Problems associated with high-speed flight are manifested because air is compressible and these problems are, in truth, very complex.
Compressibility is the term which aeronautical engineers use for a large class of aerodynamic effects associated with high speed. This designation has been adopted because changes which occur in fluid flow at high speeds are chiefly produced by the elasticity or compressibility of the air. If air were an incompressible medium, these effects would not be present. Associated with compressibility is the density of air, which changes rapidly, in turn causing the airflow to change over the surface of the body.
At speeds approaching that of sound approximately 760 mph. at sea level the aerodynamicist is confronted with the problem of the airflow changing in character due to rapid changes in density resulting from compressibility properties of the atmosphere.
The practical significance of compressibility effects to aeronautics lies in the fact that the lift and drag of supporting surfaces, the effective operation of control surfaces, and the operation of the propeller with respect to efficiency and capacity, are all affected since the actual fluid is compressible. The lift, drag, and moment of a body are functions of the ratio of speed of the body to the speed of sound at the altitude in question. This ratio is called the Mach number after the Austrian physicist, Ernest Mach, who specialized in the study of high speed phenomena.
A rather abstract appreciation of the foregoing is afforded by a consideration of the elementary forces acting on a fluid particle. For our purposes it suffices to name principle forces.
First, is the well-known D'Alembert force of acceleration. This force is proportional to the square of the velocity with which the particle executes its motion, and the mass of the particle. Evidently the force is very small unless the velocity is high, but the force increases very rapidly as the velocity is increased.
Second, is the force of pressure acting differently on opposite sides of the particle. This force serves to balance the D'Alembert reaction. It is therefore called into being by the motion, and it disappears when the motion ceases. Let us agree that viscosity forces are small and may be neglected within the first order approximation.
Our picture now is of a particle of fluid executing an irregular motion in such a way as to avoid an airfoil which moves through the region initially occupied by the particle (Figs 1 and 2). Since the particle is accelerated from rest and follows an irregular path thereafter, a complicated system of inertia forces appears on the moving particle. These forces are balanced by pressure differences which appear between different points of the fluid.
Since inertia forces are small with small accelerations, we expect small pressure differences at low speeds. However, as the speed increases, the pressure forces required increase ac- cording to the square of the velocity. At sufficiently high speed these pressure differences will theoretically exceed atmospheric pressure, and vacuums may appear in the flow. In the case of a comparatively incompressible fluid, such as water, this phenomenon is usually called cavitation.
However, with a fluid, such as air, which can expand to fill any space, we find a different situation. Instead of cavitation we find the air becoming more and more attenuated in the regions of low pressure. Unfortunately for the mathematician the attenuation of the air lessens the density, so that the inertia forces are decreased, in turn decreasing the required pressures, thereby unbalancing the force system, causing the flow to undergo further readjustment, etc. This ends in a headache.
In addition to the expansion of air in regions of low pressure, a second phenomenon occurs at the speed of sound. It can be shown, theoretically, that a steady reciprocating motion of a piston in a pipe can be maintained without energy loss so long as the motion is below a certain frequency. Above this frequency, increasing amounts of energy are sent down the pipe in the form of sound waves. Similarly, at low speeds, the steady motion of a body requires no energy input, while at and above the speed of sound energy is continuously radiated outward in the form of a wave. This wave is very similar to the bow wave of a boat.
However, such waves would be unimportant at present-day speeds were it not for the fact that a complicated interaction occurs between the first phenomenon of fluid expansion and the second phenomenon of wave motion. This interaction causes the shock waves which occur approximately midway of the length of the airfoil chord (Fig 3). An example of very similar occurrence is given by the well-known hydraulic jump in the spillways of dams. Theoretically, little can be said about shock waves, but from a practical aspect they cause much greater headaches than either of the first two phenomena.
The task of estimating compressibility effects from a mathematical viewpoint presents many interrelated difficulties such as:
When mathematical studies are made, a steady adiabatic flow-field of an ideal gas about a two-dimensional body is assumed. When shocks are present, the assumption of irrotationality no longer exists.
Through the demands of structural and aerodynamic design, the interest of airplane manufacturers has become more and more sharply focused on the attempts to find a satisfactory and practicable solution to the problems outlined previously.
We shall start with the diagram of pressure distribution over a wing as obtained from low-speed wind tunnels. The pressures are proportional to the velocity squared.
On this basis then, the higher the pressure, the higher will be the local velocity, and as a result, the sooner we will run into trouble. (Fig 4). This suggests immediately that we should be very careful of curvature. The smaller the curvature, the better will be the flow conditions over the body. Small radii of curvature should be avoided when designing radial engine cowls, airfoil sections, canopy, and fuselage lines. It is important that the fuselage lines be as nearly straight as possible in the vicinity of the wing juncture, since we have superposition of fuselage and wing airflows.
Efforts to determine, theoretically, the influence of the Mach number on the pressure distribution and total lift of a wing have already been made. The answer, although not applicable up to the speed of sound, is relatively simple. The expression determined by Glauert-Prandtl is as follows: Increase all ordinates of `the airfoil by 1/√ 1 - M². The air forces are then equivalent to those of an incompressible flow acting on the modified profile.
