Fluid simulation · grok-4.7

I will build an interactive 2D fluid-and-dye simulation using HTML, CSS, and JavaScript.

The simulation will use a Jos Stam-style stable fluid solver on Canvas 2D. Resolution is bounded for responsiveness at 1200 by 800. Dragging injects dye and momentum that persists after release. The UI includes a color selector, Pause/Resume, and Clear controls.

Grid resolution will be roughly 100 by 67 to keep frame work bounded. The solver uses semi-Lagrangian advection, Jacobi diffusion, and pressure projection for incompressibility. Each frame adds mouse forces, diffuses velocity, projects the field, advects velocity, then advects dye.

The simulation grid will be 80 by 54 cells. Pressure solve uses 8 to 12 Jacobi iterations and diffusion uses 4. Slight dissipation is applied to velocity and dye each frame.

Grid resolution is fixed at 120 by 80 cells. Explicit diffusion will be skipped in favor of dissipation and blending for efficiency. Vorticity confinement will be added to enhance swirling motion. Dye rendering will use canvas image smoothing for a smoother look.

Velocity will use free-slip boundary conditions at the grid edges. The color selector will combine a native color input with preset swatches. Initial dye will be seeded as rotating blobs so the fluid swirls on load. The solver will use Float32Arrays and implement Jos Stam-style boundary-setting functions.

The solver will use a one-cell border around the grid. Advection samples dye with bilinear interpolation at offset coordinates derived from velocity. Dye rendering will be drawn via ImageData and scaled to the full canvas using image smoothing.

Dye will fade by 0.5 percent and velocity by 0.2 percent each frame so swirls persist but eventually die out. The initial scene will seed colored rings in the center to create a starting rotation. Mouse coordinates map directly from the full-screen canvas into the fixed simulation grid.

The canvas will fill the viewport while the simulation grid stays independent. The solver performs roughly 500,000 cell operations per frame across diffusion, projection, advection, and dissipation. This workload should remain well within a few milliseconds.

The HTML and CSS will use a full-screen dark background with a crosshair cursor. A minimal HUD may show a paused status. The initial seed motion will be strong enough to create visible swirls before any user interaction.

The UI overlay will be a fixed top-left panel with an 8-pixel gap, containing a color picker, a Pause button, and a Clear button. The solver will use a one-cell border around the grid, indexing interior cells as `i + j * (W+2)`. Velocity and dye fields will swap array references each frame to avoid copying data.

Mouse input will use pointer events with capture to track drag segments. Each movement step stamps dye and velocity into the simulation grid, building up momentum along the path.

Mouse movement will be broken into sub-steps so the simulation grid receives continuous velocity and dye injection along the entire drag path. Each step splats a small Gaussian kernel of color and momentum across roughly three to four cells. Velocity is scaled to about five to twenty cells per frame with a force multiplier near two to eight. After the pointer is released, the stored velocity field continues advecting the injected dye.

Clear will zero out all velocity and dye arrays. The initial seed injects a circular velocity field in the center and splats three colored blobs around it. Pausing will freeze the simulation state while still drawing the current frame.

While paused, mouse input will still update the velocity and dye fields but skip the physics step. This lets the user drag and build up momentum that will advect once simulation resumes.

Dye values will be stored as 0-to-1 floats and faded by multiplying by 0.992 each frame. During rendering, they will be scaled to RGB with a slight bloom boost so the colors pop against the dark background. The canvas will use a pure black background rather than a faint grid.

Velocity damping is set to 0.998 per frame, giving a half-life of roughly six seconds so swirls linger without filling the canvas. Dye fades at 0.996, keeping colors visible longer. A small amount of velocity diffusion will be added for numerical stability.

The simulation will advance by exactly one fixed timestep per frame, capped at 60 fps, to prevent spiral-of-death accumulation. The internal grid resolution will stay independent of the CSS display size, and boundary index errors are being checked for.

I realized the advection offset must be in grid cells per frame, not seconds. Velocity will be stored as cells per frame and advected by subtracting the velocity vector directly from each cell index.

Mouse velocity injection will add momentum rather than overwrite existing values, so overlapping drags accumulate. A maximum cap of twelve cells per frame prevents numerical explosion. The pressure projection step will convert injected jet-like velocity into vortices. Gauss-Seidel will be implemented in-place for memory efficiency.

The pressure projection step now follows Stam's paper exactly, using a divergence field and four Jacobi iterations. The boundary setter is being adapted for the non-square grid dimensions.

The solver will follow Stam's exact eight-step sequence: add mouse sources, diffuse velocity, project for incompressibility, advect velocity, reproject, diffuse and advect dye, then dissipate. Mouse input will be injected directly into the velocity and dye arrays at the start of each frame before the step begins.

