Vol.:(0123456789) Experiments in Fluids (2025) 66:64 https://doi.org/10.1007/s00348-025-03962-w RESEARCH ARTICLE Vortex force map method to estimate unsteady forces from snapshot flowfield measurements Shūji Ōtomo1 · Pascal Gehlert2 · Holger Babinsky2 · Juan Li3 Received: 23 May 2024 / Revised: 17 November 2024 / Accepted: 9 January 2025 © The Author(s) 2025 Abstract An 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 fields obtained from the particle image velocimetry (PIV) flow measurement. This study is the first application of the VFM method to PIV velocity data. The VFM method is applied to three different kinematic families for surging flat plates and pitching NACA 0018 aerofoils at Reynolds numbers of O ( 10 4 )  , where flowfields are characterised by massive flow 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 significant feature in experimental fluid mechanics) and can be applied to snapshot data. List of Symbols a � Dimensionless pitching axis location AR � Aspect ratio c � Chord length C D � Drag coefficient C L � Lift coefficient f � Pitching frequency k � Reduced frequency Re � Chord-based Reynolds number ta � Acceleration/deceleration time U(t) � Wing velocity in surging kinematics u = (u, v) � Fluid velocity U∞ � Freestream velocity for pitching/target velocity for surging case � � Angle of attack 𝛼̇ � Pitching rate 𝛼̈ � Pitching acceleration � k � Vortex pressure force factor � � Fluid density � k � Hypothetical potential � � Spanwise vorticity � � Vorticity vector Abbreviations AM � Added mass Circ � Circulatory LEV � Leading-edge vortex PIV � Particle image velocimetry TEV � Trailing-edge vortex VFM � Vortex force map 1  Introduction In experimental fluid 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 difficult for living animals and microair vehicles. Furthermore, executing direct force measurements is challenging for low Reynolds number flows, where the loads are typically small. These measure- ments are prone to significant errors and can be substantially affected by resonance effects. In such circumstances, there is a need for a technique to accurately estimate forces from instantaneous velocity field data, which is typically obtained from particle image velocimetry (PIV). The PIV-based force * Juan Li juan.li@kcl.ac.uk 1 Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Tokyo 184‑8588, Japan 2 Engineering Department, University of Cambridge, Cambridge CB2 1PZ, UK 3 Department of Engineering, King’s College London, Strand, London WC2R 2LS, UK http://crossmark.crossref.org/dialog/?doi=10.1007/s00348-025-03962-w&domain=pdf Experiments in Fluids (2025) 66:64 64   Page 2 of 13 estimation is also attractive because it provides insights into the connection between forces and flowfields. When performing PIV for living animals, (non-phase-averaged) instantaneous PIV data need to be used for computing forces since the animals never make exactly the same motions. This requirement makes it more difficult for PIV-based force measurement because the PIV data inherently contain some experimental noises, which propagate errors in the force calculation due to either time derivative ( �∕�t ) or spatial derivatives ( ∇,∇2). The classical method uses a momentum balance for a control volume that encloses a body (e.g. Unal et al. 1997; Van Oudheusden et al. 2006, 2007; Jardin et al. 2009; David et al. 2009; Rival et al. 2010; Mohebbian and Rival 2012; Dabiri et al. 2014; Tronchin et al. 2015). This method ben- efits from not needing velocity information near the body surface—a challenging task in PIV. It, however, requires a pressure along the control surface/volume, which is nor- mally computed by integrating the Navier–Stokes equations. In addition, this method contains a time derivative ( �∕�t ), requiring a time-resolved velocity field measurement. Another approach to estimating forces is integrating sur- face pressures and skin friction, as is widely used in com- putational fluid dynamics (CFD). Typically, a pressure field is obtained from PIV velocity data by solving a pressure Poisson equation with appropriate boundary conditions. This method allows users to compute the forces from the instantaneous velocity field. Some researchers success- fully obtained forces and torque acting on a body using a pressure field obtained from PIV (e.g. Fujisawa et al. 2005; Van Oudheusden et al. 2007; Murai et al. 2007). Since the publication of the review paper on PIV-based pressure calcu- lation (Van Oudheusden 2013), pressure measurement using the Poisson equation has become more common. Now most commercial PIV software incorporates the pressure field reconstruction features. However, in PIV, the outer flowfield boundaries are constrained to be much closer to the body than those in CFD, which can introduce errors. The impulse method developed by Burgers (1920), Light- hill (1986), and Wu (1981) is another approach to estimating forces from PIV. It computes forces from the vorticity field and therefore