_{i-\frac{1}{2},j}^n\right)+2u_{i-\frac{1}{2},j}^n\left(u_{i-\frac{3}{2},j}^n-u_{i+\frac{1}{2},j}^n\right)\right]-\frac{1}{2\Delta x\Delta y}\left[\left(u_{i+\frac{1}{2},j+1}^n+u_{i+\frac{1}{2},j}^n\right)\left(v_{i+1,j+\frac{1}{2}}^n-v_{i,j+\frac{1}{2}}^n\right)+\left(u_{i-\frac{1}{2},j}^n+u_{i-\frac{1}{2},j-1}^n\right)\left(v_{i,j-\frac{1}{2}}^n+v_{i-1,j-\frac{1}{2}}^n\right)\right]-\frac{1}{{4\left(\Delta y\right)}^2}\left[\left(v_{i,j+\frac{3}{2}}^n\right)^2-\left(v_{i,j+\frac{1}{2}}^n\right)^2-\left(v_{i,j-\frac{1}{2}}^n\right)^2+\left(v_{i,j-\frac{3}{2}}^n\right)^2+2v_{i,j+\frac{1}{2}}^n\left(v_{i,j+\frac{3}{2}}^n-v_{i,j-\frac{1}{2}}^n\right)+2v_{i,j-\frac{1}{2}}^n\left(v_{i,j-\frac{3}{2}}^n-v_{i,j+\frac{1}{2}}^n\right)\right]+\frac{\mu}{\left(\Delta x\right)^2}\left(D_{i+1,j}^n-2D_{i,j}^n+D_{i-1,j}^n\right)+\frac{\mu}{\left(\Delta y\right)^2}\left(D_{i,j+1}^n-2D_{i,j}^n+D_{i,j-1}^n\right)\right\}其中 D_{i,j}^n=\frac{1}{\Delta x}\left(u_{i+\frac{1}{2},j}^n-u_{i-\frac{1}{2},j}^n\right)+\frac{1}{\Delta y}\left(v_{i,j+\frac{1}{2}}^n-v_{i,j-\frac{1}{2}}^n\right)固壁條件為： u_{i-1/2,j}^n=v_{i-1/2,j}^n=0進而 (i-1,j)，(i-3/2,j) 等 x<=i-1/2 得點上的物理量的值均為零，代入得： u_{i+1/2,j}^{n+1}=u_{i+1/2,j}^n-\frac{\Delta t}{4\Delta x}\left[u_{i+\frac{3}{2},j}^n\left(u_{i+\frac{3}{2},j}^n+2u_{i+\frac{1}{2},j}^n\right)-0\right]-\frac{\Delta t}{4\Delta y}\left[\left(u_{i+\frac{1}{2},j}^n+u_{i+\frac{1}{2},j+1}^n\right)\left(v_{i,j+\frac{1}{2}}^n+v_{i+1,j+\frac{1}{2}}^n\right)-\left(u_{i+\frac{1}{2},j-1}^n+u_{i+\frac{1}{2},j}^n\right)\left(v_{i,j-\frac{1}{2}}^n+v_{i+1,j-\frac{1}{2}}^n\right)\right]-\frac{1}{\rho}\frac{\Delta t}{\Delta x}\left(p_{I+1,J}^n-p_{I,J}^n\right)+\frac{\mu\Delta t}{\left(\Delta x\right)^2}\left(u_{i+\frac{3}{2},j}^n-2u_{i+\frac{1}{2},j}^n+0\right)+\frac{\mu\Delta t}{\left(\Delta y\right)^2}\left(u_{i+\frac{1}{2},j+1}^n-2u_{i+\frac{1}{2},j}^n+u_{i+\frac{1}{2},j-1}^n\right) v_{i,j+1/2}^{n+1}=v_{i,j+1/2}^n-\frac{\Delta t}{4\Delta x}\left[\left(u_{i+\frac{1}{2},j+1}^n+u_{i+\frac{1}{2},j}^n\right)\left(v_{i+1,j+\frac{1}{2}}^n+v_{i,j+\frac{1}{2}}^n\right)-0\right]-\frac{\Delta t}{4\Delta y}\left[v_{i,j+\frac{3}{2}}^n\left(v_{i,j+\frac{3}{2}}^n+2v_{i,j+\frac{1}{2}}^n\right)-v_{i,j-\frac{1}{2}}^n\left(v_{i,j-\frac{1}{2}}^n+2v_{i,j+\frac{1}{2}}^n\right)\right]-\frac{1}{\rho}\frac{\Delta t}{\Delta y}\left(p_{I,J+1}^n-p_{I,J}^n\right)+\frac{\mu\Delta t}{\left(\Delta x\right)^2}\left(v_{i+1,j+\frac{1}{2}}^n-2v_{i,j+\frac{1}{2}}^n+0\right)+\frac{\mu\Delta t}{\left(\Delta y\right)^2}\left(v_{i,j+\frac{3}{2}}^n-2v_{i,j+\frac{1}{2}}^n+v_{i,j-\frac{1}{2}}^n\right) \frac{1}{\left(\Delta x\right)^2}\left(p_{i+1,j}^n-2p_{i,j}^n+0\right)+\frac{1}{\left(\Delta y\right)^2}\left(p_{i,j+1}^n-2p_{i,j}^n+p_{i,j-1}^n\right)=\left(S_p\right)_{i,j}^n \left(S_p\right)_{i,j}^n=\rho\left\{\frac{D_{i,j}^n}{\Delta t}-\frac{1}{{4\left(\Delta x\right)}^2}\left[\left(u_{i+\frac{3}{2},j}^n\right)^2-\left(u_{i+\frac{1}{2},j}^n\right)^2-0+0+2u_{i+\frac{1}{2},j}^n\left(u_{i+\frac{3}{2},j}^n-0\right)+0\right]-\frac{1}{2\Delta x\Delta y}\left[\left(u_{i+\frac{1}{2},j+1}^n+u_{i+\frac{1}{2},j}^n\right)\left(v_{i+1,j+\frac{1}{2}}^n-v_{i,j+\frac{1}{2}}^n\right)+0\right]-\frac{1}{{4\left(\Delta y\right)}^2}\left[\left(v_{i,j+\frac{3}{2}}^n\right)^2-\left(v_{i,j+\frac{1}{2}}^n\right)^2-\left(v_{i,j-\frac{1}{2}}^n\right)^2+\left(v_{i,j-\frac{3}{2}}^n\right)^2+2v_{i,j+\frac{1}{2}}^n\left(v_{i,j+\frac{3}{2}}^n-v_{i,j-\frac{1}{2}}^n\right)+2v_{i,j-\frac{1}{2}}^n\left(v_{i,j-\frac{3}{2}}^n-v_{i,j+\frac{1}{2}}^n\right)\right]+\frac{\mu}{\left(\Delta x\right)^2}\left(D_{i+1,j}^n-2D_{i,j}^n+D_{i-1,j}^n\right)+\frac{\mu}{\left(\Delta y\right)^2}\left(D_{i,j+1}^n-2D_{i,j}^n+D_{i,j-1}^n\right)\right\}其中 D_{i,j}^n=\frac{1}{\Delta x}\left(u_{i+\frac{1}{2},j}^n-0\right)+\frac{1}{\Delta y}\left(v_{i,j+\frac{1}{2}}^n-v_{i,j-\frac{1}{2}}^n\right)

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