Skip to main content

Putting It All Together

  • 1144 Accesses


We examine the lift and drag characteristics of wings as these are affected by the two physical aspects of flight: friction effects and for higher speed, air compressibility. We encounter the phenomenon of stall, shock waves, and visible condensation. These make flight phenomena interesting and challenging. Friction effects in supersonic flight are shown to be an almost negligibly small part of the heating of aircraft skin surfaces.


  • Force coefficients
  • Stall
  • Adverse pressure
  • Transonic
  • Shock wave
  • Expansion wave
  • Shadowgraph
  • Condensation
  • Mach number
  • Aerodynamic heating
  • Drag polar

If it were not for friction and compressibility, the analysis of flow phenomena would be pretty simple. Friction is a necessary ingredient for lift and it controls most of the price we pay in propelling an airplane through air. Knowledge of drag as a resistive force acting on airplane surfaces is important in allowing the efficiency of the airfoil or wing to be determined and has important consequences for the performance of the airplane. Compressibility is a matter we must deal with—if we wish to fly fast.

So, how do engineers describe the performance of airplane or their wings? The first requirement is to make the presentation of such information as simple, universal, and as compact as possible.

Since the lift force scales with dynamic pressure (q),Footnote 1 it is customarily expressed as a lift coefficient, that is proportional to lift. Because the lift force is the result of pressure acting over an area, the lift coefficient is written in terms of the only area in the problem, namely the chord length of the two-dimensional airfoil or the wing area for a three-dimensional wing. Thus, we have for a two-dimensional airfoil or a wing

$$ \begin{aligned} & {\text{Section}}\,{\text{lift}}\,{\text{coefficient}} = c_l = {\text{Lift}}/\left( q \right)({\text{chord}})\,\,{\text{or}} \\ & {\text{Wing}}\,{\text{lift}}\,{\text{coefficient}} = C_L = {\text{Lift}}/\left( q \right)\left( {{\text{wing}}\,{\text{area}}} \right) \\ \end{aligned} $$

There are a similar definitions for airfoil, wing or airplane drag coefficient, CD. Similarly, the section moment coefficients shown in Fig. 6.2 are defined by

$$ {\text{Moment}}\,{\text{coefficient}} = c_{mz} = {\text{Moment}}\,{\text{at}}\,{\text{z}}/\left( q \right)\left( {{\text{chord}}} \right)^{2} $$

The moment must be specified as measured at a specific point which is usually the aerodynamic center or the quarter chord (see Fig. 6.2). For a wing or an airplane, the normalizing parameter is the product of wing area and the mean aerodynamic chord.

These are the section coefficients displayed in Fig. 6.2 for air at low Mach number, i.e., for incompressible flow. Let’s examine the role compressibility plays in determining lift coefficient. The variation of this measure of lift with angle of attack is what the inviscid theory with incompressible air correctly predicts for small angles. An example for a wing (CL) tested in a wind tunnel at various Mach numbers is shown in Fig. 7.1. This particular airfoil is cambered because it provides lift at zero angle of attack. The Mach number plot line tagged with M = 0.2 is a good approximation of incompressible flow. That line agrees very well with calculations we might make using the analysis tools developed so far. Extension of such a plot to include compressibility effects involves the higher freestream Mach numbers as noted.

Fig. 7.1
figure 1

Representative lift curve for an airfoil in incompressible flow (M ~ 0.2) and compressible (subsonic, M indicated) flow. Note the departure from linearity and greater steepness at the highest Mach number

A couple of observations are noteworthy. The maximum lift coefficient is of order 1.0 and could be larger, especially for airfoils designed with high lift devices. The slopes of the various curves at transonic Mach numbers differ somewhat because the flow is compressible. From this it can be concluded that lift is only modestly affected by compressibility, when we limit our attention to the angle of attack range and transonic flight speeds where airplanes like commercial airliners operate.

7.1 Wing Lift Performance with Viscous Air

What about the lift performance of wings in air with a greater role played by friction effects? Fig. 7.2 is the pressure distribution one might find on a simple 12% thick symmetric airfoil (NACA 0012) at 10 degrees angle of attack. The leading edge is at x/c = 0 and the trailing edge is at x/c = 1. The Mach number is low enough for the flow to be incompressible. The pressure is measured by ports (small holes) in the airfoil each connected to a pressure sensing instrument. At the various high Reynolds numbers, the role of viscosity (or friction effects) should be modest. The experimental data displays fairly consistent flow behavior even as various means are employed to examine the role of boundary layers allowed to transition to turbulent on their own (free transition) or forced to do so by a layer of roughness on the airfoil (fixed transition). This aspect is a detail that is beyond the level of this discussion. Note the very high flow acceleration rate around the leading edge to the upper surface. The use of the coefficient to represent pressures also coalesces all the data for various speeds at which the wind tunnel was run to nearly identical curves, simplifying the engineers’ work.

