學習自F. Moukalled, L. Mangani, M. Darwish所著The Finite Volume Method in Computational Fluid Dynamics - An Advanced Introduction with OpenFOAM and Matlab html
前面幾章講解的關於變量
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因爲本章內容繁雜，篇幅較長，故分紅了四部分來說解，各部分主要內容分別爲：交錯網格、同位網格、邊界條件、SIMPLE家族算法。redis
這裏是第四部分，主要講解在SIMPLE算法基礎上衍生出的其餘SIMPLE家族算法、最佳欠鬆弛因子、Rhie-Chow插值對不一樣項的處理、代碼講解。算法
在SIMPLE算法中，速度和壓力是用分離式（順序式）方式來處理的，壓力場是經過壓力修正方程算出的，而壓力修正方程的推導則是利用離散的動量方程，將連續方程中的速度場替換成壓力項來獲得的（即，從離散動量方程推導出速度和壓力的關係，而後把該關係代入到連續方程中，獲得用壓力項而非速度項表示的連續方程，即壓力修正方程）。在推導過程當中，速度修正項，
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H f [ v ′ ] 架構
拋掉該項並不會影響到最終的解，由於該項在收斂的時候值爲零。只是，這麼作影響了收斂的過程，由於在最初的迭代過程當中該項的值是至關大的。這個很大的值可能會致使發散或是減緩收斂速度，由於高估了壓力修正場。爲了平衡該做用，在SIMPLE算法中，壓力要作欠鬆弛處理，即
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儘管使用了欠鬆弛處理，SIMPLE算法的收斂速率仍然依據問題的不一樣而不一樣，所以學者們尋求各類替代方法來作更好的改進。他們的努力最終發展了一系列SIMPLE類型的家族算法，例如SIMPLEC、SIMPLER、PISO、SIMPLEX、PRIME、SIMPLEM和SIMPLEST。Moukalled和Darwish統一了針對不可壓縮和可壓縮流動的這些算法的公式，而Darwish等人和Jang等人則評估了它們的性能。這裏並不打算給出這些算法的完整描述，而是，聚焦在兩種最經常使用的算法上，它們是Van Doormal和Raithby的SIMPLEC（SIMPLE Consistent）算法，以及Issa的PISO（Pressure-Implicit Split Operator）算法。這兩種算法展現了對於
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在PRIME算法中，動量方程是顯式求解的。對動量方程的該顯式處理是合適的，由於其對總體流場的收斂貢獻很小。另外一方面，找尋壓力場的正確解則是影響整場收斂的重要因素。ide
在SIMPLEST算法中，動量方程的係數分紅對流部分和擴散部分，對流部分顯式處理，擴散部分隱式處理，這樣影響着
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在SIMPLEM（SIMPLE-Modified）算法中，壓力修正方程是先於動量方程求解的，代表壓力場是使用舊的速度場求解的。這樣會比速度修正產生更好的壓力修正，相較於SIMPLE算法，二者的優勢和缺點是互換的。性能
在SIMPLER（SIMPLE-Revised）算法中，提出了一個額外的方程，從中可直接解出壓力，而相似SIMPLE的壓力修正方程則用於更新速度場。採用分離的壓力方程的緣由是，一旦速度場使用預測壓力修正場更新後，其再也不知足動量方程。所以，壓力須要從另外一個知足速度的方程來求解，這樣可同時知足動量方程。
SIMPLEX算法的發展目的是確保收斂速度不會隨着網格的加密而衰減，其與SIMPLE算法的不一樣是
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儘管全部上述算法最初是用於交錯網格的，然而他們在同位網格框架下依然適用。
SIMPLEC（SIMPLE-Consistent）算法是SIMPLE算法的修改變型，其改動在於，簡單地假設在點
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\bold v'_C\approx\frac{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v \bold v'_F}{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v} \Rightarrow \displaystyle\sum_{F\sim NB(C)}a_F^\bold v \bold v'_F \approx \bold v'_C \displaystyle\sum_{F\sim NB(C)}a_F^\bold v
v C ′ ≈ F ∼ N B ( C ) ∑ a F v F ∼ N B ( C ) ∑ a F v v F ′ ⇒ F ∼ N B ( C ) ∑ a F v v F ′ ≈ v C ′ F ∼ N B ( C ) ∑ a F v
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\sum_{F\sim NB(C)}\frac{a_F^\bold v}{a_C^\bold v} \bold v'_F \approx \bold v'_C \sum_{F\sim NB(C)}\frac{a_F^\bold v}{a_C^\bold v} \Rightarrow \bold H_C[\bold v'] \approx \bold v'_C \bold H_C[1]
F ∼ N B ( C ) ∑ a C v a F v v F ′ ≈ v C ′ F ∼ N B ( C ) ∑ a C v a F v ⇒ H C [ v ′ ] ≈ v C ′ H C [ 1 ]
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\bold v_C'+\bold H_C[\bold v'] = -\bold D_C^{\bold v}(\nabla p')_C