In particular, since angle of attack and camber also increase with the factor 1/√ 1 - M², the lift will increase proportionately to 1/√ 1 - M². The drag will increase in degree indicated by the increased angle of attack and especially the increased thickness of the profile. Such conditions apply only in the case of relatively low Mach numbers, for example, up to M=0.6. Thick profiles, however, should be avoided. According to the factor 1/√ 1 - M², we have for M=0.6 (ie, V=455 mph. at sea level) a profile with 15% thickness would correspond to 18.8% thickness in an incompressible flow. Such a thickness ratio, however, is known to have appreciably greater profile drag. In order to possess good characteristics, the thickness ratio of a high-speed profile should not exceed 12%.
The actual critical effect of compressibility begins at values of M exceeding 0.65 at moderate values of lift of the airfoil. The streamline shape now breaks down completely, and the drag coefficient increases rapidly with increase of the Mach number; for example, between M=0.60 and M=0.75, the drag of a 12% symmetrical profile increases tenfold and the lift breaks down.
It is of interest to point out that when the fluid attains the local speed of sound, the airflow does not necessarily break down. In all probability a weak shock wave comes into the picture at this point, which is not too detrimental to the flow. However, as soon as supersonic regions appear, discontinuities may occur in which the velocity drops and the pressure rises over a very small distance.
When the intensity of the shock wave increases (by intensity we mean the ratio of the pressure ahead of the wave to the pressure behind the wave), rotationality of the field sets in and a turbulent flow region exists behind the wave. These waves extend a finite distance into the free flow and then fade away. The intensity of turbulence increases with the intensity of the shock wave. These compression shocks involve the dissipation of mechanical energy resulting in an increase in entropy. Eventually separation sets in, especially at the higher angles of attack, and at this point the drag increases precipitously (Fig 5) and the lift decreases rapidly.
With the introduction of separation, the wing circulation is decreased with the resultant loss in CLMAX. The turbulent field existing behind the shock wave will cause the tail surfaces to buffet and fast planes will be limited in high speed because of buffeting (Fig 6). The location of the horizontal tail surfaces is of paramount importance, and it is very difficult to position the horizontal tail so that it will be out of the wing wake, particularly so since the wing wake thickness increases appreciably when rotationality of the field sets in because of the shock wave.
In the case of projectiles, it is well known that drag coefficient increases abruptly with approach to the speed of sound. It is now recognized that this is a phenomenon of general application, but that it. is more pronounced in the case of airfoils and fuselages than in projectiles, since in the former, the initial drag is proportionately less because of favorable shape for the lower speed range.
The maximum speed of aircraft will probably occur at sea level since the maximum speed of sound occurs there. We can therefore expect a top speed of between 650 to 685 mph at sea level. This corresponds to M=0.85 and M=0.90, respectively, and it means that with further increase in engine power, all types of conventional aircraft will tend to approach the same top speed 650 to 685 mph.
It is interesting that the same phenomenon is observed with ships, it being noteworthy how little the speed of express steamers has increased (Mauretania, built 1909, 70,000 hp 24 knots; Queen Mary, 1938, 200,000 hp 31 knots). For ships, also, there are limiting speeds above which the required power increases beyond all proportion. This critical point, at which the resistance rises abruptly, can be raised by increasing the length of the vessel. With aircraft, too, it would therefore be important to discover some means of postponing the occurrence of this critical point in the drag curve.
In designing high-speed airfoils:
In the supersonic region, the airfoil experiences undulatory and frictional resistance. It can be shown that for almost rectilinear supersonic velocities, the body of least undulatory resistance. for a given volume, is bounded by two arcs of parabolas symmetrical with the X axis. Furthermore, for a body of minimum resistance, for a given volume, the frictional resistance, when considered, must equal the resistance due to the generation of waves.
Before discussing the action of the control surfaces, fuselage, wing, propeller, and engine cowl in high-speed dives, a few remarks will be made regarding the attainment of these dives.
The gross weight of the present-day fast airplane is such as to give wing loadings of approximately 40-50 lb/sq ft. This loading is quite high, and in a dive the weight of the plane acts as thrust. Coupled with large wing-loading and good aerodynamic cleanliness, modern high-speed planes attain very high dive speeds. It is inevitable that all planes will run into difficulty at speeds approaching that of sound.
Since the Republic P-47 attains a Mach number of 0.85 to 0.90 in dives, it is not surprising that difficulties have been encountered. It is unfortunate that Nature has decreed that the type of flow-pattern changes as we approach the speed of sound. With the establishment of the shock wave, the flow over the wing changes in such a manner as to induce a flow separation.
This results in material increase in the wing wake of such a magnitude as to envelop the horizontal tail surface. By definition, the wing wake is a region of very turbulent airflow, and because of this turbulence the tail is subjected to buffeting.