Fig. 7.2
figure 2

Variation of surface pressure coefficient along the chord for a symmetric (uncambered) airfoil (NACA 0012) at high Reynolds numbers indicated and with various boundary layer transitions to turbulent flow employed. The airfoil shape is shown at the bottom of the figure. Note the stagnation point on the lower surface where Cp = 1.0. Note also the increasing pressure on aft portion of the upper surface (falling Cp) behind the point of minimum pressure (data: NASA Langley Research Center; airfoil image: Michael Belisle: Wikimedia Commons, file: Streamlines around a NACA 0012-hu.svg)

When Reynolds number is under half a million, we might expect the flow to be primarily laminar. The airfoil performance at modest Reynolds numbers is illustrated in Fig. 7.3. Under such conditions we have fairly consistent lift behavior at small angles of attack that agrees well with an inviscid flow analysis. At higher angles of attack, the situation worsens because the lift curve no longer follows the linear portion of the curve.

Fig. 7.3
figure 3

(Data adapted from

Variation of lift (as a lift coefficient) with angle of attack and modest Reynolds number based on chord. The data is experimental for the NACA 0012 airfoil. The dashed red line is to indicate the theoretical line to be expected with inviscid flow. The arrow pointing downward is in the direction of decreasing Reynolds number. The values of Re (in thousands) varies from 290 (top curve), 200, 120, 80 and 50 at the bottom

7.2 Stall

When an airfoil’s angle of attack is increased beyond the linear portion of the lift curve (Fig. 7.1), the lift will eventually fail to increase and then decrease. The departure of such a curve from linearity is called stall. At the lower Reynolds numbers shown in Fig. 7.3, stall occurs around 10–12°. At Reynolds numbers in the millions, with a lesser importance of friction, the stall angle of attack for the same airfoil turns out to be about 16°. The data for NACA 0012 is not presented but similar trends are noted for NACA 2412 shown in Fig. 6.2). At high angles of attack, one should expect deleterious consequences not only for drag but also on lift characteristics.

In practice, flight is normally at lower angles, but it is important to know how the wing behaves near stall.

7.3 Why Does a Wing Stall?

The boundary layer is important for its role in influencing the behavior of the freestream flow. Consider that by shaping a surface to generate a lifting force, we are imposing boundaries to influence the direction and speed of the air. We try to insist that the flow accelerate to high speed (for low pressure, see Fig. 6.1 and 7.2, low negative Cp). We then motivate the flow to decelerate again to reach a pressure close to that of the original incoming flow at the trailing edge. This is evident in Fig. 7.2 because Cp is near zero at the trailing edge where it is, in fact, difficult to measure because of the local thinness. It is this deceleration to the trailing edge that is the principal challenge to the design and operation of wings and wing-like structures such as, for example, propellers or compressor blades in a jet engine.

Note that, even when operating nicely, the measured pressure coefficient is positive near the trailing edge. This means that the nearly stagnant air slowed by the boundary layers on both sides of the airfoil allows for transmission of the high-pressure information from the bottom to be shared with flow over the upper surface to about the 90% chord length point in this example.

7.4 Adverse Pressure

The air in the boundary layer along the upper surface of an airfoil is challenged by the reality that it does not have the energy required to rise back up to the high pressure that the inviscid freestream air outside this layer can. Friction has robbed it of momentum and therefore also of energy. The increasing pressure in the flow direction is an adverse pressure gradient; adverse meaning unfavorable. Air would prefer to flow into a region of lower pressure.

The freestream air can handle this change because it has sufficient kinetic energy that it can convert back to thermal energy (technically stated as enthalpy, which in incompressible flow leads to a rise in pressure). The boundary layer air cannot. When the point is reached where all kinetic energy in the boundary layer air is used up, the flow is stationary. With the pressure ahead still higher, the natural reaction is that the stationary air will simply flow backwards, i.e., upstream. With that, the flow fails to follow the prescribed flow direction and is said to separate from the boundary prescribing the flow geometry. At even larger angles of attack, useful lift is lost entirely.

