Hostname: page-component-857557d7f7-cmjwd Total loading time: 0 Render date: 2025-12-10T04:03:53.422Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted integral methods for fluid force diagnostics in incompressible flows

Published online by Cambridge University Press:  10 December 2025

An-Kang Gao Open the ORCID record for An-Kang Gao [Opens in a new window] ,
Chenyue Xie Open the ORCID record for Chenyue Xie [Opens in a new window]  and
Xi-Yun Lu Open the ORCID record for Xi-Yun Lu [Opens in a new window]
An-Kang Gao*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China , Hefei, Anhui 230026, PR China
Chenyue Xie
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China , Hefei, Anhui 230026, PR China
Xi-Yun Lu*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China , Hefei, Anhui 230026, PR China
*
Corresponding authors: An-Kang Gao, ankanggao@ustc.edu.cn; Xi-Yun Lu, xlu@ustc.edu.cn
Corresponding authors: An-Kang Gao, ankanggao@ustc.edu.cn; Xi-Yun Lu, xlu@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Whilst surface-stress integration remains the standard approach for fluid force evaluation, control-volume integral methods provide deeper physical insights through functional relationships between the flow field and the resultant force. In this work, by introducing a second-order tensor weight function into the Navier–Stokes equations, we develop a novel weighted-integral framework that offers greater flexibility and enhanced capability for fluid force diagnostics in incompressible flows. Firstly, in addition to the total force and moment, the weighted integral methods establish, for the first time, rigorous quantitative connections between the surface-stress distribution and the flow field, providing potential advantages for flexible body analyses. Secondly, the weighted integral methods offer alternative perspectives on force mechanisms, through vorticity dynamics or pressure view, when the weight function is set as divergence-free or curl-free, respectively. Thirdly, the derivative moment transformation (DMT)-based integral methods (Wu et al., J. Fluid Mech. vol. 576, 2007, 265–286) are generalised to weighted formulations, by which the interconnections among the three DMT methods are clarified. In the canonical problem of uniform flow past a circular cylinder, weighted integral methods demonstrate advantages in yielding new force expressions, improving numerical accuracy over original DMT methods, and enhancing surface-stress analysis. Finally, a force expression is derived that relies solely on velocity and acceleration at discrete points, without spatial derivatives, offering significant value for experimental force estimation. This weighted integral framework holds significant promise for flow diagnostics in fundamentals and applications.

JFM classification

Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex dynamics Aerodynamics: Aerodynamics

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1024 , 10 December 2025 , A57
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alekseenko, S.V., Kuibin, P.A. & Okulov, V.L. 2007 Theory of Concentrated Vortices: an Introduction. Springer Science & Business Media.Google Scholar
Andersen, A., Bohr, T., Schnipper, T. & Walther, J.H. 2017 Wake structure and thrust generation of a flapping foil in two-dimensional flow. J. Fluid Mech. 812, R4.10.1017/jfm.2016.808CrossRefGoogle Scholar