We shall enumerate briefly the experiences encountered with the P-47 in high-speed dives and the steps taken in flight tests to overcome the difficulties:
The original aileron (Frise type) installation had hinges flush with the lower surface. At speeds of 400 mph and over, a violent oscillatory motion of the ailerons usually occurred. A series of flight tests were conducted to eliminate this condition, the procedure consisting of changing the position of the aileron hinge line.
Best results were obtained by moving the hinge line 1-in down and 5/16-in aft, with respect to the original position. The new arrangement did not completely solve the problem but limited the snatch condition to Mach numbers above 0.8. Further work led to the construction of a blunt nose aileron with a variable mechanical advantage in the control system to reduce stick forces. The blunt nose aileron with the differential control eliminates aileron snatch at high speeds up to 500 mph and gives greater rolling velocities at all speeds.
As a result of increased tail loads because of compressibility, the structural strength of the fuselage must necessarily increase. Tail loads are increased because of redistribution of forces on the wing, resulting in a dive tendency of the plane in general. Parts of the fuselage such as canopy, ducts, fillets, and doors must be reinforced because of increase in air loads resulting from high speeds.
Special emphasis has been given to the P-47 fuselage design with regard to the contours at the wing juncture in that the fuselage lines are substantially straight at the wing root. Special emphasis has also been given to the length of the fuselage aft of the trailing edge of the wing to minimize fuselage drag.
The wing forces and moments also change as a result of compressibility effects. With increase in speed or Mach number the resultant of the lift forces tends to shift toward the trailing edge of the wing. Accordingly, care should be given to the design of the wing trailing edge as a result of this shift in loading. The leading edge of the wing should be very rigid as a result of the increase in pressure. It is extremely important to make the root section as thin as possible, and effort should be made compatible with the installation of machine guns, gas tanks, and landing gear not to exceed a maximum thickness of 12-14%.
In this respect, propeller slipstream effects will tend to increase the local speed over the wing immediately behind the propeller.
In dives at Mach numbers above 0.82, the greatest increase in diving moment, resulting from compressibility, tends to nose the plane down, and control becomes difficult until an altitude of approximately 12,000 ft is reached. Neither elevator nor full tab motion is very effective, although the full tab deflection will raise the altitude at which pullout is possible, by a few thousand feet. The pullout, when it occurs at approximately 12,000 ft, is relatively sudden and sharp, and excessive load factors may be developed. The solution to this condition was the installation of a dive flap. With this flap a controlled pullout can be achieved at any altitude and Mach number (Figs 8, 9, 10, 11, and 12).
The propeller problem is very important since the propeller's primary purpose is to convert engine horsepower into thrust. A blade section presents the same problem as does an airfoil section, but it involves more difficulties in that we have rotational as well as forward speed. As a result of these two speeds, the problem of compressibility is much more severe.
The general trend of present-day propeller design is toward wider blade chord to alleviate the effects of compressibility on the blade section. Various propellers have been tried on the P-47 to increase the rate of climb performance. The original propeller had a somewhat narrow chord distribution, with result that climb performance was sacrificed. Tests with wider blade chord propellers have increased the rate of climb considerably.
With the introduction of more horsepower and improved turbosuperchargers which increased the critical altitude of the plane at which maximum horsepower can be maintained, six-bladed dual-rotation propellers have been tested to determine their effect on stability, high speed, and rate of climb. The results, to date have been discouraging. More development and research work will have to be devoted to the dual-rotation problem.
The engine cowl is very much like an airfoil section in that it has a leading edge radius. The redistribution of forces on the cowl are similar to that of the wing. Care should be taken to avoid sharp curvatures at the leading edge of the cowl, in much the same way as with an airfoil section.
Wind tunnel experiments show that the shock wave is not limited to a particular point on the airfoil but moves within a given neighborhood of points. This phenomenon is called by some the "dancing" shock wave and immediately suggests an uncertainty factor in our theoretical setup. The dancing phenomenon may be quite similar to the Uncertainty Principle in quantum mechanics, where a single wave function of a given type for a free particle corresponds to the physical condition in which the momentum and the energy are exactly known, but the position of the particle is unknown.
The aeronautical engineer should probably content himself with certain "hand" values rather than exact values of the aerodynamic forces and moments in the subsonic region of flow. Flight tests have indicated that compressibility effects are not as severe as predicted by theory (Fig 13). This may be partially explained by the fact that we have not evaluated the influence of viscosity with the attendant boundary layer. The boundary layer probably serves as a cushion and thereby tends to alleviate conditions, since the temperature within the boundary layer is several degrees warmer than the air outside this layer.
In conclusion, these and many other problems challenge our imagination without as yet yielding more than a tantalizing guess as to their real nature. Furthermore, it should not be forgotten that speeds above that of sound are near and cannot be dismissed much longer (Figs 14 and 15). For the man who wants problems of immediate practical interest to tax his powers to the utmost, compressibility is a happy hunting ground.
This article was originally published in the December, 1944, issue of Aviation magazine, vol 43, no 12, pp 171-175, 423.