Aerodynamic versus Ballistic Flight

We consider, compare, and contrast various aspects of aerodynamic and ballistic flight. We compare the energy efficiency of aerodynamic level flight at a given altitude versus that of ballistic flight beginning and ending at this same altitude. We show that for flights short compared to Earth’s radius, aerodynamic level flight with lift-to-drag ratio L/D > 2?is more energy-efficient than ballistic flight, neglecting air resistance or drag in the latter. Smaller L/D suffices if air resistance in ballistic flight is not neglected. For a single circumnavigation of Earth, we show that aerodynamic flight with L/D > 4π is more energy-efficient than minimum-altitude circular-orbit ballistic spaceflight. We introduce the concept of gravitational scale height, which may in an auxiliary way be helpful in understanding this result. For flights traversing N circumnavigations of Earth, if then even minimum-altitude circular-orbit ballistic spaceflight is much more energy-efficient than aerodynamic flight because even at minimum circular-orbit spaceflight altitude air resistance is very small. For higher-altitude spaceflight air resistance is even smaller and the energy-efficiency advantage of spaceflight over aerodynamic flight traversing the same distance is therefore even more pronounced. We distinguish between the energy efficiency of flight per se and the energy efficiency of the engine that powers flight. Next we consider the effects of air density on aerodynamic level flight and provide a simplified view of drag and lift. We estimate the low-density/high-altitude limits of aerodynamic level flight (and for comparison also of balloons) in Earth’s and Mars’ atmospheres. Employing Mars airplanes and underwater airplanes on Earth (and hypo-thetically also on Mars) as examples, we consider aerodynamic level flight in rarefied and dense aerodynamic media, respectively. We also briefly discuss hydrofoils. We appraise the optimum range of air densities for aerodynamic level flight. We then consider flights of hand-thrown projec-tiles that are unpowered except for the initial throw. We describe how aerodynamically efficient ones (i.e., with large L/D) such as Frisbees, Aerobies, and boomerangs not only can traverse record horizontal distances, but (along with discuses) also can—since lift exceeds weight at achievable throwing speeds—maintain altitude farther if thrown horizontally against the wind than with it. Then we compare the energy efficiency of surface transportation versus that of both aerodynamic and ballistic flight.

Pressure Gradient, Power, and Energy of Vortices

We consider small vortices, such as tornadoes, dust devils, whirlpools, and small hurricanes at low latitudes, for which the Coriolis force can be neglected. Such vortices are (at least approximately) cylindrically symmetrical about a vertical axis through the center of a calm central region or eye of radius . In the region fluid (gas or liquid) circulates about the eye with speed . We take to be the outer periphery of the vortex, where the fluid speed is reduced to that of the surrounding wind field (in the cases of tornadoes, dust devils, and small hurricanes at low latitudes) or deemed negligible (in the case of whirlpools). If , angular momentum is conserved within the fluid itself;if , angular momentum must be exchanged with Earth to ensure conservation of total angular momentum. We derive the steepness and upper limit of the pressure gradients in vortices. We then discuss the power and energy of vortices. We compare the kinetic energy of atmospheric vortices and the power required to maintain them against frictional dissipation with the same quantities for Earth’s atmosphere as a whole. We explain why the kinetic energy of atmospheric vortices must be replaced on much shorter timescales than is the case for Earth’s atmosphere as a whole. Brief comparisons of cyclostrophic flow with geostrophic and friction-balanced flows are then provided. We then consider an analogy that might be drawn, at least to some extent, with gravitational systems, considering mainly spherically-symmetrical and cylindrically-symmetrical ones. Generation of kinetic energy at the expense of potential energy in fluid vortices, in geostrophic and friction-balanced flows, and in gravitational systems is then briefly discussed. We explain the variations of pressure and gravitational gradients corresponding to generation of kinetic energy exceeding, equaling, and falling short of frictional dissipation. In the Appendix, we describe a simple method for maximizing power extraction from environmental fluid (air or water) flows. In summary, we provide an overview of features and energetics of Earth’s environmental fluid flows and of gravitational analogies thereto that, even though mostly semiquantitative, hopefully may be helpful.