Start of MHD2
...In the divergent section of a supersonic nozzle (for example, two-dimensional), the gas is accelerated:
..Above, the Mach waves in the divergent of a rocket engine. These waves seem to reflect on the wall. This is related to a compatibility condition regarding the flow at the wall: the internal bisector of the two Mach waves, or characteristics, represents the direction of the flow, which must be tangent to the wall.
...The speed of sound is reached at the throat. Upstream, the gas is subsonic. It is accelerated in the divergent and the pressure decreases. The rocket engine will operate with a better efficiency the higher the exhaust velocity. However, there is a drawback: the gases must also be ejected at a pressure equal to the ambient pressure, at the altitude where these engines operate. This is called "nozzle matching". If the divergent is too pronounced, the gas will be ejected at a pressure lower than the ambient (atmospheric) pressure and recompression shock waves will appear. The Mach waves are also called the "characteristics" of the flow. In a rocket engine with rotational symmetry, these Mach waves are conical. This means that at any point of the nozzle, if there was a fixed obstacle the size of a grain of sand, 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 the Mach cone.
...Jet engines of airplanes are equipped with variable area nozzles, whose divergent opens as the altitude increases and the ambient pressure decreases.
..The divergents of jet engine nozzles are equipped with "petals" that open using hydraulic rams 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's go back to our channel. What happens when the turn looks like this:
...On the left, the convergent is not too pronounced. The characteristics (Mach waves) tend to converge, but they do not intersect (unless outside the flow itself). There is a decrease in the "local Mach number", the velocity and an increase in the water height (equivalent to the pressure in a gas).
...On the right, the turn is too pronounced. The Mach waves tend to intersect. A "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". If the ship moves at a low speed, less than the speed of propagation of surface waves (thus in "subsonic"), Mach waves do not exist. Correlatively, the water level remains constant.
...At higher speed (V > a), one can, using a computer, calculate in a two-dimensional flow the geometry of the theoretical Mach waves. It is observed that they intersect, tend to focus:
...In the drawing above, the theoretical Mach waves in a gaseous flow have been calculated, by solving the equations of fluid mechanics (Navier-Stokes) around a lenticular airfoil immersed in a supersonic gas flow, using a computer (1979). It is observed that the Mach waves tend to focus. Only one family of waves has been represented. These focusing locations are "birth places" of shock waves. Indeed, these Mach waves are compression waves. Therefore, the flow above is not physically realistic. Note the presence of two expansion fans on the sides of the airfoil. Therefore, two systems of shock waves will appear:
..Downstream of the shock waves, the gas is "shocked", recompressed, and its velocity decreases. This phenomenon occurs over a very small thickness: a few hundredths of a millimeter.
..Downstream of the frontal shock wave, after having been abruptly decelerated, the gas is continuously reaccelerated along an "expansion fan". It is even "overaccelerated", to the point that a second shock wave, called a "tail shock", must form at the trailing edge of the airfoil to restore the ambient pressure downstream, according to the principle mentioned in my comic strip "Le Mur du Silence" (see the "CD-Lanturlu"):
The gas is to be left in the state it was found upon entry.
..The velocity vector also experiences a directional discontinuity if the leading edge is a dihedral:
(phenomenon analogous to the trailing edge, if it is also in the form of a dihedral).
...Let's see what this looks like in hydraulic analogy.
...It can be seen that the overacceleration of the water on the sides of the ship makes the part of the hull that is at rest appear below the waterline.
...These wave systems (in gaseous flow or in these free surface liquid flows) modify the pressure distribution around the airfoil or the hull. This results in wave drag which adds to the friction drag. When supersonic cruising (such as, for example, during a Concorde flight), the wave drag becomes so significant that it far exceeds the friction drag. Supersonic flight is therefore a major energy consumer and the aircraft must then be equipped with powerful engines. Similarly, these supersonic flights can only be carried out at high altitude, otherwise the wave drag would become prohibitive. A jet aircraft cannot generally exceed Mach 1.2 near the ground.
Where does this energy go? It is dissipated in two ways. Supersonic vehicles create a very intense "bang" that distributes this energy far from them, just as the shock wave created by an explosive dissipates this energy over a large distance. The shock wave also causes heating of the air, but the dissipation in the form of sound is dominant.
..We have here presented a "attached wave" system. If the front part of the vehicle is blunted (nose or leading edges of the wings and tail of the space shuttle, for example), the shock wave forms 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 shock, 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). Therefore, supersonic aircraft experience strong thermal stresses on their front parts (nose, leading edges). Although this leads to an increase in drag, the nose or leading edges must be blunted at very high Mach numbers (hypersonic regime) to distribute the heat influx. Remember the very blunted nose of an experimental vehicle like the X-15.
..In the case of re-entry bodies, this is not a problem since braking is desired. Russian re-entry capsules are simply spherical. American capsules have a significant "thermal shield", or a partial ablation of the material occurs (they are not designed to last more than a few minutes and must be replaced after each re-entry, if the capsule is to be reused.
...In 1975, we posed the question of a possible supersonic evolution, even hypersonic...