Irrotational And Rotational Contributions Of Material Acceleration To Unsteady Hydrodynamic Force
A. R. Khojasteh(1), E. Overmars(1), A. Nila(2), T. Rockstroh(3), J. Westerweel(1)
1: Laboratory for Aero- and Hydrodynamics, Delft University of Technology, 2628 CD Delft, The Netherlands
2: LaVision UK Ltd., Bicester, United Kingdom
3: LaVision GmbH, Göttingen, Germany
DOI:
Unsteady hydrodynamic loading on accelerating bodies can be interpreted as the sum of an irrotational pressure response associated with exchange of kinetic energy with the surrounding fluid and a rotational contribution associated with vorticity generation and viscous diffusion (Lamb, 1932). Force reconstruction is commonly studied by recovering pressure from velocity fields, which is sensitive to differentiation noise and boundary conditions (van Oudheusden, 2013), or by obtaining the drag from the flux of Eulerian vorticity via the Josephson–Anderson relation (Savelli et al., 2026). Thanks to advances in volumetric Lagrangian Particle Tracking, the left-hand side of the incompressible Navier–Stokes momentum equation is directly available from time-resolved trajectories as the material acceleration a = Du/Dt, enabling a decomposition by applying divergence and curl operators. Using the Helmholtz–Hodge theorem (Caltagirone, 2024), we split a into a curl-free part airr = −∇ψ and a divergence-free (solenoidal) part arot . Because the pressure gradient is irrotational, airr isolates the component of a that determines the instantaneous pressure loading; the associated force captures the added-mass dominated response and remains relevant in viscous and vortical conditions. The complementary solenoidal component arot removes pressure effects and is linked to viscous diffusion and vortex dynamics. The key idea is that applying divergence and curl to Navier–Stokes separates the pressure-controlled (irrotational) and viscous/vortical (rotational) mechanisms (Caltagirone, 2024). We apply this framework to accelerating disc and sphere experiments with force sensors and time-resolved volumetric PTV. We reconstruct Firr and Frot , and compare Ftotal = Firr +Frot to the sensor signal. The results show time-resolved agreement between the reconstructed total force and the sensor signal, while resolving the evolution of irrotational and rotational contributions during vortex formation.
