【CFD理论】梯度项-01

fvSchemes
- Gauss gradient scheme
- - Interpolation schemes
- Least-squares gradient scheme
- 理论
- - 正交 computing gradient in cartesian grids
  - 非正交 green-gauss gradient
  - - 1 face-based graient computation
    - example1
    - 2.vertex-based gradient computation
- 补充学习

Gradient schemes
∇ϕ=e1∂∂x1ϕ+e2∂∂x2ϕ+e3∂∂x3ϕ\nabla \phi=\boldsymbol e_1\frac{\partial}{\partial x_1}\phi+\boldsymbol e_2\frac{\partial}{\partial x_2}\phi+\boldsymbol e_3\frac{\partial}{\partial x_3}\phi∇ϕ=e1∂x1∂ϕ+e2∂x2∂ϕ+e3∂x3∂ϕ

fvSchemes

gradSchemes
{default none;grad(p) <optional limiter> <gradient scheme> <interpolation scheme>;
}

Gauss gradient scheme

gradSchemes
{default         none;grad(U)         Gauss <interpolation scheme>;
}

Interpolation schemes

linear: cell-based linear
pointLinear: point-based linear
leastSquares: Least squares

Least-squares gradient scheme

gradSchemes
{default         none;grad(U)         leastSquares;
}

理论

正交 computing gradient in cartesian grids

(∂ϕ∂x)C=ϕE−ϕWXE−XW=ϕN−ϕSXN−XS(\frac{\partial \phi}{\partial x})_C=\frac{\phi_E-\phi_W}{X_E-X_W}=\frac{\phi_N-\phi_S}{X_N-X_S} (∂x∂ϕ)C=XE−XWϕE−ϕW=XN−XSϕN−ϕS

非正交 green-gauss gradient

两种计算方法ϕf\phi_fϕf

face-based
vertex-based

体心处梯度：
散度定理⇒\Rightarrow⇒
(∇ϕ)C=1VC∑fϕf⋅Sf(\nabla \phi)_C=\frac{1}{V_C}\sum_f \phi_f\cdot \boldsymbol S_f(∇ϕ)C=VC1∑fϕf⋅Sf
ϕf?\phi_f?ϕf?

fig. 9.3a，no skewness,f=f′f=f'f=f′
ϕf=γϕC+(1−γ)ϕF\phi_f=\gamma\phi_C+(1-\gamma)\phi_Fϕf=γϕC+(1−γ)ϕF
γ=∣∣rF−rf∣∣∣∣rF−rC∣∣=dFfdFC\gamma =\frac{\left||\boldsymbol r_F-r_f\right||}{\left||\boldsymbol r_F-r_C\right||}=\frac{d_{Ff}}{d_{FC}}γ=∣∣rF−rC∣∣∣∣rF−rf∣∣=dFCdFf
二阶精度：γ=0.5,ϕf=ϕP+ϕN2\gamma=0.5,\phi_f=\frac{\phi_P+\phi_N}{2}γ=0.5,ϕf=2ϕP+ϕN

fig. 9.3c，skewness,f≠f′f\neq f'f=f′
泰勒公式：
f(x)=f(x0)+f′(x0)(x−x0)+f′′(x−x0)2+...f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x-x_0)^2+...f(x)=f(x0)+f′(x0)(x−x0)+f′′(x−x0)2+...
ϕf=ϕf′+correction=ϕf′+(∇ϕ)f′(rf−rf′)≈f(x0)+f′(x0)(x−x0)\begin{aligned} \phi_f&=\phi_{f'}+correction\\ &=\phi_{f'}+(\nabla \phi)_{f'}(\boldsymbol r_f-\boldsymbol r_{f'})\\ &\approx f(x_0)+f'(x_0)(x-x_0)\\ \end{aligned} ϕf=ϕf′+correction=ϕf′+(∇ϕ)f′(rf−rf′)≈f(x0)+f′(x0)(x−x0)

