Numerical Methods
AUSM+up for All Speeds, Single Phase
This algorithm is used to calculate the convective fluxes \(F_c\) in the Governing Equations.
\[M_{L/R} = \frac{u_{L/R}}{a_{1/2}}\]where \(a_{1/2}\) is an average of \(a_L\) and \(a_R\)
\[\bar{M}^2 = \frac{(u^2_L+u^2_R)}{2 a^2_{1/2}} \\ M_o^2=\min(1,\max(\bar{M}^2,M_\infty^2))\\\]where \(M_\infty \neq 0\) and is a specified parameter describing the representative Mach number for the given problem.
\[\begin{eqnarray} f_a = M_o(2-M_o)\\ \mathcal{M}_{(1)}^\pm (M) = \frac{1}{2} (M \pm |M|)\\ \mathcal{M}_{(2)}^\pm (M) = \pm \frac{1}{4}(M \pm 1)^2\\ \mathcal{M}_{(4)}^\pm (M)= \begin{cases} \mathcal{M}_{(1)}^\pm & \text{if } |M| \geq 1 \\ \mathcal{M}_{(2)}^\pm (1 \mp 16\beta \mathcal{M}_{(2)}^\mp ) & \text{otherwise} \end{cases}\\ M_{1/2}=\mathcal{M}_{(4)}^+(M_L)+\mathcal{M}_{(4)}^-(M_R)+M_p\\ M_p = -\frac{K_p}{f_a} \max(1-\sigma \bar{M}^2,0) \frac{p_R-p_L}{\rho_{1/2}a_{1/2}^2} \end{eqnarray}\]where \(\rho_{1/2}\) is also an average of \(\rho_L\) and \(\rho_R\).
\[\begin{eqnarray}\dot{m} = a_{1/2}M_{1/2} \begin{cases} \rho_L & \text{if } M_{1/2}>0\\ \rho_R & \text{otherwise} \end{cases} \\ \mathcal{P}_{(5)}^\pm = \begin{cases} \frac{1}{M} \mathcal{M}^\pm_{(1)} & \text{if } |M| \geq 1 \\ \mathcal{M}^\pm_{(2)}[(\pm 2 -M) \mp 16 \alpha_{coeff}M \mathcal{M}^\mp_{(2)}] & \text{otherwise} \end{cases}\\ p_{1/2}= \mathcal{P}_{(5)}^+(M_L)p_L+\mathcal{P}_{(5)}^-(M_R)p_R+P_u\\ P_u= -K_u \mathcal{P}_{(5)}^+ \mathcal{P}_{(5)}^- (\rho_L+\rho_R)(f_a a_{1/2})(u_R-u_L)\\ \alpha_{coeff} = \frac{3}{16} (-4 + 5 f_a^2) \end{eqnarray}\]| Coefficient | Value |
|---|---|
| \(K_p\) | 0.25 |
| \(K_u\) | 0.75 |
| \(\sigma\) | 1.0 |
| \(\beta\) | 1/8 |
AUSM+up for two phases
The below calculations are for gas-gas or liquid-liquid interfaces. Gas-liquid or liquid-gas interfaces must use a Riemann solver. Any definitions not listed in this section are the same as the single phase definitions.
| Coefficient | Value |
|---|---|
| \(K_p\) | 1.0 |
| \(K_u\) | 1.0 |
| \(\sigma\) | 2.0 |
where \(\bar{M}\) is the average between \(M_L\) and \(M_R\).
Riemann Flux
Riemann problem is a classic 1D initial value problem. The exact solution can be used as a means to calculate the fluxes in each cell. Based on the adjacent cell’s state, \(W_{L/R} = \{\rho_{L/R},\ u_{L/R},\ p_{L/R}\}\), an iterative scheme can be used to achieve exact solution. The initial condition is set as
\[\begin{equation} W(x,0) = \begin{cases} W_L, & \text{ if } 1 < x \\ W_R, & \text{ if } x \geq 1 \end{cases} \end{equation}\]The pressure \(p^*\) can be found by solving the root for the following equation
\[f(p, W_L, W_R) \equiv f_L(p, W_L) + f_R(p, W_R) + (u_R-u_L) =0\]where \(f_L\) is given as \(f_L(p, W_L) = \begin{cases} (p-p_L)\left[ \frac{A_L}{p+B_L}\right]^{1/2}, & \text{ if } p>p_L (Shock) \\ \frac{2a_L}{\gamma-1}\left[\left(\frac{p}{p_L}^\frac{\gamma-1}{2\gamma}\right) -1\right], &\text{ if }p\leq p_L (Rarefaction) \end{cases}\)
\(f_R\) is given as
\[f_R(p, W_R) = \begin{cases} \frac{2a_R}{\gamma-1}\left[\left(\frac{p}{p_R}^\frac{\gamma-1}{2\gamma}\right) -1\right], &\text{ if }p\leq p_R (Rarefaction) \\ (p-p_R)\left[ \frac{A_R}{p+B_R}\right]^{1/2}, &\text{ if }p> p_R (Shock) \end{cases}\] \[A_L=\frac{2}{(\gamma+1)\rho_L},\ B_L=\frac{\gamma-1}{\gamma+1}p_L\\ A_R=\frac{2}{(\gamma+1)\rho_R},\ B_R=\frac{\gamma-1}{\gamma+1}p_R \\ u_*=\frac{1}{2}(u_L+u_R)+\frac{1}{2}[f_R(p_*)-f_L(p_*)]\]After solving for the stared region properties, we can then back out the flow properties via the following logics.