does not require an iterative computation such as pressure. Several investigators succeeded in extracting force from the unsteady PIV field using the impulse-based method, including Noca et al. (1997) who tested it on an incompressible cross-flow in a finite and arbitrarily chosen region enclosing the circular cylinder. They confirmed an accurate prediction from fully resolved computational results while capturing the trend for the under-resolved experimen- tal data. Graham et al. (2017) test the impulse method on a two-dimensional translating wing flow. By representing the influence of the missing vorticity near the wing surface by a vortex sheet, they obtained a good agreement between the estimated force from the impulse method and the direct force measurements. Limacher et al. (2018) derived the added mass force term through impulse theory, and later in Limacher et al. (2019), they validated it against PIV data on a two-dimensional flow around a cylinder accelerated from rest. Meanwhile, Gehlert and Babinsky (2019) and Gehlert et al. (2023) successfully recovered and validated the lift and drag from PIV data on accelerating and rotating cylinder flows using the impulse method. Here the added mass forces are linked to a vortex sheet by applying the rotational and translational boundary conditions to the cylinder surface. However, these impulse-based methods, applied on full domain or truncated domain, include a time derivative and therefore require a time-resolved PIV. In addition, all the vorticity generated by a body has to be inside the PIV win- dow and hence its use is limited to the starting flow  (Lin and Rockwell 1996; Poelma et al. 2006; DeVoria and Ringuette 2013; Graham et al. 2017). Noca et al. (1999) extended the impulse method for a finite domain by including the surface integrals along the control volume and evaluated against a flow around a cylinder undergoing unsteady motions. Later Kang et al. (2017) and Kang et al. (2018) extended this method for a minimum domain containing a body and body- connected vortices. Siala and Liburdy (2019) applied this method to estimate forces on an aerofoil undergoing a com- bination of pitching and plunging motion, confirming agree- ment with the result of direct force measurement. Limacher et al. (2020) compare the accuracy of the momentum and impulse-based methods for a finite domain for a cylinder accelerating from rest, reporting that the impulse-based method exhibited greater errors. Some of these aforemen- tioned PIV-based force estimation techniques are reviewed in Rival and Van Oudheusden (2017). On the other hand, the vortex force map (VFM) method is another vorticity-based force estimation technique. This method holds significant potential for analysing unsteady flows without explicitly including the pressure and calculating the time derivatives that can introduce significant errors using experimental data. Although the high-speed PIV is popular nowadays and the snapshot (non-time-resolved) PIV is getting less common, the high-speed PIV still requires expensive equipment and data processing. In addition, the VFM method allows users to visualise the contribution of vortices to forces, which is a useful feature for biological or bioinspired flight where vortices play key roles in force generation. The formulae was originally developed by Howe (1995) for a rigid body in unsteady motion in an incompressible and viscous flow. Later, Li and Wu (2015) rederived and applied it to analyze the starting flow of a flat plate at moderate angles of attack. In 2016, they further developed it into the VFM method to address high-angle-of-attack flow problems. Further, the VFM method is extended to general aerofoils (Li and Wu 2018), three-dimensional wings (Li et al. 2020b), viscous Experiments in Fluids (2025) 66:64 Page 3 of 13  64 flows (Li et al. 2021), and multiple bodies (Wang et al. 2022). In addition, Li et al. (2020a) developed a vortex moment map method to estimate the aerodynamic moment acting on a body. Li et al. (2021) have tested the VFM method on computing aerodynamic forces from the sparse sampling and small truncated domain of converged CFD simulation results. A satisfactory level of force estimation accuracy compared with the surface integral of pressure and shear stress from CFD shows the eligibility of the VFM in processing PIV data. Another vorticity-based force estimation, which is analo- gous to the VFM method, is the physics-based force and moment partitioning method (FMPM). This method was originally formulated by Quartapelle and Napolitano (1983). Later, Menon and Mittal (2021a, 2021b, 2021c) applied FMPM to CFD results to investigate the contribution of aerodynamic forces from different sources. Recently, this method was applied to PIV velocity data to estimate forces on a steady NACA 0015 aerofoil at an angle of attack of 30 degrees (Díaz-Arriba et al. 2022). Though the lift estimation was in agreement, the drag had an error of 17%, which is