Whilst a Postdoctoral Researcher at LadHyX, Ecole Polytechnique, I worked with Michael Benzaquen and Christophe Clanet on several topics in Sports Physics. Here are two brief summaries of our work on boat sports and cycling.

The physics of kayaks, canoes and rowing

Have you ever wondered why a kayak or a rowing boat is pointed at both the front and back? Whilst the point at the back makes the kayak more streamlined (similar to the tapered end of an airplane wing), the point at the front helps the boat slice through the water with minimal wave disturbance. Each of these aspects of the design are made to minimise wasted energy that could otherwise be used to make the boat go faster. But because these two sharp points have origins in two different physical phenomena, there is no reason for them to be symmetric. In other words, why not design a kayak with some front-back asymmetry?

There is a classical theory for predicting the drag due to wave generation, derived more than a century ago by J.H. Michell [1]. The major shortcoming of this formula is that, due to the reversibility of the steady potential flow formulation, it predicts no difference in the wave drag when an object with front-back asymmetry moves forwards or backwards. However, anyone who has tried to row a dinghy in the wrong direction would argue differently! In a recent paper [2], I presented several experimental observations investigating the effects of body asymmetry on wave drag, and showed that these effects can be replicated by modifying Michell’s theory to include the growth of a symmetry-breaking boundary layer. These results showed that for the paramter ranges relevant to rowing sports, it is advantageous to have a slightly more pointed leading edge than trailing edge, which is in accordance with modern sporting design.

Another factor which strongly affects the wave drag is the water depth. Olympic race courses have a minimum depth requirement of 3m, but with boats as long as 18m, rowers are likely to generate waves in both the deep (dispersive) and shallow (non-dispersive) regimes at various moments during a race. Entering from deep to shallow water is accompanied by a focusing of the wave drag near the shallow wave speed – and hence the emergence of co-existing fast and slow solution branches. In another recent paper [3] I described the non-linear dynamics of such motion, including possible bifurcation patterns and hysteresis routes during a race. Such effects have important implications in boat races, where different solution branches correspond to either competitive or non-competitive race performance.

[1] Michell, J.H. The wave-resistance of a ship, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 45.272: 106-123, (1898).

[2] Benham, G.P., Boucher, J.P., Labbé, R., Benzaquen, M., Clanet, C. Wave drag on asymmetric bodies. Journal of Fluid Mechanics, 878, pp.147-168. (2019).[pdf]

[3] Benham, G.P., Bendimerad, R., Benzaquen, M., Clanet, C. Hysteretic wave drag in shallow water. Physical Review Fluids (2020).[pdf]

Brachistochrone on a velodrome

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the fastest time, is one of the most famous studies in classical physics. There is a similar problem in track cycling, where a cyclist aims to find the trajectory on the curved sloping surface of a velodrome that results in the minimum lap time. In a recent paper [3], I extended the classical Brachistochrone problem to find the optimum cycling trajectory in a velodrome, treating the cyclist as an active particle. Starting with two canonical cases of cycling on a sloping plane and a cone, where analytical solutions were found, I then solved the problem numerically on the reconstructed surface of Velodrome de Montigny le Bretonneux in France, comparing with real cyclist data.

[3] Benham, G.P., Cohen, C., Brunet, E., Clanet, C. Brachistochrone on a velodrome. Proceedings of the Royal Society A (2020).[pdf]