Accurate fluid force measurement based on control surface integration

Nonintrusive 3D fluid force measurements are still challenging to conduct accurately for freely moving animals, vehicles, and deforming objects. Two techniques, 3D particle image velocimetry (PIV) and a new technique, the aerodynamic force platform (AFP), address this. Both rely on the control volume integral for momentum; whereas PIV requires numerical integration of flow fields, the AFP performs the integration mechanically based on rigid walls that form the control surface. The accuracy of both PIV and AFP measurements based on the control surface integration is thought to hinge on determining the unsteady body force associated with the acceleration of the volume of displaced fluid. Here, I introduce a set of non-dimensional error ratios to show which fluid and body parameters make the error negligible. The unsteady body force is insignificant in all conditions where the average density of the body is much greater than the density of the fluid, e.g., in gas. Whenever a strongly deforming body experiences significant buoyancy and acceleration, the error is significant. Remarkably, this error can be entirely corrected for with an exact factor provided that the body has a sufficiently homogenous density or acceleration distribution, which is common in liquids. The correction factor for omitting the unsteady body force, \({{{ {\rho _{\text{f}}}} \mathord{\left/ {\vphantom {{1 - {\rho _{\text{f}}}} {\left( {{\rho _{\text{b}}}\;+\;{\rho _{\text{f}}}} \right)}}} \right. \kern-0pt} {\left( {{{{\rho }}_{\text{b}}}\;+\;{\rho _{\text{f}}}} \right)}}} ,\) depends only on the fluid, \({\rho _{\text{f}}}\), and body, \({{\rho }}_{\text{b}}\), density. Whereas these straightforward solutions work even at the liquid–gas interface in a significant number of cases, they do not work for generalized bodies undergoing buoyancy in combination with appreciable body density inhomogeneity, volume change (PIV), or volume rate-of-change (PIV and AFP). In these less common cases, the 3D body shape needs to be measured and resolved in time and space to estimate the unsteady body force. The analysis shows that accounting for the unsteady body force is straightforward to non-intrusively and accurately determine fluid force in most applications.