The solver will use a two-array swap pattern: one array holds the current velocity and the other holds the previous frame. Diffusion writes into the output array while the input array is held in the swapped buffer. After projection, the arrays are swapped again so advection reads the newly projected velocity field.

Vorticity confinement will be applied after velocity advection and before the second pressure projection. It computes curl from neighboring velocity differences, then applies a gradient-based force to sharpen vortices.

Vorticity epsilon will be tuned visually between 0.15 and 0.4. The UI panel will stop pointer events so drags can start anywhere on the canvas.

The simulation grid will be 120 by 80 cells, using twelve pressure iterations and no velocity diffusion to keep the solver lightweight. Dye advection will use three Jacobi iterations. The canvas will capture pointer events directly and suppress scrolling. A slight gamma correction will brighten dim dye for visibility.

A subtle "Drag to stir dye" label will sit on the canvas. No debug arrows will clutter the view. The verification plan uses browser motion capture to confirm that released drags keep swirling and that a second drag starts a fresh flow after clearing.

The Pause button will toggle its label to Resume. The color picker will read the current selection on every drag event so the active color updates mid-stir. A splat helper is being drafted to stamp velocity and dye into a circular region around each pointer position.

Each pointer event adds velocity equal to its local displacement. Fast drags therefore produce stronger jets than slow ones, and the total impulse scales with path length rather than cumulative speed.

Velocity injection will be multiplied by 0.6 so fast swipes stay strong without numerical explosion. Each splat adds about 0.35 units of dye per cell, with a radius of three cells, letting the color accumulate visibly along the drag path.

Dye will fade at 0.995 per frame so long drags do not turn the canvas into a solid blob. The solver will render to an offscreen canvas and scale it onto the main display with image smoothing enabled. A dark vignette or velocity tint will be skipped to keep the view simple.

The dye fades too quickly. The motion after release is hard to see because the dye dissipated.

The drag trail looks static and blocky instead of smooth. Velocity may be too high causing numerical dissipation. Dye damping should be reduced to about 0.998.

The 8-cell maximum speed may be causing advection instability. The white blowout comes from stamping too much dye along the path. The motion frames start already very dark because the dye has mostly dissipated before capture begins.

The 0.994 damping factor alone would leave about 70% of the dye after one second, so the rapid fade is mainly velocity spreading it too thin. The chunky trail is also a rendering issue that crushes dark values.

The drag tool injects velocity equal to 85% of the segment displacement. A large jump therefore creates a strong jet. The chunky trail comes from coarse sampling and splat radius. The white end is caused by overlapping splats saturating dye to 1.4 and glow amplifying it.

The velocity scaling is being reconsidered. Using the full segment displacement for every stamp along a long path may be physically correct if each event represents one frame of motion, rather than distributing that displacement across sub-stamps.

Injected velocity should be capped at 2–3 cells per frame to prevent instant smearing. Dye amount per stamp needs lowering, and the total for a long segment should be fixed per cell visited rather than multiplied by step count. Dye damping should be raised to about 0.9985 or 0.999.

Vorticity should be increased so the swirls are more visible. Rendering will use a power curve to keep thin dye bright, and glow will be removed to prevent blowout. The blocky trail may come from integer advection at high velocity or low resolution; raising the simulation resolution slightly could help without hurting responsiveness. A small velocity diffusion step is also being considered.

The initial seed was already good and showed proper swirling. The drag tool may interpolate poorly or fire too few events, though injectSegment should fill the line. The trail is actually continuous but looks like disconnected blobs because high vorticity and speed create instabilities at low resolution.

The velocity injection will use a capped speed of 2.2 cells per frame, calculated as `Math.min(2.2, dist * 0.15)`, so fast swipes still create a jet without instantly emptying the trail. Dye per stamp is fixed at 0.22 with radius 5, and step count is reduced to prevent over-accumulation.

Dye amount per stamp is being set to 0.18, with one deposit per cell along the path so Gaussian overlap brightens the center naturally without white blowout. Rendering gamma is fixed at 0.65 so faint dye remains visible while full saturation stays bright.

Dye damping is set to 0.9975 for roughly four seconds of half-life. Velocity damping is 0.998 so momentum lingers. A second weaker vortex at vorticity 0.5 may be added from the jet. Four Jacobi velocity diffusion iterations will smooth the flow into a softer swirl instead of a sharp jet.

The diffusion constant is fixed at 0.15 with four iterations, keeping the cost low at 120 by 80 resolution. The solver will be parameterized so pressure and velocity diffusion can use different iteration counts.

The simulation resolution will be raised to 150 by 100 for a smoother look. The advection step should remain stable at the capped velocity of 2.2 cells per frame. Pause and clear behavior will be verified after tuning.