基于体心的需要修正，基于节点的不需要修正，因为需要求ϕf\phi_fϕf
用Gauss求体心的梯度，就要知道面心值，而面心值要修正就要知道体心的梯度

1 face-based graient computation

1.计算ϕf′=γϕP+(1−γ)ϕN2.计算(∇ϕ)P=1VP∑fϕf′⋅Sf3.计算更新的ϕf=ϕf′+γ(∇ϕ)P⋅(rf−rP)+(1−γ)(∇ϕ)N⋅(rf−rN)4.更新(∇ϕ)P=1VP∑fϕf⋅Sf5.重复3\begin{aligned} &1.计算 \phi_{f'}=\gamma\phi_P+(1-\gamma)\phi_N\\ & 2.计算(\nabla \phi)_P=\frac{1}{V_P}\sum_f\phi_{f'}\cdot\boldsymbol S_f\\ & 3.计算更新的\phi_f=\phi_{f'}+\gamma(\nabla \phi)_P\cdot(\boldsymbol r_f-\boldsymbol r_P)+(1-\gamma)(\nabla\phi)_N \cdot (\boldsymbol r_f - \boldsymbol r_N)\\ & 4.更新(\nabla \phi)_P=\frac{1}{V_P}\sum_f\phi_f\cdot\boldsymbol S_f\\ & 5.重复3 \end{aligned} 1.计算ϕf′=γϕP+(1−γ)ϕN2.计算(∇ϕ)P=VP1f∑ϕf′⋅Sf3.计算更新的ϕf=ϕf′+γ(∇ϕ)P⋅(rf−rP)+(1−γ)(∇ϕ)N⋅(rf−rN)4.更新(∇ϕ)P=VP1f∑ϕf⋅Sf5.重复3

example1

a.Green-Gauss method with no correction

(∇ϕ)C=1VC∑fϕf⋅Sf(\nabla \phi)_C=\frac{1}{V_C}\sum_f \phi_f\cdot \boldsymbol S_f(∇ϕ)C=VC1∑fϕf⋅Sf

face centrolids:
f1(8.5,11),f2(10,6.5),f3(14.5,7),f4(17.25,11.5),f5(14.75.15.5),f6(10.5,15.5)f_1(8.5,11),f_2(10,6.5),f_3(14.5,7),f_4(17.25,11.5),f_5(14.75.15.5),f_6(10.5,15.5)f1(8.5,11),f2(10,6.5),f3(14.5,7),f4(17.25,11.5),f5(14.75.15.5),f6(10.5,15.5)

surface vectors:
Sf1=−6i+jS_{f1}=-6\boldsymbol i+\boldsymbol jSf1=−6i+j
Sf2=−3i−4jS_{f2}=-3\boldsymbol i-4\boldsymbol jSf2=−3i−4j
Sf3=4i−5jS_{f3}=4\boldsymbol i-5\boldsymbol jSf3=4i−5j
Sf4=5i−0.5jS_{f4}=5\boldsymbol i-0.5\boldsymbol jSf4=5i−0.5j
Sf5=3i+5.5jS_{f5}=3\boldsymbol i+5.5\boldsymbol jSf5=3i+5.5j
Sf6=−3i+3jS_{f6}=-3\boldsymbol i+3\boldsymbol jSf6=−3i+3j

interpolation factor (gc)n(gc)_n(gc)n:
(gc)1=0.487(gc)_1=0.487(gc)1=0.487
(gc)2=0.427(gc)_2=0.427(gc)2=0.427
(gc)3=0.502(gc)_3=0.502(gc)3=0.502
(gc)1=0.538(gc)_1=0.538(gc)1=0.538
(gc)1=0.492(gc)_1=0.492(gc)1=0.492
(gc)1=0.455(gc)_1=0.455(gc)1=0.455

ϕf\phi_fϕf value:
ϕf1=(gc)1ϕC+(1−(gc)1)ϕF=110.442\phi_{f1}=(gc)_1\phi_C+(1-(gc)_1)\phi_F=110.442ϕf1=(gc)1ϕC+(1−(gc)1)ϕF=110.442
ϕf2=91.364\phi_{f2}=91.364ϕf2=91.364
ϕf3=123.674\phi_{f3}=123.674ϕf3=123.674
ϕf4=206.27\phi_{f4}=206.27ϕf4=206.27
ϕf5=263.012\phi_{f5}=263.012ϕf5=263.012
ϕf5=158.28\phi_{f5}=158.28ϕf5=158.28