This is the mechanism that typically leads to wing stall. It is also the mechanism that determines the lift and drag of the wing at all angles of attack. The flow separation near the trailing edge is central to the drag experienced. As stall becomes more pronounced, the separation point moves forward on the wing resulting in a wider wake. Figure 7.4 illustrates the relevant flow phenomena near the onset of stall. Figure 6.2 shows the general form of the variation of drag of a simple airfoil (NACA 2412) at various angles of attack (at low Mach number: incompressible flow). The camber on this airfoil leads to a minimum drag at a non-zero lifting condition, which is desirable for an airplane. The drag minimum is rather broad over a range of lift but, increases rapidly for high angles of attack.

Fig. 7.4
figure 4

An airfoil at an angle of attack sufficient to cause flow separation at the point noted (S). Regions of low (L) and high (H) pressures are noted. The dotted line is the boundary layer edge. The lift is also reduced by virtue of the downward flow deflection [black arrow showing direction of the wake (W)] not being as demanded by the trailing edge direction (green arrow). The green streamlines shown would be as shown without flow separation

Stall occurring at the trailing edge is the common mechanism for wings or airfoil at high angles of attack. It is, however, not the only mechanism. The leading edge can also be the culprit with the formation of a bubble of separated flow near the point of minimum pressure. Good design of the leading-edge geometry, the use of leading-edge flaps or slats, and, as we shall see later, the use of devices designed to forestall (pun intended) stall (called vortex generators) are very effective in making stall a manageable aspect of operating a wing for high lift.

7.5 Transonic Flight

Modern airliners cruise at speeds, called transonic, because the flow over the upper surface of the wing is locally supersonic. In this flight regime, there is a bubble of supersonic, low pressure, flow that must return to the subsonic, higher-pressure region near and aft of the wing’s trailing edge. In this supersonic “bubble”, and outside the boundary layer, no information can travel upstream from the trailing edge to tell the flow to slow down because the flow speed is faster than the speed with which information can travel.

The only way the matter can be reconciled is with a wave that suddenly changes to supersonic flow to subsonic. This is a shock wave. Across this wave, all descriptive parameters of the flow change: velocity, pressure, temperature, density, and speed of sound. The velocity decreases while all the others increase. The flow energy is conserved so that the total temperature is conserved, but since this situation cannot be reversible, entropy increases and consequently, the total pressure decreases. These changes are modest when the Mach number is above and close to one and they become more substantial when the flow Mach number reached at the front of the shock wave is larger.

That entropy should not be conserved though a shock wave is to be expected since the reverse situation of a low-speed flow suddenly accelerating to higher speed for no good reason is not likely (not ever) to be observed. Flow through a shock wave is not reversible.

Flight speeds of commercial jets are around 0.8–0.85 Mach number. A passenger sitting over the wing might, under the right circumstances, be able to see the stationary shock wave on the wing because the density change across the shock will result in a change of the refractive index of air. The change is quite small and could be visible when we look toward the horizon. However, under most circumstances, the sky on the horizon has no structure against which optical distortion could be detected by the human eye. We have to “see” the shock by other means. For example, on a flight at low latitudes or an east–west flight near noon with the sun directly overhead, the sun’s rays will also be refracted by the shock because of the abrupt change in density. Such refraction will result in a shadow on the wing surface. When visible, this shadow will typically dance fore and aft as the atmospheric conditions of the flight oscillate ever so slightly. The physics of this phenomenon is utilized in the generation of a shadowgraph for the purpose of flow visualization. Figure 7.5 illustrates the technique with a photo of a supersonic bullet with its shock wave system and the wake that follows it.

Fig. 7.5
figure 5

Left: A shadowgraph of a bullet in supersonic flight. Weak oblique waves from the nose coalesce into an oblique shock that turns into a normal shock further out. Note the turbulence in the wake (Photo: Daniel P.B. Smith, Creative Commons, 1962; Right: Shock image generated by a low-level fly-by of a US Navy “Blue Angels” F/A-18. Note the wing recompression shock above and below the wing trailing edges (above and below the number “5”) as well as shocks at the leading edge and above the cockpit. Its visibility is due to the refraction from the air density change across the shock. There is also a faint amount of condensation above the low-pressure, upper side of the wing (Image courtesy Barry Latter)

To illustrate the optics of a shock wave in circumstances that might be more readily observable, the right image in Fig. 7.5 is of an airplane flying close to the speed of sound past nearby houses. These provide a good background against which the waves become visible. The US Navy’s Blue Angels fly-by over Lake Washington waters in Seattle was entertaining and very loud - for a very short time! Note also the recompression shock aft of and above the cockpit.

