有限体积法（2）——二维、三维扩散方程的离散推导
稳态扩散方程：
∇ ⋅ ( Γ ∇ ϕ ) + S ϕ = 0 (1) \nabla \cdot ( \Gamma \nabla \phi) + S_\phi =0 \tag{1} ∇⋅(Γ∇ϕ)+Sϕ=0(1)
在有限控制体内积分，并由高斯散度定理有，
∫ C V ∇ ⋅ ( Γ ∇ ϕ ) d V + ∫ C V S ϕ d V = ∫ A ~ n ⋅ ( Γ ∇ ϕ ) d A + ∫ C V S ϕ d V = 0 (2) \begin{aligned} \int_{CV} \nabla \cdot (\Gamma \nabla \phi ) dV + \int_{CV} S_\phi dV \\ \\ =\int_{\tilde A} \bold n \cdot (\Gamma \nabla \phi)dA + \int_{CV} S_\phi dV =0 \tag{2} \end{aligned} ∫CV∇⋅(Γ∇ϕ)dV+∫CVSϕdV=∫A~n⋅(Γ∇ϕ)dA+∫CVSϕdV=0(2)
其中 n \bold n n为边界面 A ~ \tilde A A~的法向量， Γ \Gamma Γ为扩散系数。
二维扩散方程的离散
根据式 ( 1 ) (1) (1)，二维模型的扩散方程为，
∂ ∂ x ( Γ ∂ ϕ ∂ x ) + ∂ ∂ y ( Γ ∂ ϕ ∂ y ) + S ϕ = 0 (3) \frac{\partial}{\partial x} \left( \Gamma \frac{\partial \phi}{\partial x} \right) + \frac{\partial }{\partial y} \left( \Gamma \frac{\partial \phi}{\partial y} \right) + S_\phi =0 \tag{3} ∂x∂(Γ∂x∂ϕ)+∂y∂(Γ∂y∂ϕ)+Sϕ=0(3)
与一维扩散方程推导类似，先将二维计算域划分网格，
在二维空间中，梯度矢量有两个分量， ∂ ϕ ∂ x i \frac{\partial \phi}{\partial x} \bold i ∂x∂ϕi和 ∂ ϕ ∂ y j \frac{\partial \phi}{\partial y} \bold j ∂y∂ϕj，方向分别指向 x x x和 y y y轴的正方向。
散度的离散
在二维空间，单元P的边界包括 w 、 e 、 s w、e、s w、e、s和 n n n四个边界面，由于网格和坐标轴是平行或垂直的， ∂ ϕ ∂ y \frac{\partial \phi}{\partial y} ∂y∂ϕ在边界面 w 、 e w、e w、e上的通量为零，同理 ∂ ϕ ∂ x \frac{\partial \phi}{\partial x} ∂x∂ϕ在边界面 s 、 n s、n s、n上的通量为零。所以，
∫ C V ∇ ⋅ ( Γ ∇ ϕ ) d V + ∫ C V S ϕ d V = ∫ A ~ n ⋅ ( Γ ∇ ϕ ) d A + ∫ C V S ϕ d V = ∫ A ~ n ⋅ [ ∂ ∂ x ( Γ ∂ ϕ ∂ x ) i + ∂ ∂ y ( Γ ∂ ϕ ∂ y ) j ] d A + ∫ C V S ϕ d V = [ ( Γ A ∂ ϕ ∂ x ) e − ( Γ A ∂ ϕ ∂ x ) w ] + [ ( Γ A ∂ ϕ ∂ y ) n − ( Γ A ∂ ϕ ∂ y ) s ] + S ˉ ϕ Δ V = 0 (4) \begin{aligned} & \int_{CV} \nabla \cdot (\Gamma \nabla \phi ) dV + \int_{CV} S_\phi dV \\ \\ &=\int_{\tilde A} \bold n \cdot (\Gamma \nabla \phi)dA + \int_{CV} S_\phi dV \\ \\ &=\int_{\tilde A} \bold n \cdot \left[ \frac{\partial}{\partial x} \left( \Gamma \frac{\partial \phi}{\partial x} \right) \bold i + \frac{\partial}{\partial y} \left( \Gamma \frac{\partial \phi}{\partial y} \right) \bold j \right] dA + \int_{CV}S_\phi dV \\ \\ &=\left[ \left( \Gamma A \frac{\partial \phi}{\partial x} \right)_e - \left(\Gamma A \frac{\partial \phi}{\partial x} \right)_w \right] + \left[ \left( \Gamma A \frac{\partial \phi}{\partial y} \right)_n - \left(\Gamma A \frac{\partial \phi}{\partial y} \right)_s \right] + \bar S_\phi \Delta V\\ \\ &=0 \tag{4} \end{aligned} ∫CV∇⋅(Γ∇ϕ)dV+∫CVSϕdV=∫A~n⋅(Γ∇ϕ)dA+∫CVSϕdV=∫A~n⋅[∂x∂(Γ∂x∂ϕ)i+∂y∂(Γ∂y∂ϕ)j]dA+∫CVSϕdV=[(ΓA∂x∂ϕ)e−(ΓA∂x∂ϕ)w]+[(ΓA∂y∂ϕ)n−(ΓA∂y∂ϕ)s]+SˉϕΔV=0(4)