(∇ϕ)C=1VC∑fϕf⋅Sf=11.889i+12.433j(\nabla \phi)_C=\frac{1}{V_C}\sum_f \phi_f\cdot \boldsymbol S_f=11.889\boldsymbol i+12.433\boldsymbol j(∇ϕ)C=VC1∑fϕf⋅Sf=11.889i+12.433j

b.Green-Gauss method with correction

ϕf′\phi_{f'}ϕf′ value:
ϕf1′=ϕC+ϕF12=111.875\phi_{f1'}=\frac{\phi_C+\phi_{F1}}{2}=111.875ϕf1′=2ϕC+ϕF1=111.875
ϕf2′=101\phi_{f2'}=101ϕf2′=101
ϕf3′=123.5\phi_{f3'}=123.5ϕf3′=123.5
ϕf4′=209.5\phi_{f4'}=209.5ϕf4′=209.5
ϕf5′=261.5\phi_{f5'}=261.5ϕf5′=261.5
ϕf6′=158.5\phi_{f6'}=158.5ϕf6′=158.5
(∇ϕ)C=1VC∑fϕf′⋅Sf=11.53i+11.826j(\nabla \phi)_C=\frac{1}{V_C}\sum_{f} \phi_{f'}\cdot \boldsymbol S_f=11.53\boldsymbol i+11.826\boldsymbol j(∇ϕ)C=VC1∑fϕf′⋅Sf=11.53i+11.826j
ϕf\phi_{f}ϕf value:
ϕf1=ϕf1′+0.5∗[(∇ϕ)C+(∇ϕ)F1]⋅[rf−0.5∗(rC+rF)]=111.875+3.7435=115.6185\phi_{f1}=\phi_{f1'}+0.5*[(\nabla\phi)_C+(\nabla \phi)_{F1}]\cdot[\boldsymbol r_f-0.5*(\boldsymbol r_C+\boldsymbol r_F)]=111.875+3.7435=115.6185ϕf1=ϕf1′+0.5∗[(∇ϕ)C+(∇ϕ)F1]⋅[rf−0.5∗(rC+rF)]=111.875+3.7435=115.6185
ϕf2=91.911\phi_{f2}=91.911ϕf2=91.911
ϕf3=115.764\phi_{f3}=115.764ϕf3=115.764
ϕf4=224.097\phi_{f4}=224.097ϕf4=224.097
ϕf5=265.566\phi_{f5}=265.566ϕf5=265.566
ϕf6=176.046\phi_{f6}=176.046ϕf6=176.046
(∇ϕ)C=1VC∑fϕf⋅Sf=11.614i+13.76j(\nabla \phi)_C=\frac{1}{V_C}\sum_{f} \phi_{f}\cdot \boldsymbol S_f=11.614\boldsymbol i+13.76\boldsymbol j(∇ϕ)C=VC1∑fϕf⋅Sf=11.614i+13.76j

2.vertex-based gradient computation

ϕn=∑k=1NB(n)ϕFk∣∣rn−rFk∣∣∑k=1NB(n)1∣∣rn−rFk∣∣\phi_n=\frac{\sum_{k=1}^{NB(n)}\frac{\phi_{F_k}}{\left||\boldsymbol r_n-\boldsymbol r_{F_k}\right||}}{\sum_{k=1}^{NB(n)}\frac{1}{\left||\boldsymbol r_n-\boldsymbol r_{F_k}\right||}}ϕn=∑k=1NB(n)∣∣rn−rFk∣∣1∑k=1NB(n)∣∣rn−rFk∣∣ϕFk
ϕf=ϕn1+ϕn22\phi_f=\frac{\phi_{n1}+\phi_{n2}}{2}ϕf=2ϕn1+ϕn2

补充学习

fluid mechanics 101 26-28

GreenGaussGradient
LeastSquaresGradient
NodeBasedGradient

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【CFD理论】梯度项-01

【CFD理论】梯度项-01

fvSchemes

Gauss gradient scheme

Interpolation schemes

Least-squares gradient scheme

理论

正交 computing gradient in cartesian grids

非正交 green-gauss gradient

1 face-based graient computation

example1

2.vertex-based gradient computation

补充学习

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