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。算法
這裏是第三部分，主要講解在同位網格SIMPLE算法中，在組裝動量方程和壓力修正方程時，不一樣類型的邊界條件是如何考慮和添加（處理）的。app
如上圖，邊界單元 是那些至少有一個面位於邊界片上的單元，這種面被稱爲邊界面 。邊界面處邊界條件的處理，對於CFD代碼的精確性和健壯性來講十分重要。爲了讓壓力基求解代碼（pressure-based code，即SIMPLE算法類，用壓力基求解器來求解不可壓縮流動）成功實現，動量方程和壓力修正方程二者邊界條件的正確添加是不可或缺的環節。這一章節就詳細探討下不一樣類型邊界條件的添加方法。（邊界條件也稱爲「定解條件」，不添加邊界條件的方程是無解的，或者說是有無窮多解的，邊界條件添加錯誤的話，方程將得出錯誤的不符合物理意義的解，或者壓根就得不到解）svg
首先，來關注下在邊界面處的Rhie-Chow插值表達式，由於邊界面處沒法像內部面那樣子來作平均，因此邊界面處的平均將直接寫成單元值的形式，即
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\begin{aligned} \overline{\bold v_b^*}&=\bold v_C^* \\ \overline{\nabla p_b^{(n)}}&=\nabla p_C^{(n)} \\ \overline{\bold D_b^{\bold v}}&=\bold D_C^{\bold v} \\ \underbrace{\bold v_b^*}_{boundary~face} &= \underbrace{\overline{\bold v_b^*} - \overline{\bold D_b^{\bold v}}\left( \nabla p_b^{(n)}-\overline{\nabla p_b^{(n)}} \right)}_{standard~Rhie-Chow} \\ &= \underbrace{\bold v_C^*-\bold D_C^{\bold v}\left( \nabla p_b^{(n)}-\nabla p_C^{(n)} \right)}_ {boundary~Rhie-Chow} \\ \dot m_b^* &=\rho_b\bold v_b^*\cdot \bold S_b \\ &= \rho_b \left[ \bold v_C^*-\bold D_C^{\bold v}\left( \nabla p_b^{(n)}-\nabla p_C^{(n)} \right) \right] \cdot \bold S_b \end{aligned}
v b ∗ ∇ p b ( n ) D b v b o u n d a r y f a c e
v b ∗ m ˙ b ∗ = v C ∗ = ∇ p C ( n ) = D C v = s t a n d a r d R h i e − C h o w
v b ∗ − D b v ( ∇ p b ( n ) − ∇ p b ( n ) ) = b o u n d a r y R h i e − C h o w
v C ∗ − D C v ( ∇ p b ( n ) − ∇ p C ( n ) ) = ρ b v b ∗ ⋅ S b = ρ b [ v C ∗ − D C v ( ∇ p b ( n ) − ∇ p C ( n ) ) ] ⋅ S b 學習
接下來先講解動量方程邊界條件的添加方法，而後是壓力修正方程邊界條件的添加方法。對於動量方程和壓力修正方程二者邊界條件相互關聯的情形，它們的處理方法將在壓力修正方程的章節詳解。ui
動量方程的半離散形式爲
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\underbrace{\frac{(\rho\bold v)_C-(\rho\bold v)_C^\circ}{\Delta t}V_C}_{element~discretization} + \underbrace{\sum_{f\sim nb(C)}\left(\dot m_f \bold v_f\right)}_{face~discretization}= -\underbrace{\sum_{f\sim nb(C)}\left(p_f\bold S_f\right)}_{face~discretization} +\underbrace{\sum_{f\sim nb(C)}\left(\tau_f\cdot\bold S_f\right)}_{face~discretization} +\underbrace{\bold B}_{\footnotesize{\begin{matrix}element\\discretization\end{matrix}}}
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Δ t ( ρ v ) C − ( ρ v ) C ∘ V C + f a c e d i s c r e t i z a t i o n
f ∼ n b ( C ) ∑ ( m ˙ f v f ) = − f a c e d i s c r e t i z a t i o n
f ∼ n b ( C ) ∑ ( p f S f ) + f a c e d i s c r e t i z a t i o n
f ∼ n b ( C ) ∑ ( τ f ⋅ S f ) + e l e m e n t d i s c r e t i z a t i o n
