Effects of Compressibility

By Dr G L Shue Development Engineer,
Consolidated Vultee Aircraft Corporation

As a design consideration, compressibility becomes increasingly important with higher aircraft speeds

In aerodynamics, it is customary to consider air an incompressible fluid. However, it has long been a known fact that the density of air varies with pressure changes occurring in the flow over an airfoil. Therefore, aircraft engineers have recently encountered one of the most striking dilemmas in the history of heavier-than-air flight.

The dilemma is compressibility, and it may be generally defined as the effects produced by variations in the density of flow of air over an airfoil.

Compressibility becomes increasingly important as the speed of an aircraft exceeds 300 mph, and when speeds in excess of 400 mph are desired it is the foremost design consideration. This is true because the local speed of the air passing some points of an airplane may be greater than (or equal to) the speed of sound.

The speed of sound is such an important item that it has been found convenient to name the quotient of air speed and the speed of sound. Thus we have the "Mach number". The critical Mach number is reached when the flow of air attains the speed of sound at some point on an airplane.

The following effects of compressibility have been observed:

  1. —The slope of the lift curve increases with an increase of Mach number up to the critical speed; then, at super-critical Mach numbers, it decreases.
  2. —The drag coefficient remains practically constant at sub-critical Mach numbers and then increases rapidly as the critical Mach number is approached and exceeded.
  3. —The pitching moment coefficient may increase, de crease, or remain the same when the critical Mach number is reached.
  4. —When the critical Mach number is exceeded, shock zones are formed in the air stream. A shock zone is a narrow region at right angles to the air stream, across which the air suffers an abrupt decrease in velocity. The loss in energy appears as heat, and this probably accounts for most of the increase in drag. The shock increases in intensity and moves rearward on the airfoil as the speed increases.
  5. —Both the stability and control of an airplane are affected by compressibility. Since it is rather common to use a thin section on the tail surface, the tail critical Mach number is usually higher than the wing critical Mach number. Therefore, it has been frequently observed that the stability of an airplane at high speed increases to several times its low-speed value, and the control forces become exceedingly large. With some pursuit planes, the combination of stability and control has been such that the pilot could not pull the airplane out of a high-speed dive; other cases have been observed where the natural pull-out tendency was so great that the pilot could not prevent recovery.

The speed of sound in air depends only on temperature, and it is proportional to the square root of the absolute temperature. At 60° F, this speed is about 764 mph; but at -60° F, it is only 670 mph. Therefore, at higher altitudes, where air has low density and offers less drag at a given speed, it is easier to reach the critical Mach number since the speed of sound is considerably lower. This is particularly detrimental to military aviation.

Contrary to popular beliefs, compressibility burbles are not the result of air currents moving at supersonic speeds; they are created when air currents moving at supersonic speeds are forced to decelerate to subsonic speeds. If it were possible to maintain supersonic speeds in the entire flow of air over a wing or past the blades of a propeller, many problems now being encountered by manufacturers of high-speed aircraft might be solved. However, thoughts along this line are, for the present, somewhat irrelevant.

Wing compressibility has received little attention thus far, because the problems of propeller compressibility have been more urgent. However, the development of jet propulsion promises to eliminate the latter while accentuating the former.

The quantitative prediction of the behavior of an airfoil at high speed has not been too successful up to the present time. The commonly-used correction factor is that of Glauert,

 1 - M²
where M is the Mach number. This factor represents the compressible and incompressible coefficients. It is reasonably accurate at the lower Mach numbers, but it does not agree with experiment at the critical Mach number or above. On the basis of this equation, the speed of sound has been selected as the limiting speed. (Note: if M is greater than unity, the result is the square root of a negative quantity).

Von Kármán and Tsien obtained an equation which is a better approximation than the Glauert equation. It is usually given in the following form:

CPM =  √ 1 - M² + _____M²______ _CP0  _
1 + √ 1 - M²  2  
Here CP0 represents the incompressible fluid pressure coefficient, and CPM represents the compressible fluid pressure coefficient. To find the pressure coefficient, first find the difference between the free stream static pressure and the static pressure at the point in consideration; then divide this difference by the free-stream dynamic pressure. The resulting quotient is the pressure coefficient. The incompressible pressure coefficient CP0 is obtained at low speed, since air then acts similar to an incompressible fluid. The compressible pressure coefficient CPM is found at various speeds high enough for compressibility effects.

CPM and CP0 are used as coordinate axes for curves representing the conditions in the flow of air over airfoils. Similar curves are shown in Fig 1. The curve for the low Mach number represents the condition in which the air velocity does not approach the speed of sound at any point. The curve for the high Mach number represents the condition where the local air speed is greater than the speed of sound during a portion of the flow over an airfoil. In either curve, the lower left end denotes the stagnation point while the origin represents free-stream conditions and the upper right quadrant indicates a local air speed higher than the free stream speed.

At point B, the local speed of the air is equal to the speed of sound. At pointA, the maximum speed without shock has been reached; and, if this speed is exceeded, a shock zone will be formed at some point downstream. Point C represents the peak velocity and, following it, the speed drops almost instantly to a subsonic speed. This is the shock zone.

The above explanation is further illustrated by using the airfoil chord as the abscissa and plotting both CPM and CP0, as ordinates, as shown in Fig 2. Points A, B, and C are plotted so that they correspond with the points in Fig 1. Probably the best equation for approximating high-speed flow conditions was recently published by L M Greene. As it is now being interpreted at Consolidated Vultee Aircraft Corp, this equation agrees with experimental results at free-stream Mach numbers as high as 0.70. It is indicated that the flow is definitely not reversible adiabatic at the higher speeds.

Estimation of the maximum local supersonic speed without shock is possible with Greene's equation; and, when local temperature surveys have been made in a high-speed tunnel, it will no doubt become possible to make better estimates of compressibility effects at the higher Mach numbers.

This article was originally published in the October, 1944, issue of Industrial Aviation magazine, vol 1, no 5, pp 59, 97.