Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

Lipatov, I. I.; Teshukov, V. V.

doi:10.1023/b:flui.0000024816.05354.0f

Cited by 6 publications

(8 citation statements)

References 6 publications

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“…This makes it possible to construct piecewise-smooth solutions with discontinuities in the derivatives in the direction normal to characteristics. Following [14], we show that a weak discontinuity may exist only on the characteristics. Let = {(t, x, λ) | x = x(t), 0 ≤ λ ≤ 1} be the surface of a weak discontinuity, given by the equation x (t) = k(t, x).…”

Section: Propagation Of a Weak Discontinuitymentioning

confidence: 75%

“…Following , we show that a weak discontinuity may exist only on the characteristics. Let

Γ = {(t, x, λ) | x = x (t), 0 \leq λ \leq 1}

be the surface of a weak discontinuity, given by the equation

x^{'} (t) = k (t, x)

.…”

Section: Propagation Of a Weak Discontinuitymentioning

confidence: 78%

“…If the hyperbolicity conditions (14) hold, the equations of motion (5) can be transformed to the form (11), which includes derivatives only in the directions tangent to the characteristic surfaces. This makes it possible to construct piecewise-smooth solutions with discontinuities in the derivatives in the direction normal to characteristics.…”

Section: Propagation Of a Weak Discontinuitymentioning

confidence: 99%

“…A generalized method of characteristics for equations with operator coefficients [12,13] is suitable for theoretical study of wave propagation in a thin layer of viscous liquid in the long-wave approximation. This approach has already been used to study the wave propagation in a supersonic boundary layer within the framework of the model of viscous-inviscid interaction [14].…”

Section: Introductionmentioning

confidence: 99%

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Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Чесноков

Kovtunenko

2013

Stud Appl Math

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Nonlinear integrodifferential equations describing the propagation of disturbances in a thin layer of viscous liquid with free surface are studied. These equations admit solutions with weak discontinuities, which are located on the characteristics. The possibility of an unbounded increase in the amplitude of the weak discontinuity and the formation of the shock in the process of flow evolution is established. Differential balance laws approximating the integrodifferential model are proposed. These laws are used to perform numerical simulation of wave propagation in a fluid.

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Section: Propagation Of a Weak Discontinuitymentioning

confidence: 75%

“…Following , we show that a weak discontinuity may exist only on the characteristics. Let

Γ = {(t, x, λ) | x = x (t), 0 \leq λ \leq 1}

be the surface of a weak discontinuity, given by the equation

x^{'} (t) = k (t, x)

.…”

Section: Propagation Of a Weak Discontinuitymentioning

confidence: 78%

Section: Propagation Of a Weak Discontinuitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Чесноков

Kovtunenko

2013

Stud Appl Math

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“…However, the viscous regularization does not always prevent from the existence of discontinuous solutions. For example, in the theory of supersonic boundary layer, the viscosity is only present in the direction transverse to the main stream, and it is not sufficient to prevent from the shock formation [7]. Analogous results can be found in the theory of long waves in viscous shear flows down an inclined plane [8], and even in compressible flows of a non-viscous but heat-conductive gas [9].…”

Section: Introductionmentioning

confidence: 76%

Rankine–Hugoniot conditions for fluids whose energy depends on space and time derivatives of density

Gavrilyuk

Gouin

2020

Wave Motion

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By using the Hamilton principle of stationary action, we derive the governing equations and Rankine-Hugoniot conditions for continuous media where the specific energy depends on the space and time density derivatives. The governing system of equations is a time reversible dispersive system of conservation laws for the mass, momentum and energy. We obtain additional relations to the Rankine-Hugoniot conditions coming from the conservation laws and discuss the well-founded of shock wave discontinuities for dispersive systems.

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Kink stability of isothermal spherical self-similar flow revisited

Wang

2006

Gen Relativ Gravit

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The problem of kink stability of isothermal spherical self-similar flow in newtonian gravity is revisited. Using distribution theory we first develop a general formula of perturbations, linear or non-linear, which consists of three sets of differential equations, one in each side of the sonic line and the other along it. By solving the equations along the sonic line we find explicitly the spectrum, k, of the perturbations, whereby we obtain the stability criterion for the self-similar solutions. When the solutions are smoothly across the sonic line, our results reduce to those of Ori and Piran. To show such obtained perturbations can be matched to the ones in the regions outside the sonic line, we study the linear perturbations in the external region of the sonic line * E-mail: Anzhong Wang@baylor.edu † E-mail: Yumei Wu@baylor.edu; yumei@im.ufrj.br 1 (the ones in the internal region are identically zero), by taking the solutions obtained along the line as the boundary conditions. After properly imposing other boundary conditions at spatial infinity, we are able to show that linear perturbations, satisfying all the boundary conditions, exist and do not impose any additional conditions on k. As a result, the complete treatment of perturbations in the whole spacetime does not alter the spectrum obtained by considering the perturbations only along the sonic line.

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Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

Cited by 6 publications

References 6 publications

Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Rankine–Hugoniot conditions for fluids whose energy depends on space and time derivatives of density

Kink stability of isothermal spherical self-similar flow revisited

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