Today’s commercial airliners routinely fly with supersonic flow on the upper surface of the wing but only up to the speed where the waves are weak and don’t lead to a significant drag rise, see Fig. 7.6. The sketch (Fig. 7.7) illustrates the flow separation with a shock wave strong enough to cause flow separation and an associated drag increase. The separation is caused by the shock establishing regions of low and high pressure that are easily connected by a segment of the boundary layer air that allows flow upstream, from high pressure behind the wave to low pressure ahead of it. Flow separation occurs if the wave is strong enough, a situation generally avoided in commercial aviation. The subject of shock-boundary layer interaction has been a matter of intense study by government, academic research laboratories, and airplane builders for many decades. It plays an especially strong role in determining the performance of inlets on high-speed military jets. On wings, lift will also be affected by separation as suggested in Figs. 7.4 and 7.7.

Fig. 7.6
figure 6

Drag behavior of a NACA 2312 airfoil at speed near the speed of sound at a number of angles of attack. The power required to overcome drag varies as M3 which implies a sharper rise in power required than shown in this plot (NACA)

Fig. 7.7
figure 7

source of increased drag. L and H refer to the static pressures near the shock wave and imposed on the boundary

Flow separation (point “S”) due to the recompression shock on a wing at transonic speed. The low-pressure region on the upper surface is locally supersonic with a shock strong enough to cause the boundary layer to detach from the surface. The creation of the wake (W) is the

An airfoil drag test at various Mach numbers approaching M = 1 is shown in Fig. 7.6. While this and the other airfoils described in this text, are not likely to be suitable in practice, their characteristics are representative of all airfoils in a qualitative, rather than quantitative, sense. The steep drag rise above M ~ 0.6 (for this airfoil) is termed the transonic drag rise. Realistic airfoil designs for airliners, for example, have a significantly higher drag rise Mach numbers.

The rapid wing and airplane drag rise with increasing speed associated with the effects of shock waves and shock/boundary layer interaction was, for a time, a difficult barrier for flying at speeds close to that of sound. In the days during and after WW II when serious attempts at high-speed flight were pondered, the power available from the piston engines of the day was insufficient for the propeller to provide the thrust required. A second and more important reason for the difficulty of reaching near sonic airplane flight speeds is that the propeller blade always deals with air at relative speeds greater than the flight speed. The turning blade sees oncoming air as the vector sum of flight and blade rotation speeds. Consequently, it is the propeller blade that encounters a serious drag rise before the airplane does as it tries to increase speed. The prospects for flying faster looked better with the new jet engines. The first supersonic flight was, however, achieved by a rocket powered airplane. The jet engine was not yet ready.

The people making early attempts at flying very fast termed the drag increase associated with high-speed flight (Fig. 7.6) as running into the sound barrier. It is not a barrier as such and holds the flyer back from flying faster only if the power or thrust necessary to overcome the drag is not available. It turns out that on the other side of the sound barrier, further from Mach one, things get a little easier and flight in the regime is now commonplace—for military airplanes.

7.6 Shock Waves

Just what is a shock wave? It is the sudden, albeit partial, conversion of organized flow kinetic energy to disorganized thermal energy at the molecular level. Such a process cannot occur in reverse and hence is irreversible.

The sudden conversion of kinetic energy to another energy form is a possibility whenever a mechanism exists that can accept the converted energy. An interesting example closer to everyday experience is the hydraulic jump where the energy absorbing mechanism is potential energy by a liquid medium, nominally water, in a gravity field. The similarity between shock waves and hydraulic jumps is quite astounding and interesting. In both cases, the flow velocity ahead of the jump is faster than the relevant wave propagation velocity. In the case of hydraulics, the propagation is by surface waves (not compression sound waves) for which the relevant speed is proportional to the depth of the surface of a stream. In a sense, liquid depths and gas pressure play similar descriptive roles. A simple experiment in the kitchen sink (Fig. 7.8) is illustrative of the “shock” wave phenomenon. The flow ahead of the “jump” is supercritical (in the language of hydraulics, the equivalent to supersonic for us) and the flow behind the wave is subcritical. This assertion can be tested by seeing whether the flow does or does not support steady waves: a standing wave is supported by a slim object held into the water ahead of the jump, but not behind it. The flow ahead of the jump is shallow and rapid while is slow and deep behind it. Similar flow phenomena can be seen in a stream flowing between rocks or at the end of a spillway chute of a dam.

