Validation of the moving-boundary Lagrangian-mean formulation

Establish, using a deforming-mesh computation for the circular cavity with an oscillating boundary, whether transferring the wall condition to the reference circle and imposing no slip on the Lagrangian mean correctly represents the second-order moving-boundary problem.

Background

The finite-element verification in the paper tests the closed-form reference-boundary problem, while tracer integration checks the Stokes-drift decomposition within the prescribed formulation. It does not directly solve the deforming-domain problem.

The authors identify two premises that remain untested: transferring the wall condition from the displaced boundary to the reference circle, and imposing no slip on the Lagrangian mean as the appropriate second-order condition. They state that a deforming-mesh computation would settle these issues, but report that it was not performed.

References

Two premises stay untested: transferring the wall condition to the reference circle, and no slip on the Lagrangian mean as the correct second-order condition. A deforming-mesh computation would settle both, and what it costs is set by the ratio of the mean to the oscillation, $\varepsilon\,\mathrm{Wo}2$. At the operating point of Sec.~\ref{sec:scope}, where $\varepsilon = 0.01$ and $\mathrm{Wo}2 = 0.053$, that ratio is $5.3\times10{-4}$. It compares the two velocity scales rather than the two measured speeds: the peak Lagrangian speed there is $0.00401$ in the Eulerian scale, which against the wall speed $U$ is $2.1\times10{-6}$, and how finely a computation must resolve the mean follows from its own error analysis rather than from the scale ratio. At $\varepsilon = 0.05$ and $\mathrm{Wo}2 = 1$ the scale ratio is $5\times10{-2}$, about ninety times larger, and Sec.~\ref{sec:finiteWo} supplies the closed form there as well, so that point tests the same premises. The computation is therefore affordable at finite Womersley number, and it was not performed here.

— A closed-form solution for streaming and Lagrangian transport in a deforming circular cavity  (2609.24368 - Liu et al., 21 Sep 2026) in Section 4, “Numerical verification”