v C ′ + H C [ v ′ ] = − D C v ( ∇ p ′ ) C
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(1+\bold H_C[1])\bold v'_C=-\bold D_C^\bold v(\nabla p')_C \Rightarrow \bold v'_C=-\widetilde{\bold D_C^\bold v}(\nabla p')_C
( 1 + H C [ 1 ] ) v C ′ = − D C v ( ∇ p ′ ) C ⇒ v C ′ = − D C v
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經過在動量方程中添加和減去
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\left(a_C^\bold v+\sum_{F\sim NB(C)}a_F^\bold v \right)\bold v_C+ \sum_{F\sim NB(C)}a_F^\bold v(\bold v_F+\bold v_C)= -\bold v_C(\nabla p)_C+\hat{\bold b}_C^\bold v
⎝ ⎛ a C v + F ∼ N B ( C ) ∑ a F v ⎠ ⎞ v C + F ∼ N B ( C ) ∑ a F v ( v F + v C ) = − v C ( ∇ p ) C + b ^ C v
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\bold v_C + \widetilde{\bold H_C}[\bold v - \bold v_C]=-\widetilde{\bold D_C^\bold v}(\nabla p)_C+\hat{\bold B}_C^\bold v
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\bold v'_C = -\widetilde{\bold H_C}[\bold v' - \bold v'_C]-\widetilde{\bold D_C^\bold v}(\nabla p')_C
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在SIMPLEC算法中，得益於更好的近似估算（即，忽略的項更小），與SIMPLE算法相比，壓力的鬆弛處理再也不須要了，所得速度也更加符合動量方程。因而乎，可獲得更快的收斂速度。如此看來，忽略掉
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在PRIME（Pressure Implicit Momentum Explicit）算法中，動量方程是顯式求解的，該顯式處理是恰當的，由於在迭代求解過程當中動量方程對於總體流場的收斂貢獻較小，而另外一方面，壓力場的正確解則對總體收斂的貢獻相當重要。基於該認知，將PRIME算法彙總以下：
動量方程顯式求解，得到新的速度場
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\bold v_C^*=-\bold H_C[\bold v^{(n)}]-\bold D_C^\bold v(\nabla p^{(n)})_C+\bold B_C^\bold v
v C ∗ = − H C [ v ( n ) ] − D C v ( ∇ p ( n ) ) C + B C v
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\bold v_C^{**}=\bold v_C^* + \bold v_C' \quad\quad p_C^*=p_C^{(n)}+p_C'
v C ∗ ∗ = v C ∗ + v C ′ p C ∗ = p C ( n ) + p C ′
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\begin{aligned} \bold v_C^{**} &= -\bold H_C[\bold v^{**}]-\bold D_C^\bold v(\nabla p^{*})_C+\bold B_C^\bold v \\ &= -\bold H_C[\bold v^* + \bold v']-\bold D_C^\bold v[\nabla (p^{(n)}+p')]_C+\bold B_C^\bold v \end{aligned}
v C ∗ ∗ = − H C [ v ∗ ∗ ] − D C v ( ∇ p ∗ ) C + B C v = − H C [ v ∗ + v ′ ] − D C v [ ∇ ( p ( n ) + p ′ ) ] C + B C v
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\bold v_C'=\underline{-\left(\bold H_C[\bold v^*- \bold v^{(n)}]+ \bold H_C[\bold v']\right)} -\bold D_C^\bold v\nabla p'_C
v C ′ = − ( H C [ v ∗ − v ( n ) ] + H C [ v ′ ] ) − D C v ∇ p C ′
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-\sum_{f\sim nb(C)}\rho_f\overline{\bold D_f}\nabla p_f'\cdot\bold S_f= -\sum_{f\sim nb(C)}\dot m_f^*+\underline{\sum_{f\sim nb(C)}\left[ \rho_f\left(\overline{\bold H_f}\left[\bold v^* - \bold v^{(n)}\right]+\overline{\bold H_f}[\bold v']\right)\cdot \bold S_f\right]}