Fig. 7.8
figure 8

A kitchen sink water flow with a hydraulic jump. The flow away from the impact point can be described as a source flow (Photo: Wikimedia Commons: James Kilfiger, File:Hydraulic jump in sink.jpg)

7.7 Condensation at High Speed

This book is primarily about subsonic aerodynamics but high-speed flight invariably involves supersonic flow. The next few paragraphs focus on that aspect of aerodynamics because it is necessary to explain some interesting observed phenomena. Figure 5.1 shows condensation of water vapor above the wing of an airplane at modest speed. Similar images are available on airplanes at transonic speeds and are often used as illustrations of the shock wave system on the wing of an airplane under these flight conditions. For example, Fig. 7.9 shows an F/A-18 flying in air that is close to saturated with water vapor. Evident is a cloud of vapor that follows the airplane. The viewer should not interpret this as visualization of the shock wave system associated with the wing and airplane. Clarification of this phenomenon is an opportunity to delve into the nature of shock and other waves on wings that play a role here but are not visible. This introduction of the topic will also serve to aid the discussion of flows into inlets and from nozzles.

Fig. 7.9
figure 9

Photo of a F/A-18 in transonic flight in saturated air. The text outlines the argument that the cloud boundary is not a representation of the invisible shock wave system (Photo credit:

7.8 Supersonic Wings

The flow physics at M = 1 is rather complicated because parts of the field are subsonic while others are supersonic. We can make a few simplifying assumptions to better understand what is seen in Fig. 7.9. First, we note that the F/A-18 airplane has a wing that is swept to a very small degree so that looking at it as a straight the wing in two dimensions is appropriate. The wing can be modeled as a flat plate because it is also rather thin. Second, the aircraft in the figure is flying at a Mach number that is not specified but probably rather close to M = 1. Another useful assumption is to take the flight Mach number to be somewhat larger so that the behavior of waves can be illustrated more clearly.

In mildly supersonic flight, say at Mach number of 1.4, the wing creates a wave pattern that is rather simple. At supersonic speeds, the wing cannot transmit information about its presence to the flow ahead of the wing. A wave is the sudden manifestation of the interaction between the wing and its environment. This contrasts starkly with the situation when the flow is subsonic and adjusts gently to the geometry demanded by the airfoil. At a positive (lifting) angle of attack, the leading edge of a supersonic flow generates two kinds of waves. The flow under the wing is turned into itself and is compressed. This is similar to the physics that affixes bound vorticity to the airfoil with incompressible flow as illustrated in Fig. 2.5. The wave is a compression or shock wave. When the turning angles are small, and the wave is referred to as an oblique wave. Shock waves are sometimes normal shock waves that are stronger and lead to a greater total pressure loss.

The wave encountered by the flow above the wing causes the flow to occupy a greater space and is therefore expanded. This expansion wave is not sudden but spread over a region through such an angle that the flow follows the direction imposed by the upper surface. Such a wave is called an expansion (or Prandtl-Meyer) fan. The flow through it is gentle enough to be reversible.

Figure 7.10 illustrates the wave pattern when a very thin flat plate is set at an angle of attack in supersonic flow. Because the pressure above the wing is lower than ambient pressure and higher below the wing, it generates lift. Because the flow velocity is uniform along both surfaces, the center of pressure is at the half chord point and firmly located there, irrespective of small changes in angle of attack. In this fact resides the vexing challenge of the early flyers attempting to fly faster than the speed of sound, because the airplane control mechanics change in character at transonic speeds.

Fig. 7.10
figure 10

The wave pattern on a lifting flat plate airfoil in supersonic flow. Note the two kinds of waves: oblique shocks and expansion fans. The values of approximate local Mach number are noted

At the flat plate airfoil’s trailing edge is another set of waves. The upper surface flow is slowed with its locally higher (than freestream or lower surface) Mach number. A compression wave allows departure from the trailing edge parallel the lower surface flow. The lower surface flow undergoes an expansion wave. The two shock waves (lower leading edge and rear trailing edge), having dealt with differing Mach numbers in their oncoming flows are different in strengths so that the entropy levels of the flows on either side of the wake differ. The trailing edge shock wave is stronger than the leading edge one. This difference in entropy is manifest in the existence of a wake. Its orientation is in a downward direction, consistent with Newton’s laws.

A real wing is not a flat plate but would consist of a sharp leading (w)edge and curved upper and lower surfaces to close at the trailing edge. Figure 7.11 illustrates the (exaggerated) geometry. The curved surfaces send information out into the flow to tell it to adjust its flow direction. These are weak expansion waves causing the flow to increase in speed and drop in temperature. When this temperature reduction takes the air below the dew point the result is water condensation. The condensation region is not collocated with the shock wave, nor directly initiated by the wave.

