Sources of Drag in Supersonic Flow

We all know that when an object is placed in a flow, it experiences drag - force directed along the freestream flow direction.  In general, it is much harder to predict the drag coefficient accurately, than it is to predict the lift coefficient.  This is primarily because we design flying objects to have small drag compared to lift. For example, the lift-to-drag ratio of an aircraft may be 10. That means that the drag coefficient is only 1/10-th of the lift coefficient.  If our method is accurate enough to calculate the lift coefficient to within 5% error, that implies that the method can predict the drag coefficient to within 50% error.

Where does the drag come from, and how can it be minimized? We can think of several sources.
 
 Type  Mechanism Subsonic flow Supersonic Flow

1 a) Skin friction: laminar
 Viscosity in the boundary layer: Molecules bouncing off surface transfer their momentum change to the molecules in the stream, slowing them down.  Yes.   Yes (boundary layer is still subsonic!!). Also, compressibility leads to substantial density change and temperature change: worsens drag.
1 b)  Skin friction: turbulent  Momentum transfer becomes chaotic as entire packets of fluid start moving across the boundary layer: High-speed flow comes closer to surface, leading to greater shear, hence greater drag.  Yes  Yes (b.l. still subsonic). Also, compressibility leads to substantial density change and temperature change: worsens drag.

2. Pressure drag: 
Wakes and recirculation regions of low pressure  Yes  Yes (subsonic inside these regions)
3. Lift-induced drag.  Tip vortices and vortex sheets cause downwash, tilting force vector back  Yes  No. Wake cannot be felt upstream
4. Wave drag   Pressure disturbances not recovered by cancellation  NO  Yes.
5. Shocks  Entropy rise due to irreversible sudden compression Ds is roughly proportional to the cube of the static pressure change, or the cube of turning angle for oblique shocks: q3  No  Yes

 
 

Of course, in real flows, supersonic and subsonic, there are boundary layers. The relative velocity at the surface of a solid body is always zero. This leads to shear and rotation.  The rate of shearing strain (velocity difference between adjacent layers of fluid) is shearing stress; the proportionality constant  (shear stress / rate of shear strain) is viscosity.  So, the fluid does work against shearing force to move over the surface. (Or, the aircraft does work against the shearing force from the air, to move through the air.) The force needed to balance the shear force is what we call the skin friction drag.  The work done by the fluid on the airplane gets dissipated as heat, and in turn heats up the fluid close to the surface, which transfers the heat to the surface.

The shearing strain also causes rotation (imagine a hockey puck sliding along the ice, until it rubs against a wall: it now starts rotating as it moves along the wall.)  This rotation shows up as the formation of vortices in the wake behind the body. The momentum loss due to wall friction can be captured by measuring the decrease in velocity and stagnation pressure across this wake, but remember that this momentum difference is already felt at the body surface.

Note:

Even if the external flow is supersonic, inside the boundary layer the velocity decreases to zero.

Lets see each of those drag sources again:

(1) Boundary Layer drag: laminar and turbulent
The fluid molecules hitting the surface, and thus losing their momentum, then bouncing off and hitting other molecules zipping along in the flow. This is called "viscous drag".  Clearly this is a bigger deal if this "momentum transfer" is a big part of the momentum of the flow. This momentum transfer is confined to a region near the surface, called the "boundary layer". This is why the "Reynolds Number" is important: it gives a measure of how large the inertial forces of the flow are, with respect to the viscous forces. When the Reynolds number is low, viscous drag becomes very important. Generally, the Reynolds number for the flow over an airplane wing is quite large, so viscous drag is not very significant in subsonic flight. It does become significant when other kinds of drag are minimized, and at high speeds every source of drag becomes important. We generally speak of "skin friction drag" in high speed flight, because the high temperatures created at the skin worry us. The temperature becomes high because the molecules which were going smoothly along in the flow now start getting bounced all over the place when they collide with the molecules bouncing off the surface, and their motion becomes chaotic: the kinetic energy is converted to heat, and the flow near the surface becomes hot, and transfers heat to the surface.

Now, at high Reynolds number, something worse happens: the boundary layer becomes turbulent. Imagine a highway where the cars were all zipping along at 65 mph, and the right lane moves slower because there's a police car parked in the emergency lane. OK, laminar flow. Right lane moves at 30mph, left 3 lanes move at 65. Everybody moves along a straight line; verfy few lane changes. The second lane from the right moves at about 55 because they sense that they are passing the right lane traffic too fast otherwise. The left lane people are oblivious to any relative motion at all (no shear strain across the left two lanes).

Now think of a case where there is no police car, but there are still a few people moving at, say, 55 mph in the right lane, scarred by the recent shock of seeing a police car. Some of the folks who were delayed for microseconds by this slowness zip out into the second lane, going so fast that they end up in the 3rd lane. Some monster SUVs from the 4th lane zip right into the right lane, still going at 80, and hit their brakes, dissipating lots of energy and hot air. Some of the 55mph people coming behind these braking SUVs (can't see beyond them) swerve out of the right lane, to the left, to avoid collision.

A typical day on the Downtown Connector.

