An Application of 3-D Kinematical Conservation Laws: Propagation of a 3-D Wavefront

Arun, Koottungal Revi; Lukáčová-Medviďová, Mária; Prasad, Phoolan; Rao, S. V. Raghurama

doi:10.1137/080732742

Cited by 7 publications

(9 citation statements)

References 24 publications

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“…It has to be noted that the nonlinear theory of Choquet-Bruhat [9] is valid over a distance smaller than the smaller of the radii of curvatures of the front at t = 0. However, for WNLRT and SRT, the distance L could be far beyond the caustic region as evident from the numerical results presented in [3,7,18,24].…”

Section: Governing Equations Of 3-d Srtmentioning

confidence: 88%

“…Hence, the KCL is ideally suited to study the evolution of a surface having kink type of singularities. However, the KCL being purely a geometric result 3 , it forms an incomplete system of equations and additional closure relations are necessary to get a completely determined set of equations. When Ω t is a weakly nonlinear wavefront in a polytropic gas [20], the KCL system is closed by a single equation representing the conservation of total energy in a ray tube; see [3,4] for more details.…”

Section: Introductionmentioning

confidence: 99%

“…However, the KCL being purely a geometric result 3 , it forms an incomplete system of equations and additional closure relations are necessary to get a completely determined set of equations. When Ω t is a weakly nonlinear wavefront in a polytropic gas [20], the KCL system is closed by a single equation representing the conservation of total energy in a ray tube; see [3,4] for more details. Throughout this paper we shall refer to the resulting complete system of conservation laws, governing the evolution of a nonlinear wavefront, as KCL based weakly nonlinear ray theory (WNLRT), or briefly 3-D WNLRT in later sections.…”

Section: Introductionmentioning

confidence: 99%

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Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws

Arun,

Prasad

2017

Preprint

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In this paper we present a mathematical theory and a numerical method to study the propagation of a three-dimensional (3-D) weak shock front into a polytropic gas in a uniform state and at rest, though the method can be extended to shocks moving into nonuniform flows. The theory is based on the use of 3-D kinematical conservation laws (KCL), which govern the evolution of a surface in general and a shock front in particular. The 3-D KCL, derived purely on geometrical considerations, form an under-determined system of conservation laws. In the present paper the 3-D KCL system is closed by using two appropriately truncated transport equations from an infinite hierarchy of compatibility conditions along shock rays. The resulting governing equations of this KCL based 3-D shock ray theory, leads to a weakly hyperbolic system of eight conservation laws with three divergence-free constraints. The conservation laws are solved using a Godunov-type central finite volume scheme, with a constrained transport technique to enforce the constraints. The results of extensive numerical simulations reveal several physically realistic geometrical features of shock fronts and the complex structures of kink lines formed on them. A comparison of the results with those of a weakly nonlinear wavefront shows that a weak shock front and a weakly nonlinear wavefront are topologically same. The major important differences between the two are highlighted in the contexts of corrugational stability and converging shock fronts.

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Section: Governing Equations Of 3-d Srtmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws

Arun,

Prasad

2017

Preprint

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“…Since these discontinuities can appear not only on a shock front but on any moving surface Ω t , we call them kinks. The 2-D KCL and 3-D KCL have been extensively used to solve many practical problems, see [2,4,5,6,7,8,9] and the older references in these. Hence, it is worth developing the KCL theory for a space of arbitrary dimensions for a mathematical completeness.…”

Section: Introductionmentioning

confidence: 99%

Kinematical conservation laws in a space of arbitrary dimensions

Prasad

2016

Indian J Pure Appl Math

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In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ω t in d-dimensional x-space, where x = (x 1 , x 2 , . . . , x d ) ∈ R d . The KCL are derived in a specially defined ray coordinates (ξ = (ξ 1 , ξ 2 , . . . , ξ d−1 ), t), where ξ 1 , ξ 2 , . . . , ξ d−1 are surface coordinates on Ω t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω t when Ω t is a curve in R 2 and is a curve on Ω t when it is a surface in R 3 . Across a kink the normal n to Ω t and normal velocity m on Ω t are discontinuous.

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“…They reduce to ν 1 and ν 2 in equation (8.2) of [1] when u and v are orthogonal. The analytical expressions for nonzero eigenvalues of KCL are essential to determine a suitable CFL stability condition in the numerical integration of conservation laws of KCL based WNLRT, see [2] for more details.…”

Section: Calculation Of the Nonzero Eigenvaluesmentioning

confidence: 99%

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

Arun

Prasad

2010

Applied Mathematics and Computation

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Abstract. Kinematical conservation laws (KCL) is a system of conservation laws governing the evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [1] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7×7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained in [1]. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.

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An Application of 3-D Kinematical Conservation Laws: Propagation of a 3-D Wavefront

Cited by 7 publications

References 24 publications

Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws

Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws

Kinematical conservation laws in a space of arbitrary dimensions

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

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