− f ∼ n b ( C ) ∑ ρ f D f ∇ p f ′ ⋅ S f = − f ∼ n b ( C ) ∑ m ˙ f ∗ + f ∼ n b ( C ) ∑ [ ρ f ( H f [ v ∗ − v ( n ) ] + H f [ v ′ ] ) ⋅ S f ]
若是
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在PISO算法中，修正過程由兩步或者更多的步驟構成，而
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\begin{aligned} \bold v_C^{****} &= \bold v_C^{***} + \bold v''_C \\ &= -\bold H_C^{**}[\bold v^{**}]-(\bold D_C^\bold v)^{**}(\nabla p^{*})_C+\bold v''_C \\ &= -\bold H_C^{**}[\bold v^{*}+\bold v']-(\bold D_C^\bold v)^{**}(\nabla p^{*})_C+\bold v''_C \\ &= -\bold H_C^{**}[\bold v^{*}]-\bold H_C^{**}[\bold v'] -(\bold D_C^\bold v)^{**}(\nabla p^{*})_C+\bold v''_C \\ &= \underbrace{-\bold H_C^{**}[\bold v^{*}]-(\bold D_C^\bold v)^{**}(\nabla p^{*})_C}_{\approx\bold v_C^{**}} -\bold H_C^{**}[-\bold D_C^\bold v(\nabla p')_C]+\bold v''_C \\ &\approx \bold v_C^{**}+\underline{\bold v''_C -\bold H_C^{**}[-\bold D_C^\bold v(\nabla p')_C]} \end{aligned}
v C ∗ ∗ ∗ ∗ = v C ∗ ∗ ∗ + v C ′ ′ = − H C ∗ ∗ [ v ∗ ∗ ] − ( D C v ) ∗ ∗ ( ∇ p ∗ ) C + v C ′ ′ = − H C ∗ ∗ [ v ∗ + v ′ ] − ( D C v ) ∗ ∗ ( ∇ p ∗ ) C + v C ′ ′ = − H C ∗ ∗ [ v ∗ ] − H C ∗ ∗ [ v ′ ] − ( D C v ) ∗ ∗ ( ∇ p ∗ ) C + v C ′ ′ = ≈ v C ∗ ∗
− H C ∗ ∗ [ v ∗ ] − ( D C v ) ∗ ∗ ( ∇ p ∗ ) C − H C ∗ ∗ [ − D C v ( ∇ p ′ ) C ] + v C ′ ′ ≈ v C ∗ ∗ + v C ′ ′ − H C ∗ ∗ [ − D C v ( ∇ p ′ ) C ]
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\bold v_C''=-\underline{\bold H_C^{**}[\bold v'']}-(\bold D_C^\bold v)^{**}(\nabla p'')_C
v C ′ ′ = − H C ∗ ∗ [ v ′ ′ ] − ( D C v ) ∗ ∗ ( ∇ p ′ ′ ) C
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-\sum_{f\sim nb(C)}\rho_f\overline{\bold D_f}\nabla p''_f\cdot \bold S_f = -\sum_{f\sim nb(C)}\dot m_f^* + \underline{\sum_{f\sim nb(C)}\left(\rho_f\overline{\bold H_f}[\bold v'']\cdot \bold S_f\right)}
− f ∼ n b ( C ) ∑ ρ f D f ∇ p f ′ ′ ⋅ S f = − f ∼ n b ( C ) ∑ m ˙ f ∗ + f ∼ n b ( C ) ∑ ( ρ f H f [ v ′ ′ ] ⋅ S f )
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經過對該流程的分析，不難發現，PISO能夠視爲一個SIMPLE步和一個或多個PRIME步的結合，所以結合了SIMPLE算法的隱式特性和PRIME算法的穩定性。同位網格上的PISO算法的流程可彙總以下：
爲計算時刻
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SIMPLE 一步
隱式求解動量方程得到新速度場
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\bold v_C^*+\bold H_C[\bold v^*] = -\bold D_C^{\bold v}(\nabla p^{(n)})_C + \bold B_C^{\bold v}
v C ∗ + H C [ v ∗ ] = − D C v ( ∇ p ( n ) ) C + B C v
更新單元面處的質量流量，使用Rhie-Chow差值技術得到知足動量方程的質量流量場
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\dot m_f^* = \rho_f \bold v_f^* \cdot \bold S_f = \rho_f \overline{\bold v_f^*} \cdot \bold S_f - \rho_f\overline{\bold D_f^{\bold v}}\left( \nabla p_f^{(n)}-\overline{\nabla p_f^{(n)}} \right) \cdot \bold S_f
