OPTIMIZING FORMATION FLIGHT VIA THE CONSTRUCTAL LAW

Abstract

This paper explores 2 cases of flyers in formation: birds and aircraft, both seeking to save energy by flying in the wake of preceding flyers. The V-formation design occurs naturally with birds and is used in practice as a fundamental aircraft formation – aiming to boost lift/reduce induced drag felt by the flyers. The extent of the energy savings depends on the design configuration of the formation. The Constructal Law is applied to this formation flight problem to analyze how the distribution of drag among flyers in a V-formation is related to the optimal formation flight configuration. An analytical model that predicts this optimal configuration is developed and expressed regarding the fundamental formation parameters (velocity, number of aircraft, wingspan, weight, and air density). 

Model results show that there is not a fixed optimal V-formation design for all formation systems. Instead, the optimal configuration adapts as parameters change. The optimal configuration becomes narrower as the flyers' speed and spacing increase. As the number of flyers increases, the optimal configuration approaches a finite maximum angle. Trends predicted by the model are substantiated by observations from nature, including those of birds, boats, and other wake-generating flow systems. In line with predictions of the Constructal Law, induced drag along the optimal V-formation is distributed as uniformly as possible among flyers (depending on the longitudinal spacing). The model suggests the formation flight system constantly evolves towards a state that maximizes collective access to energy savings by changing the formation design accordingly.