| if | \(u_{*} \geq 0\) | ||||
|---|---|---|---|---|---|
| \(p_* > p_L\) (Left Shock) | \(p_*\leq p_L\) (Left Expansion) | ||||
| \(S_L \geq 0\) | \(S_L < 0\) | \(S_{TL} \geq 0 \& S_{HL} \geq 0\) | \(S_{TL} \geq 0 \& S_{HL} < 0\) | \(S_{TL} < 0\) | |
| \(u(x=0)\) | \(u_L\) | \(u_{*L}\) | \(u_L\) | \(\frac{2}{\gamma+1}\left[ a_L+\frac{\gamma-1}{2}u_L\right]\) | \(u_{*L}\) |
| \(\rho(x=0)\) | \(\rho_L\) | \(\rho_{*L}\) | \(\rho_L\) | \(\rho_L\left[\frac{2}{\gamma+1}+\frac{\gamma-1}{(\gamma+1)a_L}u_L \right]^{\frac{2}{\gamma-1}}\) | \(\rho_{*L}\) |
| \(p(x=0)\) | \(p_L\) | \(p_{*L}\) | \(p_L\) | \(p_L\left[ \frac{2}{\gamma+1}+\frac{\gamma-1}{(\gamma+1)a_L}u_L\right]^{\frac{2\gamma}{\gamma-1}}\) | \(p_{*L}\) |
| if | \(u_{*} < 0\) | ||||
| \(p_* < p_R\) (Right Shock) | \(p_*\geq p_R\) (Right Expansion) | ||||
| \(S_R \geq 0\) | \(S_R < 0\) | \(S_{HR} \geq 0 \& S_{TR} \geq 0\) | \(S_{HR} \geq 0 \& S_{TR} < 0\) | \(S_{HR} < 0\) | |
| \(u(x=0)\) | \(u_{*R}\) | \(u_R\) | \(u_{*R}\) | \(\frac{2}{\gamma+1}\left[-a_R+\frac{\gamma-1}{2}u_R\right]\) | \(u_R\) |
| \(\rho(x=0)\) | \(\rho_{*R}\) | \(\rho_R\) | \(\rho_{*R}\) | \(\rho_R\left[\frac{2}{\gamma+1}-\frac{\gamma-1}{(\gamma+1)a_R}u_R \right]^{\frac{2}{\gamma-1}}\) | \(\rho_R\) |
| \(p(x=0)\) | \(p_{*R}\) | \(p_R\) | \(p_{*R}\) | \(p_R\left[ \frac{2}{\gamma+1}-\frac{\gamma-1}{(\gamma+1)a_R}u_R\right]^{\frac{2\gamma}{\gamma-1}}\) | \(p_R\) |
Pressure Gradient Scaling (PGS)
Pressure Gradient Scaling (PGS) is a technique to address the very small-time steps sometimes associated with compressible low-speed flows. In this approach the acoustic wave speeds are artificially reduce thereby increasing the allowable step size. Implementation details and the affects on the LODI and AUSM+ flux vector splits methods are available in DesJardin et al. [4].
References
- [1] Liou, M. S. (2006). “A sequel to AUSM, Part II: AUSM+-up for all speeds.” Journal of Computational Physics, 214, 137-170.
- [2] Chang, C. H. and Liou, M. S. (2007), “A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+up scheme.” Journal of Computational Physics, 225, 840-873.
- [3] Toro, E. F. (2009). “Riemann Solvers and Numerical Methods for Fluid Dynamics - A PracticalIntroduction. International series of monographs on physics.” Springer. ISBN: 978-3-540-49834-6
- [4] DesJardin, Paul E., Timothy J. O’Hern, and Sheldon R. Tieszen. “Large eddy simulation and experimental measurements of the near-field of a large turbulent helium plume.” Physics of fluids 16.6 (2004): 1866-1883.