m ˙ f ∗ = ρ f v f ∗ ⋅ S f = ρ f v f ∗ ⋅ S f − ρ f D f v ( ∇ p f ( n ) − ∇ p f ( n ) ) ⋅ S f
使用新的質量流量
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\begin{aligned} a_F^{p'} &= FluxF_f = -\rho_f \mathcal D_f \\ a_C^{p'} &= -\sum_{f\sim nb(c)}FluxF_f=-\sum_{F\sim NB(C)}a_F^{p'} \\ b_C^{p'} &= -\sum_{f\sim nb(C)} FluxV_f+\underline{\sum_{f\sim nb(C)}\left(\rho_f \overline{\bold H_f}[\bold v'] \cdot \bold S_f\right) } \\ &= -\sum_{f\sim nb(C)} \dot m_f^*+\underline{\sum_{f\sim nb(C)}\left(\rho_f \overline{\bold H_f}[\bold v'] \cdot \bold S_f\right) } \end{aligned}
a F p ′ a C p ′ b C p ′ = F l u x F f = − ρ f D f = − f ∼ n b ( c ) ∑ F l u x F f = − F ∼ N B ( C ) ∑ a F p ′ = − f ∼ n b ( C ) ∑ F l u x V f + f ∼ n b ( C ) ∑ ( ρ f H f [ v ′ ] ⋅ S f ) = − f ∼ n b ( C ) ∑ m ˙ f ∗ + f ∼ n b ( C ) ∑ ( ρ f H f [ v ′ ] ⋅ S f )
基於壓力修正場，更新單元形心處的壓力和速度場，以及單元面處的質量流量場，以得到知足連續方程的場，這些場分別用
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\begin{aligned} \bold v_C^{**} &= \bold v_C^* + \bold v'_C & \bold v'_C &= -\bold D_C^{\bold v}(\nabla p')_C \\ \dot m_f^{**} &= \dot m_f^* + \dot m'_f & \dot m'_f &= -\rho_f \overline{\bold D_f^{\bold v}}\nabla p'_f \cdot \bold S_f \\ p_C^* &= p_C^{(n)}+\lambda^p p'_C \end{aligned}
v C ∗ ∗ m ˙ f ∗ ∗ p C ∗ = v C ∗ + v C ′ = m ˙ f ∗ + m ˙ f ′ = p C ( n ) + λ p p C ′ v C ′ m ˙ f ′ = − D C v ( ∇ p ′ ) C = − ρ f D f v ∇ p f ′ ⋅ S f
PRIME一步或多步
使用最新可用的速度和壓力場，計算動量方程的係數並顯式求解該方程；
使用Rhie-Chow差值技術，更新單元面處的質量流量；
使用新的質量流量，組裝壓力修正方程，並求解以得到壓力修正場；
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-\sum_{f\sim nb(C)}\rho_f\overline{\bold D_f}\nabla p''_f\cdot \bold S_f = -\sum_{f\sim nb(C)}\dot m_f^* + \underline{\sum_{f\sim nb(C)}\left(\rho_f\overline{\bold H_f}[\bold v'']\cdot \bold S_f\right)}
− f ∼ n b ( C ) ∑ ρ f D f ∇ p f ′ ′ ⋅ S f = − f ∼ n b ( C ) ∑ m ˙ f ∗ + f ∼ n b ( C ) ∑ ( ρ f H f [ v ′ ′ ] ⋅ S f )
更新壓力、速度、質量流量場；
返回第6步，依據想要的修正步數，屢次重複該流程；
將最新的速度、壓力、質量流量值做爲初值猜想值；
返回第2步，並重復該流程直到收斂；
將收斂值設爲
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PISO算法的流程圖以下圖所示
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p ′ 爲推進SIMPLE算法的收斂，動量和連續方程要作欠鬆弛處理，分別使用欠鬆弛因子
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\begin{aligned} & -\bold D_C(\nabla p')_C=-\bold H_C[\bold v']-\lambda^p\bold D_C(\nabla p')_C \\ \Rightarrow & \lambda^p=1+\frac{\bold H_C[\bold v']}{\bold v'_C}= 1+\frac{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v \bold v'_F}{a_C^\bold v \bold v'_C} \end{aligned}
⇒ − D C ( ∇ p ′ ) C = − H C [ v ′ ] − λ p D C ( ∇ p ′ ) C λ p = 1 + v C ′ H C [ v ′ ] = 1 + a C v v C ′ F ∼ N B ( C ) ∑ a F v v F ′
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\bold v_C'\approx \frac{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v \bold v'_F}{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v }