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\begin{aligned} \underbrace{\sum_{f\sim nb(C)}\left(\dot m_f \bold v_f\right)}_{face~discretization}&= \sum_{f\sim interior~nb(C)}\left(\dot m_f \bold v_f\right)+\underbrace{\dot m_b \bold v_b}_{boundary~face} \\ \underbrace{\sum_{f\sim nb(C)}\left(\tau_f\cdot\bold S_f\right)}_{face~discretization}&= \sum_{f\sim interior~nb(C)}\left(\tau_f\cdot\bold S_f\right)+\underbrace{\tau_b\cdot\bold S_b}_{boundary~face} \\ &= \sum_{f\sim interior~nb(C)}\left(\tau_f\cdot\bold S_f\right)+\underbrace{\bold F_b}_{boundary~face} \\ \underbrace{\sum_{f\sim nb(C)}\left(p_f\bold S_f\right)}_{face~discretization} &= \sum_{f\sim interior~nb(C)}\left(p_f\bold S_f\right)+\underbrace{p_b\bold S_b}_{boundary~face} \end{aligned}
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f ∼ n b ( C ) ∑ ( m ˙ f v f ) f a c e d i s c r e t i z a t i o n
f ∼ n b ( C ) ∑ ( τ f ⋅ S f ) f a c e d i s c r e t i z a t i o n
f ∼ n b ( C ) ∑ ( p f S f ) = f ∼ i n t e r i o r n b ( C ) ∑ ( m ˙ f v f ) + b o u n d a r y f a c e
m ˙ b v b = f ∼ i n t e r i o r n b ( C ) ∑ ( τ f ⋅ S f ) + b o u n d a r y f a c e
τ b ⋅ S b = f ∼ i n t e r i o r n b ( C ) ∑ ( τ f ⋅ S f ) + b o u n d a r y f a c e
F b = f ∼ i n t e r i o r n b ( C ) ∑ ( p f S f ) + b o u n d a r y f a c e
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\displaystyle\sum_{f\sim nb(C)}\rho_f\bold S_f
f ∼ n b ( C ) ∑ ρ f S f spa
通常來講，無滑移或滑移邊界條件能夠用於移動或固定壁面。添加過程包括計算和線性化壁面處的剪切應力，這與Dirichlet條件是不一樣的，儘管對於笛卡爾網格兩種條件在外部表現上是相同的。3d
無滑移壁面邊界 （
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p_b=?;~\dot m_b=0;~\bold v_b=\bold v_{wall}
p b = ? ; m ˙ b = 0 ; v b = v w a l l code
無滑移邊界條件意味着壁面處的流體速度
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上圖代表可經過確保剪切應力與壁面相切，以及邊界速度方程與壁面速度相等，來知足這些條件。由壁面施加到流體上的力
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\bold F_b=\bold F_{\parallel}=\tau_{wall}S_b
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\begin{aligned} \bold n &= \frac{\bold S_b}{S_b}=n_x\bold i+ n_y \bold j + n_z \bold k \\ d_{\perp} &= \bold d_{Cb}\cdot n=\frac{\bold d_{Cb}\cdot\bold S_b}{S_b} \end{aligned}
n d ⊥ = S b S b = n x i + n y j + n z k = d C b ⋅ n = S b d C b ⋅ S b
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\bold v_{\parallel}=\bold v-(\bold v\cdot \bold n)\bold n=\left\{ \begin{matrix} u-(un_x+vn_y+wn_z)n_x \\ v-(un_x+vn_y+wn_z)n_y \\ w-(un_x+vn_y+wn_z)n_z \end{matrix} \right\}
v ∥ = v − ( v ⋅ n ) n = ⎩ ⎨ ⎧ u − ( u n x + v n y + w n z ) n x v − ( u n x + v n y + w n z ) n y w − ( u n x + v n y + w n z ) n z ⎭ ⎬ ⎫
