This is a 2D field-native counterpart to the 3D particle-tracer version of this simulator. Instead of flying discrete markers through a 3D perspective scene, this page solves the identical steady velocity field as an Eulerian dye-concentration field on a fixed grid: transverse "lanes" of dye stream in from the upstream edge and are advected cell-by-cell every frame using the exact same flow equations, so the whole stir zone's material history is visible at once as continuous banding rather than a scatter of dots.
r = √(x²+y²), ω signed (rad/s, + = CCW)
Rotational term (solid body inside the pin,
decaying vortex outside it — the classic two-zone
approximation of the stirred/TMAZ region):
DF(r) = 1 for r ≤ R
= (R/r)·exp(−(r−R)/L) for r > R
(u_rot, v_rot) = (−ω·y, ω·x) · DF(r)
Small radial term (models forging-driven outward/
rearward extrusion that builds the onion-ring bands):
(u_out, v_out) = k·(x, y)·exp(−r/2L)
Total field: u = v_travel + u_rot + u_out
v = v_rot + v_out
Dye transport (each grid cell, per frame):
φ(x,y,t+dt) = φ(x − u·dt, y − v·dt, t) (semi-Lagrangian
back-trace + bilinear sample)
φ = stripe(y) = 0.5+0.5·sin(2πy/period) at the upstream inflow
Because dye is a passively-advected scalar (no diffusion term added beyond the small, honest numerical smoothing that bilinear sampling introduces), each lane's colour stays attached to the same fluid parcel along its pathline — exactly the same physics the 3D version's coloured spheres show, just rendered as a continuous field that reveals the whole stir zone's banding structure simultaneously instead of one marker at a time.
- Advance ratio λ = ωR/v — the standard FSW parameter comparing tool surface speed to travel speed; higher λ means more material rotations per millimetre of weld travel, i.e. more thorough stirring (and tighter, more numerous bands in the dye field).
- Advancing side (orange tint) is where the pin's rotational surface velocity adds to the traverse velocity; retreating side (blue tint) is where it opposes it — tinted directly onto the dye field so the asymmetry in band spacing between the two sides is visible at a glance.
- The pin-radius and RPM sliders reproduce the tradeoff real welding engineers face: too little stirring (low λ) leaves unbonded "kissing-bond" defects, too much (very high λ) overheats and thins the joint.