Fluid Mechanics

 Photo of a Prandtl-Glauert condensation cloud on a transonic F-18A. It is a US Navy photo and was found at the Chamorrobible site. Click on the photo to learn more about it.

Sonic Boom, Sound Barrier, and Condensation Clouds
Prandtl-Glauert Condensation Clouds

Nowadays there are many photographs on the web which depict condensation clouds similar to that seen at the right. Excellent sources of such photos include the sites by Peter Steehouwer, Evert Konigsveld, Jeff Wilkinson, and the ChamorroBible site and, of course, my own Gallery of Fluid Mechanics. Unfortunately, there is also a great deal of confusion (on other sites and forums) as to the physical origin of these clouds. Because these clouds tend to be seen when the aircraft flies at near-sonic speeds, it is frequently said that they "visualize shock waves" or are due to the aircraft "bursting through the sound barrier". Neither statement is true, although the second at least suggests that the phenomena occurs in the near sonic or, in the language of aerodynamics, transonic, regime. The point of this page is to give a short, non-technical discussion of the physical origins of this phenomenon.

The clouds appear for the same reason that clouds always form, namely, that the air has cooled to the point that the ambient water vapor condenses. Flows around bodies and wings always change the temperature and pressure of the fluid. It is well known that lift is caused by pressure differences on top or bottom of a wing or body so that it ought to be obvious that the pressure varies from point to point in a flow around an object. The fact that the temperature changes can be seen by noting that most fluid flows and nearly every aerodynamic flow are frictionless. In the language of thermodynamics, the flow is said to be reversible or loss-free. As a result the entropy of the flow is a constant and the temperature (T) at each point in the flow is necessarily related to the pressure (p) as follows:

 T a p(g - 1)/g, (Pg1)

for low pressure gases. The constant g is the ratio of specific heats. For air, g = 1.4 and the exponent on the pressure (p) in the above equation is approximately 0.29. Thus, the temperature of the air will increase and decrease as the pressure increases and decreases. Regions of high pressure will necessarily correspond to regions of high temperature and regions of low pressure will correspond to regions of low temperature.

Lift-Induced Condensation
An example of condensation due to the pressure variations over wings is illustrated by the various forms of lift-induced condensation found in my Gallery of Fluid Mechanics. This type of condensation cloud is formed when an aircraft undergoes a high-lift maneuver resulting in very low pressures on the upper surface of the wings. The corresponding temperatures are so low that the water vapor condenses on the upper (low-pressure) side of the wing. A characteristic of lift-induced condensation is that it is asymmetric, i.e., there will be much more condensation on the upper side of the wing than on the lower side, and that it is usually associated with high-g turns.

Condensation due to the Prandtl-Glauert Singularity
In the case of the lift-induced condensation the temperature variations found in normal flight are exaggerated by the large pressure differences required to generate high lift. At speeds near that of sound, the temperature and pressure variations occurring at every speed can also be exaggerated in steady level flight. The mechanism for this near-sonic exaggeration of the temperature variations is the so-called Prandtl-Glauert singularity which requires that pressure and temperature perturbations approach ±¥ as the flight speed approaches the ambient sound speed. In terms of the temperature, the Prandtl-Glauert singularity takes the form:

 T - T¥ constant (Pg2) ----------- »  ------------------, U2/2cp¥ | 1 - (M¥)2 |½

where M¥= the Mach number = the ratio of the flight or far-field flow speed (denoted by U) to the sound speed of the undisturbed atmosphere. The quantities T¥ and cp¥ are the temperature and specific heat at constant pressure in the undisturbed atmosphere. The constant in (Pg2) depends on the specific shape and orientation of the wing or body, but not on the Mach number or aircraft speed. Equation (Pg2) holds for gases at atmospheric pressure.

 Remark Unfortunately, I don't know of a simple physical explanation for the existence of the Prandtl-Glauert singularity which is easily understood by laypeople and even undergraduates in science and engineering. Perhaps the best route is just to take my word (or, rather, the word of the great men and women of aerodynamics) that it exists and is of the form given. Of course, if you are an upperclassman in engineering, math, or physics, my discussion of aerodynamic similarity laws might help guide you through the mathematical explanation.
A more general form of (Pg2) be found by combining the small disturbance approximation for the temperature with the general expression for the Prandtl-Glauert singularity, cast in terms of the pressure coefficient.