Anderson, J. 2010 Fundamentals of Aerodynamics, 5th edition., McGraw hill.Google Scholar
Bandyopadhyay, P.R. 2016 Highly maneuverable biorobotic underwater vehicles. In Springer Handbook of Ocean Engineering, pp. 281–300. Springer.10.1007/978-3-319-16649-0_11CrossRefGoogle Scholar
Bréguet, L. 1923 Calcul du poids de combustible consummé par un avion en vol ascendant. Comptes Rendus Hebdomadaires Des Séances De l’Académie Des Sciences 177, 870–872.Google Scholar
Burgers, J.M. 1920 On the resistance of fluid and vortex motion. In Proc. K. Akad. Wet, vol. 23, pp. 774–782.Google Scholar
Cantwell, C.D. et al. 2015 Nektar++: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219.10.1016/j.cpc.2015.02.008CrossRefGoogle Scholar
Chang, C.-C. 1992 Potential flow and forces for incompressible viscous flow. Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 437(1901), 517–525.Google Scholar
Chiu, T.-Y., Tseng, C.-C., Chang, C.-C. & Chou, Y.-J. 2023 Vorticity forces of coherent structures on the NACA0012 aerofoil. J. Fluid Mech. 974, A52.10.1017/jfm.2023.815CrossRefGoogle Scholar
Chorin, A.J., Marsden, J.E. & Marsden, J.E. 1993 A Mathematical Introduction to Fluid Mechanics. 3rd edn. Springer.10.1007/978-1-4612-0883-9CrossRefGoogle Scholar
Chu, B.-T. & Kovásznay, L.S.G. 1958 Non-linear interactions in a viscous heat-conducting compressible gas. J. Fluid Mech. 3 (5), 494–514.CrossRefGoogle Scholar
Díaz-Arriba, D., Protas, B., Acher, G., Pons, F. & David, L. 2022 Application of a variational approach to the computation of forces around a wing. Exp. Fluids 63 (1), 31.10.1007/s00348-021-03376-4CrossRefGoogle Scholar
Ding, H., Shu, C., Yeo, K.S. & Xu, D. 2004 Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method. Comput. Meth. Appl. Mech. Engng 193 (9), 727–744.10.1016/j.cma.2003.11.002CrossRefGoogle Scholar
Gao, A.-K., Cantwell, C.D., Son, O. & Sherwin, S.J. 2023 Three-dimensional transition and force characteristics of low-Reynolds-number flows past a plunging airfoil. J. Fluid Mech. 973, A43.10.1017/jfm.2023.735CrossRefGoogle Scholar
Gao, A.-K. & Wu, J. 2019 A note on the Galilean invariance of aerodynamic force theories in unsteady incompressible flows. Acta Mech. Sinica 35 (6), 1150–1154.10.1007/s10409-019-00896-5CrossRefGoogle Scholar
Gao, A.-K., Zou, S.-F., Shi, Y. & Wu, J.Z. 2019 Passing-over leading-edge vortex: the thrust booster in heaving airfoil. AIP Adv. 9 (3), 035314.10.1063/1.5064696CrossRefGoogle Scholar
Gehlert, P., Andreu-Angulo, I. & Babinsky, H. 2023 Vortex force decomposition–forces associated with individual elements of a vorticity field. Exp. Fluids 64 (6), 112.10.1007/s00348-023-03654-3CrossRefGoogle Scholar
Graham, M. & Li, J. 2024 Vortex shedding and induced forces in unsteady flow. Aeronaut. J. 128 (1328), 2149–2190.10.1017/aer.2024.82CrossRefGoogle Scholar
Howe, M.S. 1989 On unsteady surface forces, and sound produced by the normal chopping of a rectilinear vortex. J. Fluid Mech. 206, 131–153.10.1017/S0022112089002259CrossRefGoogle Scholar
Jiang, X.Y., Lee, C.B., Smith, C.R., Chen, J.W. & Linden, P.F. 2020 Experimental study on low-speed streaks in a turbulent boundary layer at low Reynolds number. J. Fluid Mech. 903, A6.10.1017/jfm.2020.617CrossRefGoogle Scholar
Joukowski, N. 1906 De la chute dans l’air de corps légers de forme allongée, animés d’un mouvement rotatoire. Bull. Inst. Aérodyn. Koutchino 1, 51–65.Google Scholar
