SP-367 Introduction to the Aerodynamics of Flight

 

V. TRANSONIC FLOW.

 

 

[103] Up to this point the airplane was considered to be in motion at subsonic speeds. The air was treated as though it were incompressible and a study of the aerodynamics involved using this simplifying assumption was made. As the airplane speed increases, however, the air loses its assumed incompressibility and the error in estimating, for example, drag, becomes greater and greater.

 

The question arises as to how test an airplane must be moving before one must take into account compressibility. One important quantity which is an indicator is the speed of sound of the air through which the airplane is flying.

 

A disturbance in the air will send pressure pulses or waves out into the air at the speed of sound. Consider the instance of a cannon fired at sea level. An observer situated some distance from the cannon will see the flash almost instantaneously but the sound wave is heard (or the pressure wave is felt) some time later. The observer can easily compute the speed of sound by dividing the distance between him and the cannon by the time it takes the sound to reach him. The disturbance propagates out away from the cannon in an expanding hemispherical shell as shown in figure 83.

 

As was shown in figure 7, the speed of sound varies with altitude. To be more precise, it depends upon the square root of the absolute temperature. At sea level under standard conditions (To = 288.15 K) the speed of sound is 340.3 m/sec but at an altitude of 15 kilometers where the temperature is down to 216.7 K the speed of sound is only 295.1 m/sec. This difference indicates that an airplane flying at this altitude encounters the speed of sound at a slower speed, and, therefore, comes up against compressibility effects sooner.

 

An airplane flying well below the speed of sound creates a disturbance in the air and sends out pressure pulses in all directions as shown in figure 84(a). Air ahead of the airplane receives these "messages" before the airplane arrives and the flow separates around the airplane. But as the airplane approaches the speed of sound, the pressure pulses merge closer and closer together (fig. 84(b)) in front of the airplane and little time elapses between the time the air gets a warning of the airplane approach and the airplane's actual arrival time At the speed of sound (fig. 84(c)) the pressure pulses move at the same speed as the airplane. They merge together ahead of the airplane into a "shock wave" which is an almost instantaneous line of change in pressure, temperature, and density. The air has no warning of the impending approach of the airplane and abruptly passes through the shock system. There is a tendency for the air to break away from the airplane and not flow smoothly about it; as a result, there is a change in the aerodynamic forces from those experienced at low incompressible flow speeds.

 


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Figure 83 - Speed of sound of a disturbance

 Figure 83.- Speed of sound of a disturbance.

 

 

The Mach number is a measure of the ratio of the airplane speed to the speed of sound. In other words, it is a number that may relate the degree of warning that air may have to an airplane approach. The Mach number is named after Ernst Mach, an Austrian professor (1838 to 1916). Figure 85 shows the names given to various flight regimes. For Mach numbers less than one, one has subsonic flow, for Mach numbers greater than one, supersonic flow, and for Mach numbers greater than 5 the name used is hypersonic flow. Additionally, transonic flow pertains to the range of speeds in which flow patterns change from subsonic to supersonic or vice versa, about Mach 0.8 to 1.2. Transonic flow presents a special problem area as neither equations describing subsonic flow nor those describing supersonic flow may be accurately applied to he regime.

 


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Shock wave formation

(a) Zero and low-speed disturbance.
(b) Nearing Mach 1.
(c) Mach 1.
 
Figure 84.- Shock-wave formation.

Flight engine terminology

Figure 85.- Flight engine terminology.

 

[106] With these definitions in mind, one may now examine in a little more detail the flow at high speeds. Up to now, the airplane was considered to be in motion at subsonic speeds. Drag was composed of three main components- skin-friction drag, pressure drag, and induced drag (or drag due to lift). At transonic and supersonic speeds there is a substantial increase in the total drag of the airplane due to fundamental changes in the pressure distribution.

