EffectModel drag-swap — method differences & resolution behaviour#

Companion to the EffectModel concept-proof (tests/verification/test_effectmodel_stokes_drag_swap.py). It records how far apart the HydrodynamicModel backends are on the same prescribed-motion sphere, why, and whether the gap closes with resolution.

Scope: the concept-proof runs Stokeslet and FVM (a free-space translating sphere). IBLBM is compile-only here — it is hardcoded to the rotating UMR helix and cannot resolve a free-space translating sphere without new code; its production use (de Boer) runs at 192³.

Measured drag / analytical Stokes (6πμaV)#

Stokeslet (free-space sphere) — converges monotonically with surface refinement (continuum-exact BEM):

surface points

drag / 6πμaV

80

0.928

320

0.972

1280

0.988

FVM (sphere in a pipe, confinement a/R ≈ 0.30) — erratic at the coarse grids runnable in a test, because the sphere spans only 1–2.5 cells:

grid

sphere radius

|drag| / 6πμaV

8³

~1.0 cell

8.83

12³

~1.5 cells

1.25

16³

~2.0 cells

1.14

20³

~2.5 cells

0.71

So in the concept-proof: Stokeslet ≈ 1.00× analytical (exact); FVM ranges ~0.7–9× depending on resolution. This is why the concept-proof asserts Stokeslet to ~5% but FVM only to within an order of magnitude — at 8³ the FVM drag is not a converged quantity.

Why they differ#

  • Stokeslet (BEM) solves the unbounded Stokes equations with the exact Green’s function on the body surface — continuum-exact; the only error is surface-mesh density + the regularization ε, both → 0 with refinement.

  • FVM (Navier–Stokes + IBM, Cartesian grid) has several errors, dominant at coarse resolution:

    1. Under-resolved sphere — at 1–2.5 cells/radius the immersed boundary is barely represented (the 8.83 at ~1 cell is essentially noise).

    2. IBM diffuse-interface smearing (ibm_eps ~ dx) mislocates the no-slip surface by ~1 cell.

    3. Finite domain / confinement — a/R ≈ 0.30 is a real physical drag enhancement (Haberman wall correction ≈ 2–2.5×), not a discretization error; it does not vanish with grid refinement.

    4. Transient (finite step count) and finite Re (full NS, not pure Stokes).

  • IBLBM / LBM (same family) add: lattice resolution, an O(dx) staircased / bounce-back wall position, and weak compressibility (Mach).

The deep point: the methods solve different boundary-value problems — unbounded Stokes (Stokeslet) vs. a walled, finite-Re box (FVM/LBM) — so they converge to different limits in general.

Does the gap close at higher resolution?#

  • Stokeslet: yes, cleanly — a convergent BEM (1% at 1280 points).

  • FVM / LBM: the coarse-grid scatter disappears, but not the whole gap. With ~8–16+ cells/radius and enough steps the under-resolution + IBM-smearing errors collapse and the drag stabilises to a well-defined value — but that value is the confined drag (≈2–2.5× free-space here), not Stokeslet’s free-space 1.0. Matching Stokeslet additionally requires a large domain (a/R → 0) and low Re. Agreement only holds in the shared regime — which is exactly what the de Boer / de Jongh cross-validation establishes at production resolution, not what the (coarse, swap-surface) concept-proof attempts.

Implication: quantitative FVM drag at microrobot scale needs a far better-resolved / higher-fidelity FVM path than the coarse pilot grid (a slender helix in a vessel spans an even wider scale range than this sphere) — the convergence behaviour here is why the swap-out roadmap (E6) flags FVM fidelity as a dedicated work item.