Kang, L., Gao, A.-K., Han, F., Cui, W. & Lu, X.-Y. 2023 Propulsive performance and vortex dynamics of jellyfish-like propulsion with burst-and-coast strategy. Phys. Fluids 35 (9), 091904.CrossRefGoogle Scholar
Kang, L.L., Liu, L.Q., Su, W.D. & Wu, J.Z. 2018 Minimum-domain impulse theory for unsteady aerodynamic force. Phys. Fluids 30 (1), 016107.10.1063/1.5010008CrossRefGoogle Scholar
von Kármán, T.& Burgers, J.M. 1935 General aerodynamic theory. In Perfect Fluids, vol. 2, Springer.Google Scholar
von Kármán, T.H. & Sears, W.R. 1938 Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5 (10), 379–390.10.2514/8.674CrossRefGoogle Scholar
Karniadakis, G.E., Israeli, M. & Orszag, S.A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414–443.10.1016/0021-9991(91)90007-8CrossRefGoogle Scholar
Kutta, W. 1902 Lift forces in flowing fluids. Illustrated Aeronaut. Commun 3, 133–135.Google Scholar
Li, G.-J. & Lu, X.-Y. 2012 Force and power of flapping plates in a fluid. J. Fluid Mech. 712, 598–613.CrossRefGoogle Scholar
Li, J., Wang, Y., Graham, M. & Zhao, X. 2020 Vortex moment map for unsteady incompressible viscous flows. J. Fluid Mech. 891, A13.CrossRefGoogle Scholar
Li, J. & Wu, Z.-N. 2018 Vortex force map method for viscous flows of general airfoils. J. Fluid Mech. 836, 145–166.10.1017/jfm.2017.783CrossRefGoogle Scholar
Lighthill, J. 1986 An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press.Google Scholar
Liu, C., Zheng, X. & Sung, C.H. 1998 Preconditioned multigrid methods for unsteady incompressible flows. J. Comput. Phys. 139 (1), 35–57.10.1006/jcph.1997.5859CrossRefGoogle Scholar
Liu, H., Wang, S. & Liu, T. 2024 Vortices and forces in biological flight: insects, birds, and bats. Annu. Rev. Fluid Mech. 56 (2024), 147–170.10.1146/annurev-fluid-120821-032304CrossRefGoogle Scholar
Liu, L.Q., Shi, Y.P., Zhu, J.Y., Su, W.D., Zou, S.F. & Wu, J.Z. 2014 Longitudinal–transverse aerodynamic force in viscous compressible complex flow. J. Fluid Mech. 756, 226–251.CrossRefGoogle Scholar
Lynch, K.P. & Scarano, F. 2014 Material acceleration estimation by four-pulse tomo-PIV. Meas. Sci. Technol. 25 (8), 084005.10.1088/0957-0233/25/8/084005CrossRefGoogle Scholar
Mao, F., Kang, L.L., Wu, J.Z., Yu, J.-L., Gao, A.K., Su, W.D. & Lu, X.-Y. 2020 A study of longitudinal processes and interactions in compressible viscous flows. J. Fluid Mech. 893, A23.10.1017/jfm.2020.213CrossRefGoogle Scholar
Marongiu, C. & Tognaccini, R. 2010 Far-field analysis of the aerodynamic force by Lamb vector integrals. Aiaa J. 48 (11), 2543–2555.10.2514/1.J050326CrossRefGoogle Scholar
Martín-Alcántara, A. & Fernandez-Feria, R. 2019 Assessment of two vortex formulations for computing forces of a flapping foil at high Reynolds numbers. Phys. Rev. Fluids 4, 024702.10.1103/PhysRevFluids.4.024702CrossRefGoogle Scholar
Menon, K. & Mittal, R. 2021 Quantitative analysis of the kinematics and induced aerodynamic loading of individual vortices in vortex-dominated flows: a computation and data-driven approach. J. Comput. Phys. 443, 110515.10.1016/j.jcp.2021.110515CrossRefGoogle Scholar
Munk, M. 1919 Isoperimetrische Aufgaben Aus Der Theorie Des Fluges. Dieterichsche Universitäts-Buchdruckerei.Google Scholar
Noca, F. 1996 On the evaluation of instantaneous fluid-dynamic forces on a bluff body, Tech. Rep. Report FM96-5.GALCIT.Google Scholar