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\begin{aligned} \tau_{wall} &\approx -\mu_b\frac{(\bold v_C-\bold v_b)_{\parallel}}{d_{\perp}}= -\mu_b\frac{(\bold v_C-\bold v_b)-[(\bold v_C-\bold v_b)\cdot\bold n]\bold n}{d_{\perp}} \\ &=-\frac{\mu_b}{d_{\perp}}\left\{ \begin{matrix} (u_C-u_b)-[(u_C-u_b)n_x+(v_C-v_b)n_y+(w_C-w_b)n_z]n_x \\ (v_C-v_b)-[(u_C-u_b)n_x+(v_C-v_b)n_y+(w_C-w_b)n_z]n_y \\ (w_C-w_b)-[(u_C-u_b)n_x+(v_C-v_b)n_y+(w_C-w_b)n_z]n_z \end{matrix} \right\} \end{aligned}
τ w a l l ≈ − μ b d ⊥ ( v C − v b ) ∥ = − μ b d ⊥ ( v C − v b ) − [ ( v C − v b ) ⋅ n ] n = − d ⊥ μ b ⎩ ⎨ ⎧ ( u C − u b ) − [ ( u C − u b ) n x + ( v C − v b ) n y + ( w C − w b ) n z ] n x ( v C − v b ) − [ ( u C − u b ) n x + ( v C − v b ) n y + ( w C − w b ) n z ] n y ( w C − w b ) − [ ( u C − u b ) n x + ( v C − v b ) n y + ( w C − w b ) n z ] n z ⎭ ⎬ ⎫
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\bold F_b==-\frac{\mu_bS_b}{d_{\perp}}\left\{ \begin{matrix} (u_C-u_b)-[(u_C-u_b)n_x+(v_C-v_b)n_y+(w_C-w_b)n_z]n_x \\ (v_C-v_b)-[(u_C-u_b)n_x+(v_C-v_b)n_y+(w_C-w_b)n_z]n_y \\ (w_C-w_b)-[(u_C-u_b)n_x+(v_C-v_b)n_y+(w_C-w_b)n_z]n_z \end{matrix} \right\}
F b = = − d ⊥ μ b S b ⎩ ⎨ ⎧ ( u C − u b ) − [ ( u C − u b ) n x + ( v C − v b ) n y + ( w C − w b ) n z ] n x ( v C − v b ) − [ ( u C − u b ) n x + ( v C − v b ) n y + ( w C − w b ) n z ] n y ( w C − w b ) − [ ( u C − u b ) n x + ( v C − v b ) n y + ( w C − w b ) n z ] n z ⎭ ⎬ ⎫
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\begin{aligned} a_C^u &\leftarrow \underbrace{a_C^u}_{interior~faces~contribution} + \underbrace{\frac{\mu_bS_b}{d_{\perp}}(1-n_x^2)}_{boundary~face~contribution} \\ 0 &\leftarrow a_{F=b}^{u} \\ b_C^u &\leftarrow \underbrace{b_C^u}_{interior~faces~contribution} + \underbrace{\frac{\mu_bS_b}{d_{\perp}}[u_b(1-n_x^2)+(v_C-v_b)n_yn_x+(w_C-w_b)n_zn_x]-p_bS_b^x}_{boundary~face~contribution} \end{aligned}
a C u 0 b C u ← i n t e r i o r f a c e s c o n t r i b u t i o n
a C u + b o u n d a r y f a c e c o n t r i b u t i o n
d ⊥ μ b S b ( 1 − n x 2 ) ← a F = b u ← i n t e r i o r f a c e s c o n t r i b u t i o n
b C u + b o u n d a r y f a c e c o n t r i b u t i o n
d ⊥ μ b S b [ u b ( 1 − n x 2 ) + ( v C − v b ) n y n x + ( w C − w b ) n z n x ] − p b S b x
v
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−
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\begin{aligned} a_C^v &\leftarrow \underbrace{a_C^v}_{interior~faces~contribution} + \underbrace{\frac{\mu_bS_b}{d_{\perp}}(1-n_y^2)}_{boundary~face~contribution} \\ 0 &\leftarrow a_{F=b}^{v} \\ b_C^v &\leftarrow \underbrace{b_C^v}_{interior~faces~contribution} + \underbrace{\frac{\mu_bS_b}{d_{\perp}}[(u_C-u_b)n_xn_y+v_b(1-n_y^2)+(w_C-w_b)n_zn_y]-p_bS_b^y}_{boundary~face~contribution} \end{aligned}
