Airplane configurations optimized for achieving high aerodynamic efficiency in supersonic cruising flight typically feature a thin and slender wing with sharp trailing edges and a slender fuselage body with a smooth variation of the cross section area between the pointed nose and tail. In order to avoid unfavorable aerodynamic interactions, the arrangement of aircraft components is carefully optimized, and at small angles of attack such a configuration produces weak shock waves, thin boundary layers, and attached flows. An accurate method for computing the pressure distribution on a two‐dimensional airfoil is the shock‐expansion technique treated in Section 4.8. An alternative and widely used method is the linear theory for thin airfoils, which is applicable to two‐dimensional potential flow, and was first published in 1925 by the Swiss scientist J. Ackeret (1898–1981) [11]. This flow model replaces shock waves with Mach waves, disregarding variations in the local Mach number.
Linear theory yields a good approximation of the pressure distribution at locations where flow separation is not dominant and can be used to compute the pressure distribution of thin airfoil sections at a small angle of attack in flows at transonic and low‐supersonic Mach numbers. For subsonic as well as supersonic flight at small angles of attack, the two‐dimensional flow in which a flying vehicle is immersed can be treated as predominantly isentropic with small perturbations imposed by flow deflections. The analysis of this type of flow is based on the conservation laws of mass, momentum, and energy. Combination of the associated equations yields a set of non‐linear partial differential equations that must usually be solved numerically. However, in the absence of viscosity, rotational flows, and shock waves, the external air stream can be represented as a potential flow field. If only small perturbations are manifest in the flow, the velocity potential can be modified and solved in terms of a linearized Laplace equation. The properties of such a flow can be computed fairly accurately by means of small perturbation theory. Its solution can then be represented as an approximation for the pressure distribution on the vehicle's surface exposed to the flow. Lift and wave drag are obtained by computing the resultant of the normal forces due to pressure, whereas shear forces are often approximated by means of quasi‐empirical methods for predicting friction drag due to viscosity.
The fundamentals of supersonic linearized aerodynamic theory were derived as early as the 1920s. Measurements have proven that in many applications linearized solutions for slender bodies at small incidences to the flow give accurate predictions of experimental results. Linear theory has been successfully applied to computation of the pressure distribution on airfoils during the development of supersonic airplanes in the period 1950–1960, but cannot be used for transonic flow. For certain applications, more complicated solutions obtained from second‐order theories or non‐linear computational fluid dynamics (CFD) analysis developed since the 1970s have to be preferred. This applies in particular to lift and drag of lifting surfaces intended to realize a significant percentage of the theoretical leading‐edge suction (cf. Chapter 9).
Introductions to the velocity potential equation in isentropic flow in which there is no mechanism to start vorticity of the fluid elements can be found in [8] and other publications mentioned in the bibliography of this chapter. A planar wall with a (very small) perturbation immersed in a potential flow field is depicted in Figure 5.1(a). The ‐axis is in the direction of the uniform oncoming flow, the ‐axis is normal to it. A velocity potential function can be defined that satisfies the equation . At an arbitrary point in the flow the local velocity has components in the ‐direction and in the ‐direction, where and are called perturbation velocities. Introduction of these concepts into the velocity potential leads to the perturbation velocity potential function
In the case of two‐dimensional supersonic flow this leads to Laplace's equation,
a non‐linear partial differential equation, which mostly cannot be solved analytically. Fortunately, practical solutions have been derived by accepting an approximate solution for the case of slender components of supersonic aircraft in cruising flight. For this case it is assumed that the velocity perturbations and are small in comparison with the oncoming flow velocity and it can be shown that in such cases several terms of the exact velocity potential equation can be ignored relative to the more essential ones. The result is a linear partial differential equation for the perturbation velocities,
which appears to be reasonably accurate for slender bodies at small angles of incidence to the flow moving at supersonic speeds not close to the sonic velocity. Equation (5.3) is a hyperbolic differential equation that has a general solution in the form of the functional relation , indicating the property that the velocity potential is constant along straight lines of constant . Since a Mach wave has a slope relative to the free flow equal to , the velocity potential is constant along Mach lines. This implies that over a surface featuring a very small ramp angle relative to the oncoming flow a Mach wave is generated which is propagated downstream and away from the wall with a slope .
