Weakly Compressible SPH

WeaklyCompressibleSPHSystem(initial_condition,
                            density_calculator, state_equation,
                            smoothing_kernel, smoothing_length;
                            viscosity=nothing, density_diffusion=nothing,
                            acceleration=ntuple(_ -> 0.0, NDIMS),
                            buffer_size=nothing,
                            correction=nothing, source_terms=nothing,
                            surface_tension=nothing, surface_normal_method=nothing,
                            reference_particle_spacing=0.0))

System for particles of a fluid. The weakly compressible SPH (WCSPH) scheme is used, wherein a stiff equation of state generates large pressure changes for small density variations. See Weakly Compressible SPH for more details on the method.

Arguments

• initial_condition: InitialCondition representing the system's particles.
• density_calculator: Density calculator for the system. See ContinuityDensity and SummationDensity.
• state_equation: Equation of state for the system. See StateEquationCole.
• 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). See ArtificialViscosityMonaghan or ViscosityAdami.
• density_diffusion: Density diffusion terms for this system. See DensityDiffusion.
• acceleration: Acceleration vector for the system. (default: zero vector)
• buffer_size: Number of buffer particles. This is needed when simulating with OpenBoundarySPHSystem.
• correction: Correction method used for this system. (default: no correction, see Corrections)
• 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.
• surface_tension: Surface tension model used for this SPH system. (default: no surface tension)
• surface_normal_method: The surface normal method to be used for this SPH system. (default: no surface normal method or ColorfieldSurfaceNormal() if a surface_tension model is used)
• reference_particle_spacing: The reference particle spacing used for weighting values at the boundary, which currently is only needed when using surface tension.

StateEquationCole(; sound_speed, reference_density, exponent,
                  background_pressure=0.0, clip_negative_pressure=false)

Equation of state to describe the relationship between pressure and density of water up to high pressures.

Keywords

• sound_speed: Artificial speed of sound.
• reference_density: Reference density of the fluid.
• exponent: A value of 7 is usually used for most simulations.
• background_pressure=0.0: Background pressure.
• clip_negative_pressure=false: Negative pressure values are clipped to 0, which prevents spurious surface tension with SummationDensity but allows unphysical rarefaction of the fluid.

StateEquationIdealGas( ;sound_speed, reference_density, gamma, background_pressure=0.0,
                       clip_negative_pressure=false)

Equation of state to describe the relationship between pressure and density of a gas using the Ideal Gas Law.

Keywords

• sound_speed : Artificial speed of sound.
• reference_density : Reference density of the fluid.
• gamma : Heat-capacity ratio
• background_pressure=0.0 : Background pressure.
• clip_negative_pressure=false: Negative pressure values are clipped to 0, which prevents spurious surface tension with SummationDensity but allows unphysical rarefaction of the fluid.

DensityDiffusion

An abstract supertype of all density diffusion formulations.

Currently, the following formulations are available:

See Density Diffusion for a comparison and more details.

DensityDiffusionAntuono(initial_condition; delta)

The commonly used density diffusion terms by Antuono (2010), also referred to as δ-SPH. The density diffusion term by Molteni (2009) is extended by a second term, which is nicely written down by Antuono (2012).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = 2\left(\rho_a - \rho_b - \frac{1}{2}\big(\nabla\rho^L_a + \nabla\rho^L_b\big) \cdot r_{ab}\right) \frac{r_{ab}}{\Vert r_{ab} \Vert^2},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively and $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$. The symbol $\nabla\rho^L_a$ denotes the renormalized density gradient defined as

\[\nabla\rho^L_a = -\sum_b (\rho_a - \rho_b) V_b L_a \nabla_{r_a} W(\Vert r_{ab} \Vert, h)\]

with

\[L_a := \left( -\sum_{b} V_b r_{ab} \otimes \nabla_{r_a} W(\Vert r_{ab} \Vert, h) \right)^{-1} \in \R^{d \times d},\]

where $d$ is the number of dimensions.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

DensityDiffusionFerrari()

A density diffusion term by Ferrari (2009).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = \frac{\rho_a - \rho_b}{2h} \frac{r_{ab}}{\Vert r_{ab} \Vert},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively, $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$ and $h$ is the smoothing length.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.

DensityDiffusionMolteniColagrossi(; delta)

The commonly used density diffusion term by Molteni (2009).

The term $\psi_{ab}$ in the continuity equation in DensityDiffusion is defined by

\[\psi_{ab} = 2(\rho_a - \rho_b) \frac{r_{ab}}{\Vert r_{ab} \Vert^2},\]

where $\rho_a$ and $\rho_b$ denote the densities of particles $a$ and $b$ respectively and $r_{ab} = r_a - r_b$ is the difference of the coordinates of particles $a$ and $b$.

See DensityDiffusion for an overview and comparison of implemented density diffusion terms.