Fig. 7.11
figure 11

The wave pattern associated with a more realistic airfoil (exaggerated geometry) and operating at supersonic Mach numbers. The values of M along the expansion waves are meant to be illustrative and depend on the airfoil geometry. The wave pattern under the wing is similar

A finer point to consider about the appearance of the cloud is that it takes time (milliseconds) for droplets to form so that the dewpoint line must be a bit ahead of the cloud. At the trailing edge, the recompression and the associated warming will cause the droplets to evaporate, again delayed by the finite rate of the process. The details are quite complicated by the additional consideration of the latent heat involved in the process, but they are beautiful to observe.

We note that a rocket ascending into space may experience the cloud formation phenomenon around its nose when the atmospheric conditions allow it. The physics is the same as described for our wing.

The above diversion into supersonic flow phenomena was necessary to explain the condensation observations shown in Fig. 7.9. We leave the subject, revisit it in Chap. 13, and return to the concluding discussion of subsonic airplanes and their performance.

7.9 All the Airplane Aerodynamics in One Place

The understanding we have built can be put in a form that is useful for description of the aerodynamic performance of a wing and/or an airplane. In addition to the lift curve described earlier, a good way of representing this summary is the so-called drag polar where lift is plotted versus drag. Figure 7.12 is such a plot for a commercial airliner. Lift and drag are both noted in non-dimensional coefficient forms using dynamic pressure and wing area. For an airplane designed to cruise efficiently, the drag per unit lift is very important because it determines the engine thrust required and the associated fuel usage. The drag polar for a commercial airliner is usually determined for a variety of flight Mach numbers. An aerodynamicist measures the efficiency of a flight vehicle or its wing by the lift to drag ratio, or L/D, this ratio is akin to the inverse of a friction coefficient determined in the high school physics experiment involving the dragging of a brick across the floor.

Fig. 7.12
figure 12

Sketch of lift and drag performance of a subsonic airliner type of airplane. This drag polar is typically obtained for various Mach numbers of interest. A genuine performance plot for a Boeing 727–100 is shown in Appendix D

The tangent line from the origin of such a graph to the curve identifies the operating condition associated with maximum L/D. Piston-engine powered aircraft are typically operated at a speed for maximum L/D for good fuel economy which is especially important if long range is to be realized. For a modern commercial jet airplane, this ratio is in the vicinity of 20 meaning that the drag (and engine thrust) is about 5% of the lift. That parameter grew from about 15 in the early days of the jet age.

It turns out that a jet airliner operated at maximum range (to minimize fuel burn) has to operate at a point where the product of flight Mach number and L/D is a maximum. This is associated with the reality that a jet engine driven airplane engine produces thrust in contrast to a piston engine that produces power. A maximum value of M·L/D exists because increasing Mach number leads to falling airplane L/D due to the drag mechanisms associated with transonic flight as discussed above. As a consequence, commercial airliners typically operate at specific Mach numbers around 0.80. Because the jet airliner’s best performance is sensitively connected to flight number (rather than speed as such), instrumentation for such airplanes includes a Mach meter.

Airplane design and operations were attempted in the past where cruise Mach number was meant to reach 0.90 for competitive advantage between airlines, but such performance goals did not meet with long-term success. For airplanes designed to operate at supersonic Mach numbers, the situation was even more dire. The modest values of L/D achievable was the Achilles heel for such airplanes. That was an issue with the now out-of-service Concorde and would likely have been a concern for the American SST (SuperSonic Transport and its contemplated derivatives) had it been built and put in service.

7.10 The Computer to the Rescue

Computers today play an enormous role in much of engineering, manufacturing, and commerce. Their role in understanding flow phenomena was and is central to build better models, to wit, mathematical models. The reality of the complexity of flow description and the motivation for overcoming it with nimble and automatic computing machines deserves a few comments.

The simple physical models that make it possible to visualize the important phenomena involved in air flow about airplanes are just that, simple. In our tale, references have been made to better analysis results from the output of computers. The reasons behind that warrant a little more exposure to daylight.

In the digital age, it is fair to say that just about all important determinations of the performance of wings and airplanes is done on computers. One could say that within the computer is a mathematical model of an airplane. So why the computer and what does it actually do? The computer is much less expensive to run than experimental tests using wind tunnels and certainly more accurate and faster than any mathematical modeling done with paper and pencil. The computer can quickly accommodate a configuration or operational change and improved to reflect a better understanding of the modeling architecture.

As far as flow analysis is concerned, the large modern computer is well suited to the task because the description regimen is based on a set of sometimes complicated differential equations and the field is large. The ‘differential’ aspect has to do with the mathematical modeling of the entirety of the flow field as an array of small elements, describable as cells, and writing Newton’s equations of motion for that cell. The equations of motion are a form of F = ma that relates pressure differences and shear stresses acting on a cell to the local change in velocity.

