Two-dimensional Descent Through a Compressible Atmosphere: Sequential Deceleration of an Unpowered Load

Equations, based on Rayleigh’s drag law valid for high Reynolds number, are derived for two-dimensional motion through a compressible atmosphere in isentropic equilibrium, such as characterizes the Earth’s troposphere. Solutions yield horizontal and vertical displacement, velocity, and acceleration as a function of altitude and ground-level temperature. An exact analytical solution to the equations linearized in the aero-thermodynamic parameter is given; in general the equations must be solved numerically. The theory, applied to the unpowered fall of a large aircraft stabilized to flat descent by symmetrical, sequential deployment of horizontal and vertical decelerators, shows that such an aircraft can be brought down with mean peak deployment and impact decelerations below 10g. Copyright c © EPLA, 2010 Introduction. – The air resistance (drag force) on an object descending through an atmosphere depends on the air density, square of the relative speed, effective area presented to the air stream, and drag coefficient [1]. The air density in turn is a function of altitude and air temperature. In an atmosphere in isentropic equilibrium, such as characterizes the Earth’s troposphere (depth 8–16 km from poles to tropics) [2], the density varies adiabatically with altitude. The drag coefficient is largely independent of size, but depends weakly on Reynolds number and sensitively on shape and origin (i.e. from pressure or friction). In this paper I derive and investigate a set of coupled nonlinear differential equations characterizing the two-dimensional descent through an ideal-gas atmosphere under adiabatic conditions of a composite object subject to a quadratic drag force as first proposed by Rayleigh for high Reynolds number (Re> 1000). The coupling of horizontal and vertical motions lead to results that can differ significantly from one-dimensional applications of Rayleigh’s equation for drag in an incompressible fluid. The theory developed here facilitates realistic modeling of the impact of temperature and density variations on air drag, serves as a model for extension to more general polytropic atmospheres, and permits exact analysis of (a)E-mail: [email protected]; http://www.trincoll. edu/∼silverma the controlled descent of fragile loads, a topic of vital concern to space agencies, cargo transporters, and general aviation. In regard to the latter, in particular, I illustrate the significance of the theory by demonstrating how a large passenger airliner, having suffered total loss of power, may be brought to ground by means of a sequentially released parachute-assisted descent with impact deceleration below 10g, where g= 9.8m/s is the acceleration of gravity near the Earth’s surface. The trend in design of modern commercial aircraft, driven in part by rising costs of fuel, construction materials, and labor, is to larger, heavier planes that transport ever greater numbers of passengers. Aerodynamicists now routinely contemplate design models capable of carrying 800 or more people [3]. Although air travel is presently considered very safe, no human-made machine is 100% reliable, and it is therefore certain that at least one of these airplanes would eventually fail in service with a huge number of fatalities. It is consequently of major importance to investigate how the laws of physics may be used to avert such a catastrophe. The maximum acceleration that a human can endure has long been of practical interest to organizations concerned with high-speed transport. Estimates have ranged from about 10g to 100g, depending on duration and orientation of impact. Particularly striking was the case of racing driver David Purley who survived a crash estimated to have produced 179g [4]. A comprehensive