Frequently asked questions
How can speed diverge while energy stays finite?
Energy depends on speed squared integrated over volume. In the paper, the region of intense motion shrinks quickly enough for its energy to decrease even as its speed grows. Near the center, fluid spirals inward and exits along the axis; the core becomes progressively thinner relative to its height.
The hard part is keeping the applied force smooth all the way through the singular time. The paper uses oscillatory pulses and successive corrections to cancel otherwise singular terms in the momentum balance. Our finite pulse families and numerical forcing do not reproduce or certify that entire construction. Tracking the target is a solver check; proving those cancellations is a separate mathematical task.
Paper: Theorem 1.1 & §2 ↗ Lean formalization ↗ What reproduction would require ↗
Do we save the full refined field?
Yes. Oliver saved all three velocity and all three force components in double precision at every cell of all five levels, for all 280 output times. The hierarchy stores 1,310,720 cells; 1,179,648 are active after coarse cells covered by finer levels are excluded. The complete fields remain on the archive machine.
The movies sample 256² pixels per plane using the finest containing native cell, without spatial interpolation. The moving horizontal plane is selected using the full active 3D field, not those display pixels. Fine structure can still be absent from the display. The mesh diagram shows refinement regions, not every cell. A native refined-grid 3D browser export is not yet prepared.
How much storage does that take?
Each full native snapshot contains 1,310,720 × 6 × 8 bytes of field data: 60 MiB. All 280 snapshots require 16.41 GiB before headers, checkpoints and derived render files. These figures include covered coarse cells because they are stored even though they are excluded from composite norms. Oliver retained a 20 GiB free-disk reserve.
How does movie time relate to simulation time?
All 280 saved states appear in order, with no interpolation or dropped frames. Ordinary states hold for 0.2 seconds, the initial rest state for one second, and the final state for half a second. GIFs and videos use the same holds. The output schedule becomes denser toward the endpoint, so five saved frames per second is not a uniform simulation-time clock. The displayed timestamp identifies the actual state. Frame times and playback record
Which planes do the movies show?
Both movies show a vertical x–z slice at y = 0 and a horizontal x–y slice at z = 0. These planes remain fixed throughout the run. The velocity and forcing movies use the same planes.
How much computation does the run require?
Oliver's completed refined run and its completion audit took about 9.02 hours on one Mac, with four solver threads and up to four forcing threads. We have not measured a FLOP count for this AMReX calculation.
The earlier 100–400 TFLOP planning budget applies only to the previous uniform 192³ PhiFlow run through t = 0.985. It must not be carried over to a different integrator and multilevel projection. A TFLOP is a trillion floating-point operations; FLOP/s would be a processing rate.
The configured output clocks, maximum timestep, diffusion limit, and forcing-phase limit give about 4,113 active steps. Each RK2 step makes two pressure projections, each requiring three forward and three inverse complex 3D Fourier transforms. Using the conventional approximation 5 N log₂(N) per transform, with N = 192³, gives about 40 TFLOP for those transforms. This is an operation-count convention, not a hardware measurement. FFT benchmark methodology ↗
We allow 3–10 times that FFT work for the full calculation, including repeated target and force evaluations and finite-difference kernels. That multiplier is an explicit planning assumption, not a profile or a rigorous bound. Velocity-dependent CFL limits may require more steps. Restarts, validation runs, and rendering are excluded. Estimate script ↗ · Inputs and calculation
How is peak speed measured?
The homepage number is the native solver's full-field peak-speed diagnostic at the completed endpoint, t = 0.995, reported in model units. It is not the maximum of the reduced display pixels and does not change as the movie plays.