 

This drag increase encountered at these high speeds is called wave drag. The drag of the airplane wing, or for that matter, any part of the airplane, rises sharply and large increases in thrust are necessary to obtain further increases in speed. This wave drag is due to the unstable formation of shock waves which transforms a considerable part of the available propulsive energy into heat, and to the induced separation of the flow from the airplane surfaces. Throughout the transonic range, the drag coefficient of the airplane is greater than in the supersonic range because of the erratic shock formation and general flow instabilities. Once a supersonic flow has been established, however, the flow stabilizes and the drag coefficient is reduced. Figure 86 shows the variation of an airplane wing drag coefficient with Mach number. The total drag at transonic and supersonic speeds can be divided into two categories: (1) zerolift drag composed of skin-friction drag and wave (or pressure-related) drag of zero lift and (2) lift-dependent drag composed of induced drag (drag due to lift) and wave (or pressure-related) drag due to lift. The infamous "sound barrier" shows up rather clearly in figure 86 since to fly supersonically the airplane must provide enough thrust to exceed the maximum transonic drag that is encountered. In the early days of transonic flight, the sound barrier represented a real barrier to higher speeds. Once past the transonic regime, the drag coefficient and the drag decreases, and less thrust is required to fly supersonically. However, as it proceeds toward higher supersonic speeds, the drag increases (even though the drag coefficient may show a decrease).

 


Variation of wing drag coeeficient with Mach number

Figure 86.- Variation of wing drag coeeficient with Mach number.

 

[107] There is a famous little story, untrue of course, of the pilot who flew his plane beyond the sound barrier and then got trapped there because of insufficient reverse thrust to get back below the speed of sound. Another case of perpetual motion.

 

It is a large loss in propulsive energy due to the formation of shocks that causes wave drag; figure 87 shows this shock formation about an airfoil. Up to a free-stream Mach number of about 0.7 to 0.8, compressibility effects have only minor effects on the flow pattern and drag. The flow is subsonic everywhere (fig. 87(a)). As the flow must speed up as it proceeds about the airfoil, the local Mach number at the airfoil surface will be higher than the free-stream Mach number. There eventually occurs a freestream Mach number called the critical Mach number at which a sonic point appears somewhere on the airfoil surface, usually near the point of maximum thickness and indicates that the flow at that point has reached Mach 1 (fig. 87(b)). As the freestream Mach number is increased beyond the critical Mach number and closer to Mach 1, larger and larger regions of supersonic flow appear on the airfoil surface (fig. 87(c)). In order for this supersonic flow to return to subsonic flow, it must pass through a shock (pressure discontinuity). This loss of velocity is accompanied by an increase in temperature, that is, a production of heat. This heat represents an expenditure of propulsive energy which may be presented as wave drag. These shocks appear anywhere on the airplane (wing, fuselage, engine nacelles, etc.) where, due to curvature and thickness, the localized Mach number exceeds 1.0 and the airflow must decelerate below the speed of sound (fig. 88(a)). For transonic flow the wave drag increase is greater than would be estimated from a loss of energy through the shock. In fact, the shock wave interacts with the boundary layer so that a separation of the boundary layer occurs immediately behind the shock. This condition accounts for a large increase in drag which is known as shock-induced (boundary-layer) separation.

 

The free-stream Mach number at which the drag coefficient of the airplane increases markedly is called the drag-divergence Mach number. Large increases in thrust are required to produce any further increases in airplane speed. If an airplane has an engine of insufficient thrust, its speed will be limited by the drag-divergence Mach number. The prototype Convair F- 102A was originally designed as a supersonic interceptor but early flight tests indicated that because of high drag, it would never achieve this goal. It will be explained later how success was achieved for this airplane through proper redesigning.

 

Figure 87(d) shows the character of the flow at a free-stream Mach number close to one where large regions are in supersonic flow and the shocks are very strong. At a free-stream Mach number greater than 1, a bow shock appears around the airfoil nose. Most of the airfoil is in supersonic flow. The flow begins to realine itself parallel to the body surface and stabilize, and the shock-induced separation is reduced.

 


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Shock formation

Figure 87.- Shock formation.

 

This condition results in lower drag coefficients. Supersonic flow is more wellbehaved than transonic flow and there are adequate theories that can predict the aerodynamic forces and moments present. Often, in transonic flow, the flow is unsteady and the shock waves on the body surface may jump back and forth along the surface, thus disrupting and separating the flow over the wing surface. This sends pulsing unsteady flow back to the tail surfaces of the airplane. The result is that the pilot feels a buffeting and vibration of both wing and tail controls. This condition occurred especially in the first airplane types to probe the sound barrier. With proper design, however, airplane configurations gradually evolved to the point where flying through the transonic region posed little or no difficulty in terms of wing buffeting or loss of lift (fig. 88(b)).

 


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Supersonic characteristics

(a) Total aircraft shocks,
(b) Improving transonic flight.
 
Figure 88.- Supersonic characteristics.