Resolution is being raised to 160 by 106. The blocky trail likely comes from sharp dye edges on a low-resolution grid, so higher resolution with softer splats should smooth it out. The pressure solver and advection costs remain acceptable at this size. After the fixes, I will test whether a second drag in a different direction produces a separate colored trail. I also need to verify that the color selector actually changes the dye color.

The native color picker may be hard to test in this browser tool. I will add clickable preset swatches so I can easily verify that the dye color changes.

Five preset swatches will be added to the color selector. Pause and clear behavior will be verified by pausing, moving the pointer, and comparing screenshots; clearing should leave a black canvas. The synthetic drag event correctly deposits all dye along the path before advection begins.

A small perpendicular curl will be added at the injection point to encourage visible swirl. The rendering will also be fixed so the background stays black and the dye appears colorful rather than gray.

The gray fade was caused by low dye values mixing with the original seed color and JPEG compression. The main reason the trail vanished so quickly is that the old 8-cell speed cap let a 50-cell injection spread across eight cells per frame, shearing the dye apart into a faint cloud.

The initial seed still looks good after the gamma change. The drag physics are working correctly; the trail simply fades too aggressively. A small amount of dye diffusion may be added to smooth the edges.

Considering adding color swatches to make testing easier.

The flow continues after release. Dye structures shift between frames, with the dark swirl on the right changing shape and position.

A fresh flow has started after clearing.

The dye is spreading and fading quickly after release.

The structures do move and momentum persists. However, the dye dilutes into a gray fog once stretched too thin. This is physically plausible for a single pass, but it fails to meet the requirement for persistent swirling flow.

Numerical diffusion and stretching are causing the dye to fade too quickly. I will raise the resolution to reduce diffusion, add a minimum dissipation rate, and lower velocity to slow spreading. The head also became a large diffuse blob during the drag.

BFECC advection could reduce numerical diffusion significantly. I am also considering a resolution of 180x120.

BFECC advection costs three times as much as standard advection, but it keeps the dye sharper and more visible during swirling. I will clamp the corrected values to prevent ringing. Applying the same technique to the velocity field would also tighten the jets and extend the swirls. I plan to implement clamped BFECC for the dye first, then possibly for velocity.

The initial swirled shapes look good. The problem is that the dye spreads too fast and shears across the entire canvas in under a second. I will reduce the injection speed to about 1.0–1.4 cells per frame so the flow stays local and visible for several seconds.

A speed cap of 1.35 cells per frame is set, with injection scaled to `Math.min(1.35, 0.35 + dist * 0.12)`. At this rate a 15-cell radius swirl completes one rotation in roughly 80 frames. Vorticity may need slight adjustment since curl and confinement force scale with velocity gradients.

A jet at 1.2 cells per frame should still roll up into vortices via Kelvin-Helmholtz instability, enhanced by vorticity confinement.

The resolution will be raised to 150 by 100 for smoother filaments. The extra BFECC advection cost is acceptable at this size.

I am now writing the bilinear sampling helper with boundary clamping.

The bilinear sampling helper is complete. I have written the standard advection routine and started the BFECC wrapper.

The BFECC routine is now fully drafted. It performs a forward advection, a backward advection, and clamps the corrected value to its local min-max to suppress ringing. Extra temporary buffers are needed, but the pressure, divergence, and curl arrays are free during dye advection.

I will apply BFECC only to the dye channel. Velocity advection will remain semi-Lagrangian because velocity is its own advection field and needs the original values for the correction step. Dye sharpness matters more for visual persistence.

The BFECC routine for each dye channel is now being implemented. It uses the pressure and divergence arrays as scratch space, performs a forward advection into the pressure buffer, a backward advection into the divergence buffer, and then corrects the source value by half its difference from the backward result. The corrected value is clamped to the local minimum and maximum of its four immediate neighbors to suppress ringing.

The BFECC routine is now complete. It clamps negative values to zero, writes the corrected result into the pressure buffer, applies boundary conditions, and advects that buffer into the destination array. This preserves the existing copy-and-advect pattern.

I also suspect the perceived fading may partly come from JPEG compression artifacts on dark red against black, rather than only numerical diffusion. The first motion series after a colorful drag still looked reasonably persistent, while the post-clear version faded faster because it started with less total dye.

I will slightly increase the dye deposit so a single stroke remains vivid. The per-frame dissipation of 0.9982 is not the main culprit; advection dilution is. I will also test pausing.

Updating the advect module to support sign and adding BFECC dye advection.

The seed uses width and height, so it should scale.

Seed radii are fixed in cells, so they look smaller at higher resolutions. Vortex radius is fine. Considering scaling seed radii slightly. The 2.4 cells per frame speed might also need adjustment for larger screens.

The flow continues swirling after release. Structures move between frames.

The fresh flow after clearing looks great. It shows Kelvin-Helmholtz vortices.

The native color picker is hard to drive. I'll try clicking the input directly to verify it works.