v C ′ ≈ F ∼ N B ( C ) ∑ a F v F ∼ N B ( C ) ∑ a F v v F ′
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\bold v_C'\approx -\frac{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v \bold v'_F}{\lambda^\bold v a_C^\bold v} \Rightarrow a_C^\bold v\bold v_C'\approx -\frac{\displaystyle\sum_{F\sim NB(C)}a_F^\bold v \bold v'_F}{\lambda^\bold v }
v C ′ ≈ − λ v a C v F ∼ N B ( C ) ∑ a F v v F ′ ⇒ a C v v C ′ ≈ − λ v F ∼ N B ( C ) ∑ a F v v F ′
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基於Rhie-Chow插值，使用同位網格所獲得的解是依賴於動量方程中的欠鬆弛因子的。爲了消除該依賴性，須要對Rhie-Chow插值作修改，欠鬆弛動量方程爲
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\frac{1}{\lambda^\bold v}a_C^\bold v \bold v_C = -\sum_{F\sim NB(C)}a_F^\bold v \bold v_F + \bold b_C^\bold v-V_C\nabla p_C+\left(\frac{1-\lambda^\bold v}{\lambda^\bold v}\right)a_C^\bold v\bold v_C^{(n)}
λ v 1 a C v v C = − F ∼ N B ( C ) ∑ a F v v F + b C v − V C ∇ p C + ( λ v 1 − λ v ) a C v v C ( n )
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\frac{1}{\lambda^\bold v}a_f^\bold v \bold v_f = -\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v-V_f\nabla p_f+\left(\frac{1-\lambda^\bold v}{\lambda^\bold v}\right)a_f^\bold v\bold v_f^{(n)}
λ v 1 a f v v f = − n b ∼ N B ( C ) ∑ a n b v v n b + b f v − V f ∇ p f + ( λ v 1 − λ v ) a f v v f ( n )
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\frac{1}{\lambda^\bold v}\overline{a_f^\bold v} \bold v_f = -\overline{\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v}-\overline{V_f}\nabla p_f+\left(\frac{1-\lambda^\bold v}{\lambda^\bold v}\right)\overline{a_f^\bold v}\bold v_f^{(n)}
λ v 1 a f v v f = − n b ∼ N B ( C ) ∑ a n b v v n b + b f v − V f ∇ p f + ( λ v 1 − λ v ) a f v v f ( n )
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\begin{aligned} -\overline{\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v} =& -g_C\left(\sum_{F\sim NB(C)}a_{F}^\bold v \bold v_{F} + \bold b_C^\bold v\right) -g_F\left(\sum_{N\sim NB(C)}a_{N}^\bold v \bold v_{N} + \bold b_F^\bold v\right) \\ =&g_C\left[\frac{1}{\lambda^\bold v}a_C^\bold v \bold v_C +V_C\nabla p_C-\left(\frac{1-\lambda^\bold v}{\lambda^\bold v}\right)a_C^\bold v\bold v_C^{(n)}\right] + \\ &g_F\left[\frac{1}{\lambda^\bold v}a_F^\bold v \bold v_F +V_F\nabla p_F-\left(\frac{1-\lambda^\bold v}{\lambda^\bold v}\right)a_F^\bold v\bold v_F^{(n)}\right] \\ =&\frac{1}{\lambda^\bold v}\overline{a_f^\bold v \bold v_f} +\overline{V_f\nabla p_f}-\left(\frac{1-\lambda^\bold v}{\lambda^\bold v}\right)\overline{a_f^\bold v\bold v_f^{(n)}} \end{aligned}
− n b ∼ N B ( C ) ∑ a n b v v n b + b f v = = = − g C ⎝ ⎛ F ∼ N B ( C ) ∑ a F v v F + b C v ⎠ ⎞ − g F ⎝ ⎛ N ∼ N B ( C ) ∑ a N v v N + b F v ⎠ ⎞ g C [ λ v 1 a C v v C + V C ∇ p C − ( λ v 1 − λ v ) a C v v C ( n ) ] + g F [ λ v 1 a F v v F + V F ∇ p F − ( λ v 1 − λ v ) a F v v F ( n ) ] λ v 1 a f v v f + V f ∇ p f − ( λ v 1 − λ v ) a f v v f ( n )
