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Aerodynamic Wave Drag AccessScience from McGraw-Hill Education Page 1 of 5 www.accessscience.com Aerodynamic wave drag Contributed by: Richard T. Whitcomb Publication year: 2019 The force retarding an airplane, especially in supersonic flight, as a consequence of the formation of shock waves. Although the physical laws governing flight at speeds in excess of the speed of sound are the same as those for subsonic flight, the nature of the flow about an airplane and, as a consequence, the various aerodynamic forces and moments acting on the vehicle at these higher speeds differ substantially from those at subsonic speeds. Basically, these variations result from the fact that at supersonic speeds the airplane moves faster than the disturbances of the air produced by the passage of the airplane. These disturbances are propagated at roughly the speed of sound and, as a result, primarily influence only a region behind the vehicle. Causes of wave drag The primary effect of the change in the nature of the flow at supersonic speeds is a marked increase in the drag, resulting from the formation of shock waves about the configuration. These strong disturbances, which may extend for many miles from the airplane, cause significant energy losses in the air, the energy being drawn from the airplane. At supersonic flight speeds these waves are swept back obliquely, the angle of obliqueness decreasing with speed (Fig. 1). For the major parts of the shock waves from a well-designed airplane, the angle of obliqueness is equal to sin,−1 (1∕M ), where M is the Mach number, the ratio of the flight velocity to the speed of sound. See also: SHOCK WAVE; SUPERSONIC FLIGHT. The shock waves are associated with outward diversions of the airflow by the various elements of the airplane. This diversion is caused by the leading and trailing edges of the wing and control surfaces, the nose and aft end of the fuselage, and other parts of the vehicle. Major proportions of these effects also result from the wing incidence required to provide lift. Magnitude of wave drag Usually the aerodynamic forces acting on a vehicle, such as drag, are considered as coefficients. The wave-drag coefficient for a vehicle depends on many parameters, including the thickness-to-chord ratios and shapes of the airfoil sections of wings and fins, the planform shapes of such surfaces, the length-to-diameter ratio of the body, and the shape of the engine housing. It also depends on the Mach number. For an unswept surface with a thin airfoil whose shape is designed for wave-drag reduction (as discussed below), the coefficient of wave drag is AccessScience from McGraw-Hill Education Page 2 of 5 www.accessscience.com WIDTH:BFig. 1 Shock waves about an airplane at supersonic speeds. μ = angle of obliqueness. given approximately by Eq. (1), (1) Image of Equation 1 where t∕c is the airfoil thickness-to-chord ratio and α is the angle of attack in radians. For surfaces with more complex airfoil shapes and planforms, the computation of wave drag is more complex. For complete vehicles, the wave drag at zero lift is given by the area rule. For supersonic flight, the cross-sectional areas used are obtained in planes inclined at the Mach angle. At the lower supersonic speeds, the wave drag at the zero lift condition is usually more significant than the drag due to wing incidence. But when the Mach number is increased, the relative magnitude of wave drag at the zero lift condition gradually decreases, and the drag associated with wing incidence progressively increases, so at the higher supersonic speeds wave drag due to lift is usually more important than the zero lift value. For a well-designed vehicle, wave drag is usually roughly equal to the sum of the basic skin friction and the induced drag due to lift. See also: AERODYNAMIC FORCE; AIRFOIL; TRANSONIC FLIGHT. AccessScience from McGraw-Hill Education Page 3 of 5 www.accessscience.com WIDTH:BFig. 2 Comparison of airfoil sections for subsonic flight and supersonic flight. Reduction of wave drag The wave drag at the zero lift condition is reduced primarily by decreasing the thickness-chord ratios for the wings and control surfaces and by increasing the length-diameter ratios for the fuselage and bodies. Also, the leading edge of the wing and the nose of the fuselage are made relatively sharp (Fig. 2). With such changes, the severity of the diversions of the flow by these elements is reduced, with a resulting reduction of the strength of the associated shock waves. Also, the supersonic wave drag can be reduced by shaping the fuselage and arranging the components on the basis of the area rule. See also: WING; WING STRUCTURE. The wave drag can also be reduced by sweeping the wing panels (Fig. 3a). Some wings intended for supersonic flight have large amounts of leading-edge sweep and little or no trailing-edge sweep (Fig. 3b). Such planforms are referred to as delta or modified delta. For a simple infinite-span, constant-chord airfoil, the effective Mach number determining the aerodynamic characteristics is the component of the flight Mach number normal to the swept elements (Fig. 4). This Mach number is defined by Eq. (2), Image of Equation 2 (2) where Λ is the angle of the sweep. If the sweep is such that M,normal is less than the subsonic Mach number at which the initial onset of wave drag occurs for the unswept airfoil, the swept airfoil will have no wave drag for the flight Mach number. For flight Mach numbers for which the normal component is greater than that for the onset of a shock on the unswept airfoil, the wave drag is not eliminated but is significantly reduced. For practical finite-span swept or delta wings, the reductions of wave drag are usually less than for the ideal infinite-span case. These reductions can be substantially improved by the proper chordwise and spanwise variations of thickness, camber, and incidence. The shape changes required are now determined using very complex fluid-dynamic relationships and supercomputers. See also: COMPUTATIONAL FLUID DYNAMICS. When the speed is increased to supersonic values, an airplane at a given attitude and altitude experiences large increases in drag, in addition to those associated with the different nature of the flow, because of the higher dynamic pressure at these higher speeds. To offset this effect, supersonic airplanes usually fly at considerably higher altitudes than subsonic vehicles. For example, for efficient flight at Mach 2, an airplane must fly at an altitude of about 60,000 ft (18,000 m). AccessScience from McGraw-Hill Education Page 4 of 5 www.accessscience.com WIDTH:BFig. 3 Wing panels. (a) Sweep-back. (b)Delta. WIDTH:BFig. 4 Geometry that determines the effective Mach number for an infinite-span swept airfoil. A major problem associated with supersonic flight, particularly at the higher supersonic speeds, is that of taking air into the engines. This air must be decelerated from the flight velocity to a relatively low speed at the compressor of the engine, without excessive energy losses. With a simple inlet, such as that used on subsonic and transonic airplanes, a strong normal shock wave forms ahead of the forward face at supersonic speeds. This shock causes severe loss of energy in the air reaching the engine, and consequent losses of engine performance. AccessScience from McGraw-Hill Education Page 5 of 5 www.accessscience.com In addition, the drag of the airplane is increased. To reduce these losses, special inlets and diffusers which decelerate the airflow to the engine by a series of weak disturbances are used. See also: SUPERSONIC DIFFUSER. Richard T. Whitcomb Bibliography B. Mele, M. Ostieri, and R. Tognaccini, Aircraft lift and drag decomposition in transonic flows, J. Aircraft, 54(5):1–12, 2017 DOI: http://doi.org/10.2514/1.c034288 W. Yong, Study on aerodynamic characteristics of supersonic airfoil, Mod. Mech. Eng., 9(1):13–19, 2019 DOI: http://doi.org/10.4236/mme.2019.91002 Additional Readings S. Hershel (ed.), Fundamentals of Aerodynamics, Larsen & Keller Education, 2018.
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