Supersonic flight without a "bang".
...Let us imagine a lenticular profile immersed in a supersonic gas flow. Shock waves (attached) will form—plane waves. Two of these waves will originate from the leading edge, and two others will be located near the trailing edge. In perspective:
Side view:
Hydraulic analogy.
...There is a very simple way to understand, by analogy, the mechanism of shock wave formation. In a liquid flow with a free surface, the propagation of surface waves is equivalent to that of sound in a gas. Let us take, for example, a fluid mass at rest. An angler watching his float gently tugs on his line. The float will oscillate and trigger the emission of circular waves, which will propagate at a speed of the order of a few centimeters per second:
...In your bathtub, you will obtain the same result by moving a toothpick or a matchstick up and down.
If the fluid is in motion (top view), these circular waves, which were not emitted at the same time, will become off-center:
...The image on the right corresponds to "sonic" motion. Sound waves are the image of a pressure disturbance in a gas. These hydraulic analogy experiments, formerly taught at the National Superior School of Aeronautics (Supaéro), from which I graduated, were part of practical laboratory work.
...What happens when the fluid moves at a speed greater than the propagation speed of the surface waves? We obtain the diagram below:
..The disturbances emitted by the object will tend to accumulate along two lines originating from the object.
...Not everyone has a waterway at hand. Therefore, instead of considering a stationary object placed in a liquid flow, one can just as well move the object and obtain the same result. You can do the same in your bathtub water by moving your matchstick, or the sharper tip of your wooden toothpick. What will then form are what are called "Mach waves." If we know the propagation speed a of the surface waves and the speed V of the object, it is easy to calculate the Mach angle a.
...Conversely, if we measure this angle a, we can calculate the fluid speed V.
...These Mach waves will be flatter the higher this speed V is.
...A weir is an excellent hydraulic laboratory where the speed V is variable, increasing downstream. If you place your matchstick in the water, you will observe this:
..Another "hydraulic laboratory" is a gutter. The roughness present on the vertical surface of the sidewalk, in contact with the flowing water, triggers the emission of Mach waves, as do those lining the other edge of the liquid channel. The water surface is thus striated by this wave network, which indicates both the direction and speed of the flow. The fluid flows along the interior bisector of the Mach waves. If the flow in the gutter occurs at constant speed, the liquid surface is striated with ripples, Mach waves, forming "parallel hatching":
...If the water speed increases downstream because the slope increases, the waves will flatten:
..The opposite phenomenon occurs if there is a deceleration of the fluid, because the slope decreases:
..This deceleration in a gutter can be due to friction of the water on the bottom when the depth becomes sufficiently shallow. If you look closely at a flow, you will observe this:
...The Mach waves straighten up as one approaches the "edge of the water," friction on the bottom slowing down the flow. When the waves become perpendicular to the flow, the speed has dropped to that of surface wave propagation. The flow has become "subsonic." In this region, Mach waves disappear. If you dip a matchstick or a hairpin into the flow, you can verify this.
...There is thus much to learn from observing gutters.
...If the gutter curves, there are two possible scenarios. Here is the first:
..By taking this curve, the water will accelerate. This produces what is called in fluid mechanics an "expansion fan." The fact that the characteristics flatten indicates this acceleration. Correlatively, the water height, equivalent to pressure in a gas, will decrease. We can complete this drawing by including the second family of Mach waves:
....To learn more about MHD, refer to the comic book I published in 1983 with Éditions Belin, 8 rue Férou, Paris 75006. You can also obtain the CD version of this album on the "Cd-Lanturlu." Access the order form by clicking on the icon at the bottom of the page.
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