Ōtomo, S., Gehlert, P., Babinsky, H. & Li, J. 2025 Vortex force map method to estimate unsteady forces from snapshot flowfield measurements. Exp. Fluids 66 (3), 1–13.CrossRefGoogle Scholar
Palacios, R. & Cesnik, C.E.S. 2023 Dynamics of flexible aircraft: coupled flight mechanics, aeroelasticity, and control. In Cambridge Aerospace Series. Cambridge University Press.10.1017/9781108354868CrossRefGoogle Scholar
Prandtl, L. 1904 Über flüssigkeitsbewegung bei sehr kleiner reibung. Proc. Third Int. Math. Cong. 2, 484.Google Scholar
Prandtl, L. 1918 Tragflügeltheorie. i. mitteilung. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1918, 451–477.Google Scholar
Protas, B. 2007 On an attempt to simplify the Quartapelle–Napolitano approach to computation of hydrodynamic forces in open flows. J. Fluid. Struct. 23 (8), 1207–1214.10.1016/j.jfluidstructs.2007.05.002CrossRefGoogle Scholar
Quartapelle, L. & Napolitano, M. 1983 Force and moment in incompressible flows. AIAA J. 21 (6), 911–913.10.2514/3.8171CrossRefGoogle Scholar
Saffman, P.G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sahoo, A., Dwivedy, S.K. & Robi, P.S. 2019 Advancements in the field of autonomous underwater vehicle. Ocean Engng 181, 145–160.CrossRefGoogle Scholar
Shyy, W., Aono, H., Chimakurthi, S.K., Trizila, P., Kang, C.-K., Cesnik, C.E.S. & Liu, H. 2010 Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46 (7), 284–327.10.1016/j.paerosci.2010.01.001CrossRefGoogle Scholar
Taylor, G.I. 1926 Note on the connection between the lift on an aërofoil in a wind and the circulation round it. Phil. Trans. R. Soc. Lond. A 225, 238–246.Google Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter, Tech. Rep. NACA-TR-496.NACA.Google Scholar
Wang, S., Zhang, X., He, G. & Liu, T. 2013 A lift formula applied to low-Reynolds-number unsteady flows. Phys. Fluids 25 (9), 093605.CrossRefGoogle Scholar
Wu, J., Liu, L. & Liu, T. 2018 Fundamental theories of aerodynamic force in viscous and compressible complex flows. Prog. Aerosp. Sci. 99, 27–63.10.1016/j.paerosci.2018.04.002CrossRefGoogle Scholar
Wu, J.C. 1981 Theory for aerodynamic force and moment in viscous flows. AIAA J. 19 (4), 432–441.10.2514/3.50966CrossRefGoogle Scholar
Wu, J.-Z., Lu, X.-Y. & Zhuang, L.-X. 2007 Integral force acting on a body due to local flow structures. J. Fluid Mech. 576, 265–286.10.1017/S0022112006004551CrossRefGoogle Scholar
Wu, J.Z., Ma, H.Y. & Zhou, M.D. 2006 Vorticity and Vortex Dynamics. Springer.10.1007/978-3-540-29028-5CrossRefGoogle Scholar
Wu, J.-Z. & Wu, J.-M. 1993 Interactions between a solid surface and a viscous compressible flow field. J. Fluid Mech. 254, 183–211.10.1017/S0022112093002083CrossRefGoogle Scholar
Xie, C., Wang, J. & E, W. 2020 a Modeling subgrid-scale forces by spatial artificial neural networks in large eddy simulation of turbulence. Phys. Rev. Fluids 5, 054606.10.1103/PhysRevFluids.5.054606CrossRefGoogle Scholar
Xie, C., Yuan, Z. & Wang, J. 2020 b Artificial neural network-based nonlinear algebraic models for large eddy simulation of turbulence. Phys. Fluids 32 (11), 115101.10.1063/5.0025138CrossRefGoogle Scholar
Yu, Y. 2014 The virtual power principle in fluid mechanics. J. Fluid Mech. 744, 310–328.10.1017/jfm.2014.45CrossRefGoogle Scholar
Zou, S.-F., Wu, J.Z., Gao, A.-K., Liu, L., Kang, L. & Shi, Y. 2019 On the concept and theory of induced drag for viscous and incompressible steady flow. Phys. Fluids 31 (6), 065106.10.1063/1.5090165CrossRefGoogle Scholar