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This question was proposed on Physic StackExchange yesterday (https://physics.stackexchange.com/questions/830965/how-to-determine-the-alpha-value-of-artificial-viscosity-in-smoothed-particle-hy), but was closed today because it deals more with engineering instead of physics, and is thus off-topic. So I propose it here and hope someone may help me explain it.
I am confused about how to choose an appropriate value of $\alpha$ in the artificial viscosity. The value that I deduced is far from the recommended value and led to great numerical instability.
Artificial viscosity is introduced into the momentum equation of smoothed particle hydrodynamics:
$$\Pi_{ij}=-\alpha h\frac{c_i+c_j}{\rho_i+\rho_j}\frac{\boldsymbol{v}_{ij}\cdot\boldsymbol{r}_{ij}}{{r_{ij}}^2+\epsilon h^2},$$
where $\alpha$ is a dimensionless factor, $h$ is SPH kernel radius and $c$ is the speed of sound. The artificial viscosity is related to the physical dynamic viscosity (Pa*s) by
$$\mu=\frac{\rho\alpha hc}{8}$$
for two-dimensional cases [1]. Hence, if the values of $h$, $c$ and $\mu$ are given, we can estimate the value of $\alpha$ as
$$\alpha=\frac{8\mu}{\rho hc}.$$
When it comes to the sound of speed, in order to both limit the density variation within 1% ($\delta\rho/\rho\sim v^2/c^2$) and allow an acceptable timestep (by the CFL condition), $c$ is also artificial, and customary to be $10v_\mathrm{max}$, where $v_\mathrm{max}$ is the maximal fluid velocity [2,3]. As to the case of dam break with the initial water column height of $H_0$, the estimate of $v_\mathrm{max}$ is
$$v_\mathrm{max}=\sqrt{2gH_0}.$$
So Monaghan set $c$ as $\sqrt{200gH_0}$ in [2].
Assume the initial spacing between fluid particles is $H_0/N$, and the SPH kernel radius is triple the spacing,
$$h=3\frac{H_0}{N}.$$
Now we may obtain a proper value of $\alpha$:
$$\alpha=\frac{8\mu}{\rho\cdot(3H_0/N)\cdot 10\sqrt{2gH_0}}=\frac{2\sqrt{2}}{15}\frac{\mu N}{\rho\sqrt{gH_0^3}}$$
In the case of dam break, one can assume that $\mu=1\times 10^{-3}~\mathrm{Pa\cdot s}$ (water), $\rho=1000~\mathrm{kg/m^3}$, $100\leq N\leq 1000$, $0.1~\mathrm{m}\leq H_0 \leq 1~\mathrm{m}$, $g=9.81~\mathrm{m/s^2}$, and we may estimate that
$$6\times10^{-6}\leq\alpha\leq2\times10^{-3}.$$
This is way too far from the recommended range of $\alpha$ which is 0.01-1.
And when I used the estimated alpha value to simulate the dam break, it could not converge as expected. **So, I wonder whether there is any mistake in my estimation, or any misunderstanding of the SPH theory.** Any comments or advice will be appreciated!
**References**
[1] Monaghan, J. J. Smoothed particle hydrodynamics. Rep. Prog. Phys. 68, 1703–1759 (2005).
[2] Monaghan, J. J. Simulating Free Surface Flows with SPH. Journal of Computational Physics 110, 399–406 (1994).
[3] Monaghan, J. J. Smoothed Particle Hydrodynamics and Its Diverse Applications. Annual Review of Fluid Mechanics 44, 323–346 (2012).
The choice of α\alphaα depends on several factors related to the nature of the problem you are simulating. Here are some key considerations for selecting α\alphaα:
1. Shock Capturing:
Artificial viscosity is often introduced in SPH to handle shock waves and prevent interpenetration of particles. The value of α\alphaα needs to be high enough to smooth out shocks but not so large that it causes excessive dissipation in regions without shocks.
- Typical Values: α\alphaα typically ranges between 0.1 and 1.0.For simulations with strong shocks, such as astrophysical phenomena or high-speed fluid flows, values closer to α=1.0\alpha = 1.0α=1.0 may be appropriate. For weak shocks or more subtle fluid dynamics, lower values, like α=0.1\alpha = 0.1α=0.1 or α=0.01\alpha = 0.01α=0.01, might be sufficient to prevent unphysical particle interactions without excessive dissipation.
2. Physical Dissipation Balance:
Artificial viscosity adds a non-physical dissipation to prevent particle interpenetration and manage numerical artifacts. It’s important to ensure that α\alphaα is not so high that it introduces artificial damping that overwhelms the physical viscosity or energy transport in the system.
- Adaptive Techniques: Many modern SPH codes use adaptive viscosity where α\alphaα varies locally depending on the flow properties (like the local divergence of velocity). This can help reduce dissipation away from shock regions. A common approach is to initialize with a high α\alphaα (near 1) in shocked regions, and let it decay to a minimum value (e.g., α=0.01\alpha = 0.01α=0.01) in smoother regions.
3. Problem-Specific Tuning:
- In astrophysical simulations (such as star formation or galaxy dynamics), shocks are prevalent, so α\alphaα values near 1.0 might be more appropriate.
- In low-speed fluid dynamics, such as modeling gentle flows or incompressible fluids, α\alphaα should be small, to avoid over-damping the flow and distorting the physics.
Benchmarking: It's often useful to tune α\alphaα through benchmarking, comparing the SPH results against analytical or experimental data.
4. Time-Stepping and Resolution:
Finer particle resolution and smaller time steps reduce the need for high artificial viscosity values. If your simulation uses higher resolution or smaller time steps, you may opt for a smaller α\alphaα, as the particles themselves will naturally behave more stably due to the finer resolution.
5. Use of Additional Switches:
Some methods introduce additional "switches" or second-order terms (such as a β\betaβ-term in the artificial viscosity) to further control dissipation, which can help reduce the need for large values of α\alphaα. For example, the Monaghan-Balsara switch helps minimize artificial viscosity in shear flows.
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