one order higher than that of the lift. Zhu et al. (2023) applied FMPM to PIV velocity data to estimate the aerodynamic moment acting on a sinusoidally pitching NACA 0012 aero- foil in quiescent water. The estimated moment saw a quali- tative match with that of the direct force measurement, but the disagreement was large as the estimated moment was approximately 50% smaller. The aim of this paper is to apply the VFM method to PIV data for unsteady flow and assess its validity in experi- mental fluid mechanics. To accomplish this, we apply the VFM method to flowfields around aerofoils undergoing two different kinematic families (surge and pitch) characterised by coherent leading-edge and trailing-edge vortices (LEV and TEV) and estimated forces are compared with the direct force measurements. The rest of this paper consists of the following. Section 2 introduces the force estimation technique based on the vor- tex force map method. Section 3 describes the summary of experiments and aerofoil kinematics. Section 4 compares the estimated forces with direct force measurements. Finally in Sect. 5, we draw a conclusion and suggest future studies. 2 � Force estimation technique 2.1 � Vortex force map method This section describes the method to compute the lift and drag through the VFM method. Following Li et al. (2021), the total force acting on an aerofoil is partitioned into the added mass force FAM k  , the vortex force FVort k  , vortex pres- sure force FVP , and skin friction force FSF where k = {N,A} denotes the force direction (chord normal or chord tangential). In Eq. 1, the vortex pressure force FVP is due to the vorticity diffusion on the body surface. In this paper, we do not consider forces due to the viscous pressure FVP and skin friction FSF as they are inversely proportional to the Reynolds number and deemed negligible for flows at a Reynolds number of O ( 104 )  (Li et al. 2021). In fact, vortex forces are the predominant factor under these mas- sively separated flow conditions  (Li and Wu 2016; Ōtomo et al. 2021). Hence the total force is approximated to have the contributions from the added mass and the vortex forces, The forces are normalised For any body geometry, the added mass force is expressed as (Batchelor 2000; Saffman 1995; Li et al. 2021; Limacher 2021; Fernandez-Feria 2019) where �k is the hypothetical potential, which will be intro- duced later in Eqs. 11 and 12. Here � is the fluid density, ̇U is the acceleration of the body surface, and n is the unit vec- tor pointing inward from the body surface. The added mass force is independent from the flowfield, but dependent on the body geometry and acceleration. Although the added mass expression is derived for an inviscid fluid, it is applicable for viscous and massively separated flows as theoretically argued in Leonard and Roshko (2001) and Eldredge (2010) and experimentally confirmed by Corkery et al. (2019) and Gehlert and Babinsky (2019). In this study, we use the analytical solution of Eq. 4 for an ellipse undergoing pitch-surge motion (Limacher 2021), which approximates symmetric NACA aerofoils (Limacher 2021; Ōtomo et al. 2024a), where c is the chord length, b is the thickness (short axis) of the ellipse, and � is the angle of attack. Here, uA and uN are the chordwise and chord-normal components of the relative velocity of the aerofoil’s centre of mass, and 𝛼̇ is the pitching (1)Fk = FAM k + FVort k + FVP k + FSF k , (2)Fk = FAM k + FVort k . (3)Ck = Fk 1 2 �U2 ∞ c , CAM k = FAM k 1 2 �U2 ∞ c , CVort k = FVort k 1 2 �U2 ∞ c . (4)FAM k = 𝜌∫SB 𝜙kU̇ ⋅ ndS, (5)CAM A = − 𝜋c 2U2 ∞ ( 𝛼̇uN + b2 c2 u̇A ) , (6)CAM N = 𝜋c 2U2 ∞ ( b2 c2 𝛼̇uA − u̇N ) , Experiments in Fluids (2025) 66:64 64   Page 4 of 13 rate. The thickness is b = 0 for the flat plate and b = 0.18 for the NACA 0018 aerofoil. Equation 6 is the generalised form of added mass force in Theodorsen (1935). The added mass lift and drag are, In the VFM method, the vortex force is expressed as, where u = (u, v) is the fluid velocity vector and � is the spanwise vorticity. Vf is the fluid volume, which is the PIV domain in experiments. �k is the vortex pressure force fac- tor, which is expressed as Here �k is the hypothetical potential defined in the body- fixed frame of reference (x, y). It is termed ‘hypothetical’ because it differs from the classical potential flow solutions to aerofoils where the Kutta condition is satisfied. The poten- tials in chord-normal �N and chord-tangential directions �A are found by solving the Laplace equations with boundary conditions. The potential in the normal direction is, where Sb denotes the body surface. Similarly for the chord- tangential direction, The vortex lift and drag are therefore expressed as, (7)CAM L = CAM N cos � − CAM A sin �, (8)CAM D = CAM N sin � + CAM A cos �. (9)FVort k = �∫Vf �k ⋅ u�dV , (10)�k = ( ��k �y ,− ��k �x ) . (11) �2�N �x + �2�N �y = 0, ��N �x = 0, ��N �y = 1 (x, y) → Sb, ��N �x = 0, ��N �y = 0 (x, y) → ∞, (12) �2�A �x + �2�A �y = 