The Triple Point Paradox for the Nonlinear Wave System

Tesdall, Allen M.; Sanders, Richard; Keyfitz, Barbara Lee

doi:10.1137/060660758

Cited by 42 publications

(34 citation statements)

References 12 publications

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“…Recently, Tesdall et al 25 performed similar calculations using the model of the nonlinear wave system. They achieved a high numerical resolution that was sufficient for the analysis of the fine structure of the flow field near the triple point under the conditions of the von Neumann paradox.…”

Section: Introductionmentioning

confidence: 91%

Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge

2008

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The reflection of weak shock waves has been reconsidered analytically using shock polars. Based on the boundary condition across the slipstream, the solutions of the three-shock theory ͑3ST͒ were classified as "standard-3ST solutions" and "nonstandard-3ST solutions." It was shown that there are two situations in the nonstandard case: A situation whereby the 3ST provides solutions of which at least one is physical and a situation when the 3ST provides a solution which is not physical, and hence a reflection having a three-shock confluence is not possible. In addition, it is shown that there are initial conditions for which the 3ST does not provide any solution. In these situations, a four-wave theory, which is also presented in this study, replaces the 3ST. It is shown that four different wave configurations can exist in the weak shock wave reflection domain, a Mach reflection, a von Neumann reflection, a ?R ͑this reflection is not named yet!͒, and a modified Guderley reflection ͑GR͒. Recall that the wave configuration that was hypothesized by Guderley ͓"Considerations of the structure of mixed subsonic-supersonic flow patterns," Air Materiel Command

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Section: Introductionmentioning

confidence: 91%

Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge

2008

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“…It was conjectured in [11] that this sequence is infinite for an inviscid weak shock reflection. Since the detection of this multiple patch structure in [11], the same phenomenon has been found in solutions of the nonlinear wave system in [12], and, more recently, in the compressible Euler equations in [13].…”

Section: Introductionmentioning

confidence: 76%

“…However, there is a lack of agreement on the question of whether a shock terminates the supersonic patch (as occurs on a transonic airfoil), and on whether or not a sequence of such patches, each with a terminating shock, actually occurs. The multiple-patch structure has been found in solutions of the UTSD equations, the nonlinear wave system, and the full Euler equations, and it appears likely that a sequence of supersonic patches and triple points is a generic feature of some class of hyperbolic conservation laws, possibly characterized by "acoustic waves," as proposed in [12]. A natural question is whether the single-patch structure is also typical in solutions, possibly in a different range of parameter values.…”

Section: Introductionmentioning

confidence: 96%

Further Results on Guderley Mach Reflection and the Triple Point Paradox

Tesdall

Sanders

Popivanov³

2015

J Sci Comput

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Recent numerical solutions and shock tube experiments have shown the existence of a complex reflection pattern, known as Guderley Mach reflection, which provides a resolution of the von Neumann paradox of weak shock reflection. In this pattern, there is a sequence of tiny supersonic patches, reflected shocks and expansion waves behind the triple point, with a discontinuous transition from supersonic to subsonic flow across a shock at the rear of each supersonic patch. In some experiments, however, and in some numerical computations, a distinctly different structure which has been termed Guderley reflection has been found. In this structure, there appears to be a single expansion fan at the triple point, a single supersonic patch, and a smooth transition from supersonic to subsonic flow at the rear of the patch. In this work, we present numerical solutions of the compressible Euler equations written in self-similar coordinates at a set of parameter values that were used in previous computations which found the simple single patch structure described above. Our solutions are more finely resolved than these previous solutions, and they show that Guderley Mach reflection occurs at this set of parameter values. These solutions lead one to conjecture that the two patterns are not distinct: rather, Guderley reflection is actually underresolved Guderley Mach reflection.123 722 J Sci Comput (2015) 64:721-744

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“…To simplify the notations, we let F = F − U and G = G− U, and the problem in conserved form (5) can be rewritten as [4,15] U +F +G +2U = 0.…”

Section: The Numerical Methodsmentioning

confidence: 99%

“…These methods appear to capture the shock interaction of the nonlinear wave system accurately. Since our goal is to capture self-similar solutions, we rewrite the system using the self-similar variables [4] and implement the numerical schemes to the self-similar system, rather than solve the initial value problem for the unsteady nonlinear wave system in the original variables.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions to the self-similar transonic two-dimensional nonlinear wave system

Kim

Lee

2010

Math. Meth. Appl. Sci.

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We present numerical results on a two-dimensional Riemann problem governed by the self-similar nonlinear wave system that gives rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax-Liu positive schemes and Strang splitting, and obtain several numerical solutions for the model system. With the numerical results that we have obtained, we present several analyses of the transonic shock strengths and the positions of the transonic shocks with various Riemann data. Moreover, due to the presence of the corner of the object, numerical oscillations are apparent. We discuss regularity results for the solution near the corner of the object.

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The Triple Point Paradox for the Nonlinear Wave System

Cited by 42 publications

References 12 publications

Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge

Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge

Further Results on Guderley Mach Reflection and the Triple Point Paradox

Numerical solutions to the self-similar transonic two-dimensional nonlinear wave system

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