其中 A A A代表边界面的面积， S ˉ \bar{S} Sˉ是控制体单元 Δ V \Delta V ΔV内的平均源项。边界面 w w w和 s s s上的通量为负，原因与一维扩散方程情况类似。
梯度的离散
与一维方程相同，梯度项采用中心差分格式离散，
( Γ A ∂ ϕ ∂ x ) w = Γ w A w ( ϕ P − ϕ W ) δ x W P (5a) \left ( \Gamma A \frac{\partial \phi}{\partial x} \right)_w = \Gamma_w A_w \frac{ ( \phi_P - \phi_W) }{\delta x_{WP}} \tag{5a} (ΓA∂x∂ϕ)w=ΓwAwδxWP(ϕP−ϕW)(5a)
( Γ A ∂ ϕ ∂ x ) e = Γ e A e ( ϕ E − ϕ P ) δ x P E (5b) \left ( \Gamma A \frac{\partial \phi}{\partial x} \right)_e = \Gamma_e A_e \frac{ ( \phi_E - \phi_P) }{\delta x_{PE}} \tag{5b} (ΓA∂x∂ϕ)e=ΓeAeδxPE(ϕE−ϕP)(5b)
( Γ A ∂ ϕ ∂ y ) s = Γ s A s ( ϕ P − ϕ S ) δ y S P (5c) \left ( \Gamma A \frac{\partial \phi}{\partial y} \right)_s = \Gamma_s A_s \frac{ ( \phi_P - \phi_S) }{\delta y_{SP}} \tag{5c} (ΓA∂y∂ϕ)s=ΓsAsδySP(ϕP−ϕS)(5c)
( Γ A ∂ ϕ ∂ y ) n = Γ n A n ( ϕ N − ϕ P ) δ y P N (5d) \left ( \Gamma A \frac{\partial \phi}{\partial y} \right)_n = \Gamma_n A_n \frac{ ( \phi_N - \phi_P) }{\delta y_{PN}} \tag{5d} (ΓA∂y∂ϕ)n=ΓnAnδyPN(ϕN−ϕP)(5d)
将式 ( 5 a ) (5a) (5a)~ ( 5 d ) (5d) (5d)带入到式 ( 4 ) (4) (4)中，有
Γ e A e ( ϕ E − ϕ P ) δ x P E − Γ w A w ( ϕ P − ϕ W ) δ x W P + Γ n A n ( ϕ N − ϕ P ) δ y P N − Γ s A s ( ϕ P − ϕ S ) δ y S P + ( S u + S P ϕ P ) = 0 (6) \begin{aligned} \Gamma_e A_e \frac{(\phi_E - \phi_P)}{\delta x_{PE}} - \Gamma_w A_w \frac{(\phi_P - \phi_W)}{\delta x_{WP}} + \Gamma_n A_n \frac{(\phi_N - \phi_P)}{\delta y_{PN}} \\ \\ - \Gamma_s A_s \frac{(\phi_P - \phi_S)}{\delta y_{SP}} + (S_u + S_P \phi_P) =0 \tag{6} \end{aligned} ΓeAeδxPE(ϕE−ϕP)−ΓwAwδxWP(ϕP−ϕW)+ΓnAnδyPN(ϕN−ϕP)−ΓsAsδySP(ϕP−ϕS)+(Su+SPϕP)=0(6)
式中 ( S u + S P ϕ P ) = S ˉ ϕ Δ V (S_u + S_P \phi_P)=\bar S_\phi \Delta V (Su+SPϕP)=SˉϕΔV，整理得，
( Γ w A w δ x W P + Γ e A e δ x P E + Γ s A s δ y S P + Γ n A n δ y P N − S P ) ϕ P = ( Γ w A w δ x W P ) ϕ W + ( Γ e A e δ x P E ) ϕ E + ( Γ s A s δ y S P ) ϕ S + ( Γ n A n δ y P ) ϕ N + S u (7) \begin{aligned} &\left( \frac{\Gamma_w A_w}{\delta x_{WP}} + \frac{\Gamma_e A_e}{\delta x_{PE}} + \frac{\Gamma_s A_s}{\delta y_{SP}} +\frac{\Gamma_n A_n}{\delta y_{PN}} - S_P \right) \phi_P \\ \\ &=\left( \frac{\Gamma_w A_w}{\delta x_{WP}} \right) \phi_W + \left( \frac{\Gamma_e A_e}{\delta x_{PE}} \right) \phi_E + \left( \frac{\Gamma_s A_s}{\delta y_{SP}} \right) \phi_S + \left( \frac{\Gamma_n A_n}{\delta y_{P}} \right) \phi_N + S_u \end{aligned} \tag{7} (δxWPΓwAw+δxPEΓeAe+δySPΓsAs+δyPNΓnAn−SP)ϕP=(δxWPΓwAw)ϕW+(δxPEΓeAe)ϕE+(δySPΓsAs)ϕS+(δyPΓnAn)ϕN+Su(7)
简化之，
a P ϕ P = a W ϕ W + a E ϕ E + a S ϕ S + a N ϕ N + S u (8) a_P \phi_P = a_W \phi_W + a_E \phi_E + a_S \phi_S + a_N \phi_N + S_u \tag{8} aPϕP=aWϕW+aEϕE+aSϕS+aNϕN+Su(8)
其中各系数为
a W = Γ w A w δ x W P (9a) a_W = \frac{\Gamma_w A_w}{\delta x_{WP}} \tag{9a} aW=δxWPΓwAw(9a)
a E = Γ e A e δ x P E (9b) a_E = \frac{\Gamma_e A_e}{\delta x_{PE}} \tag{9b} aE=δxPEΓeAe(9b)
a S = Γ s A s δ y S P (9c) a_S = \frac{\Gamma_s A_s}{\delta y_{SP}} \tag{9c} aS=δySPΓsAs(9c)
a N = Γ n A n δ y P N (9d) a_N = \frac{\Gamma_n A_n}{\delta y_{PN}} \tag{9d} aN=δyPNΓnAn(9d)
a P = a W + a E + a S + a N = S P (9e) a_P = a_W + a_E + a_S + a_N = S_P \tag{9e} aP=aW+aE+aS+aN=SP(9e)
边界处的扩散系数 Γ \Gamma Γ可以通过线性插值计算，见一维扩散方程推导中公式 ( 12 ) (12) (12)。
三维离散的扩散方程
三维扩散方程：
∂ ∂ x ( Γ ∂ ϕ ∂ x ) + ∂ ∂ y ( Γ ∂ ϕ ∂ y ) + ∂ ∂ z ( Γ ∂ ϕ ∂ z ) + S ϕ = 0 (10) \frac{\partial}{\partial x} \left( \Gamma \frac{\partial \phi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \Gamma \frac{\partial \phi}{\partial y} \right) + \frac{\partial}{\partial z} \left( \Gamma \frac{\partial \phi}{\partial z} \right) + S_\phi = 0 \tag{10} ∂x∂(Γ∂x∂ϕ)+∂y∂(Γ∂y∂ϕ)+∂z∂(Γ∂z∂ϕ)+Sϕ=0(10)
三维扩散方程的离散推导和二维的套路是一毛一样的，无非就是多了两个边界面 b b b和 t t t，散度离散和梯度离散时多了两项，梯度离散方法和方程式 ( 5 ) (5) (5)一样也用中心差分格式，然后带入，整理，简化之。
散度离散
∫ C V ∇ ⋅ ( Γ ∇ ϕ ) d V + ∫ C V S ϕ d V = ∫ A ~ n ⋅ [ ∂ ∂ x ( Γ ∂ ϕ ∂ x ) i + ∂ ∂ y ( Γ ∂ ϕ ∂ y ) j + ∂ ∂ z ( Γ ∂ ϕ ∂ z ) k ] d A + ∫ C V S ϕ d V = [ ( Γ A ∂ ϕ ∂ x ) e − ( Γ A ∂ ϕ ∂ x ) w ] + [ ( Γ A ∂ ϕ ∂ y ) n − ( Γ A ∂ ϕ ∂ y ) s ] + [ ( Γ A ∂ ϕ ∂ z ) t − ( Γ A ∂ ϕ ∂ z ) b ] + S ˉ ϕ Δ V = 0 (11) \begin{aligned} &\int_{CV} \nabla \cdot ( \Gamma \nabla \phi ) dV + \int_{CV} S_\phi dV \\ \\ &=\int_{\tilde A} \bold n \cdot \left[ \frac{\partial}{\partial x}\left( \Gamma \frac{\partial \phi}{\partial x} \right) \bold i + \frac{\partial}{\partial y}\left( \Gamma \frac{\partial \phi}{\partial