Weighted integral methods for fluid force diagnostics in incompressible flows

TL;DR

This work develops a unified weighted integral framework for diagnosing fluid forces in incompressible flows by introducing a second-order tensor weight into the Navier–Stokes equations, producing a weak form that retains full momentum transport information. It clarifies how different weight choices recover, extend, or couple existing control-volume methods, including divergence-free and curl-free projections and three weighted DMT formulations (advection, diffusion, boundary). Through canonical 2-D cylinder flow tests, it demonstrates improved numerical accuracy, richer force decompositions, and the ability to relate surface-pressure distributions to local flow structures via Q-based diagnostics. The framework accommodates force estimation from velocity/acceleration data without spatial derivatives or pressure measurements, paving the way for experimental and data-driven force diagnostics and enabling more insightful flow-control analyses. Overall, the weighted integral methods offer a flexible, Galilean-invariant toolkit that links surface stresses to underlying flow dynamics and enhances force diagnostics in both fundamentals and applications.

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.

Figures (17)

• Figure 1: Schematic diagram of the control volume enclosing the solid body.
• Figure 2: Summary of the roles of control-volume integral methods for force and moment diagnostics.
• Figure 3: Time-dependent dimensionless force on the circular cylinder at $Re=100$. (a) Drag. (b) Lift.
• Figure 4: Vorticity field in the flow past a circular cylinder at $Re=100$. (a) Instantaneous flow at the instant of maximum lift. Black lines are contour lines of $Q/(U_\infty^2d^{-2})=0.1$. V0, V2 and V4 label the most recently formed vortices from the upper surface. V1 and V3 are the most recently formed vortices from the lower surface. (b) Time-averaged flow. Black lines with arrows show the velocity profiles.
• Figure 5: Instantaneous kinetic energy of the disturbance flow and rescaled dissipation rate at the instant of maximum lift. (a) Kinetic energy, $0.5\rho|\hbox{\boldmath $u$}'|^2/(\rho U_\infty^2)$. (b) Rescaled dissipation rate, $\mathsfbi{D}:\mathsfbi{D}/(U_\infty^2d^{-2})$. Black contour lines represent dimensionless vorticity, with negative values (-0.1 and -0.03) as dashed lines and positive values (0.03 and 0.1) as solid lines.