a C v 0 b C v ← i n t e r i o r f a c e s c o n t r i b u t i o n
a C v + b o u n d a r y f a c e c o n t r i b u t i o n
d ⊥ μ b S b ( 1 − n y 2 ) ← a F = b v ← i n t e r i o r f a c e s c o n t r i b u t i o n
b C v + b o u n d a r y f a c e c o n t r i b u t i o n
d ⊥ μ b S b [ ( u C − u b ) n x n y + v b ( 1 − n y 2 ) + ( w C − w b ) n z n y ] − p b S b y
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\begin{aligned} a_C^w &\leftarrow \underbrace{a_C^w}_{interior~faces~contribution} + \underbrace{\frac{\mu_bS_b}{d_{\perp}}(1-n_z^2)}_{boundary~face~contribution} \\ 0 &\leftarrow a_{F=b}^{w} \\ b_C^w &\leftarrow \underbrace{b_C^w}_{interior~faces~contribution} + \underbrace{\frac{\mu_bS_b}{d_{\perp}}[(u_C-u_b)n_xn_z+(v_C-v_b)n_yn_z+w_b(1-n_z^2)]-p_bS_b^z}_{boundary~face~contribution} \end{aligned}
a C w 0 b C w ← i n t e r i o r f a c e s c o n t r i b u t i o n
a C w + b o u n d a r y f a c e c o n t r i b u t i o n
d ⊥ μ b S b ( 1 − n z 2 ) ← a F = b w ← i n t e r i o r f a c e s c o n t r i b u t i o n
b C w + b o u n d a r y f a c e c o n t r i b u t i o n
d ⊥ μ b S b [ ( u C − u b ) n x n z + ( v C − v b ) n y n z + w b ( 1 − n z 2 ) ] − p b S b z
p
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p_b
p b
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p
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+
∇
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n
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⋅
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b
p_b=p_C+\nabla p_C^{(n)}\cdot \bold d_{Cb}
p b = p C + ∇ p C ( n ) ⋅ d C b
m
˙
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∗
=
ρ
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v
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⋅
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∇
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)
⋅
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b
\dot m_b^* = \rho_b\bold v_b^*\cdot \bold S_b - \rho_b\bold D_C^{\bold v}(\nabla p_b^{(n)}-\nabla p_C^{(n)})\cdot \bold S_b
m ˙ b ∗ = ρ b v b ∗ ⋅ S b − ρ b D C v ( ∇ p b ( n ) − ∇ p C ( n ) ) ⋅ S b
v
b
∗
\bold v_b^*
v b ∗
v
b
∗
‾
\overline{\bold v_b^*}
v b ∗
v
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∗
\bold v_C^*
v C ∗
由於壁面邊界處的質量流量和速度爲零，上述方程變爲
0
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b
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∇
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b
0=0-\rho_b\bold D_C^{\bold v}(\nabla p_b^{(n)}-\nabla p_C^{(n)})\cdot \bold S_b
0 = 0 − ρ b D C v ( ∇ p b ( n ) − ∇ p C ( n ) ) ⋅ S b
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∇
