SmoothedParticleHydrodynamics

Documentation for SmoothedParticleHydrodynamics.

The fluid is represented by a set of discrete particles. Each particle is centered at the location $\mathbf r_i$. The position varies in time:

\[\frac{d \mathbf r_i}{dt} = \mathbf v_i\]

where $\mathbf v_i$ is the velocity of the i-th particle. Each particle can have a property $A$:

\[A(\mathbf r) = \int A(\mathbf r') W(|\mathbf r - \mathbf r'|,h) d^n r'\]

The function $W$ is the kernel function which has a characteristic length-scale $h$ and it normalized:

\[\int W(|\mathbf r - \mathbf r'|,h) d^n r' = 1\]

and converges to the Dirac function $\delta(\mathbf r - \mathbf r')$ if $h$ tends to zero.

\[\lim_{h \rightarrow 0} W(|\mathbf r - \mathbf r'|,h) = \delta(\mathbf r - \mathbf r')\]

For small $h$ enought and sufficiently many particles, we can therefore approximate the integral using the discrete sum:

\[A(\mathbf r) = \sum_i V_i A_i W(|\mathbf r - \mathbf r_i|,h)\]

where $V_i$ is the volume associated to the i-th particle. Knowing the mass $m_j$ of every particle, we can calculate the density as:

\[\rho(r) = \sum_j m_j W(|\mathbf r - \mathbf r_j|,h)\]

since $m_i = V_i \rho_i$. The density at the particle location $\mathbf r_i$ becomes:

\[\rho_i = \rho(\mathbf r_i) = \sum_j m_j W(|\mathbf r_i - \mathbf r_j|,h)\]

If we differentiate the density relative to time:

\[\frac{d\rho_i}{dt} = \sum_j m_j (\mathbf v_i - \mathbf v_j) \nabla W_{ij}\]

which can be seen as a discretized form of:

\[\frac{d\rho}{dt} = -\rho \nabla \cdot v\]

The pressure gradient is computed using the symmetric form by analogy of the evolution equation of density:

\[\nabla p(r_i) = \rho_i \sum_j m_j \left(\frac{p_i}{\rho_i^2} + \frac{p_j}{\rho_j^2}\right) ∇W(|\mathbf r - \mathbf r_i|,h)\]

Similarly the viscosity force is given by:

\[F_i = \mu \nabla^2 \mathbf v(r_i) \mu \sum_j m_j \frac{\mathbf v_j - \mathbf v_i}{\rho_j} \left(\frac{p_i}{\rho_i^2} + \frac{p_j}{\rho_j^2}\right) ∇²W\]

SmoothedParticleHydrodynamics.KernelSpiky — Method
k = KernelSpiky(n,h)

Kernel in n dimension with characteristic radius h given by the following expression:

\[W(r,h) = c \; (h - r)^3\]

for $r \le h$ where $r$ is the radial distance and where the normalization coefficient $c$ is given by:

\[\frac{1}{S_n \, c} = \int_0^h\left(h-r\right)^3 r^{n-1}dr =\frac{h^{n+3}}{n}-\frac{3h^{n+3}}{n+1}+\frac{3h^{n+3}}{n+2}-\frac{h^{n+3}}{n+3}\]

where $S_n$ is the surface of the unit n-sphere.

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