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v f = v f − D f v ( ∇ p f − ∇ p f ) + ( 1 − λ v ) ( v f ( n ) − v f ( n ) )
當使用後向Euler瞬態格式求解瞬態問題時，離散動量方程爲
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a_C^\bold v \bold v_C = -\sum_{F\sim NB(C)}a_F^\bold v \bold v_F + \bold b_C^\bold v-V_C\nabla p_C+a_C^\circ\bold v_C^\circ
a C v v C = − F ∼ N B ( C ) ∑ a F v v F + b C v − V C ∇ p C + a C ∘ v C ∘
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a_f^\bold v \bold v_f = -\sum_{nb\sim NB(f)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v-V_f\nabla p_f+a_f^\circ\bold v_f^\circ
a f v v f = − n b ∼ N B ( f ) ∑ a n b v v n b + b f v − V f ∇ p f + a f ∘ v f ∘
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\overline{a_f^\bold v} \bold v_f = -\overline{\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v}-\overline{V_f}\nabla p_f+\overline{a_f^\circ}\bold v_f^\circ
a f v v f = − n b ∼ N B ( C ) ∑ a n b v v n b + b f v − V f ∇ p f + a f ∘ v f ∘
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\begin{aligned} -\overline{\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v} =& -g_C\left(\sum_{F\sim NB(C)}a_{F}^\bold v \bold v_{F} + \bold b_C^\bold v\right) -g_F\left(\sum_{N\sim NB(C)}a_{N}^\bold v \bold v_{N} + \bold b_F^\bold v\right) \\ =&g_C\left[a_C^\bold v \bold v_C +V_C\nabla p_C-a_C^\circ\bold v_C^\circ\right] + \\ &g_F\left[a_F^\bold v \bold v_F +V_F\nabla p_F-a_F^\circ\bold v_F^\circ\right] \\ =&\overline{a_f^\bold v \bold v_f} +\overline{V_f\nabla p_f}-\overline{a_f^\circ\bold v_f^\circ} \end{aligned}
− n b ∼ N B ( C ) ∑ a n b v v n b + b f v = = = − g C ⎝ ⎛ F ∼ N B ( C ) ∑ a F v v F + b C v ⎠ ⎞ − g F ⎝ ⎛ N ∼ N B ( C ) ∑ a N v v N + b F v ⎠ ⎞ g C [ a C v v C + V C ∇ p C − a C ∘ v C ∘ ] + g F [ a F v v F + V F ∇ p F − a F ∘ v F ∘ ] a f v v f + V f ∇ p f − a f ∘ v f ∘
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\bold v_f=\overline{\bold v_f}-\overline{\bold D_f^\bold v}(\nabla p_f-\overline{\nabla p_f})+ \frac{\overline{a_f^\circ\bold D_f^\circ}}{V_f}(\bold v_f^{\circ}-\overline{\bold v_f^{\circ}})
v f = v f − D f v ( ∇ p f − ∇ p f ) + V f a f ∘ D f ∘ ( v f ∘ − v f ∘ )
當在交錯網格上處理體積力的時候，體積力項的架構和壓力梯度項的架構是同樣的。在同位網格狀況下，體積力、速度、動量變量是在相同位置上計算的。這樣，爲了讓體積力的離散保持和壓力一樣的架構，須要把體積力作從新分配。離散動量方程爲
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a_C^\bold v \bold v_C = -\sum_{F\sim NB(C)}a_F^\bold v \bold v_F + \bold b_C^\bold v-V_C\nabla p_C+V_C\overline{\overline{\bold B_C^\bold v}}
a C v v C = − F ∼ N B ( C ) ∑ a F v v F + b C v − V C ∇ p C + V C B C v
