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Sandbox Physics

Fluid dynamics · a manipulable numerical field

Interactive Jet

A two-dimensional incompressible velocity field is extruded into luminous depth inside a transparent flow chamber. Tune viscosity, dye diffusion, pressure iterations, and vorticity enhancement to compare entrainment and the wake behind a cylinder; arm the fluid brush to inject dye and momentum by dragging through the test section.

Interactive modelInteractive Jet
Last solver-step cost0 ms0\,\mathrm{ms}
Simulation time0.0000 s0.0000\,\mathrm{s}
Current field DliveD_{\mathrm{live}}0.0000 s−10.0000\,\mathrm{s^{-1}}
Last solve: before DbeforeD_{\mathrm{before}}n/a\text{n/a}
Last solve: after DafterD_{\mathrm{after}}n/a\text{n/a}
Divergence reductionn/a\text{n/a}
Last solve iterationsn/a\text{n/a}
Projection snapshotNo projection yet
Mean kinetic energy K/ρK/\rho0.00 J kg−10.00\,\mathrm{J\,kg^{-1}}
Peak vorticity ∣ω∣max⁡\lvert\omega\rvert_{\max}0.00 s−10.00\,\mathrm{s^{-1}}

Physics tutorial

Interactive Jet: what advection, diffusion, and projection each do

BackgroundA real-time canvas stores velocity and dye on a fixed grid. Each step alternates velocity diffusion, advection, and pressure projection before transporting dye. The two physical cell spacings are used separately; slow playback does not change the time units of the readouts.

Why it mattersSeparating the solver stages turns swirling motion into more than an effect: divergence can be measured before and after pressure projection, exposing the exchange between stability and numerical dissipation.

Start with the essentials

Focus question
Why can one pressure projection return an arbitrarily pushed velocity field toward incompressibility?
One-sentence intuition
Projection corrects an intermediate velocity with a potential gradient. Paint while paused and project only to compare the same field. Finite iterations and a collocated grid leave a measured residual, not a claim of ∇h⋅u=0\nabla_h\cdot\boldsymbol{u}=0.

Core mathematical model

Incompressible velocity evolution

∂u∂t+(u⋅∇)u=−∇p+ν∇2u+f,∇⋅u=0\frac{\partial\boldsymbol{u}}{\partial t}+\left(\boldsymbol{u}\cdot\nabla\right)\boldsymbol{u}=-\nabla p+\nu\nabla^2\boldsymbol{u}+\boldsymbol{f},\qquad \nabla\cdot\boldsymbol{u}=0

Advection transports momentum, viscosity diffuses velocity gradients, the nozzle and brush supply force, and pressure enforces incompressibility.

Semi-Lagrangian backtrace

ϕn+1(x)≈ϕn ⁣(x−Δt un(x))\phi^{n+1}(\boldsymbol{x})\approx\phi^n\!\left(\boldsymbol{x}-\Delta t\,\boldsymbol{u}^n(\boldsymbol{x})\right)

Trace backward from each grid point and interpolate the old field. The step is robust, but repeated interpolation smooths detail.

Pressure projection

∇h2φ=∇h⋅u∗,u′=u∗−∇hφ\nabla_h^2\varphi=\nabla_h\cdot\boldsymbol{u}^{*},\qquad \boldsymbol{u}'=\boldsymbol{u}^{*}-\nabla_h\varphi

The stored velocity potential absorbs the time-step and density factors; it is not pressure in Pascals. Rectangular-grid relaxation uses different horizontal and vertical weights. A finite-iteration projection does not guarantee zero divergence.

Live divergence root mean square

Dlive=1Nf∑i∈fluid(∇h⋅u)i2,R=DbeforeDafterD_{\mathrm{live}}=\sqrt{\frac{1}{N_f}\sum_{i\in\mathrm{fluid}}\left(\nabla_h\cdot\boldsymbol{u}\right)_i^2},\qquad R=\frac{D_{\mathrm{before}}}{D_{\mathrm{after}}}

The live value is recalculated from the current velocity. The last before/after pair is a historical snapshot, marked stale after painting or parameter edits. A zero-field reduction ratio is undefined, not a claimed improvement.

Common difficulties

Stable does not mean accurate

Typical misconceptionIf the algorithm never blows up, every small vortex must be preserved correctly.

Better mental modelSemi-Lagrangian advection remains stable at large time steps but introduces interpolation dissipation. Vorticity enhancement restores visual detail, not the true scales that were lost.

Spatial particles do not make a three-dimensional solve

Typical misconceptionBecause particles occupy depth, the solver must resolve three-dimensional turbulence.

Better mental modelEvery particle samples the same two-dimensional velocity section. Depth makes structure easier to inspect but cannot create three-dimensional vortex stretching.

Run the experiment

  1. 01

    Build a cylinder wake

    Choose Cylinder wake and let the dye pass the obstacle.

    What to observe: The jet slows and splits upstream of the cylinder, leaving a low-speed wake and alternating rotational structure downstream.
  2. 02

    Inspect pressure projection

    Pause, arm the brush, and paint a disturbance. Enable Divergence diagnostic and click Project only. Change the iteration count, then project again.

    What to observe: The before value matches the live residual before the click; the after value matches the new live residual. Time and dye do not advance. More iterations usually reduce residual, but do not prove exact incompressibility.
  3. 03

    Add momentum by hand

    Arm the fluid brush, drag sideways through the wake, then disable it to orbit the chamber.

    What to observe: Local dye and velocity change together; the new disturbance is advected, diffused, and projected again.

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