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\begin{aligned} \bold D_C^{\bold v}\nabla p_b^{(n)}\cdot \bold S_b=\nabla p_b^{(n)}\cdot \bold S'_b=\nabla p_C^{(n)}\cdot \bold S'_b \end{aligned}
D C v ∇ p b ( n ) ⋅ S b = ∇ p b ( n ) ⋅ S b ′ = ∇ p C ( n ) ⋅ S b ′
S
b
′
\bold S'_b
S b ′
S
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′
=
E
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+
T
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\bold S'_b=\bold E_b+\bold T_b
S b ′ = E b + T b
∇
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n
)
⋅
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+
T
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)
=
∇
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⋅
S
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′
\nabla p_b^{(n)}\cdot (\bold E_b+\bold T_b)=\nabla p_C^{(n)}\cdot \bold S'_b
∇ p b ( n ) ⋅ ( E b + T b ) = ∇ p C ( n ) ⋅ S b ′
E
b
\bold E_b
E b
C
b
\bold{Cb}
C b
D
C
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=
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∇
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−
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b
)
\mathcal D_C(p_b-p_C)=(\nabla p_C^{(n)}\cdot \bold S'_b-\nabla p_b^{(n)}\cdot \bold T_b)
D C ( p b − p C ) = ( ∇ p C ( n ) ⋅ S b ′ − ∇ p b ( n ) ⋅ T b )
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p_b=p_C+\frac{\nabla p_C^{(n)}\cdot \bold S'_b-\nabla p_b^{(n)}\cdot \bold T_b}{\mathcal D_C}
p b = p C + D C ∇ p C ( n ) ⋅ S b ′ − ∇ p b ( n ) ⋅ T b
滑移壁面邊界 （
p
b
=
?
;
m
˙
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0
;
F
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0
p_b=?;~\dot m_b=0;~\bold F_b=\bold 0
p b = ? ; m ˙ b = 0 ; F b = 0
如上圖所示，對於這類邊界條件，壁面剪切應力爲零，所以邊界力也是零。邊界壓力值的計算和前面無滑移邊界中的同樣，可採用兩種方法來計算。動量方程（矢量形式）的係數修改以下：
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\begin{aligned} a_C^{\bold v} &\leftarrow \underbrace{a_C^{\bold v}}_{interior~faces~contribution}\\ 0 &\leftarrow a_{F=b}^{\bold v} \\ \bold b_C^{\bold v} &\leftarrow \underbrace{\bold b_C^{\bold v}}_{interior~faces~contribution} - \underbrace{p_b\bold S_b}_{boundary~face~contribution} \end{aligned}
a C v 0 b C v ← i n t e r i o r f a c e s c o n t r i b u t i o n
a C v ← a F = b v ← i n t e r i o r f a c e s c o n t r i b u t i o n
b C v − b o u n d a r y f a c e c o n t r i b u t i o n
p b S b
考慮三種類型的進口邊界條件：（i）給定速度；（ii）給定靜壓和速度方向；（iii）給定總壓和速度方向。對於壓力邊界條件的處理將放到後續的壓力修正方程邊界條件小節中詳細闡述，本節就先不涉及了。
給定速度（Specified Velocity） （
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p_b=?
p b = ?
