通量差分分裂格式 FDS-Roe 基本思想、推导过程、基本形式等
Roe’s approximate Riemann solver is based on the decomposition of the flux difference (FDS ) over a face of the control volume into a sum of wave contributions
F i + 1 2 = 1 2 [ F ( Q R ) + F ( Q L ) − ∣ A ˉ R o e ∣ i + 1 2 ( Q R − Q L ) ] \begin{equation}
\bold{F}_{i+\frac{1}{2}} = \frac{1}{2} \begin{bmatrix}
\bold{F}(\bold{Q}_R) + \bold{F}(\bold{Q}_L) -
|\bar{A}_{Roe}|_{i+\frac{1}{2}}
(\bold{Q}_R - \bold{Q}_L)
\end{bmatrix}
\end{equation} F i + 2 1 = 2 1 [ F ( Q R ) + F ( Q L ) − ∣ A ˉ R oe ∣ i + 2 1 ( Q R − Q L ) ]
The Roe matrix A ˉ R o e \bar{A}_{Roe} A ˉ R oe convective flux Jacobian A ˉ c \bar{A}_c A ˉ c Roe-averaged variables
A ˉ c = ∂ F c ∂ Q \begin{equation}
\bar{A}_c = \frac{\partial{\bold{F_c}}}{\partial{\bold{Q}}}
\end{equation} A ˉ c = ∂ Q ∂ F c
Conservative variables and convective fluxes
Q = [ ρ ρ u ρ v ρ w ρ E ] , F c = [ ρ V ρ u V + p n x ρ v V + p n y ρ w V + p n z ρ H V ] , \begin{gather}
\bold{Q} = \begin{bmatrix}
\rho \\
\rho u \\
\rho v \\
\rho w \\
\rho E \\
\end{bmatrix}, \quad
\bold{F_c} = \begin{bmatrix}
\rho V \\
\rho u V + p n_x \\
\rho v V + p n_y \\
\rho w V + p n_z \\
\rho H V \\
\end{bmatrix},
\end{gather} Q = ρ ρ u ρ v ρw ρE , F c = ρ V ρ u V + p n x ρ v V + p n y ρw V + p n z ρ H V ,
where total enthalpy and velocity (n x , n y , n z n_x, n_y, n_z n x , n y , n z
H = E + p / ρ , V = u n x + v n y + w n z \begin{gather}
H = E + p/\rho, \quad
V = u n_x + v n_y + w n_z
\end{gather} H = E + p / ρ , V = u n x + v n y + w n z
Performing eigen decomposition on Roe matrix
∣ A ˉ R o e ∣ = L − 1 ∣ Λ ~ ∣ L \begin{equation}
|\bar{A}_{Roe}| = L^{-1} |\tilde{\Lambda}|L
\end{equation} ∣ A ˉ R oe ∣ = L − 1 ∣ Λ ~ ∣ L
Thus (3D Case)
∣ A ˉ R o e ∣ i + 1 2 ( Q R − Q L ) = ∣ Δ F 1 ∣ + ∣ Δ F 2 , 3 , 4 ∣ + ∣ Δ F 3 ∣ \begin{equation}
|\bar{A}_{Roe}|_{i+\frac{1}{2}}(\bold{Q}_R - \bold{Q}_L) =
|\Delta \bold{F}_1| +
|\Delta \bold{F}_{2,3,4}| +
|\Delta \bold{F}_3|
\end{equation} ∣ A ˉ R oe ∣ i + 2 1 ( Q R − Q L ) = ∣Δ F 1 ∣ + ∣Δ F 2 , 3 , 4 ∣ + ∣Δ F 3 ∣
where
∣ Δ F 1 ∣ = ∣ V ~ − c ~ ∣ ( Δ p − ρ ~ c ~ Δ V 2 c ~ 2 ) [ 1 u ~ − c ~ n x v ~ − c ~ n y w ~ − c ~ n z H ~ − c ~ V ~ ] ∣ Δ F 2 , 3 , 4 ∣ = ∣ V ~ ∣ { ( Δ ρ − Δ p c ~ 2 ) [ 1 u ~ v ~ w ~ q ~ 2 / 2 ] + ρ ~ [ 0 Δ u − Δ V n x Δ v − Δ V n y Δ w − Δ V n z u ~ Δ u + v ~ Δ v + w ~ Δ w − V ~ Δ V ] } ∣ Δ F 5 ∣ = ∣ V ~ + c ~ ∣ ( Δ p + ρ ~ c ~ Δ u 2 c ~ 2 ) [ 1 u ~ + c ~ n x v ~ + c ~ n y w ~ + c ~ n z H ~ + c ~ V ~ ] \begin{align}
|\Delta \bold{F}_1| &=
|\tilde{V} - \tilde{c}|\begin{pmatrix}\displaystyle
\frac{\Delta p - \tilde{\rho}\tilde{c}\Delta V}{2\tilde{c}^2}
\end{pmatrix} \begin{bmatrix}
1 \\
\tilde{u} - \tilde{c} n_x \\
\tilde{v} - \tilde{c} n_y \\
\tilde{w} - \tilde{c} n_z \\
\tilde{H} - \tilde{c} \tilde{V}
\end{bmatrix} \\
|\Delta \bold{F}_{2,3,4}| &=
|\tilde{V}| \begin{Bmatrix}
\begin{pmatrix}\displaystyle
\Delta \rho - \frac{\Delta p}{\tilde{c}^2}