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\overline{\bold B_f^\bold v}=g_C\bold B_C^\bold v+(1-g_C)\bold B_F^\bold v
B f v = g C B C v + ( 1 − g C ) B F v
第二步是對這些面值進行平均，算得單元形心值，以下圖所示
單元形心的平均值的推導，最好是經過考慮以下圖所示的一維問題來實現。
對於靜止流體，壓力梯度應該和體積力平衡，推出
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\begin{aligned} \nabla p_C &= \frac{\displaystyle\sum_f p_f \bold S_f}{V_C} \\ &= \frac{\displaystyle\sum_f (g_Cp_C+(1-g_C)p_F) \bold S_f}{V_C} \end{aligned}
∇ p C = V C f ∑ p f S f = V C f ∑ ( g C p C + ( 1 − g C ) p F ) S f
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\begin{aligned} \nabla p_C &= \frac{\displaystyle\sum_f (g_C(p_F+\overline{\bold B_f^\bold v}\cdot \bold d_{CF})+(1-g_C)p_F) \bold S_f}{V_C} \\ &=\frac{\displaystyle\sum_f p_F\bold S_f}{V_C}+ \frac{\displaystyle\sum_f g_C(\overline{\bold B_f^\bold v}\cdot \bold d_{CF})\bold S_f}{V_C} \\ &=p_F\underbrace{\frac{\displaystyle\sum_f \bold S_f}{V_C}}_{=0}+ \frac{\displaystyle\sum_f g_C(\overline{\bold B_f^\bold v}\cdot \bold d_{f})\bold S_f}{V_C} \\ &= \frac{\displaystyle\sum_f g_C(\overline{\bold B_f^\bold v}\cdot \bold d_{f})\bold S_f}{V_C} \\ &= \overline{\overline{\bold B_C^\bold v}} \end{aligned}
∇ p C = V C f ∑ ( g C ( p F + B f v ⋅ d C F ) + ( 1 − g C ) p F ) S f = V C f ∑ p F S f + V C f ∑ g C ( B f v ⋅ d C F ) S f = p F = 0
V C f ∑ S f + V C f ∑ g C ( B f v ⋅ d f ) S f = V C f ∑ g C ( B f v ⋅ d f ) S f = B C v
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\overline{\overline{\bold B_C^\bold v}} = \frac{\displaystyle\sum_f g_C(\overline{\bold B_f^\bold v}\cdot \bold d_{f})\bold S_f}{V_C}
B C v = V C f ∑ g C ( B f v ⋅ d f ) S f
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\overline{a_f^\bold v} \bold v_f = \overline{-\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v}-\overline{V_f}\nabla p_f+\overline{V_f\bold B_f^\bold v}
a f v v f = − n b ∼ N B ( C ) ∑ a n b v v n b + b f v − V f ∇ p f + V f B f v
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\begin{aligned} -\overline{\sum_{nb\sim NB(C)}a_{nb}^\bold v \bold v_{nb} + \bold b_f^\bold v} =& -g_C\left(\sum_{F\sim NB(C)}a_{F}^\bold v \bold v_{F} + \bold b_C^\bold v\right) -g_F\left(\sum_{N\sim NB(C)}a_{N}^\bold v \bold v_{N} + \bold b_F^\bold v\right) \\ =&g_C\left[a_C^\bold v \bold v_C +V_C\nabla p_C-V_C\overline{\overline{\bold B_C^\bold v}}\right] + \\ &g_F\left[a_F^\bold v \bold v_F +V_F\nabla p_F-V_F\overline{\overline{\bold B_F^\bold v}}\right] \\ =&\overline{a_f^\bold v \bold v_f} +\overline{V_f\nabla p_f}-\overline{V_f\overline{\overline{\bold B_f^\bold v}}} \end{aligned}
− n b ∼ N B ( C ) ∑ a n b v v n b + b f v = = = − g C ⎝ ⎛ F ∼ N B ( C ) ∑ a F v v F + b C v ⎠ ⎞ − g F ⎝ ⎛ N ∼ N B ( C ) ∑ a N v v N + b F v ⎠ ⎞ g C [ a C v v C + V C ∇ p C − V C B C v