The geometry of an airplane or wing being examined is a boundary of the flow field, as often are the conditions very far away from the object. In internal flows, such as those through a nozzle or diffuser, the boundaries are hard surfaces external to the flow. Together, specification of such descriptive parameters forms what are called boundary conditions. Dominant among these is that the local flow velocity cannot be normal to a rigid surface, nor can it be anything but zero along the surface when friction effects are considered.

The basic calculational cells are small. Imagine, for example, the number of cells with a characteristic dimension like one centimeter or half an inch required to describe the flow about an airliner. It is huge. Further complicating the matter is that within flow features like boundary layers and wakes, the cell size there has to be even smaller. On the other hand, the cells far from the object can be larger. There changes in cell properties (velocity, etc.) vary gently, meaning slightly over larger distances.

Keeping track of properties in a large field requires large computers with high information storage capacity and rapid computation speed. Historically, the demands of computational capability for the analysis of airflow investigations motivated the development of heavy-duty computers. Another motivating and related application demanding such computers is the analysis of weather systems. The computer is well-suited to do the repetitive calculations required until adjacent air flow elements form the nice continuum nature requires.

The flow phenomena that influence wing performance such as viscosity effects and waves markedly complicate the descriptive equations so that dissimilar sets of equations govern the behavior of cells. These regions are the ones that may be heavily influenced by entropy generation mechanisms. In boundary layers, for example, the effects of friction involve the necessary inclusion of terms in Newton’s equations of motion that account for friction. These unfortunately increase the difficulty of managing the mathematics, especially so when the flow is sufficiently energetic so that proper accounting of energy is critical. Luckily for the engineer, under many circumstances, the viscous terms can be safely omitted from descriptions in the far field. When the field consist of regions analyzed by locally different means, the computer analysis is often carried out in subregions. In the end, the results have to be stitched together to ensure a realistic variation of properties as required by nature.

As far as friction effects are concerned, an important challenge is the correct description of friction effects, be they laminar or unsteady and turbulent. Wave phenomena also present computational challenges because waves reach into the far field. Quite challenging is the situation where the effects of waves and friction merge into the need to properly describe the shock/boundary layer interactions that play such an important role in flow separation from external or internal flow boundaries.

A lot of progress has been made in computational fluid dynamics to allow the determination of good performance estimates. By good is meant, the calculation of integrated results like lift and drag, with narrow ranges of uncertainty under the many realistic conditions that might be encountered in practice.

7.11 Aerodynamic Heating is Not Due to Friction

Aircraft like the Concorde, the SR-71, or the Space Shuttle come in to land with an elevated skin temperature. The Concorde just spent 3 h cruising at M = 2, the SR-71 spent time at more than M = 3 and the re-entering Space Shuttle, who knows? Peak Mach number reached is said to be around 25 through all layers of the atmosphere. The skin temperatures reached by the two supersonic cruise aircraft (195 °F (90 °C) and 500 °F (260 °C), respectively) have a lot to say about the aircrafts’ design. The skin and structure of the Concorde is made with aluminum and the SR-71 with titanium. The Space Shuttle was covered with ceramic tiles that can stand the hellish temperatures of re-entry from space and are replaced as necessary after landing. Are these material choices associated with the flight Mach numbers? Very definitely. Are they due to friction? The answer is No, with a little bit of Yes.

Let’s look at the nature of friction. A friction force is experienced by a solid body (as our high school physics experiment again) whose motion is retarded by stationary surface like a brick sliding on the floor or a rope slipping through our hands. When a force, including a friction force, acts with a speed, there is an expenditure of mechanical power. In the case of friction forces, the mechanical power is converted to thermal power, heat. That is the nature of our experience with frictional heating and why automobile or aircraft wheel brakes get hot when used aggressively.

For fluids, liquids and gases, the situation is more complicated. Shearing forces acting on moving elements of fluid exert mechanical power that ends up in the fluid itself. For example, the oil in a journal bearing experiences such work expenditure and, as a result, the oil warms up. In many applications, such heating has to be dealt with by cooling of the liquid involved.

For unbounded fluids flowing past a surface, friction is manifest in the creation of a boundary layer on the stationary surface. Within the boundary layers, there are shear forces acting on adjacent moving elements of fluid. The heat generated is the product of this shear and the local velocity. At the bottom of the boundary layer, the shear is relatively large, but the velocities are small while the opposite is true near the edge of the boundary layer. There is, consequently, a zone of peak heat generation somewhere in the central region of the boundary layer. This heat is distributed by the motion of molecules and/or eddies within the boundary layer and plays a role (a very small one as it turns out) in heating the surfaces of our flight vehicles.