The pressure change of the flow caused by a kink in the surface due to a small perturbation is derived using Figure 5.1(a). Compared with Figure 4.5, the oblique shock with wave angle is replaced by a Mach wave at an angle with the upstream flow. The Mach wave causes a pressure increment when the surface is inclined into the flow, in which case the deflection angle is defined as positive. The disturbed velocity behind the Mach wave is the combination of the oncoming flow velocity and its increments in the ‐direction and in the ‐direction. The velocity component normal to the Mach wave has changed by , which is decomposed into and . For an infinitely small it can be assumed that . Using , the pressure change due to the flow deflection follows from Euler's equation
and the pressure coefficient becomes
The sign of is positive where the surface is inclined into the free stream flow, leading to an increased wall pressure, and negative where the surface is inclined away from the flow, leading to a reduced wall pressure. Since Equation (5.3) is linear, the pressure distributions due to the airfoil's incidence and the variation of the section thickness and camber along the chord can be calculated separately and then added and the lift, drag, and pitching moment of the airfoil section are found by integration of the resulting pressure distribution along the airfoil contour.
The important Equation (5.5) has the consequence that at an arbitrary point of the airfoil surface the pressure coefficient is proportional to the local inclination angle which is determined by the angle of attack and the distribution of thickness and camber. Substitution of the airfoil geometry depicted in Figure 5.1(b) into the pressure coefficient according to Equation (5.5) yields the pressure distribution along the airfoil surfaces as follows:
where and are the coordinates of the upper and lower surface, respectively. This result implies that the pressure distribution depends only on the airfoil geometry and the angle of attack. As an example, Figure 5.1(b) depicts a circular‐arc airfoil at an angle of attack . A positive determines a negative pressure coefficient on the upper surface and a positive pressure coefficient on the lower surface, resulting in a normal force coefficient .
When carrying out the integration of the pressure distribution along the contour it is observed that the thickness distributions of the upper surface and the lower surface as well as profile camber do not contribute to the lift. Ackeret's theory discovered in [11] suggests that thickness and camber do not improve the lift/drag ratio. The lift coefficient depends on the angle of attack as follows:
In other words, the airfoil experiences lift only due to its incidence to the flow, which is equal to that of a flat plate, and Figure 5.2 compares the lift gradient of an airfoil at supersonic and subsonic Mach numbers. The wing of an aircraft in subsonic flight generates the majority of the lift by low pressures acting on the upper wing surface, whereas the zero‐lift pressure drag is near‐zero on the provision that most of the leading edge suction acting on the nose is fully realized. However, a wing in supersonic flow generates more than half of the lift by a compressive force acting on the lower surface at the cost of an equivalent amount of pressure drag. Figure 5.2 suggests that, according to the Prandtl–Glauert equation, the subsonic lift gradient is considerably higher than the supersonic lift gradient according to Ackeret's theory. On the other hand, the design condition in cruising flight of a supersonic airliner is typically , compared to for a high‐subsonic airplane. In other words, for a specified airfoil lift, both airplane categories require a similar angle of attack.
When linearized theory is applied to sharp‐edged thin airfoils with a smooth distribution of the upper and lower geometry, the results are qualitatively correct and accurate enough to be used during the initial design stages of a supersonic cruising aircraft. In particular, the predicted pressure distribution along the chord can be useful for the initial stages of aerodynamic design in order to compare different airfoil sections with respect to their contribution to the aerodynamic efficiency of the airfoil. Application of Equation (5.6) yields the coefficient of pressure drag due to lift,
Addition of the pressure drag at zero lift of the upper and lower airfoil parts with thickness and as denoted in Figure 5.1(b) yields the coefficient of pressure drag due to thickness:
where and the factor depends on the details of the airfoil contour, as noted in Equation (5.9). For diamond and lenticular airfoils depicted on ;Figures 4.9 and 5.1(b) we can write
which amounts to for a symmetric diamond‐shape airfoil and for a circular‐arc airfoil [9]. The total pressure drag found from addition of the drag due to lift and the thickness drag and the maximum aerodynamic efficiency is obtained for
Combined with the lift according to Equation (5.6) this yields the maximum aerodynamic efficiency
Equation (5.10) demonstrates that the maximum aerodynamic efficiency of a diamond airfoil with surrounded by potential flow amounts to . This suggests that a two‐dimensional sharp‐edge thin airfoil experiences a much higher pressure drag for given lift than the pressure drag of a typical low‐speed airfoil in subsonic flow.