0, ��A �x = 1, ��A �y = 0 (x, y) → Sb, ��A �x = 0, �A y = 0 (x, y) → ∞. (13)LVort = �∫Vf �L ⋅ u�dV , where vortex pressure lift factor and vortex pressure drag factor are, The VFM method is applied to a surging flat plate at � = 45◦ and a periodically pitching NACA 0018 aerofoil. The total lift and drag can be computed from Eqs. 7, 8, 13, and 14. Moreover, VFMs are computed from Eqs. 10–15. Figure 1 shows a VFM for a flat plate at � = 45◦ . The VFMs are obtained by solving Laplace Eqs. 11 and 12 under specific boundary conditions and depend solely on the geo- metric parameters of the body, but independent on the flow. The contours represent the magnitude of �k , indicating the intensity of the vortex force contribution at different loca- tions. The white lines with arrows depict the direction of the vortex movement that contributes either positively or negatively to the force. For instance, a counterclockwise vor- tex moving along the arrows generates positive force, while movement in the opposite direction results in negative force. The opposite holds true for a clockwise vortex. For a zero- thickness flat plate, sin(45◦) = cos(45◦) , and from Eqs. 12, �A = 0 , consequently, the VFMs for lift and drag are iden- tical and axisymmetric about the chord line (see Eq. 15). Figure 2 shows VFMs for a NACA 0018 aerofoil at two selected angles of attack ( � = 10◦ and � = 62◦ ), correspond- ing to two typical instants during the pitching motion. Note that this is the first time that the VFM method is applied to a system with a body in motion. While applying the VFM method on an inertial frame of reference, the transient VFM for a moving aerofoil is the same as the VFM for a stationary aerofoil at the same angle of attack. In Fig. 2, the magnitude (14)DVort = �∫Vf �D ⋅ u�dV , (15) �L = �N cos � − �A sin �, �D = �N sin � + �A cos �. Fig. 1   VFM for a flat plate at � = 45◦ . At this angle of attack, the VFMs for lift and drag are identical and exhibit symmetry about the chord line Experiments in Fluids (2025) 66:64 Page 5 of 13  64 of � is larger for lift than for drag at a low angle of attack, indicating that a vortex in the flow field contributes more to lift than to drag. Conversely, at a high angle of attack, the vortex contributes more to drag than to lift. A vanishing value of � at infinity indicates a zero force contribution of the vortex at infinity. The decrease in |�| with distance from the body demonstrates a minimal force contribution from far-field vorticity. 2.2 � Impulse method The impulse method  (Burgers 1920; Lighthill 1986; Wu 1981) is also used to compare our VFM method for the surg- ing flat plate (referred as Case 2 later in the paper) in which all the vorticity is contained in the PIV window (see Sect. 3). Since the detailed description of this method is available in Graham et al. (2017), it will be abbreviated here. In the impulse theory, the forces F are expressed as the rate of change of the impulse I, where I is the fluid impulse, which is the first moment of vorticity �, Here n = 2 and 3 in two and three dimensions, respectively. x is the position vectors. (16)F = � dI dt , (17)I = 1 n − 1 ∫Vf x × �dV . 3 � Experimental details and aerofoil kinematics The experimental data used for this paper stem from two sets of measurements. The first experiment (Case 1) is the surg- ing flat plate at a high angle of attack. The second experi- ment (Case 2 and 3) is the pitching aerofoil at a high ampli- tude. Table 1 summarises the test cases used in this study. In all cases, both direct force and PIV measurements are performed. The following subsections provide a summary of these experiments. Although the dual-sheet laser illuminates both the suc- tion and pressure sides of the aerofoil, optical access to the pressure side near the surface is restricted due to the camera- wing configuration. Consequently, these regions are masked in grey in the flowfield plots presented in Sect. 4.2. 3.1 � Case 1: surging flat plate at ̨ = 45◦ The experiments for Case 1 are performed in the towing tank facilities of the University of Cambridge. The same experimental facility was used in, for example, Pitt-Ford and Babinsky (2013); Stevens and Babinsky (2017); Corkery et al. (2019), so it will be briefly described here. The tow- ing tank has a 7-m-length and 1-m-square cross section. A rectangular carbon-fibre flat plate wing is surged by a car- riage. The plate has a chord of c = 0.12 m, an immersed span of 0.48 m, and a thickness of 4 mm. The PIV plane is at the mid-span. The dual-laser sheets using the beam split- ter enable the capture of velocity vectors on both the suction and pressure sides of the plate. A skim plate on the carriage end removes influences of the free surface and provides a symmetry plane, so the effective aspect ratio is 8. Due to the difficulty of experimentally achieving two-dimensional flow, the flowfield inherently exhibits three-dimensionality, and the VFM method is applied to a single plane within this