The low speed drag level is often defined at a Mach number of 0.5, below which the airplane drag coefficient at a given lift coefficient is generally invariant with Mach number. The increase in the airplane drag coefficient at higher Mach numbers is called compressibility drag. The compressibility drag includes any variation of the viscous and vortex drag with Mach number, shock-wave drag, and any drag due to shock-induced separations. The incremental drag coefficient due to compressibility is designated CDc.
In exploring compressibility drag, we will first limit the discussion to unswept wings. The effect of sweepback will then be introduced. For aspect ratios above 3.5 to 4.0, the flow over much of the wing span can be considered to be similar to two-dimensional flow. Therefore, we will be thinking at first in terms of flow over two-dimensional airfoils.
When a wing is generating lift, velocities on the upper surface of the wing are higher than the freestream velocity. As the flight speed of an airplane approaches the speed of sound, i.e., M>0.65, the higher local velocities on the upper surface of the wing may reach and even substantially exceed M= 1.0. The existence of supersonic local velocities on the wing is associated with an increase of drag due to a reduction in total pressure through shockwaves and due to thickening and even separation of the boundary layer due to the local but severe adverse pressure gradients caused by the shock waves. The drag increase is generally not large, however, until the local speed of sound occurs at or behind the 'crest' of the airfoil, or the 'crestline' which is the locus of airfoil crests along the wing span. The crest is the point on the airfoil upper surface to which the freestream is tangent, Figure 1. The occurrence of substantial supersonic local velocities well ahead of the crest does not lead to significant drag increase provided that the velocities decrease below sonic forward of the crest.
Fig. 1 Definition of the Airfoil Crest
A shock wave is a thin sheet of fluid across which abrupt changes occur in p, r, V and M. In general, air flowing through a shock wave experiences a jump toward higher density, higher pressure and lower Mach number. The effective Mach number approaching the shock wave is the Mach number of the component of velocity normal to the shock wave. This component Mach number must be greater than 1.0 for a shock to exist. On the downstream side, this normal component must be less than 1.0. In a two-dimensional flow, a shock is usually required to bring a flow with M > 1.0 to M < 1.0. Remember that the velocity of a supersonic flow can be decreased by reducing the area of the channel or streamtube through which it flows, When the velocity is decreased to M = 1.0 at a minimum section and the channel then expands, the flow will generally accelerate and become supersonic again. A shock just beyond the minimum section will reduce the Mach number to less than 1.0 and the flow will be subsonic from that point onward.
Whenever the local Mach number becomes greater than 1.0 on the surface of a wing or body in a subsonic freestream, the flow must be decelerated to a subsonic speed before reaching the trailing edge. If the surface could be shaped so that the surface Mach number is reduced to 1.0 and then decelerated subsonically to reach the trailing edge at the surrounding freestream pressure, there would be no shock wave and no shock drag. This ideal is theoretically attainable only at one unique Mach number and angle of attack. In general, a shock wave is always required to bring supersonic flow back to M< 1.0. A major goal of transonic airfoil design is to reduce the local supersonic Mach number to as close to M = 1.0 as possible before the shock wave. Then the fluid property changes through the shock will be small and the effects of the shock may be negligible. When the Mach number just ahead of the shock becomes increasingly larger than 1.0, the total pressure losses across the shock become greater, the adverse pressure change through the shock becomes larger, and the thickening of the boundary layer increases.
Near the nose of a lifting airfoil, the streamtubes close to the surface are sharply contracted signifying high velocities. This is a region of small radius of curvature of the surface, Figure 1, and the flow, to be in equilibrium, responds like a vortex flow, i.e. the velocity drops off rapidly as the distance from the center of curvature is increased. Thus the depth, measured perpendicular to the airfoil surface, of the flow with M > 1.0 is small. Only a small amount of fluid is affected by a shock wave in this region and the effects of the total pressure losses caused by the shock are, therefore, small. Farther back on the airfoil, the curvature is much less, the radius is larger and a high Mach number at the surface persists much further out in the stream. Thus, a shock affects much more fluid. Furthermore, near the leading edge the boundary layer is thin and has a full, healthy, velocity profile. Toward the rear of the wing, the boundary layer is thicker, its lower layers have a lower velocity and it is less able to keep going against the adverse pressure jump of a shock. Therefore, it is more likely to separate.
For the above reasons supersonic regions can be carried on the forward part of an airfoil almost without drag. Letting higher supersonic velocities create lift forward allows the airfoil designer to reduce the velocity at and behind the crest for any required total lift and this is the crucial factor in avoiding compressibility drag on the wing.
The unique significance of the crest in determining compressibility drag is largely an empirical matter although many explanations have been advanced. One is that the crest divides the forward facing portion of the airfoil from the aft facing portion. Supersonic flow, and the resulting low pressures (suction) on the aft facing surface would contribute strongly to drag. Another explanation is that the crest represents a minimum section when the flow between the airfoil upper surface and the undisturbed streamlines some distance away is considered, figure 2. Thus, if M= >1.0 at crest, the flow will accelerate in the diverging channel behind the crest, this leads to a high supersonic velocity, a strong suction and a strong shock.
Fig. 2 One View of the Airfoil Crest
The freestream Mach number at which the local Mach number on the airfoil first reaches 1.0 is known as the critical Mach number. The freestream Mach number at which M= 1.0 at the airfoil crest is called the crest critical Mach number, Mcc The locus of the airfoil crests from the root to the tip of the wing is known as the crestline.
Empirically it is found that the drag of conventional airfoils rises abruptly at 2 to 4% higher Mach number than that at which M= 1.0 at the crest (supercritical airfoil are a bit different as discussed briefly later). The Mach number at which this abrupt drag rise starts is called the drag divergence Mach number, MDiv. This is a major design parameter for all high speed aircraft. The lowest cost cruising speed is either at or slightly below MDiv depending upon the cost of fuel.
Since Cp at the crest increases with CL, MDiv generally decreases at higher CL. At very low CL, the lower surface becomes critical and MDiv decreases, as shown in Figure 3.
Fig. 3 Typical Variation of Airfoil MDiv with CL
The drag usually rises slowly somewhat below MDiv due to the increasing strength of the forward, relatively benign shocks and to the gradual thickening of the boundary layer. The latter is due to the shocks and the higher adverse pressure gradients resulting from the increase in airfoil pressures because Cp at each point rises with (1-M02)-1/2. The nature of the early drag rise is shown in Figure 4.
Figure 4. Typical Variation Of CDc with Mach Number
There is also one favorable drag factor to be considered as Mach number is increased. The skin friction coefficient decreases with increasing Mach number as shown in figure 5. Below Mach numbers at which waves first appear and above about M= 0.5, this reduction just about increased drag from the higher adverse pressure gradient due to Mach Therefore, the net effect on drag coefficient due to increasing Mach M = 0.5 is usually negligible until some shocks occur on the wing or favorable effect of Mach number on skin friction is very significant sonic Mach number, however.
Figure 5. The Ratio of the Skin Friction Coefficient in Compressible Turbulent Flow to the Incompressible Value at the Same Reynolds Number