Choose the declared model.
r(u)=R·u · θ(u)=d·2π·n·u+phase · x(u)=pitch·n·u
y(u)=r·cos θ · z(u)=r·sin θ · 0 ≤ u ≤ 1
The spiral begins at one canonical centre and winds outward around its axis. Circulation sign selects handedness; diameter controls its radial extent. This is an explicit geometric model. The retained rotational field remains separately selectable and is not presented as this spiral.
The separate rotational-field reference uses a finite core, differential rotation and axial flow. Its material trajectories are retained for comparison.
Retained rotational-field diagnostics · transverse coordinate over time
Rotational material trajectory
Selected sample · y coordinate . The graph uses the same angular velocity, identity and time as the 3D view. Its white endpoint is the current sample value.
How is this achieved? Open the mathematics and configure the field
vθ(r) = Γ[1−exp(−r²/a²)]/(2πr)
ω(0) = Γ/(2πa²) · θ(t)=θ₀+ω(r)t
x(t)=x₀+ut · y(t)=r cos θ(t) · z(t)=r sin θ(t)
This declared regularized vortex uses a fixed core a, circulation Γ and uniform axial speed u. Its trajectories are evaluated analytically, rather than accumulated with Euler steps. It is a reference field, separate from a native QZL physical-law or tensor-shell derivation. Position, velocity, vorticity, observer frame and event history remain separate record fields.
XY, XZ and YZ planes share one origin identity U=1. The symbolic layer label (−1+) and numerical coordinate projection are retained as separate fields.
Helical streamlines show rotational and axial flow. The gold marker is the selected material identity; the graph below follows its transverse coordinate. Drag to orbit, or use the view buttons.
Paired action moments · −2 / −2
The two repeated labels identify distinct forward and inverse events. The inverse restores the sampled field parameters; its later execution record remains. Field phase time and event chronology are separate, as required by the source correction on timestamped inversions.
No sampled actions yet.
Local rendering cost: .
Time progression starts paused.
The export contains the declared frame, parameters, time and every sample’s identity, 3D position, envelope, phase and amplitude. Maximum radius-invariant error: . Calculated coordinates and vectors remain in the record. Camera orbit changes the view; frame controls change B. The two are recorded and interpreted separately.
Shared time · Independent frames · Retained material identities
Retained rotational fields.
Independent orientations.
Start with two regularized rotational fields in the same frame and opposite circulation. Expand to four orientations and compare their directions at the same moment. The gold marker identifies the same material address within each independent field. Use the time controls above; core radius, diameter and axial flow remain shared.
Same material address. Different signed and oriented readings.
Selected address: VORTEX-4-6-0 in each independent field. The gold trail evaluates its previous reference second analytically.
| Field | Signed Γ | Yaw / pitch | Angular velocity rad/s | World x / y / z | Velocity x / y / z |
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Comparison calculates when enabled.
These are independently evaluated analytical fields with explicit frame bases. Their comparison does not calculate mutual fluid interaction. Rendering recentres axial travel for a stable comparison; exported world positions retain that travel.
Accuracy you can inspect.
For sector context, NASA’s Review of Idealized Aircraft Wake Vortex Models (AIAA 2014-0927) examines circulation, core size, vorticity and the use of model profiles in simulation and lidar analysis. This tool declares its fixed-core profile separately from diffusion, wall effects and turbulent wake evolution.
Each particle retains its ID and initial radius. The live radius-error metric compares every generated trajectory with that invariant. Zero circulation keeps all identities and axial flow; reversing circulation reverses rotation while keeping frame and field types. This finite-core model does not introduce a singular velocity at the axis.
Inspect the research derivations