\end{pmatrix} \begin{bmatrix}
1 \\
\tilde{u} \\
\tilde{v} \\
\tilde{w} \\
\tilde{q}^2/2
\end{bmatrix} + \tilde{\rho} \begin{bmatrix}
0 \\
\Delta{u} - \Delta V n_x \\
\Delta{v} - \Delta V n_y \\
\Delta{w} - \Delta V n_z \\
\tilde{u} \Delta{u} + \tilde{v} \Delta{v} + \tilde{w} \Delta{w} - \tilde{V} \Delta{V}
\end{bmatrix}
\end{Bmatrix} \\
|\Delta \bold{F}_5| &=
|\tilde{V} + \tilde{c}|\begin{pmatrix}\displaystyle
\frac{\Delta p + \tilde{\rho}\tilde{c}\Delta u}{2\tilde{c}^2}
\end{pmatrix} \begin{bmatrix}
1 \\
\tilde{u} + \tilde{c} n_x \\
\tilde{v} + \tilde{c} n_y \\
\tilde{w} + \tilde{c} n_z \\
\tilde{H} + \tilde{c} \tilde{V}
\end{bmatrix} \\
\end{align} ∣Δ F 1 ∣ ∣Δ F 2 , 3 , 4 ∣ ∣Δ F 5 ∣ = ∣ V ~ − c ~ ∣ ( 2 c ~ 2 Δ p − ρ ~ c ~ Δ V ) 1 u ~ − c ~ n x v ~ − c ~ n y w ~ − c ~ n z H ~ − c ~ V ~ = ∣ V ~ ∣ ⎩ ⎨ ⎧ ( Δ ρ − c ~ 2 Δ p ) 1 u ~ v ~ w ~ q ~ 2 /2 + ρ ~ 0 Δ u − Δ V n x Δ v − Δ V n y Δ w − Δ V n z u ~ Δ u + v ~ Δ v + w ~ Δ w − V ~ Δ V ⎭ ⎬ ⎫ = ∣ V ~ + c ~ ∣ ( 2 c ~ 2 Δ p + ρ ~ c ~ Δ u ) 1 u ~ + c ~ n x v ~ + c ~ n y w ~ + c ~ n z H ~ + c ~ V ~
conservative variables difference
Δ Q = Q R − Q L \begin{equation}
\Delta \bold{Q} = \bold{Q_R} - \bold{Q_L}
\end{equation} Δ Q = Q R − Q L
Roe’s averages are computed from the left and the right state
ρ ~ = ρ L ρ R u ~ = u L ρ L + u R ρ R ρ L + ρ R v ~ = v L ρ L + v R ρ R ρ L + ρ R w ~ = w L ρ L + w R ρ R ρ L + ρ R H ~ = H L ρ L + H R ρ R ρ L + ρ R c ~ = ( γ − 1 ) ( H ~ − q ~ 2 / 2 ) V ~ = u ~ n x + v ~ n y + w ~ n z q ~ 2 = u ~ 2 + v ~ 2 + w ~ 2 \begin{align}
\tilde{\rho} &= \sqrt{\rho_L \rho_R} \\
\tilde{u} &= \frac{u_L \sqrt{\rho_L} + u_R \sqrt{\rho_R}}
{\sqrt{\rho_L} + \sqrt{\rho_R}} \\
\tilde{v} &= \frac{v_L \sqrt{\rho_L} + v_R \sqrt{\rho_R}}
{\sqrt{\rho_L} + \sqrt{\rho_R}} \\
\tilde{w} &= \frac{w_L \sqrt{\rho_L} + w_R \sqrt{\rho_R}}
{\sqrt{\rho_L} + \sqrt{\rho_R}} \\
\tilde{H} &= \frac{H_L \sqrt{\rho_L} + H_R \sqrt{\rho_R}}
{\sqrt{\rho_L} + \sqrt{\rho_R}} \\
\tilde{c} &= \sqrt{(\gamma - 1) (\tilde{H} - \tilde{q}^2/2)} \\
\tilde{V} &= \tilde{u} n_x + \tilde{v} n_y + \tilde{w} n_z \\
\tilde{q}^2 &= \tilde{u}^2 + \tilde{v}^2 + \tilde{w}^2 \\
\end{align} ρ ~ u ~ v ~ w ~ H ~ c ~ V ~ q ~ 2 = ρ L ρ R = ρ L + ρ R u L ρ L + u R ρ R = ρ L + ρ R v L ρ L + v R ρ R = ρ L + ρ R w L ρ L + w R ρ R = ρ L + ρ R H L ρ L + H R ρ R = ( γ − 1 ) ( H ~ − q ~ 2 /2 ) = u ~ n x + v ~ n y + w ~ n z = u ~ 2 + v ~ 2 + w ~ 2
the left and right state can be reconstructed by MUSCL scheme
∣ Λ c ∣ = { ∣ Λ c ∣ if ∣ Λ c ∣ > δ Λ c 2 + δ 2 2 δ if ∣ Λ c ∣ ≤ δ \begin{equation}
|\Lambda_c| = \begin{cases}
|\Lambda_c| & \text{if } |\Lambda_c| > \delta \\
\displaystyle{\frac{\Lambda_c^2 + \delta^2}{2 \delta}} & \text{if } |\Lambda_c| \leq \delta \\
\end{cases}
\end{equation} ∣ Λ c ∣ = ⎩ ⎨ ⎧ ∣ Λ c ∣ 2 δ Λ c 2 + δ 2 if ∣ Λ c ∣ > δ if ∣ Λ c ∣ ≤ δ
Stationary expansion question
use entropy correction and increase dissipation