Fluid dynamics · simulating a splash with particles

SPH Free-Surface Splash

How can particles that only sense their neighbors work out the density and pressure of splashing water? Each particle estimates the local density by adding up its nearby neighbors with a smooth weighting function. An equation of state then turns that density into pressure. The water is treated as slightly compressible: giving it a higher artificial sound speed keeps the density closer to constant, but forces smaller time steps.

A two-dimensional, weakly compressible SPH simulation builds a free surface out of moving particles. In six scenarios, the particle field, the compact kernel, the pressure–depth profile and the histories of energy and density error update together on one canvas.

Simulation time

t\sqrt{g/H}

$0$

Fluid particles

N

$260$

Bulk density RMS

\varepsilon_\rho

$0\%$

Normalized mass

M/M_0

$1.000$

Acoustic time step

\Delta t_{\mathrm{CFL}}

$0$

Scenario validation $x_f/H=0$

Numerical experiment

Look at individual particles

3D volume solverParticles also move in depth, with a 3D kernel and particle mass. The spacing is coarse so the browser can keep up.Numerical cohesionA simplified attraction between particle pairs, labeled here as a numerical model term.Color by pressureTurn off to color by normalized density instead.Selected kernel neighborhoodTap a particle to move the compact-support inspection circle.Kernel smoothing length

h/\Delta x

1.55

Particle spacing

\Delta x/H

0.040

Numerical sound speed

c_0/\sqrt{gH}

10

Kinematic viscosity

\nu/\sqrt{gH^3}

0.008

Gravity scale

g/g_0

1.00

The 3D mode uses the same Tait equation of state with a cubic kernel normalized for volume. The impact case sends a spherical droplet into a shallow pool. Pressure is clipped at zero. The walls are simplified: their ghost particles count only toward density, and contact with them is inelastic. The numerical cohesion is not a calibrated surface-tension coefficient, and a coarse splash is not a validated prediction of how a crown breaks up.

Use Hydrostatic check to see how pressure grows with depth, and Free fall check to compare the centroid with its analytic path. Changing the particle spacing shows how sensitive the result is to resolution.

References: Monaghan (1994), DOI 10.1006/jcph.1994.1034; Morris, Fox & Zhu (1997), DOI 10.1006/jcph.1997.5776.

This is a two-dimensional weakly compressible SPH teaching model. Density comes from the sum $\rho_i=\sum_j m_j W_{ij}$ , and pressure from the Tait form $p_i=B\left[\left(\rho_i/\rho_0\right)^7-1\right]$ . Numerical viscosity, simplified walls and an optional numerical cohesion are stated simplifications of the model, so the splash shape is not a high-fidelity engineering prediction.

Physics tutorial

SPH free surfaces: reconstructing a pool from neighbors

BackgroundSmoothed particle hydrodynamics (SPH) splits a fluid into moving particles, each carrying mass, position and velocity. A particle looks only at neighbors within a finite radius, estimates the density from a weighted sum, and then moves under pressure gradients, viscosity and gravity. Where the particles end, a free surface appears on its own.

Why it mattersDam breaks, sloshing and droplet impacts all have interfaces that move violently. A fixed grid needs a separate method to track the interface, but Lagrangian particles simply move with the fluid. The price is noisy density, errors at the boundaries and a strict limit on the explicit time step.

Start with the essentials

Focus question: If each particle knows only its nearby neighbors, how do they together produce pressure, nearly incompressible water and the motion of a free surface?
One-sentence intuition: A kernel that is nonzero only over a short range turns a handful of discrete neighbors into a smooth estimate. Raising the numerical sound speed $c_0$ reduces density variation but shortens the stable time step. In weakly compressible SPH (WCSPH), water that compresses less costs more computing time.

Core mathematical model

SPH density summation

\rho_i=\sum_j m_j W\!\left(\lVert\boldsymbol r_i-\boldsymbol r_j\rVert,h\right)

The cubic-spline kernel is nonzero only within a finite support radius. A particle inside the bulk has neighbors almost all around it; a particle on the free surface has none on the air side, so its neighborhood is cut short.

Symmetric pressure force and viscosity

\frac{\mathrm d\boldsymbol v_i}{\mathrm dt}=-\sum_j m_j\left(\frac{p_i}{\rho_i^2}+\frac{p_j}{\rho_j^2}\right)\nabla_i W_{ij}+\nu\nabla^2\boldsymbol v_i+\boldsymbol g

The pressure term is symmetric in each pair of particles, so the forces come in pairs. The viscous Laplacian, wall reflection and cohesion used here are deliberately low-order teaching approximations.

Tait equation of state and acoustic CFL

p_i=\frac{\rho_0c_0^2}{\gamma}\left[\left(\frac{\rho_i}{\rho_0}\right)^\gamma-1\right],\qquad \Delta t\leq C\frac{h}{c_0+\max_i\lVert\boldsymbol v_i\rVert},\qquad \gamma=7

The equation of state lets the water compress by a finite amount, and in return the simulation can step forward explicitly. A higher numerical sound speed holds down the density error, but it also shrinks the allowed time step.

Common difficulties

Particles are not water molecules

Typical misconceptionEach blue point in the rendering is one real water molecule.

Better mental modelAn SPH particle stands for a small but finite parcel of fluid. It carries a smoothed mass, velocity and thermodynamic state; it is not an individual molecule.

A pretty splash is not convergence

Typical misconceptionIf the crown-shaped splash looks right, the solver must be correct.

Better mental modelFirst check the total mass, the density error in the bulk, the hydrostatic relation $p=\rho g d$ , the free-fall motion of the centroid and how much the result changes with particle spacing. Looking right is only the last layer of evidence.

Run the experiment

01
See how a kernel neighborhood estimates density
Pause the dam break and tap a bulk particle, then a free-surface particle. Compare the support circle in panel A and the neighbor count in panel B.
What to observe: A bulk particle has neighbors on all sides, spread almost evenly in every direction. A surface particle has none on the air side, so its density sum comes out short at the boundary.
02
Test hydrostatic pressure
Choose Hydrostatic check, reduce the viscosity, step forward, and compare the particle pressures with the hydrostatic line in panel C.
What to observe: Pressure rises with depth overall, but particle noise, the wall treatment and the low density at the free surface pull the points away from the analytic line.
03
Compare sound speed and time step
Reset the same scenario at lower and higher numerical sound speeds, then compare bulk-density RMS with the acoustic time-step readout.
What to observe: A higher sound speed usually reduces the density fluctuations but needs a shorter time step. This is the central computational tradeoff of weakly compressible SPH.
04
Check the resolution
Change particle spacing, reset the dam break, and compare front position, density error, and histories at similar simulation times.
What to observe: The particle count, the neighborhood sampling and the stable time step all change together. If the result shifts a lot with resolution, a single splash run is not a reliable prediction.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.

Core mathematical model

SPH density summation

Symmetric pressure force and viscosity

Tait equation of state and acoustic CFL

Common difficulties

Particles are not water molecules

A pretty splash is not convergence

Run the experiment

See how a kernel neighborhood estimates density

Test hydrostatic pressure

Compare sound speed and time step

Check the resolution