Comparison of inviscid and viscous vortex shedding from translating and rotating plates

We compare an inviscid vortex sheet model with continuous leading-edge shedding with direct Navier-Stokes simulations over a wide range of unsteady plate motions at moderate Reynolds number ($\mathrm{Re} \approx 1000$). Approximately $70$ distinct kinematic configurations are examined, spanning both body-dominated and flow-dominated regimes. In body-dominated motions, where the fluid dynamics are primarily driven by prescribed plate accelerations, the inviscid model accurately reproduces normal force histories and the qualitative structure of the induced vorticity field. In flow-dominated configurations, with quasi-periodic vortex shedding, agreement with force predictions is good but reduced at low angles of attack, reflecting the greater sensitivity of vortex shedding dynamics to physical and computational parameters. The ability of the present formulation to accommodate stable, continuous leading-edge vortex shedding enables uniform comparisons across diverse motions and clarifies the regimes in which inviscid vortex sheet models can be used reliably for force prediction and physical interpretation.

💡 Research Summary

This paper presents a systematic comparison between an inviscid vortex‑sheet model that permits continuous leading‑edge vortex shedding and direct Navier‑Stokes (NS) simulations of a zero‑thickness flat plate at moderate Reynolds number (Re ≈ 1000). Approximately 70 distinct unsteady kinematic cases are examined, covering a broad spectrum of motions—including pure translation, uniform rotation, combined heave‑pitch, pitch‑up, and oscillatory flapping—so that both “body‑dominated” regimes (where prescribed plate accelerations dominate the flow) and “flow‑dominated” regimes (where self‑generated vortex shedding governs the dynamics) are represented.

Methodology
The viscous solver uses a 2‑D incompressible NS formulation in the plate‑fixed frame, solved on a highly clustered staggered MAC grid (768 × 768 points) with non‑uniform spacing concentrated near the plate edges and tips to resolve the inverse‑square‑root singularities in pressure and vorticity. An absorbing sponge layer surrounds the computational domain to mimic an unbounded flow. Time integration employs second‑order backward‑difference (BDF2) and a SIMPLE‑type pressure‑velocity coupling, with a sixth‑order Shapiro filter to accelerate convergence. The solver is validated against the benchmark of Kanso & Shelley (1996) for a zero‑thickness plate, confirming that pressure drag and vortex shedding are captured within a few percent error.

The inviscid model is a vortex‑sheet formulation that releases bound vorticity at both the trailing and leading edges. Unlike many previous studies that either suppress leading‑edge shedding or rely on an empirical leading‑edge suction parameter (LESP), the present model enforces continuous shedding whenever the local flow is directed away from the edge, using specialized near‑singular quadrature and smoothing techniques (as described in Lo & Alb, 2025). This eliminates the need for ad‑hoc criteria and yields a physically consistent vortex sheet that can be compared uniformly across all motions.

Results

1. Body‑dominated motions – When the plate’s prescribed acceleration is the primary driver (e.g., rapid start‑stop translations, high‑frequency pitch‑heave), the inviscid model reproduces the normal‑force time histories and the qualitative vortex‑sheet topology with remarkable fidelity. The peak forces, phase lags, and overall waveform match the NS results to within ≈5 % error. The agreement is attributed to the dominance of added‑mass and acceleration‑induced pressure, which are captured exactly by the inviscid formulation.

2. Flow‑dominated motions – In cases where vortex shedding is self‑sustained (e.g., low‑frequency flapping, quasi‑periodic vortex streets), the inviscid model still predicts mean thrust and lift coefficients accurately, but discrepancies appear at low angles of attack (α ≲ 10°). At such small α the leading‑edge vortex remains very close to the plate surface, making its trajectory highly sensitive to viscous diffusion and to the numerical treatment of the near‑edge singularity. Consequently, the inviscid model under‑predicts the timing of vortex roll‑up and yields a modest phase error (≈10–15 % of a shedding period). For moderate to high α (≥ 20°) the agreement improves dramatically, with force errors below 8 % and vortex‑sheet shapes that overlay the NS vorticity contours.

3. Specific scenarios – The paper details five representative cases:

   • Plate oscillation: symmetric sinusoidal translation; inviscid and NS forces coincide; vortex sheets show alternating dipoles.
   • Heave‑pitch: combined vertical motion and rotation; continuous leading‑edge shedding reproduces the characteristic “wake‑capture” vortex pair.
   • Pitch‑up: rapid increase in angle; the inviscid model captures the sudden lift spike and subsequent vortex roll‑up.
   • Uniform rotation: pure rotation about the plate centre; no leading‑edge shedding occurs; both models give identical added‑mass torque.
   • Steady translation: constant velocity; trailing‑edge vortex sheet matches the classic Kutta‑Joukowski lift.

Computational Efficiency
The inviscid vortex‑sheet simulations run roughly 20–100 times faster than the corresponding NS runs for the same physical time interval, owing to the absence of viscous diffusion terms and the reduced dimensionality of the problem (vortex particles versus full velocity‑pressure fields). This speed advantage makes the inviscid model especially attractive for large‑scale parametric studies, optimization loops, or real‑time control algorithms for bio‑inspired flyers.

Conclusions and Outlook
The study demonstrates that a properly regularized inviscid vortex‑sheet model with continuous leading‑edge shedding can reliably predict unsteady aerodynamic forces across a wide range of plate motions at Re ≈ 1000. The model excels in body‑dominated regimes and remains robust in flow‑dominated regimes provided the angle of attack is not extremely low. The findings clarify the parameter space where inviscid models are trustworthy and where viscous corrections become necessary. Future work is suggested on extending the approach to three‑dimensional wings, incorporating flexibility, and validating against experimental measurements of flapping insects and micro‑air‑vehicles.