m
˙
b
\dot m_b
m ˙ b
v
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\bold v_b
v b
如上圖，對於給定速度邊界條件的進口來講，在邊界面處的對流項（
m
˙
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\dot m_b\bold v_b
m ˙ b v b
F
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⋅
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\bold F_b=\tau_b\cdot\bold S_b
F b = τ b ⋅ S b
v
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\bold v_b
v b
m
˙
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\dot m_b
m ˙ b
p
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∇
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⋅
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p_b=p_C+\nabla p_C^{(n)}\cdot \bold d_{Cb}
p b = p C + ∇ p C ( n ) ⋅ d C b
m
˙
b
v
b
\dot m_b\bold v_b
m ˙ b v b
F
b
\bold F_b
F b
b
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v
\bold b_C^{\bold v}
b C v
a
F
=
b
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a_{F=b}^{\bold v}
a F = b v
a
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a_C^{\bold v}
a C v
邊界單元的係數修改成
a
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←
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(15.136)
\begin{aligned} a_C^{\bold v} &\leftarrow a_C^{\bold v} \\ \bold b_C^{\bold v} &\leftarrow \bold b_C^{\bold v}- \dot m_b \bold v_b+\bold F_b-p_b \bold S_b \tag{15.136}\\ 0 &\leftarrow a_{F=b}^{\bold v} \end{aligned}
a C v b C v 0 ← a C v ← b C v − m ˙ b v b + F b − p b S b ← a F = b v ( 1 5 . 1 3 6 )
a
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←
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←
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−
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\begin{aligned} a_C^{\bold v} &\leftarrow a_C^{\bold v} \\ \bold b_C^{\bold v} &\leftarrow \bold b_C^{\bold v}-a_{F=b}^{\bold v}\bold v_b\\ 0 &\leftarrow a_{F=b}^{\bold v} \end{aligned}
a C v b C v 0 ← a C v ← b C v − a F = b v v b ← a F = b v
給定靜壓和速度方向（Specified Static Pressure and Velocity Direction） （
p
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=
p
s
p
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c
i
f
i
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d
p_b=p_{specified}
p b = p s p e c i f i e d
m
˙
b
\dot m_b
m ˙ b
e
v
\bold e_{\bold v}
e v
v
b
\bold v_b
v b
如圖，在進口處給定靜壓，即
p
b
p_b
p b
因爲邊界壓力
p
b
p_b
p b
p
b
p_b
p b
∇
p
C
\nabla p_C
∇ p C
邊界處的質量流量是由連續方程算出的（見下節）。那麼，對於指定的速度方向，即，單位矢量
e
v
\bold e_{\bold v}
e v
m
˙
b
∗
∗
=
ρ
b
v
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∗
⋅
S
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=
ρ
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∣
∣
v
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∣
∣
e
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⇒
∣
∣
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∣
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⇒
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\begin{aligned} & \dot m_b^{**}=\rho_b \bold v_b^{**} \cdot \bold S_b=\rho_b ||\bold v_b^{**}|| \bold e_{\bold v}\cdot \bold S_b \\ \Rightarrow & ||\bold v_b^{**}||=\frac{\dot m_b^{**}}{\rho_b ( \bold e_{\bold v}\cdot \bold S_b)} \\ \Rightarrow & \bold v_b^{**}=||\bold v_b^{**}|| \bold e_{\bold v} \end{aligned}
