Focusing of weak shock waves at an arête

Cramer, M. S.; Seebass, A. R.

doi:10.1017/s0022112078002062

Cited by 37 publications

(18 citation statements)

References 7 publications

(3 reference statements)

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“…Similar derivations can be found in [13] for general systems of conservation laws and in [20] for weak detonations. Equations (3.3.11) were also derived in [5] as asymptotic expansions at aretes.…”

Section: Derivation Of the Asymptotic Equationsmentioning

confidence: 99%

“…B. Keller, A. Majda and R. R. Rosales (see [19] for a detailed review). Nonlinear expansions were conjectured at singular rays, caustics and artes, based on the geometry of the linear solutions, in [5], [13] and [20]. The resulting asymptotic equations, although simpler than the full equations of gas dynamics, are still nonlinear and multidimensional and, to present knowledge, not solvable in closed form under general conditions.…”

Section: Introductionmentioning

confidence: 99%

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Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection

Tabak

Rosales

1994

Physics of Fluids

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Some phenomena involving intersection of weak shock waves at small angles are considered: the focusing of curved fronts at artes, the transition between regular and irregular reflection of oblique shock waves on rigid walls and the diffraction patterns arising behind obstacles. The asymptotic equations describing this regime, which are those of unsteady small-disturbance transonic flow, are studied with a mixed theoretical and numerical approach.The intersection of three shock waves plays a central role in most of these phenomena, giving rise to the von Neumann paradox of shock reflection and to the curious transition between linear and fully nonlinear focusing investigated experimentally by Sturtevant and Kulkarny. This "triple-point paradox" is studied in the context of the asymptotic equations, and a solution is proposed that involves an unusual kind of singularity.A numerical algorithm for the solution of the equations of transonic flow is devised, based on switching to a new set of coordinates that mixes space and time, and applying a generalization of the method of Godunov. This algorithm is used to perform numerical experiments on shock reflection, arxtes and singular rays.

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Section: Derivation Of the Asymptotic Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection

Tabak

Rosales

1994

Physics of Fluids

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“…However, the scope of application of the KZ equation is much more general, as it can model many diffraction effects localized along singularities, such as the Fresnel diffraction ͑as shown in the present study͒, the tip of finite fold caustics ͑Marchiano, 2003͒, or cusped caustics ͑Cramer andSeebass, 1978;Coulouvrat, 2000͒. A generalized version of the KZ equation models nonlinear diffraction in the shadow zone of an upward-refracting atmosphere ͑Coulouvrat, 2002͒.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Fresnel diffraction of weak shock waves

Coulouvrat

Marchiano

2003

The Journal of the Acoustical Society of America

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Fresnel diffraction at a straight edge is revisited for nonlinear acoustics. Considering the penumbra region as a diffraction boundary layer governed by the KZ equation and its associated jump relations for shocks, similarity laws are established for the diffraction of a step shock, an "N" wave, or a periodic sawtooth wave. Compared to the linear case described by the well-known Fresnel functions, it is shown that weak shock waves penetrate more deeply into the shadow zone than linear waves. The thickness of the penumbra increases as a power of the propagation distance, power 1 for a step shock, or 3/4 for an N wave, as opposed to power 1/2 for a periodic sawtooth wave or a linear wave. This is explained considering the frequency spectrum of the waveform and its nonlinear evolution along the propagation, and is confirmed by direct numerical simulations of the KZ equation. New formulas for the Rayleigh/Fresnel distance in the case of nonlinear diffraction of weak shock waves by a large, finite aperture are deduced from the present study.

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“…The liquid flow and the pressure behind the shock front can be determined either using the Poisson' integral formula, as it was done by Cramer and Seebass (1978), or in a simpler way as the self-similar solution of the problem (11)-(13) with the additional conditions (27), (28).…”

Section: Intensity Of the Shock Wavementioning

confidence: 99%

Impact on the surface of a compressible liquid

Korobkin

1995

Shock Waves

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The propagation of a weak shock wave produced by an abrupt impact on a rigid boundary of a slightly compressible liquid is considered. Initially the liquid is at rest and occupies the region enclosed by a circular cylindrical surface. At some instant of time the points of this surface obtain velocities directed radially inwards. The velocity value depends on both the time and the angular coordinate and is small compared with the sound velocity. The liquid flow is assumed to be plane. A weak shock wave is formed under the impact and propagates to the centre of the liquid region. Initially the shock front is circular and bends later on due to non-linear effects. The form of the front is very complicated near the centre and depends on the distribution of the initial impact velocity. The evolution of the shock front at this stage is described by the geometrical acoustic theory. As a result a local focusing of separate parts of the shock wave is observed. The position of the shock wave and the liquid flow behind it are described within the framework of the transonic approximation within those zones. The structure of the liquid flow close to the focusing points is shown to be unaffected by any details of the external action and to be described by the solution of the non-linear boundary problem free from parameters.

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Focusing of weak shock waves at an arête

Cited by 37 publications

References 7 publications

Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection

Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection

Nonlinear Fresnel diffraction of weak shock waves

Impact on the surface of a compressible liquid

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