velocity field. Although we do not quantify the downwash due to a tip vortex on the PIV plane, we deem it is small in starting flow, based on the investigations by Jones and Babinsky (2010), Jones et al. (2011), and Jardin et al. (2012). Nevertheless, a discrepancy still remains between the experi- mental results and the theoretical assumptions. Fig. 2   The VFMs of a NACA  0018 aerofoil for a  lift at � = 10◦ , b drag at � = 10◦ , c lift at � = 62◦ , and d drag at � = 62◦. Table 1   Summary of tested cases Cases Kinematics Aerofoil Re 1 Surge Flat plate 20000 � = 45◦ , ta∕(U∞c) = 1 2 Symmetric triangular pitch NACA 0018 32000 �0 = 64◦ , k = 0.22 , � = 0.5 3 Asymmetric triangular pitch NACA 0018 32000 �0 = 64◦ , k = 0.22 , � = 0.3 Experiments in Fluids (2025) 66:64 64   Page 6 of 13 The total of 193 × 106 velocity vectors are generated, yielding a physical resolution of 3.6 mm or 0.03c. Although this resolution does not resolve the velocity profile of the boundary layer or shear layer, the computational study by Li et al. (2021) confirmed the VFM method showed a good agreement with the CFD result even when a coarse mesh is used ( 116 × 116 mesh grids for a domain of three chord length each, which has lower resolution than our PIV). The flat plate accelerates from rest and reaches the con- stant velocity U∞ = 0.219 m s−1 at tU∞∕c = 1 and deceler- ates to halt at tU∞∕c = 3 as shown in Fig. 3b. The Reyn- olds number is Re = 20000 based on the chord and the final velocity. The angle of attack is fixed at � = 45◦ . The PIV window captures all the vorticity generated by the wing. The PIV data visualised and used for the VFM method and impulse method are instantaneous (non-phase-averaged) and unfiltered. 3.2 � Cases 2 and 3: pitching NACA 0018 aerofoil at ̨ 0 = 64◦ Since the experimental details and aerofoil kinematics for cases 2 and 3 have already been documented by Ōtomo et al. (2021), they will be briefly described here. The dataset (force and PIV) for these two cases is available in Ōtomo et al. (2024b). The experiments for cases 2 and 3 are performed in the water channel facility at UNFoLD laboratory at EPFL. The channel has a test section of 0.6 m × 0.6 m × 3 m in width, height, and length. A NACA 0018 aerofoil with the chord length c = 0.15 m is immersed in the freestream veloc- ity U∞ = 0.215 m s−1 , resulting in a chord-based Reynolds number of Re = 32000 . The aerofoil is pitched about the quarter chord from the leading edge ( a = 0.25 ) at the pitch- ing amplitude �0 = 64◦ . The reduced frequency is fixed at k = �fc∕U∞ = 0.22 , where f is the pitching frequency. As shown in Fig. 4, the asymmetric triangular kinematics is used to prescribe the pitching motion. t∗ 1 and t∗ 2 denote the locations of the maximum and minimum angles of attack, respectively. For a symmetric pitching case (Case 2), t∗ 1 = 0.25 and t∗ 2 = 0.75 , whereas t∗ 1 = 0.15 and t∗ 1 = 0.85 for the asymmetric pitching case (Case 3). Regions near t∗ 1 and t∗ 2 consist of 4th-order polynomial for smoothing for t∗ a = 0.15T  , where T = f −1 is the pitching period. Time-resolved PIV is performed with 1024 × 1024 px resolution and 1000 Hz frame rate. The total of 126 × 126 velocity vectors are generated, yielding a physical resolution Fig. 3   a Experimental setup (surging kinematics) and b recorded surging kinematics for Case 1. Markers denote the selected snapshots used for flowfield visualisations in the result section (Sect. 4). The image of the experimental setup is taken from Corkery et al. (2019) and edited Fig. 4   a The experiment setup for Cases 2 and 3 (pitching NACA 0018) and b asymmetric triangular pitching kinemat- ics for Cases 1 and 2. Shaded regions denote acceleration/ deceleration parts: t∗ a = 0.15T  . The image is taken from Ōtomo et al. (2021) and edited Experiments in Fluids (2025) 66:64 Page 7 of 13  64 of 2.4 mm or 0.016c. In this case, the resolution is also not fine enough to resolve the velocity profile of the boundary layer, but the VFM method is not sensitive to grid resolution in computing forces (Li et al. 2021). 4 � Results 4.1 � Case 1: surging flat plate at ̨ = 45◦ Figure 5 shows the result for the surging flat plate (Case 1). Lift and drag estimated by VFM method are compared with those of the direct force balance measurements and impulse method (Eq. 16) in Fig. 5a. The added mass forces are esti- mated from Eqs. 7 and  8 in which the wing acceleration U̇ is computed from the optically recorded wing kinematics. The impulse formula inherently accounts for the added mass force by directly assessing the rate of change of the impulse generated by the evolution of vorticity in the boundary layer. Since the result of the impulse method is inevitably noisy due to the time derivative ( �∕�t ) arising from the experi- mental noises in PIV (see the green shaded line in Fig. 5a), the signal is filtered using the third-order Savitzky-Golay filter with the window size of 21 points (corresponding to t = 0.075 s or tU∞∕c = 0.14 ). Figure 5b shows the vorticity field for five typical instants selected from Fig. 5a. In VFM theory, for a flat plate with zero-thickness surg- ing at an angle of attack of 45◦ , the lift and drag curves should overlap. This is evident in Fig. 5a, where the lift and drag curves calculated from the VFM method are identical. However, the force balance measurements of lift and drag in Fig. 5a show a slight discrepancy due to the thickness of the plate and the vibration of the carriage. Overall, the lift and drag values obtained from the VFM method and the impulse method successfully capture the magnitude and main trend of the lift and drag curves from direct force measurements. The impulse method qualitatively captures the initial oscillations in the force balance measure- ments better than the VFM method (Fig. 5a). However, the impulse method introduces large fluctuations, necessitating the application of a filter for accurate force prediction. Addi- tionally, it fails to accurately extract the force over the whole period. In contrast, the VFM method successfully captures the major trend of the force balance measurement through- out the entire period. Its intrinsic nature, without the need for time derivatives, enables its use with snapshot data. The added mass force appears only during the acceleration and deceleration periods (see the dashed orange line in Fig. 5). Since the VFMs for the lift and drag on a flat plate at � = 45◦ are the same (Fig. 1), the lift and drag distribution is also the same as shown in Fig. 5c. During the accelerating process (Fig. 5c. 1.), the newly formed LEV and TEV con- tribute to positive lift and drag. During the surging process (Fig. 5c 2. and 3.), the upper half and the bottom half of the LEV contribute positively and negatively to lift and drag, respectively. Meanwhile, the TEV contributes solely to posi- tive lift and drag only when it is close to the aerofoil. During the stopping process (Fig. 5c. 4. and 5.), the LEV and TEV formed on the lower surface of the plate contribute to minor negative lift and drag. In this experiment, we did not incorporate the end plate. The discrepancy between the forces measured from the force balance and those computed from the VFM method could be attributed to the three-dimensional effect, which is difficult to eliminate perfectly in the experiment. 4.2 � Cases 2 and 3: pitching NACA 0018 aerofoil at ̨ 0 = 64◦ 4.2.1 � Case 2: symmetrically pitching aerofoil Figure 6 shows the result for the symmetrically pitching aerofoil (Case 2). Lift and drag coefficients computed from the VFM method are compared with those from direct force measurements in Fig. 6a. The added mass forces are com- puted based on the prescribed kinematics. Since the recorded angle of attack shows a relative error of less than 2% com- pared to the prescribed values, this error is considered neg- ligible for the added mass calculations. The vorticity field of 5 selected time instants is shown in Fig. 6b. Vortex lift and drag contributions of the same instants are shown in Fig. 6c and d, respectively. The periodic oscillations in lift and drag are closely linked to the formation and shedding of vortices. The lift and drag increase around 0 < tU∞∕c < 2 (marked as 1. and 2.) due to the accumulation of vorticity on the upper surface of the aerofoil, primarily contributing to positive lift and drag. The lift drops around tU∞∕c ≈ 2 (between 2. and 3.), while drag increases during this period as the aerofoil pitches to a higher angle of attack, where the LEV primarily contributes to drag. Both lift and drag decrease at tU∞∕c > 3 (marked as 3. and 4.) as the primary LEV moves away from the aerofoil, resulting in a weakening of the positive force contribution. Overall, the estimated lift and drag coefficients are in excellent agreement with that of the direct force measure- ment throughout the whole period. At the initial stage, spe- cifically when tU∞∕c < 2 , the vortex lift is slightly under- estimated (e.g. at 1.), while the drag aligns well with the measurements. This might be due to the missing informa- tion in the optically inaccessible region close to the aerofoil surface. At moderate angles of attack, vorticity forms and accu- mulates along the upper surface (e.g. at 2.), contributing positively to both lift and drag, while the TEVs mainly contribute to positive lift and drag. With the rolling up of a concentrated LEV (at 3.), the upper part contributes to Experiments in Fluids (2025) 66:64 64   Page 8 of 13 Fig. 5   Results for surging flat plate (Case 1): a comparison of lift coefficients and drag coefficients from experiments, impulse method, and VFM method (the shaded green line shows the unfiltered forces from the impulse method), b instantaneous vorticity field, c lift (equals drag) contri- bution of the selected five instants. Note that, for a zero- thickness flat plate at � = 45◦ , � L ⋅ u�dV = � D ⋅ u�dV . Experiments in Fluids (2025) 66:64 Page 9 of 13  64 positive lift and