y} \right) \bold j + \frac{\partial}{\partial z}\left( \Gamma \frac{\partial \phi}{\partial z} \right) \bold k \right] dA + \int_{CV} S_\phi dV \\ \\ &=\left[ \left( \Gamma A \frac{\partial \phi}{\partial x} \right)_e - \left(\Gamma A \frac{\partial \phi}{\partial x} \right)_w \right] + \left[ \left( \Gamma A \frac{\partial \phi}{\partial y} \right)_n - \left(\Gamma A \frac{\partial \phi}{\partial y} \right)_s \right] \\ \\ & \qquad + \left[ \left( \Gamma A \frac{\partial \phi}{\partial z} \right)_t - \left(\Gamma A \frac{\partial \phi}{\partial z} \right)_b \right] + \bar S_\phi \Delta V =0 \tag{11} \end{aligned} ∫CV∇⋅(Γ∇ϕ)dV+∫CVSϕdV=∫A~n⋅[∂x∂(Γ∂x∂ϕ)i+∂y∂(Γ∂y∂ϕ)j+∂z∂(Γ∂z∂ϕ)k]dA+∫CVSϕdV=[(ΓA∂x∂ϕ)e−(ΓA∂x∂ϕ)w]+[(ΓA∂y∂ϕ)n−(ΓA∂y∂ϕ)s]+[(ΓA∂z∂ϕ)t−(ΓA∂z∂ϕ)b]+SˉϕΔV=0(11)
梯度离散
梯度项采用中心差分格式离散，边界面 e 、 w 、 s 、 n e、w、s、n e、w、s、n处的梯度离散和式 ( 5 ) (5) (5)一样，多出来的两项公式如下，
( Γ A ∂ ϕ ∂ z ) b = Γ b A b ( ϕ P − ϕ B ) δ z B P (12a) \left ( \Gamma A \frac{\partial \phi}{\partial z} \right)_b = \Gamma_b A_b \frac{ ( \phi_P - \phi_B) }{\delta z_{BP}} \tag{12a} (ΓA∂z∂ϕ)b=ΓbAbδzBP(ϕP−ϕB)(12a)
( Γ A ∂ ϕ ∂ x ) t = Γ t A t ( ϕ T − ϕ P ) δ z P T (12b) \left ( \Gamma A \frac{\partial \phi}{\partial x} \right)_t = \Gamma_t A_t \frac{ ( \phi_T - \phi_P) }{\delta z_{PT}} \tag{12b} (ΓA∂x∂ϕ)t=ΓtAtδzPT(ϕT−ϕP)(12b)
带入式 ( 11 ) (11) (11)，有
[ Γ e A e ( ϕ E − ϕ P ) δ x P E − Γ w A w ( ϕ P − ϕ W ) δ x W P ] + [ Γ n A n ( ϕ N − ϕ P ) δ y P N − Γ s A s ( ϕ P − ϕ S ) δ y S P ] + [ Γ t A t ( ϕ T − ϕ P ) δ z P T − Γ b A b ( ϕ P − ϕ B ) δ z B P ] + ( S u + S P ϕ P ) = 0 (13) \begin{aligned} &\left[ \Gamma_e A_e \frac{(\phi_E - \phi_P)}{\delta x_{PE}} - \Gamma_w A_w \frac{(\phi_P - \phi_W)}{\delta x_{WP}} \right] \\ \\ &+ \left[ \Gamma_n A_n \frac{(\phi_N - \phi_P)}{\delta y_{PN}} - \Gamma_s A_s \frac{(\phi_P - \phi_S)}{\delta y_{SP}} \right ] \\ \\ &+ \left[ \Gamma_t A_t \frac{(\phi_T - \phi_P)}{\delta z_{PT}} - \Gamma_b A_b \frac{(\phi_P - \phi_B)}{\delta z_{BP}} \right ] \\ \\ &+ (S_u + S_P \phi_P) =0 \tag{13} \end{aligned} [ΓeAeδxPE(ϕE−ϕP)−ΓwAwδxWP(ϕP−ϕW)]+[ΓnAnδyPN(ϕN−ϕP)−ΓsAsδySP(ϕP−ϕS)]+[ΓtAtδzPT(ϕT−ϕP)−ΓbAbδzBP(ϕP−ϕB)]+(Su+SPϕP)=0(13)
整理并简化之，
