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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
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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

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

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