Entropically Damped Artificial Compressibility (EDAC) for SPH

As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure

\[\frac{\mathrm{d} p_a}{\mathrm{d}t} = - \rho c_s^2 \nabla \cdot v + \nu \nabla^2 p,\]

which is derived by Clausen (2013). This equation is similar to the continuity equation (first term, see ContinuityDensity), but also contains a pressure damping term (second term, similar to density diffusion see DensityDiffusion), which reduces acoustic pressure waves through an entropy-generation mechanism.

The pressure evolution is discretized with the SPH method by Ramachandran (2019) as following:

The first term is equivalent to the classical artificial compressible methods, which are commonly motivated by assuming the artificial equation of state (StateEquationCole with exponent=1) and is discretized as

\[- \rho c_s^2 \nabla \cdot v = \sum_{b} m_b \frac{\rho_a}{\rho_b} c_s^2 v_{ab} \cdot \nabla_{r_a} W(\Vert r_a - r_b \Vert, h),\]

where $\rho_a$, $\rho_b$, $r_a$, $r_b$, denote the density and coordinates of particles $a$ and $b$ respectively, $c_s$ is the speed of sound and $v_{ab} = v_a - v_b$ is the difference in the velocity.

The second term smooths the pressure through the introduction of entropy and is discretized as

\[\nu \nabla^2 p = \frac{V_a^2 + V_b^2}{m_a} \tilde{\eta}_{ab} \frac{p_{ab}}{\Vert r_{ab}^2 \Vert + \eta h_{ab}^2} \nabla_{r_a} W(\Vert r_a - r_b \Vert, h) \cdot r_{ab},\]

where $V_a$, $V_b$ denote the volume of particles $a$ and $b$ respectively and $p_{ab}= p_a -p_b$ is the difference in the pressure.

The viscosity parameter $\eta_a$ for a particle $a$ is given as

\[\eta_a = \rho_a \frac{\alpha h c_s}{8},\]

where it is found in the numerical experiments of Ramachandran (2019) that $\alpha = 0.5$ is a good choice for a wide range of Reynolds numbers (0.0125 to 10000).

Note

The EDAC formulation keeps the density constant and this eliminates the need for the continuity equation or the use of a summation density to ﬁnd the pressure. However, in SPH discretizations, $m/\rho$ is typically used as a proxy for the particle volume. The density of the ﬂuids can therefore be computed using the summation density approach.

Ramachandran (2019)

TrixiParticles.EntropicallyDampedSPHSystem — Type

EntropicallyDampedSPHSystem(initial_condition, smoothing_kernel,
                            smoothing_length, sound_speed;
                            pressure_acceleration=inter_particle_averaged_pressure,
                            density_calculator=SummationDensity(),
                            alpha=0.5, viscosity=nothing,
                            acceleration=ntuple(_ -> 0.0, NDIMS), buffer_size=nothing,
                            source_terms=nothing)

System for particles of a fluid. As opposed to the weakly compressible SPH scheme, which uses an equation of state, this scheme uses a pressure evolution equation to calculate the pressure. See Entropically Damped Artificial Compressibility for SPH for more details on the method.

Arguments

initial_condition: Initial condition representing the system's particles.
sound_speed: Speed of sound.
smoothing_kernel: Smoothing kernel to be used for this system. See Smoothing Kernels.
smoothing_length: Smoothing length to be used for this system. See Smoothing Kernels.

Keyword Arguments

viscosity: Viscosity model for this system (default: no viscosity). Recommended: ViscosityAdami.
acceleration: Acceleration vector for the system. (default: zero vector)
pressure_acceleration: Pressure acceleration formulation (default: inter-particle averaged pressure). When set to nothing, the pressure acceleration formulation for the corresponding density calculator is chosen.
density_calculator: Density calculator (default: SummationDensity)
buffer_size: Number of buffer particles. This is needed when simulating with OpenBoundarySPHSystem.
source_terms: Additional source terms for this system. Has to be either nothing (by default), or a function of (coords, velocity, density, pressure, t) (which are the quantities of a single particle), returning a Tuple or SVector that is to be added to the acceleration of that particle. See, for example, SourceTermDamping. Note that these source terms will not be used in the calculation of the boundary pressure when using a boundary with BoundaryModelDummyParticles and AdamiPressureExtrapolation. The keyword argument acceleration should be used instead for gravity-like source terms.

source

References

Prabhu Ramachandran. "Entropically damped artiﬁcial compressibility for SPH". In: Computers and Fluids 179 (2019), pages 579–594. doi: 10.1016/j.compfluid.2018.11.023
Jonathan R. Clausen. "Entropically damped form of artificial compressibility for explicit simulation of incompressible flow". In: American Physical Society 87 (2013), page 13309. doi: 10.1103/PhysRevE.87.013309