The | 1 - (M¥)2 |½ term seen in (Pg2) will be very small when the flight Mach number M¥ is near one. As a result, the right hand side of (Pg2) will be very large when M¥ » 1 and the temperature perturbation (T - T¥) will be correspondingly large. This is the amplification referred to as the Prandtl-Glauert singularity. If the Mach number is sufficiently close to one, the temperature perturbations in the low-pressure, low-temperature portions of the flow can become large enough to cause condensation of the ambient water vapor. If condensation does occur, then the resultant cloud is referred to as a Prandtl-Glauert condensation cloud.

Characteristic Cloud Shape
The characteristic conical shape of the Prandtl-Glauert clouds is consistent with and can be predicted by the characteristic flow patterns of near-sonic, i.e., transonic, flow known to aerodynamicists. An example of the flow over a two-dimensional wing is sketched below. At flow or flight speeds near the sound speed (M¥ » 1) any bump, e.g., a wing or canopy, causes the pressures and temperatures to drop. The bump can also cause the flow to become slightly supersonic; this is the case illustrated below. As indicated in the sketch, the region of low pressure and temperature will exist over a portion of the upper and lower surfaces of the wing and will be terminated by a shock wave.

 Sketch depicting the near-sonic flow over a wing. The quantity M is the Mach number which is just the ratio of the flow speed to the sound speed and the incoming flow is taken to be subsonic. A few temperature contours can be seen by running your mouse over the image.
As pointed out in my brief discussion of shock waves, the shock decelerates the airflow and heats it. (I have also sketched the isotherms, i.e., the lines of constant temperature similar to those seen on a weather map, for the same flow. You can see these by running your mouse over the image.) Because the condensation is likely to be initiated along a constant temperature line, the front of the cloud ought to inherit the conical shape of the isotherms and ought to be terminated by a nearly flat shock surface. This, of course, is exactly what is seen in photographs such as that seen at the top of the present page.

Good examples of the very flat base of the cloud can also be seen in the photo at the top of the page, in a recently contributed photo of a transonic F-14, and in this widely distributed photo of an F4 Phantom.

The above discussion holds when the flight speed is close enough to one to ensure that the characteristic transonic flow pattern, complete with terminating shock wave, is established. However, when the flight or flow speed is just at the outer edge of the transonic regime, shock waves are not generated and the flow appears to be closer to that seen in incompressible (low speed) flows. Under these conditions, the amplification due to the Prandtl-Glauert singularity could still be important resulting in condensation. The difference between this marginal case and the fully developed transonic flow illustrated in the sketch at the above left is that the temperature and cloud shape is not likely to have the distinct "conical" shape seen in the sketch and photo above. Examples of this marginal case are likely to be this image of a cloud on an F-14 or even the spectacular image of the B2 stealth bomber.

Conclusions
The clouds formed by the Prandtl-Glauert singularity are due to the near-sonic amplification of the pressure and temperature perturbations which naturally occur whenever air passes over any bump or object. Thus, an aircraft can fly at one-half or twice the speed of sound and generate no clouds. However, if the same aircraft flies at 0.95 or 1.05 times the speed of sound, the amplification implicit in (Pg2) may be enough to cause condensation in the low-pressure, low-temperature portions of the flow.

Because Prandtl-Glauert condensation can form in both slightly supersonic and slightly subsonic flow, a sonic boom may or not be heard.

The shape of Prandtl-Glauert condensation clouds will reflect the isotherms characteristic of transonic flow. In many cases, the cloud will have the conical shape seen at the top of this page. As pointed out in the previous subsection, other patterns could also be observed which are completely consistent with the near-sonic flow patterns and temperature distributions known to aerodynamicists.

A common error is to state that the cloud "visualizes shock waves". As indicated above, the front of the cloud has nothing to do with the shock wave. As a note to aspiring aerodynamicists, the cloud also does not correspond to the flow Mach lines which tend to be nearly vertical in near-sonic flows. Although the shock does not correspond to the leading edge of the cloud, it does correspond to the termination of the cloud which gives rise to the characteristic flat base of the cone.

Other earmarks of a Prandtl-Glauert condensation is that it will be reasonably symmetrical with respect to the top and bottom of the aircraft. This fact, and the fact that Prandtl-Glauert condensation can be seen in steady level flight, can be used to distinguish it from lift-induced condensation which tends to be associated with high-g maneuvers and tends to occur primarily on the upper or low-pressure side of the aircraft.

Finally, it should be clear that Prandtl-Glauert condensation has nothing to do with "breaking the sound barrier" and is not a Star Trek-like "burst" through Mach one. An aircraft can generate a Prandtl-Glauert condensation cloud without ever exceeding the speed of sound.