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Vol.:(0123456789)Experiments in Fluids (2025) 66:64 https://doi.org/10.1007/s00348-025-03962-wRESEARCH ARTICLEVortex force map method to estimate unsteady forces from snapshot fl owf i eld measurementsShūji ?tomo 1  · Pascal Gehlert 2  · Holger Babinsky 2  · Juan Li 3Received: 23 May 2024 / Revised: 17 November 2024 / Accepted: 9 January 2025 / Published online: 28 February 2025 © The Author(s) 2025AbstractAn accurate non-intrusive force measurement is challenging in many situations, especially those involving animals and vehicles. This paper reports a non-intrusive technique based on the vortex force map (VFM) method, which computes forces from snapshot velocity and vorticity fi elds obtained from the particle image velocimetry (PIV) fl ow measurement. This study is the fi rst application of the VFM method to PIV velocity data. The VFM method is applied to three dif f erent kinematic families for surging fl at plates and pitching NACA 0018 aerofoils at Reynolds numbers of O ( 10 4 ) , where fl owf i elds are characterised by massive fl ow separation with the shedding of the coherent leading-edge and trailing-edge vortices. In all three cases, we observe an agreement between the direct force measurements and the VFM method even if a relatively small region around aerofoils is captured for PIV. Moreover, physical explanations of the linkage between the forces and vortical structures are provided based on the visualised force contribution of each vortex. The VFM method is highly robust to noise (a signif i cant feature in experimental fl uid mechanics) and can be applied to snapshot data.List of Symbolsa Dimensionless pitching axis locationAR Aspect ratioc Chord lengthC D Drag coef f i cientC L Lift coef f i cientf Pitching frequencyk Reduced frequencyRe Chord-based Reynolds numbert a Acceleration/deceleration timeU(t) Wing velocity in surging kinematicsu = (u,v) Fluid velocityU ∞ Freestream velocity for pitching/target velocity for surging case? Angle of attack? ? Pitching rate? ? Pitching acceleration? k Vortex pressure force factor? Fluid density? k Hypothetical potential? Spanwise vorticity? Vorticity vectorAbbreviationsAM Added massCirc CirculatoryLEV Leading-edge vortexPIV Particle image velocimetryTEV Trailing-edge vortexVFM Vortex force map1 IntroductionIn experimental fl uid mechanics, the most reliable and hence the most popular way to measure forces acting on a body is to use a load cell, which is dif f i cult for living animals and microair vehicles. Furthermore, executing direct force measurements is challenging for low Reynolds number fl ows, where the loads are typically small. These measure-ments are prone to signif i cant errors and can be substantially af f ected by resonance ef f ects. In such circumstances, there is a need for a technique to accurately estimate forces from instantaneous velocity fi eld data, which is typically obtained from particle image velocimetry (PIV). The PIV-based force * Juan Li juan.li@kcl.ac.uk1 Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan2 Engineering Department, University of Cambridge, Cambridge CB2 1PZ, UK3 Department of Engineering, King’s College London, Strand, London WC2R 2LS, UK