⇒ ⇒ m ˙ b ∗ ∗ = ρ b v b ∗ ∗ ⋅ S b = ρ b ∣ ∣ v b ∗ ∗ ∣ ∣ e v ⋅ S b ∣ ∣ v b ∗ ∗ ∣ ∣ = ρ b ( e v ⋅ S b ) m ˙ b ∗ ∗ v b ∗ ∗ = ∣ ∣ v b ∗ ∗ ∣ ∣ e v
指定總壓和速度方向（Specified Total Pressure and Velocity Direction） （
p
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p_{o,b}=p_{o,specified}
p o , b = p o , s p e c i f i e d
m
˙
b
\dot m_b
m ˙ b
e
v
\bold e_{\bold v}
e v
v
b
\bold v_b
v b
如上圖，進口指定總壓和速度方向，可是速度幅值和靜壓未知，但它們之間的關係可利用總壓表達式給出
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⏟
s
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a
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2
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m
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p_o=\underbrace{p}_{static~pressure}+\underbrace{\frac{1}{2}\rho \bold v \cdot \bold v}_{dynamic~pressure}
p o = s t a t i c p r e s s u r e
p + d y n a m i c p r e s s u r e
2 1 ρ v ⋅ v
p
o
p_o
p o
p
p
p
ρ
\rho
ρ
v
\bold v
v
考慮三種類型的出口邊界條件：（i）指定靜壓；（ii）指定質量流量；（iii）徹底發展流動。
指定靜壓 （
p
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=
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s
p
e
c
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f
i
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d
p_b=p_{specified}
p b = p s p e c i f i e d
m
˙
b
\dot m_b
m ˙ b
v
b
\bold v_b
v b
對於動量方程來講，徹底發展條件是假設在指定壓力出口處，沿着面矢量方向的速度梯度爲零。這等效於假設出口速度等於邊界單元速度，所以其和零通量邊界條件相似，其添加至關簡單。
動量方程的係數修改成
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\begin{aligned} a_C^{\bold v} &\leftarrow \underbrace{a_C^{\bold v}}_{interior~faces~contribution}+ \underbrace{\dot m_b}_{boundary~face~contribution} \\ 0 &\leftarrow a_{F=b}^{\bold v} \\ \bold b_C^{\bold v} &\leftarrow \underbrace{\bold b_C^{\bold v}}_{interior~faces~contribution}- \underbrace{p_b \bold S_b}_{boundary~face~contribution} \end{aligned}
a C v 0 b C v ← i n t e r i o r f a c e s c o n t r i b u t i o n
a C v + b o u n d a r y f a c e c o n t r i b u t i o n
m ˙ b ← a F = b v ← i n t e r i o r f a c e s c o n t r i b u t i o n
b C v − b o u n d a r y f a c e c o n t r i b u t i o n
p b S b
v
b
=
v
C
\bold v_b = \bold v_C
v b = v C
m
˙
b
v
b
=
m
˙
b
v
C
\dot m_b \bold v_b=\dot m_b \bold v_C
m ˙ b v b = m ˙ b v C
a
C
v
a_C^{\bold v}
a C v
m
˙
b
\dot m_b
m ˙ b
然而，爲了保證在只是在出口邊界面的法向矢量方向的通量爲零（切向通量未必是零），速度一般是使用邊界通量外插到出口的，邊界通量則是由邊界單元算出的，即（下式至關於刨掉了速度梯度的法向份量，只保留速度梯度的切向份量）
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\nabla \bold v_b=\nabla \bold v_C-(\nabla \bold v_C \cdot \bold e_b) \bold e_b
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\bold v_b = \bold v_C + \nabla \bold v_b \cdot \bold d_{Cb}
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(15.142)
\begin{aligned} a_C^{\bold v} &\leftarrow \underbrace{a_C^{\bold v}}_{interior~faces~contribution}+ \underbrace{\dot m_b}_{boundary~face~contribution} \\ 0 &\leftarrow a_{F=b}^{\bold v} \tag{15.142}\\ \bold b_C^{\bold v} &\leftarrow \underbrace{\bold b_C^{\bold v}}_{interior~faces~contribution} \underbrace{-\dot m_b(\nabla \bold v_b \cdot \bold d_{Cb})-p_b \bold S_b}_{boundary~face~contribution} \end{aligned}
a C v 0 b C v ← i n t e r i o r f a c e s c o n t r i b u t i o n
a C v + b o u n d a r y f a c e c o n t r i b u t i o n
m ˙ b ← a F = b v ← i n t e r i o r f a c e s c o n t r i b u t i o n