drag, while the lower part contributes to negative lift and drag. As the LEV detaches and moves away from the aerofoil (between 3. and 4.), its positive effect on forces decreases as shown in Fig. 6a. A square PIV window with a width and height of approximately 2c as shown in Fig. 6b, c, and d, is used to conduct the VFM method for the force estimation. As shown in Fig 6a., compared to the direct force measure- ment, the result of VFM method has small oscillations in forces due to the vortices moving away from the PIV window. Nonetheless, the effect of a limited window is small as verified in Li et al. (2021). This is an advanta- geous aspect of the VFM method because only the vortex information in the small PIV window (approximately half a chord length in the wake) needs to be captured, unlike the impulse method, which requires either capturing all the shed vortices or introducing complex terms to account for the PIV window boundaries. 4.2.2 � Case 3: asymmetrically pitching aerofoil Figure 7a shows the comparison of the lift and drag coef- ficients computed from the VFM and those from direct force measurements for the asymmetrically pitching aero- foil (Case 3). The vorticity field of 5 selected time instants is shown in Fig. 7b. Vortex lift and drag contributions are shown in Fig. 7c and d, respectively. The overall periodic oscillations in the forces are simi- larly linked to the formation and shedding of vortices as Case 2. Similar to the results of Case 2 (symmetric pitching case), the VFM method accurately reproduces the overall lift and drag coefficients obtained from direct force balance measurements. The underestimation of the lift coefficient at the initial stage ( tU∞∕c < 2 ) can be attributed to the lack of data in the optically inaccessible region near the leading edge of the aerofoil, where lift Fig. 6   Results for symmetri- cally pitching aerofoil (Case 2): a. comparison of lift and drag coefficients between experi- ments and VFM method (the angle of attack history is plotted in grey), b. instantaneous vorti- city field, c. vortex lift contribu- tion ( � L ⋅ u�dV  ), d. vortex drag contribution ( � D ⋅ u�dV  ) of the five selected instants. Experiments in Fluids (2025) 66:64 64   Page 10 of 13 contribution is substantial (see Fig. 2a). However, the drag aligns closely with the measurements during this initial phase. The significant f luctuation in drag observed at 2 < tU∞∕c < 4 (at 2. and 4. in Fig. 7a) underscores the fact that at a higher angle of attack, the absence of vortex infor- mation in the PIV shadow has a more pronounced impact on drag than on lift (see the VFM in Fig. 2d). At 4. and 5. in Fig. 7b, coherent LEV and TEV leave the PIV window, their influence on lift and drag is minimal, similar to what is observed in Case 2. A clear difference from Case 2 is the formation of a coherent TEV, which grows as large as the primary LEV. This TEV has both positive and negative contributions to lift and drag. Overall, the forces estimated through the VFM method exhibit a remarkable agreement with the direct force meas- urement for most of the period. Note that, in Figs. 6 and 7, at the start of the upstroke (small angles of attack), lift is underestimated, while drag does align well. Conversely, in the middle of the upstroke (large angles of attack), lift aligns closely with the experimental data, whereas drag is overes- timated. This discrepancy may arise from missing veloc- ity and vorticity data near the aerofoil surface, particularly in the shadowed region and the shear layer. As illustrated in Fig. 2, the dense positive (yellow) region near the aero- foil contributes significantly to lift at small angles of attack (Fig. 2a.) and to drag at large angles of attack (Fig. 2d.), whereas the impact of near-body vortices is minimal for drag at small angles of attack (Fig. 2b.) and for lift at large angles of attack (Fig. 2c.). Qualitatively, the underestimation of lift is due to the omission of positive lift-contributing vortices, while the overestimation of drag is due to the exclusion of vortices that would contribute negatively to drag. A quantita- tive analysis would require additional data near the surface or detailed CFD investigations, which could be of interest for future studies. Fig. 7   Results for asymmetri- cally pitching aerofoil (Case 3): a. comparison of lift and drag coefficients between experi- ments and VFM method (the angle of attack history is plotted in grey), b. instantaneous vorti- city field, c. vortex lift contribu- tion ( � L ⋅ u�dV  ), d. vortex drag contribution ( � D ⋅ u�dV  ) of the five selected instants. Experiments in Fluids (2025) 66:64 Page 11 of 13  64 5 � Concluding remarks This work is the first to apply the VFM method to esti- mate unsteady aerodynamic forces using snapshot velocity and vorticity field data from PIV measurements. We have tested the VFM method in two-dimensional flows, and added mass forces are computed based on the analytical expression for an