Is there another source of heat for the air passing along our high-speed airplane? Yes, and it is an important one. By its nature, the boundary layer air is forced to come to rest relative to the airplane, or at least slow down. The air that is so slowed brings with it its kinetic energy. When that kinetic energy source is large, as it is for high-speed flight, the local heating from the deceleration of the freestream air delivers kinetic energy converted to thermal energy that is not just significant, but dominant. That last statement has to be supported by plausible evidence.

Short of a detailed analysis, the best one can do is to make an order of magnitude assessment. That involves examining what physical quantities play roles and devising a figure of merit that characterizes the relative importance of two effects to be compared.

Such a figure is the ratio is of the heat generated by dynamic conversion of kinetic energy and the heat generated by the shear forces associated with friction. One can show that the ratio is closely related to and numerically similar to the Reynolds number (based on flow distance), a number that is numerically in the millions for a full-scale airplane. In short, frictional heating on a high-speed airplane is very small compared to the heating from the kinetic energy of the oncoming air. Engineers who have been concerned with this have established that the surface temperature on a surface like that of an airplane is typically somewhat less than the stagnation temperature of the air. Near a stagnation point (on the blunt nose) or line, however, like the leading edge of the wings, the temperature realized there is the full stagnation temperature. Thus, one can say, at least for conservative design purposes, that the entire vehicle is bathed with air close to the stagnation temperature.

This state of affairs is well illustrated qualitatively in the Space Shuttle sketch (Fig. 7.13) showing the leading edges white hot while the lower surfaces of the wings and body are also hot but cooler. NASA engineers equipped the leading edges as well as the bottom surface of the Shuttle with the best performance thermal protection system that they could devise.

Fig. 7.13
figure 13

Sketch of the Space Shuttle re-entering the atmosphere (NASA)

Leaving the Space Shuttle aside, we can look at the temperatures experienced by the other two airplanes as representative examples and see how the stagnation temperatures at the conditions where they fly correlates with after-flight body temperatures. It is reported in practice that the Concorde operates skin temperature is typically between about 200 °F and 260 °F (93–126 °C) while that of the SR-71’s is typically near 500 °F (260) and sometimes as high as 1000 °F (540 °C).

The stagnation temperatures relevant to these airplanes can be estimated by making a few assumptions. Consider that we are flying in an environment like the stratosphere where the absolute temperature is about 400° R (-60 °F or -51 °C). The Rankine scale is identical to the Fahrenheit scale except that it is referenced to absolute zero temperature. For practical reasons, namely that temperatures near absolute zero are not in our everyday world, the Fahrenheit scale, still used in the US, employs some diabolical thinking that the temperature of freezing water (at 14.7 psi absolute atmospheric pressure or 101,000 N/m2) is exactly 32 °F. Because this scale is familiar, at least in the United States, we will stick with it. Table 7.1 shows the Fahrenheit temperatures reached at a stagnation point. What is apparent in this table is that a Concorde flying at M near 2 will be hot on landing. More so for the SR-71 that is capable of speeds greater than M = 3. Aluminum melts near 600 °F (315 °C) while titanium melts at over 3000 °F (1650 °C). The Concorde design could safely use aluminum, but the SR-71 had to employ a titanium structure. Note that our commercial airliner’s skin temperature is 20 °F (-7 °C), cold enough to face the possibility of ice build up under some conditions. Their leading edges are equipped with means to heat them and thus avoid worries with icing issues whose buildup might change the shape of the airfoil shape that was developed with such care.

Table 7.1 Stagnation temperatures for an airplane flying in the stratosphere at various speeds characterized by Mach number

Now consider that the Space Shuttle reached Mach numbers near 25. It ‘flew’ back into the atmosphere at speeds where surface temperatures reached are said to be near 2300 °F (1260 °C) and air temperatures higher yet!

The numbers in the table are estimates. Nevertheless, the good correlation between observed vehicle temperatures with the stagnation temperatures seems a sufficiently strong to able to say, with a good deal of certainty, that friction per se plays a very small role in contributing to airframe heating of these examples.


  1. 1.

    See cautionary note in the introduction about symbols with multiple meanings.

Author information

Authors and Affiliations


Rights and permissions

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Reprints and Permissions

Copyright information

© 2022 The Author(s)

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Decher, R. (2022). Putting It All Together. In: The Vortex and The Jet. Springer, Singapore.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-16-8027-4

  • Online ISBN: 978-981-16-8028-1

  • eBook Packages: EngineeringEngineering (R0)