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2 David D. Marshall, Nov 1, 2005, I hope this helpsOblique Shock Paradox. I hope this helps I saw your posting on CFD Online about your book and was intrigued by your mention of the oblique shock paradox. I read through the section and am not sure where the paradox arises. The reason that the oblique shock angle does not go to zero as the deflection angle goes to zero is because at zero deflection the oblique shock wave becomes an isentropic pressure wave (actually when there is a small perturbation in the deflection angle). An isentropic pressure wave takes a finite amount of time to travel through the flow and the angle that the pressure wave makes with respect to the flow angle is the Mach angle (check out Anderson's "Modern Compressible Flow" pp. 131133 for a Mach angle discussion). What the oblique shocks degenerate to in the limit of zero deflection angle is simply an isentropic pressure wave which is consistent with how shocks form in the first place (as coalescing of small perturbation pressure waves that become a finite sized wave front). I hope this helps, or perhaps I missed something in the your text. Let me know if I did.
Response:
Dear Dr. David D. Marshall, Thank you for your response. The question you posted is excellent and it will give me the opportunity to explain what is the difference between the common approach and BarMeir's solution. Let me start by stating that my solution provides the exact solution to the oblique shock. My solution indeed satisfies the equation as appears in the NACA 1135 report. (equation 150a page 10.) If you disagree with the equation than I can refer you to the reference in the end of that report. You can check that the solution satisfies equation by substituting it into the equation. Now assuming that the equation and the solution are correct, examining the solution show an interesting phenomenon. "no solution" appears in two zones. One which everyone agrees and explains it by the detached shock. What we now have new, after introduction of my solution, is the discovery of new zone of "no solution". This zone is lay between zero inclination to a small angle which depends on the Mach number. The common approach to explain the connection between the weak weak shock to the normal shock was by stating that is a singular bifurcated point (at least mathematically speaking) in which a normal shock can coexist with Mach wave. Hence, the view, perhaps implicitly, was that jump occurs only at delta==zero. As you may noticed, I believe that the mathematical solution I discovered tell us more. First, the jump does not occur at zero but at different angle (though close to zero). Second, the shock does not behave as a weak weak shock approach to Mach wave. The Shock simply ``jump'' before reaching zero angle to a normal shock. This happens because when the mathematics tells that when an imaginary solution ( not real numbers, imaginary number is a square of (1) ) to our model appears different physical phenomenon occurs. Now, I always take the advise of my adviser, the late Dr. Eckert, and caution that I did not look at experimental evidence since I wasn't able to find one and I do not have the facility to carry such experiments. However, if we can study from the past in the case of shock or chock flow, the mathematicians were right ( you can check my book on the history of the shock) that discontinuity exist before Mach demonstrated that they were right. By the way, see the similarity to the explanation to the "toilet problem" (clock or counter clock rotation question) that Asher Shapiro found. It was furnished in the southern hemisphere several year after it was published. I hope that this explanation convince you that My approach is the correct to explain the paradox (of discontinuity). There are other issues that I did not totally resolve in relationship to this topic. But, they can be only question of getting used to a strange situation (it is in relationship to Mach number). Thank you again for the question. If you do not have any objections I will post this correspondence in my web site.
Thanks,
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