ellipse undergoing pitch-surge motion. Two different kinematics (pitch and surge) and aerofoils (NACA 0018 and flat plate) are tested to validate the appli- cability of the VFM method for PIV data. Overall, the estimated lift and drag of three different cases are in agreement with the result of the direct force measurement. In Case 1 (surging flat plate), the VFM method provides accurate force estimates over extended periods, whereas the impulse method introduces large fluctuations and fails when vortices move outside the PIV window. Furthermore, the intrinsic nature of the VFM method, which excludes time derivatives, makes it suitable for estimating forces from single snapshot data in unsteady flows. Minor discrepancies between VFM-based estima- tion and force balance measurements may be attributed to three-dimensional effects. In Cases 2 and 3 (pitching NACA  0018 aerofoil at �0 = 64◦ ), the lift and drag estimates from the VFM method show excellent agreement with the measured forces. The only exception is at low angles of attack, where the vorti- city close to the aerofoil surface is optically inaccessible. In addition, the vortices leaving the PIV window are trivial to the force contribution in the VFM method. Further, forces estimated by the VFM method are smooth when compared to the impulse method, showing robustness to experimental noise. We, however, also find that the lift is underestimated at small angles of attack and the drag is overestimated at large angles of attack for Cases 2 and 3 (pitching aerofoil cases). We deem the errors are associated with the missing vorticity near the aerofoil surface, which can be investigated further to improve the VFM method applied to PIV. This result represents a significant advancement towards the development of a non-intrusive PIV-based force meas- urement technique using the VFM method. For future work, the application of the vortex moment method (Li et al. 2020a) and the three-dimensional VFM method (Li et al. 2020b) for single and multiple bodies  (Wang et al. 2022) on PIV data will be further explored, with three- dimensional flowfields obtained from tomographic PIV and shake-the-box particle tracking. The VFM method for flexible bodies will be developed and applied to PIV data. The ultimate aim of this study is to estimate the unsteady forces and moments acting on freely flying/swimming objects, such as living animals, from PIV data, where direct force measurement is not possible. Acknowledgements  This research is partially funded by the Daiwa Anglo-Japanese Foundation through Daiwa Foundation Awards (14465/15310), JSPS KAKENHI (JP24K17204), and JKA and its promotion funds from KEIRIN RACE (2024 M-468). The work is also supported by the Engineering Start-up Grant of King’s College London. The experimental data of the pitching aerofoil were acquired at EPFL led by Karen Mulleners. We thank Karen for the opportunity to use the experimental data for further analysis. Authors contribution  Shuji Otomo helped in formal analysis, experi- ments, investigation, methodology, visualisation, writing. Pascal Gehlert was involved in experiments, investigation, methodology. Holger Babinsky contributed to conceptualization, review, editing, supervision. Juan Li helped in conceptualization, software, formal analysis, investigation, methodology, visualisation, writing. Data availability  The PIV and force data for cases 2 and 3 are avail- able from the following link: https://​doi.​org/​10.​7488/​ds/​7677. The PIV and force data for case 1 are available from the corresponding author upon request. Declarations  Conflict of interest  The authors declare no conflict of interest. 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Exp Fluids 64:1–18. https://​doi.​org/​10.​1007/​ s00348-​023-​03698-5 Publisher's Note  Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. https://doi.org/10.1007/s00348-019-2803-5 https://doi.org/10.1007/s00348-016-2290-x https://doi.org/10.1007/s00348-014-1870-x https://doi.org/10.1007/s00348-014-1870-x https://doi.org/10.1006/jfls.1997.0111 https://doi.org/10.1088/0957-0233/24/3/032001 https://doi.org/10.1088/0957-0233/24/3/032001 https://doi.org/10.1007/s00348-006-0149-2 https://doi.org/10.1007/s00348-007-0261-y https://doi.org/10.1017/jfm.2022.956 https://doi.org/10.1017/jfm.2022.956 https://doi.org/10.2514/3.50966 https://doi.org/10.1007/s00348-023-03698-5 https://doi.org/10.1007/s00348-023-03698-5 Vortex force map method to estimate unsteady forces from snapshot flowfield measurements Abstract 1 Introduction 2 Force estimation technique 2.1 Vortex force map method 2.2 Impulse method 3 Experimental details and aerofoil kinematics 3.1 Case 1: surging flat plate at  3.2 Cases 2 and 3: pitching NACA 0018 aerofoil at  4 Results 4.1 Case 1: surging flat plate at  4.2 Cases 2 and 3: pitching NACA 0018 aerofoil at  4.2.1 Case 2: symmetrically pitching aerofoil 4.2.2 Case 3: asymmetrically pitching aerofoil 5 Concluding remarks Acknowledgements References