# Planar momentum balance in three-dimensional flows: applications to load estimation

### Abstract

Load estimation from the momentum balance for a planar control volume is investigated, a configuration which is ubiquitous in experimental fluid mechanics. The classical formulation relies on an assumption of two-dimensional flow which is violated for instantaneous turbulent or transitional flow fields, and to varying degrees in measured mean and second order statistics. A rigorous formulation required for the exact momentum balance over the planar control volume is derived here, involving additional terms related to the three-dimensional momentum fluxes and viscous forces. The formulation is verified using instantaneous and mean control volume load estimations based on synthetic measurements extracted from direct numerical simulations and experimental particle image velocimetry measurements of flow over a cylinder in a turbulent wake regime. The main result is the demonstration of the significance of area integrals involving out-of-plane velocity and velocity gradients, which are shown to be required for accurate estimation of instantaneous sectional loads and cause a sensitivity to experimental conditions for mean sectional load estimates. If volumetric measurements are unavailable to the experimentalist, the divergence-free condition on the velocity field for incompressible flow may be invoked to account for a subset of the three-dimensional terms. This enables higher accuracy, particularly for the mean drag estimates in the statistically three-dimensional flow encountered in experiment. The findings highlight common issues responsible for inconsistent estimates of instantaneous and mean sectional loads for three-dimensional flows using control volume methods, and offer practical recipes for detecting and minimizing errors due to flow three-dimensionality.

### Graphical abstract

## Notes

### Acknowledgements

The authors gratefully acknowledge the contribution of the Natural Science and Engineering Research Council (NSERC) (Grant no. RGPIN-04222) to the funding of this research.

## Supplementary material

## References

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