 

The question as to whether one may delay the drag-divergence Mach number to a value closer to 1 is a fascinating subject of novel aerodynamic designs. What this really suggests is the ability to fly at near-sonic velocities with the same available engine thrust before encountering large wave drag. There are a number of ways of delaying the transonic wave drag rise (or equivalently, increasing the drag-divergence Mach number closer to 1). These include

 

(1) Use of thin airfoils

(2) Use of sweep of the wing forward or back

(3) Low-aspect-ratio wing

(4) Removal of boundary layer and vortex generators

[110] (5) Supercritical and area-rule technology

 

These methods are now discussed individually.

 

Thin airfoils: The wave drag rise associated with transonic flow is roughly proportional to the square of the thickness-chord ratio (t/c). If a thinner airfoil section is used, the flow speeds around the airfoil will be less than those for the thicker airfoil. Thus, one may fly at a higher free-stream Mach number before a sonic point appears and before one reaches the drag-divergence Mach number. The disadvantages of using thin wings are that they are less effective (in terms of lift produced) in the subsonic speed range and they can accommodate less structure (wing fuel tanks, structural support members, armament stations, etc.) than a thicker wing. Figure 89(a) shows the airfoil sections used by three U.S. fighters over the past three decades. As the speeds have increased, the thickness-chord ratios have decreased. The F-104 (fig. 89(b)) was designed to achieve the minimum possible wave drag but was penalized with low subsonic lift. As a result, the landing speed of this airplane was particularly high and landing mishaps were common among untrained pilots. Figure 90 illustrates the effect of using a thinner section on the transonic drag. Notice, in particular, that the drag divergence Mach number is delayed to a greater value.

 

Sweep: It was Adolf Busemann in 1935 who proposed that sweep may delay and reduce the effects of compressibility. A swept wing will delay the formation of the shock waves encountered in transonic flow to a higher Mach number. Additionally, it reduces the wave drag over all Mach numbers. Figure 91 shows experimental data confirming this result as a wing is swept from no sweep to a high degree of sweep.

 

One may view the effect of sweep of a wing as effectively using a thinner airfoil section (t/c reduced). In figure 92(a) a straight wing is shown with the airflow approaching perpendicularly to the wing. Notice a typical airfoil section. If the wing is now swept to some angle of sweep Capital Greek letter lambda, the same flow over the wing encounters new airfoil sections that are longer than previously. The maximum ratio of thickness to chord has been reduced. (See fig. 92(b).) One is effectively using a thinner airfoil section as the flow has more time in which to adjust to the situation. The critical Mach number (at which a sonic point appears) and the drag-divergence Mach number are delayed to higher values; Sweepforward or sweepback will accomplish these desired results. Figure 93 shows a modern jet airplane employing forward sweep. Forward sweep has disadvantages, however, in the stability and handling characteristics at low speeds.

 

A major disadvantage of swept wings is that there is a spanwise flow along the wing, and the boundary layer will thicken toward the tips for sweepback and toward the roots for sweepforward. In the case of sweepback, there is an, early separation and....

 


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Thin airfoils

(a) Changes in airfoil sections.
(b) F-104G airplane.
 
Figure 89.- Thin airfoils.
  

Effect of airfoil
thickness on transonic drag

Figure 90.- Effect of airfoil thickness on transonic drag. Lift = 0; q = Constant; MDD, drag divergence Mach number.

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Effects of sweep on wing
transonic drag coefficient

Figure 91.- Effects of sweep on wing transonic drag coefficient.

Sweep reduces effective
thickness-chord

Figure 92.- Sweep reduces effective thickness-chord ratio.

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113]

HFB 320 Hansa-Jet with
forward sweep

Figure 93.- HFB 320 Hansa-Jet with forward sweep.

 

...stall of the wing-tip sections and the ailerons lose their roll control effectiveness. The spanwise flow may be reduced by the use of stall fences, which are thin plates parallel to the axis of symmetry of the airplane. In this manner a strong boundarylayer buildup over the ailerons is prevented. (See fig. 94(a).) Wing twist is another possible solution to this spanwise flow condition.

 


 

Stall fences and vortex generators

Figure 94.- Stall fences and vortex generators.

 

[114] Low aspect ratio: The wing's aspect ratio is another parameter that influences the critical Mach number and the transonic drag rise. Substantial increases in the critical Mach number occur when using an aspect ratio less than about four. However from previous discussions, low-aspect-ratio wings are at a disadvantage at subsonic speeds because of the higher induced drag.