In view of the result from the previous paragraph it is not surprising that much attention has been paid to optimizing the shape of two‐dimensional supersonic airfoils. Soon after the development of the linearized supersonic flow theory, it was recognized that a two‐dimensional airfoil having minimum pressure drag for a given chord and a given thickness consists of straight lines. It is, however, obvious that in addition to the thickness ratio auxiliary conditions such as the wing volume or some structural requirement may be more important. According to [1], the minimum pressure drag is obtained for the minimum value of the integral
with notations defined in Figure 5.3(a). The auxiliary optimization condition requires the incorporation of an isoperimetric constraint defined by = constant, in which the exponent determines one of the following specifications of the optimization problem and its solution:
The resulting airfoil contours are compared in Figure 5.3, suggesting that the three optimum shapes and the corresponding minimum pressure drag are close together. For instance, for specified thickness ratio , the pressure drag for case is 8% higher than for case whereas case has 4.5% less drag than case . And compared with the double‐wedge airfoil with the same chord and thickness, the three constrained optimum profiles have between 19% and 1.3% more drag. This does not imply that the double‐wedge airfoil is always preferable, since its surface area is a dramatic 30% less than that of the other airfoils.
Ackeret's linear theory applies to inviscid and potential flow and hence skin friction drag should be added to the pressure drag due to lift and thickness. Assuming that the friction drag of a thin airfoil section equals that of a flat plate with the same chord and the same exposed area of the upper and lower surfaces, the total two‐dimensional profile drag coefficient amounts to
It is readily shown that the maximum aerodynamic efficiency amounts to
Assuming that variation of the flight speed has no effect on the friction drag coefficient, the skin friction drag appears to have a significant effect on the total airfoil drag and may reduce the maximum aerodynamic efficiency by approximately 50%. However, kinetic heating of the boundary layer in high‐speed flight causes the skin friction to decrease and the aerodynamic efficiency increases gradually with increasing Mach number. It is also worth noting that according to Equation (5.14) the attainable aerodynamic efficiency of thin, sharp‐edged airfoils with thickness ratios between 0.03 and 0.05 in two‐dimensional supersonic flow at Mach 2 has the same order of magnitude as Concorde's as well as published values of supersonic transport airplane projects in Figure 2.4.
The pressure coefficient distribution along the chord according to Equation (5.9) is also used to compute the leading‐edge pitching moment of a two‐dimensional airfoil. The following solution is provided by [9]:
The center of pressure of a symmetrical airfoil is located at = 0.50. Since this value is independent of the angle of attack, the aerodynamic center of an airfoil in supersonic flow coincides with the center of pressure: . This important characteristic differ significantly from the aerodynamic center location of airfoils in subsonic flow, for which the aerodynamic center is located at the quarter‐chord location.
It is emphasized that the airfoil properties are derived for two‐dimensional wings whereas conditions for three‐dimensional wings are quite different. Moreover, linearized potential flow theory is strictly applicable to isentropic supersonic flow with small disturbances and is not valid for transonic Mach numbers. The presence of phenomena such as shock waves, flow separation caused by shock wave/boundary layer interaction, and pockets of subsonic flow are not modeled by linear theory. In spite of these restrictions, results obtained by Ackeret's theory give good insight into the overall effect of varying basic wing shape parameters on the aerodynamic efficiency of airfoils in linearized potential flow.
Different from airfoils in subsonic flow, the thickness drag in supersonic flow is very sensitive to its thickness ratio. Thickness as well as camber contribute to drag and not to lift, and hence a flat plate can be seen as an aerodynamically ideal supersonic airfoil. Obviously, thickness is required to provide a wing with adequate volume and strength and the flat plate does not represent a practical solution. Nevertheless, airfoil sections for supersonic application are much thinner than those for subsonic aircraft: typical thickness ratios are between 3% and 5%.
3.231.146.172