This is like turbulent boundary layer motion. Think now of the cars as packets of fluid, rather than as molecules. The intelligence level of the fluid packets is nearly equivalent to that of many SUV drivers. In turbulent flow, entire packets get transported across the layer, carrying their momenum with them. Fast fluid comes very close to the surface; slower fluid reaches far out into the what used to be undisturbed flow. Note that now a lot of the momentum of the flow is directed across the flow, rather than along the main flow direction. Entire packets of fluid are now using up their momenum zipping randomly across the boundary layer like the molecules were doing before.  The "boundary layer" becomes a lot thicker, but the skin friction is still what is felt close to the surface: this goes sharply up, because now there are fast-moving packets of fluid shearing against the wall. The fluid that had been slowed down leaves the surface region and goes out to create havoc across the boundary layer. Remember, the fuel in the airplane is wasted in  decelerating every one of those packets of fluid before they go zipping out into the freestream again.

The momentum loss in the turbulent boundary layer is thus much higher than in the laminar boundary layer, although the velocity profile across the turbulent boundary layer looks nice and strong, with high velocity close to the surface. Much of this momentum loss can be computed if we know the "Reynolds stresses": the momentum wasted in traveling sideways. Note that this is NOT a viscous phenomenon, and the Reynolds stress is NOT just a shear stress. Think now of the hockey puck making less progress towards the goal because it bounces off several hockey players who are beating each other up with sticks.

Potential flow theory looks at the flow outside the boundary layer (remember that we assumed "irrotational"? The boundary layer is certainly rotational), so viscous drag is not captured by potential flow theory. We have to use "common sense" tricks like the "Kutta condition" to figure out the "right"amount of rotation in the boundary layer to solve an aerodynamics problem.

2) Pressure drag.
The flow becomes distorted so that regions of low pressure occur on the downstream side, pulling the aircraft backwards. We try our best in aircraft design to avoid such regions. Examples of such regions are flow separation zones or "recirculation bubbles", and wakes. We try to make wakes as thin as we can, by keeping the low-momentum fluid in the boundary layer as close to the surface as possible. When you think about it, pressure drag, in many cases, arises from viscosity, but its cure may be related to making the boundary layer turbulent, so that the flow remains attached. So, what you do to reduce skin friction drag (i.e., making the boundary layer less turbulent) may in fact increase your pressure drag because the flow separates over the aft portions of the airfoil. For this reason, it is a good idea to consider these types of drag separately.

A flow separation region may be carried on a supersonic configuration, though of course the velocity inside this region may be low relative to the surface. An example is the recirculation region formed over the wing of a winged re-entry vehicle when its control surfaces are deflected (Space Shuttle, X-33, or the X-38 Crew Return Vehicle)

3) Lift-Induced Drag
The third source of drag, which is a very significant portion of the drag in subsonic flight, is "lift-induced drag", or "induced drag". This is, literally, drag induced by our efforts to generate lift. As we know, lift is simply any aerodynamic force perpendicular to the freestream, and any lift generation is accompanied by induced drag. We have seen how to calculate induced drag of wings in low-speed flow.

Is there lift-induced drag in supersonic flow? Well, any time you generate lift, you turn the flow, and there must be some penalty associated with this. However, in supersonic flow, these are captured over the wing surface: what happens downstream of the wing cannot affect the flow over the wing. Hence, we will say here that lift-induced drag is not an issue in supersonic flight. Note, however, that all supersonic aircraft must fly subsonic to get to supersonic speeds, and must decelerate through the subsonic regime. (Well, exceptions: a Mars Glider may be launched from the mother ship while it is already moving at supersonic speed during atmospheric entry, and may transform itself into something else (maybe a balloon?) before reaching subsonic speed.  An anti-radar missile may be launched at supersonic speed by a fighter aircraft, and never decelerate to subsonic speed before it blows up)

4) Wave Drag:
This is a strange phenomenon which is very much felt on supersonic aircraft, but would not be anticipated without thinking of the differerence between subsonic and supersonic flows.

Calculating Wave Drag: click here to go to the section on how to calculate wave drag in linearized supersonic potential flow

As showin in the above link, we calculated drag coefficients in supersonic flow, from the pressure coefficient distribution over an airfoil in supersonic flow, after assuming that the flow was potential, and neglecting boundary layer effects. This is called "wave drag". It is a feature of supersonic flow. Disturbances in ideal supersonic flow propagate out to infinity, unchanged in amplitude. The energy of these disturbances must come from the kinetic energy of the flow. This is the source of the "wave drag".








In subsonic potential flow, on the other hand, the passage of an object leaves no trace: conditions return to what they were before the object came by. This is because the disturbances from different portions of the airfoil cancel out as one goes away from the airfoil.

4)  Drag due to shocks:
Shocks are regions where the entropy goes up, and the stagnation pressure drops. Thus they cause drag.

Shocks cause large amounts of drag: the entropy increases suddenly across them. This is because inside the sharply discontinuous shock, dissipative effects such as viscosity and heat conduction take away some of the energy of organized motion and convert it to the energy of the random, chaotic motion of gas molecules.

The entropy rise across a shock is related to the third power of the static pressure change, or the turning angle, whereas the drag coefficient in wave drag is proportional to the square of the turning angle.

The drag by a shock of given static pressure ratio is thus considerably greater than the total wave drag due to several waves, each of small pressure ratio.
 

Moral:

If you have to make waves in supersonic flow, make a series of small waves rather than one big one. This is used in designing wings, fuselages, and engine inlets.