a P ϕ P = a W ϕ W + a E ϕ E + a S ϕ S + a N ϕ N + a B ϕ B + a T ϕ T + S u (14) a_P \phi_P = a_W \phi_W + a_E \phi_E + a_S \phi_S + a_N \phi_N + a_B \phi_B + a_T \phi_T+ S_u \tag{14} aPϕP=aWϕW+aEϕE+aSϕS+aNϕN+aBϕB+aTϕT+Su(14)
各项系数，
a W = Γ w A w δ x W P (15a) a_W = \frac{\Gamma_w A_w}{\delta x_{WP}} \tag{15a} aW=δxWPΓwAw(15a)
a E = Γ e A e δ x P E (15b) a_E = \frac{\Gamma_e A_e}{\delta x_{PE}} \tag{15b} aE=δxPEΓeAe(15b)
a S = Γ s A s δ y S P (15c) a_S = \frac{\Gamma_s A_s}{\delta y_{SP}} \tag{15c} aS=δySPΓsAs(15c)
a N = Γ n A n δ y P N (15d) a_N = \frac{\Gamma_n A_n}{\delta y_{PN}} \tag{15d} aN=δyPNΓnAn(15d)
a B = Γ b A b δ z B P (15e) a_B = \frac{\Gamma_b A_b}{\delta z_{BP}} \tag{15e} aB=δzBPΓbAb(15e)
a T = Γ t A t δ z P T (15f) a_T = \frac{\Gamma_t A_t}{\delta z_{PT}} \tag{15f} aT=δzPTΓtAt(15f)
a P = a W + a E + a S + a N + a B + a T − S P (15h) a_P = a_W + a_E + a_S + a_N + a_B + a_T - S_P \tag{15h} aP=aW+aE+aS+aN+aB+aT−SP(15h)
总结
通过有限体积法离散后的稳态扩散方程可以统一写成如下形式，
a P ϕ P = Σ a n b ϕ n b + S u (16) a_P \phi_P = \Sigma a_{nb}\phi_{nb} + S_u \tag{16} aPϕP=Σanbϕnb+Su(16)
其中，“ n b nb nb”代表单元 P P P相邻的节点， Σ \Sigma Σ代表所有相邻节点之和。
系数 a P a_P aP与 a n b a_{nb} anb之间的关系为：
a P = Σ a n b − S P (17) a_P = \Sigma a_{nb} - S_P \tag{17} aP=Σanb−SP(17)
相邻节点的系数 a n b a_{nb} anb计算如下表，
| a W a_W aW | a E a_E aE | a S a_S aS | a N a_N aN | a B a_B aB | a T a_T aT | |
|---|---|---|---|---|---|---|
| 1D | Γ w A w δ x W P \frac{\Gamma_w A_w}{\delta x_{WP}} δxWPΓwAw | Γ w A e δ x P E \frac{\Gamma_w A_e}{\delta x_{PE}} δxPEΓwAe | ||||
| 2D | Γ w A w δ x W P \frac{\Gamma_w A_w}{\delta x_{WP}} δxWPΓwAw | Γ w A e δ x P E \frac{\Gamma_w A_e}{\delta x_{PE}} δxPEΓwAe | Γ s A s δ y S P \frac{\Gamma_s A_s}{\delta y_{SP}} δySPΓsAs | Γ n A n δ y P N \frac{\Gamma_n A_n}{\delta y_{PN}} δyPNΓnAn | ||
| 3D | Γ w A w δ x W P \frac{\Gamma_w A_w}{\delta x_{WP}} δxWPΓwAw | Γ w A e δ x P E \frac{\Gamma_w A_e}{\delta x_{PE}} δxPEΓwAe | Γ s A s δ y S P \frac{\Gamma_s A_s}{\delta y_{SP}} δySPΓsAs | Γ n A n δ y P N \frac{\Gamma_n A_n}{\delta y_{PN}} δyPNΓnAn | Γ b A b δ z B P \frac{\Gamma_b A_b}{\delta z_{BP}} δzBPΓbAb | Γ t A t δ z P T \frac{\Gamma_t A_t}{\delta z_{PT}} δzPTΓtAt |
参考资料：
- Versteeg H K , Malalasekera W . An introduction to computational fluid dynamics : the finite volume method = 计算流体动力学导论[M]. 世界图书出版公司, 2010.