b C v b o u n d a r y f a c e c o n t r i b u t i o n
− m ˙ b ( ∇ v b ⋅ d C b ) − p b S b ( 1 5 . 1 4 2 )
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\dot m_b \bold v_b=\dot m_b \bold v_C+\dot m_b(\nabla \bold v_b \cdot \bold d_{Cb})
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如上圖，既然流動是不可壓縮的，那麼指定質量流量的邊界條件就等效於指定邊界速度的法向份量。速度的計算是經過假設其方向是和主網格節點方向同樣，即，
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(\bold e_{\bold v})_b=(\bold e_{\bold v})_C
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\begin{aligned} &\dot m_b=\rho_b\bold v_b\cdot \bold S_b=\rho_b|\bold v_b|(\bold e_{\bold v})_C\cdot \bold S_b \\ \Rightarrow & |\bold v_b|=\frac{\dot m_b}{\rho_b(\bold e_{\bold v})_C\cdot \bold S_b} \end{aligned}
⇒ m ˙ b = ρ b v b ⋅ S b = ρ b ∣ v b ∣ ( e v ) C ⋅ S b ∣ v b ∣ = ρ b ( e v ) C ⋅ S b m ˙ b
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徹底發展出口流動（Fully Developed Outlet Flow）
對於徹底發展流動，出口面的法向速度梯度假設爲零。所以，出口速度假設已知，且根據零法向梯度算得
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\begin{aligned} \nabla \bold v_b&=\nabla \bold v_C-(\nabla \bold v_C \cdot \bold e_b) \bold e_b \\ \bold v_b &= \bold v_C + \nabla \bold v_b \cdot \bold d_{Cb} \end{aligned}
∇ v b v b = ∇ v C − ( ∇ v C ⋅ e b ) e b = v C + ∇ v b ⋅ d C b
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p_b=p_C+\nabla p_C\cdot \bold d_{Cb}
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穿過對稱邊界的標量將發生反射，如此看來，對標量而言對稱邊界條件的添加可將標量的法向梯度設爲零，就跟絕熱壁面邊界條件同樣。對於速度矢量，在如上圖所示的對稱邊界上，速度一樣發生反射，即，速度的平行份量（平行於對稱邊界）保持其幅值和方向不變，而速度的法向垂直份量（垂直於對稱邊界）則變爲零。這在對稱邊界上產生了零剪切應力和非零的法嚮應力。所以，一樣的邊界條件可用於粘性流動的滑移邊界條件的施加。
垂直於邊界方向的單位矢量
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\begin{aligned} \bold n &= \frac{\bold S_b}{S_b}=n_x\bold i+ n_y \bold j + n_z \bold k \\ d_{\perp} &= \bold d_{Cb}\cdot n=\frac{\bold d_{Cb}\cdot\bold S_b}{S_b} \end{aligned}
n d ⊥ = S b S b = n x i + n y j + n z k = d C b ⋅ n = S b d C b ⋅ S b
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\begin{aligned} \bold v_{\perp} &= \bold 0 \\ \frac{\partial \bold v_{\parallel}}{\partial \bold n} &= \bold 0 \end{aligned}
v ⊥ ∂ n ∂ v ∥ = 0 = 0
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\bold v_{\perp}=(\bold v\cdot \bold n)\bold n=\left\{ \begin{matrix} (u_Cn_x+v_Cn_y+w_Cn_z)n_x \\ (u_Cn_x+v_Cn_y+w_Cn_z)n_y \\ (u_Cn_x+v_Cn_y+w_Cn_z)n_z \end{matrix} \right\}
v ⊥ = ( v ⋅ n ) n = ⎩ ⎨ ⎧ ( u C n x + v C n y + w C n z ) n x ( u C n x + v C n y + w C n z ) n y ( u C n x + v C n y + w C n z ) n z ⎭ ⎬ ⎫
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\bold v_{\parallel}=\bold v-(\bold v\cdot \bold n)\bold n=\left\{ \begin{matrix} u_C-(u_Cn_x+v_Cn_y+w_Cn_z)n_x \\ v_C-(u_Cn_x+v_Cn_y+w_Cn_z)n_y \\ w_C-(u_Cn_x+v_Cn_y+w_Cn_z)n_z \end{matrix} \right\}
v ∥ = v − ( v ⋅ n ) n = ⎩ ⎨ ⎧ u C − ( u C n x + v C n y + w C n z ) n x v C − ( u C n x + v C n y + w C n z ) n y w C − ( u C n x + v C n y + w C n z ) n z ⎭ ⎬ ⎫
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\sigma_{\perp}\approx-2\mu_b\frac{\bold v_{\perp}}{d_{\perp}}=-2\frac{\mu_b}{d_{\perp}} \left\{ \begin{matrix} (u_Cn_x+v_Cn_y+w_Cn_z)n_x \\ (u_Cn_x+v_Cn_y+w_Cn_z)n_y \\ (u_Cn_x+v_Cn_y+w_Cn_z)n_z \end{matrix} \right\}
σ ⊥ ≈ − 2 μ b d ⊥ v ⊥ = − 2 d ⊥ μ b ⎩ ⎨ ⎧ ( u
