Start of MHD manipulation MHD1
The Supersonic Flight Without the "Boom"
...Imagine a lens-shaped profile immersed in a supersonic gas flow. Shock waves (attached) will form, in the form of planar waves. Two of these waves will originate from the leading edge, and two others will be located near the trailing edge. In perspective:
Profile 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. Take, for example, a mass of fluid at rest. A fisherman watching his float gently pulls on his line. The float will oscillate and generate circular waves that propagate at a speed of a few centimeters per second:
...In your bathtub, you can achieve the same result by moving a toothpick or a matchstick up and down.
If the fluid is in motion (viewed from above), these circular waves—emitted at different times—will become offset:
...The right-hand image corresponds to "sonic" motion. Sound waves are the image of a pressure disturbance in a gas. These hydraulic analogies, once taught at the National Superior School of Aeronautics (Supaéro), from which I graduated, were the subject of practical exercises.
...What happens when the fluid moves faster than the speed of surface wave propagation? We obtain the following pattern:
...The disturbances emitted by the object tend to accumulate along two lines originating from the object.
...Not everyone has a stream at hand. Therefore, instead of considering a stationary object in a liquid current, we can just as well move the object and obtain the same result. You can do the same in your bathtub by moving your matchstick or the tip of your wooden toothpick, which is sharper. This will generate what are known as "Mach waves." If we know the speed a of surface wave propagation and the speed V of the object's movement, it is easy to calculate the Mach angle α.
...Conversely, if we measure this angle α, we can determine the fluid's speed V.
...The Mach waves will be increasingly flattened as the speed V increases.
...A spillway is an excellent hydraulic laboratory where the speed V varies, increasing downstream. If you place your matchstick in the water, you will observe the following:
...Another "hydraulic laboratory" is a gutter. Irregularities on the vertical surface of the sidewalk, in contact with the flowing water, generate Mach waves, just as those along the other end of the liquid channel do. The water surface is thus marked by this network of waves, indicating both the direction and the speed of the flow. The fluid flows along the internal bisector of the Mach waves. If the flow in the gutter occurs at constant speed, the liquid surface becomes streaked with small waves—Mach waves—forming "parallel hatching":
...If the water speed increases downstream due to increasing slope, the waves flatten:
...The opposite phenomenon occurs if the fluid slows down due to decreasing slope:
...This deceleration in a gutter may result from water friction with the ground when the depth becomes sufficiently small. If you examine the flow closely, you will observe this:
...Mach waves straighten out as you approach the "water's edge," as friction with the ground slows the flow. When the waves become perpendicular to the flow, the speed has decreased 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.
...Thus, there is much to learn from observing gutters.
...If the gutter curves, there are two possible scenarios. Here is the first:
...As the water takes this bend, it accelerates. This produces what is known in fluid mechanics as an "expansion fan." The fact that the characteristics flatten indicates this acceleration. Correspondingly, the water height—equivalent to pressure in a gas—decreases. We can complete this diagram by adding the second family of Mach waves:
...To learn more about MHD, refer to the comic strip I published in 1983 by Belin Editions, 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 the icon at the bottom of the page.
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