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!
=========================
Happy to know that the question is reopened. These days I tested different speeds of sound ($c_0$) with Morris’s laminar viscosity model (4), and was surprised to observe that $c_0$ lower than $10v_\mathrm{max}$ can lead to a more stable pressure field!
N.B. To ensure the laminar flow, $H_0$ was set as 0.002 m such that Reynolds number=396. Other parameters are:
$\Delta L=3\times10^{-5}~\mathrm{m}, ~h=1.5\Delta L,~\Delta t=0.2h/c_0$.
$c_0$ is set to be 10, 5, 2.5 and 1.5 times of $v_\mathrm{max}$.
I compared the SPH results with those obtained by COMSOL two-phase laminar flow:
COMSOL showed that the velocity is between 0-0.18 m/s and the pressure difference is about 9 Pa. Lowering the speed of sound did not alter the velocity field significantly, but did stabilize the pressure field. This is really astonishing. It seems that $\delta\rho/\rho$ did not decrease as quickly as $v^2/c_0^2$. For example, when $c_0=1.5v_\mathrm{max}$, it showed that $\delta\rho/\rho=0.133$ while $v^2/c_0^2=0.36$.
References
-
Monaghan, J. J. Smoothed particle hydrodynamics. Rep. Prog. Phys. 68, 1703–1759 (2005).
-
Monaghan, J. J. Simulating Free Surface Flows with SPH. Journal of Computational Physics 110, 399–406 (1994).
-
Monaghan, J. J. Smoothed Particle Hydrodynamics and Its Diverse Applications. Annual Review of Fluid Mechanics 44, 323–346 (2012).
-
Morris, J. P., Fox, P. J. & Zhu, Y. Modeling low Reynolds number incompressible flows using SPH. Journal of Computational Physics 136, 214–226 (1997).