 

Removal or reenergizing the boundary layer: By bleeding off some of the boundary layer along an airfoil's surface, the drag-divergence Mach number can be increased. This increase results from the reduction or elimination of shock interactions between the subsonic boundary layer and the supersonic flow outside of it.

 

Vortex generators are small plates, mounted along the surface of a wing and protruding perpendicularly to the surface as shown in figure 94(b). They are small wings, and by creating a strong tip vortex, the generators feed high-energy air from outside the boundary layer into the slow moving air inside the boundary layer. This condition reduces the adverse pressure gradients and prevents the boundary layer from stalling. A small increase in the drag-divergence Mach number can be achieved. This method is economically beneficial to airplanes designed for cruise at the highest possible drag-divergence Mach number.

 

Supercritical and area-rule technology: One of the more recent developments in transonic technology and destined to be an important influence on future wing design is the NASA supercritical wing developed by Dr. Richard T. Whitcomb of the NASA Langley Research Center. A substantial rise in the drag-divergence Mach number is realized. Figure 95(a) shows a classical airfoil operating near the Mach 1 region (supercritical- beyond the critical Mach number) with its associated shocks and separated boundary layer. Figure 95(b) shows the supercritical airfoil operating at the same Mach number. The airfoil has a flattened upper surface which delays the formation and strength of the shocks to a point closer to the trailing edge. Additionally, the shock- induced separation is greatly decreased. The critical Mach number is delayed even up to 0.99. This delay represents a major increase in commercial airplane performance.

 

The curvature of a wing gives the wing its lift. Because of the flattened upper surface of the supercritical airfoil, lift is reduced. However, to counteract this the new supercritical wing has increased camber at the trailing edge.

 

There are two main advantages of the supercritical airfoil as shown in figure 96. First, by using the same thickness-chord ratio, the supercritical airfoil permits high subsonic cruise near Mach 1 before the transonic drag rise. Alternatively, at lower drag divergence Mach numbers, the supercritical airfoil permits a thicker wing section to be used without a drag penalty. This airfoil reduces structural weight and permits higher lift at lower speeds.

 


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115]

Classical and supercritical airfoils.

(a) Classical airfoil.
(b) Supercritical airfoil.
 
Figure 95.- Classical and supercritical airfoils.

Two uses of supercritical

Figure 96.- Two uses of supercritical wing.

 

Coupled to supercritical technology is the "area-rule" concept also developed by Dr. Richard T. Whitcomb of NASA Langley Research Center in the early 1950's for transonic airplanes and later applied to supersonic flight in general.

 

Basically, area ruling states that minimum transonic and supersonic drag is obtained when the cross-sectional area distribution of the airplane along the longitudinal axis can be projected into a body of revolution which is smooth and shows no abrupt changes in cross section along its length. Or, if a graph is made of the cross-sectional area against body position, the resulting curve is smooth. If it is not a smooth curve, [116] then the cross section is changed accordingly. Figure 97 presents the classic example of the application of this concept-the Convair F-102A.

 

The original Convair F-102A was simply a scaled-up version of the XF-92A with a pure delta wing. But early tests indicated that supersonic flight was beyond its capability because of excessive transonic drag and the project was about to be canceled. Area ruling, however, saved the airplane from this fate. Figure 97(a) shows the original form of the F-102A and the cross-sectional area plotted against body station. Notice that the curve is not very smooth as there is a large increase in cross-sectional area when the wings are encountered. Figure 97(b) shows the F-102A with a coke- bottle-waist-shaped fuselage and bulges added aft of the wing on each side of the tail to give a better area-rule distribution, as shown in the plot. The F-102A was then able to reach supersonic speeds because of the greatly reduced drag and entered military service in great numbers.

 


Area ruling of F-102A airplane

(a) YF-102 before area ruling.
(b) F-102 after area ruling.
 
Figure 97.- Area ruling of F-102A airplane.

 

Recently, the area-rule concept has been applied to design a near-sonic transport capable of cruising at Mach numbers around 0.99. In addition to area ruling, a supercritical wing is used. Figure 98 shows the configuration obtained and the resulting cross-sectional area plot. Notice this curve now is completely smooth and indicates that the shape is near optimum. The shocks and drag divergence are delayed to a near-sonic Mach number.

 


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Near-sonic transport area
ruling

Figure 98.- Near-sonic transport area ruling.

 


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