...In the divergent section of a supersonic nozzle (for example, a two-dimensional one), the gas is accelerated:
..Above are the Mach waves in the divergent section of a rocket nozzle. These waves appear to reflect off the wall. This is related to a compatibility condition concerning the flow at the wall: the internal bisector of the two Mach waves, or characteristics, represents the flow direction, which must be tangent to the wall.
...The speed of sound is reached at the throat. Upstream, the gas is at subsonic speed. It is accelerated in the divergent section and the pressure drops. A rocket engine will operate with better efficiency the higher the exhaust velocity. However, there is a trade-off: the gases must also be expelled at a pressure equal to the ambient pressure at the altitude where these nozzles operate. This is called "nozzle adaptation." If the divergent section is too pronounced, the gas will be expelled at a pressure lower than the ambient (atmospheric) pressure, and recompression shock waves will appear. Mach waves are also called the "characteristics" of the flow. In a rocket engine with axisymmetric symmetry, these Mach waves are conical. This means that at any point in the nozzle, if a fixed obstacle the size of a grain of sand were placed, it would generate a conical Mach wave.
..The velocity vector would then correspond to the half-angle of the cone. The higher the Mach number, the sharper this Mach cone would be.
...Jet engines in aircraft are equipped with variable-section nozzles, whose divergent sections open up as altitude increases and ambient pressure decreases.
..The divergent sections of jet engine nozzles are fitted with "petals" that open using hydraulic actuators as the pressure decreases, this being directly controlled by a barometric measurement. Since the exhaust velocity is higher, these engines have better efficiency at high altitude.
...But let us return to our open channel. What happens when the bend appears as follows:
...On the left, the "convergent" section is not too pronounced. The characteristics (Mach waves) tend to converge, but they do not intersect (otherwise they would be outside the flow itself). There is a decrease in the "local Mach number," a decrease in velocity, and an increase in water depth (the equivalent of pressure in a gas).
...On the right, the bend is too pronounced. The Mach waves tend to intersect. A liquid "hydraulic jump" appears, analogous to a shock wave in a gas. The flow experiences a discontinuity. Downstream of the shock wave, the velocity drops abruptly across the jump-wave.
..The bow of a ship is also a "convergent" section. If the ship moves at a low speed, less than the propagation speed of surface waves (thus in "subsonic" regime), Mach waves do not exist. Correspondingly, the water level remains constant.
...At higher speeds (V > a), using a computer, one can calculate the geometry of theoretical Mach waves in a two-dimensional flow. It is observed that they intersect and tend to focus:
...In the drawing above, theoretical Mach waves were calculated in a gas flow by solving the fluid mechanics equations (Navier-Stokes) around a lenticular airfoil profile immersed in a supersonic gas stream, using a computer (1979). It is observed that the Mach waves tend to focus. Only one family of waves is shown. These focal points are the "birthplaces" of shock waves. Indeed, these Mach waves are compression wavelets. The flow above is therefore not physically realistic. Note the presence of two expansion fans on the flanks of the profile. Two systems of shock waves will therefore appear:
..Downstream of the shock waves, the gas is "shocked," recompressed, and its velocity decreases. This phenomenon occurs over a very thin layer: a few hundredths of a millimeter.
..Downstream of the bow shock wave, after having been abruptly decelerated, the gas is continuously reaccelerated according to an "expansion fan." It is even "overaccelerated," to the point that a second shock wave, called the base shock wave, must establish itself at the trailing edge of the profile to restore the ambient pressure downstream, according to the principle mentioned in my comic strip "The Sound Barrier" (see the "CD-Lanturlu"):
Please leave the gas in the state in which you found it upon entry.
..The velocity vector also undergoes a discontinuity in direction if the leading edge is a dihedral:
(phenomenon analogous at the trailing edge if it is also dihedral-shaped).
...Let us see what this looks like in hydraulic analogy.
...We see that the overacceleration of the water on the flanks of the ship causes the part of the hull located at rest below the waterline to become visible.
...These wave systems (in gas flows or in these liquid flows with a free surface) modify the pressure distribution around the profile or hull. This results in wave drag, which adds to friction drag. During supersonic cruise (for example, during a Concorde flight), wave drag becomes so significant that it greatly exceeds friction drag. Supersonic flight is therefore a high energy consumer, and aircraft must be equipped with powerful engines. Similarly, these supersonic flights can only be conducted at altitude; otherwise, wave drag would become prohibitive. A jet aircraft can barely exceed Mach 1.2 at ground level.
Where does this energy go? It dissipates in two ways. Supersonic vehicles create a very intense "boom," which distributes this energy far away from them, just as the shock wave created by an explosive dissipates its energy over large distances. The shock wave also causes heating of the air, but dissipation in the form of sound is dominant.
..We have presented here a system of "attached waves." If the front part of the vehicle is blunt (nose or leading edges of the wings and tail, in the case of the space shuttle), the shock wave establishes itself at a certain distance from the object. Since the velocity is zero at the "stagnation point" of the flow, the flow becomes subsonic downstream of the wave, and then reacceleration occurs.
..The recompression of the gas downstream of a shock wave is accompanied by heating. The temperature at the "stagnation point" increases very rapidly with the Mach number (as its square). Supersonic aircraft therefore experience high thermal stresses on their frontal parts (nose, leading edges). Although this leads to an increase in drag, the noses or leading edges must be blunted at very high Mach numbers (hypersonic regime) to distribute the heat influx. Remember the very blunt nose of an experimental vehicle like the X-15.
..In the case of reentry bodies, this is not a problem since braking is precisely what is sought. Russian reentry capsules are simply spherical. American capsules feature a large "thermal shield," where partial ablation of the material occurs (they are not designed to last more than a few minutes and must be replaced after each reentry if the capsule is reused).
...In 1975, we posed the problem of possible supersonic, and even hypersonic (up to Mach 15) flight in dense air, skimming the rooftops. With a conventional system (rocket propellant or jet engine, ramjet style, plus thermal shield, or frontal parts cooled with liquid helium), such very high-velocity cruises are totally unthinkable. No system could withstand such a heat influx or provide the thrust necessary to overcome such significant wave drag.
...I was the first to suggest that MHD could solve this problem (note to the Comptes Rendus of the Academy of Sciences in 1975 and article in Science et Vie in 1976). It is quite shocking to see certain groups of people take this idea as their own, without the slightest citation of this pioneering work, and when a former technician, perfectly incompetent in the matter, declares on television, as recently happened:
..- We now know how to make vehicles travel at supersonic speeds without creating a shock wave.
The thing becomes outright ridiculous. But let us say that this is quite common in technical-scientific circles.
..Since 1975, we have therefore wondered how shock waves, which create both a strong heat influx and strong wave drag, could be completely eliminated. To address this problem, it is necessary to understand well why these shock waves form.
..When an object enters a gas mass at subsonic speed, it is the sound waves that "warn" the gas of its arrival. When an object, even a blunt one, enters a gas at subsonic speed, the latter begins to move aside well before the object reaches it:
