On the Hyperbolicity of Two-Layer Flows

Barros, Ricardo; Choi, Woo-Young

doi:10.1142/9789812835291_0009

Cited by 11 publications

(19 citation statements)

References 8 publications

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“…Once again, by examining the highest degree polynomial, we find that p = 0 and q = 0 are the asymptotes to curve (18). The existence of additional asymptotes depend on the number of real roots of the quadratic equation for v ≡ p/q, (19) Notice that if real roots exist for this equation, they must have opposite signs. Any line with slope 1 will then intersect the curve at least three times.…”

Section: A Taylor's Configurationmentioning

confidence: 88%

“…Then, using Fuller's root location criteria, [17][18][19] the stability diagram on the (α, J)-plane can be drawn, as shown in Figure 2, where the Richardson number J is defined for this configuration by…”

Section: A Stability Analysis For a Bounded Domainmentioning

confidence: 99%

“…We will see that this can only happen if r = ∞ and α → 0. Indeed, if we replace v = 1 in (19), one obtains (after some simplifications)…”

Section: Appendix B: Remark On the Taylor Configurationmentioning

confidence: 99%

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Elementary stratified flows with stability at low Richardson number

Barros

Choi

2014

Physics of Fluids

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We revisit the stability analysis for three classical configurations of multiple fluid layers proposed by Goldstein ["On the stability of superposed streams of fluids of different densities," Proc. R. Soc. A. 132, 524 (1931)], Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. A 132, 499 (1931)], and Holmboe ["On the behaviour of symmetric waves in stratified shear layers," Geophys. Publ. 24, 67 (1962)] as simple prototypes to understand stability characteristics of stratified shear flows with sharp density transitions. When such flows are confined in a finite domain, it is shown that a large shear across the layers that is often considered a source of instability plays a stabilizing role. Presented are simple analytical criteria for stability of these low Richardson number flows. C 2014 AIP Publishing LLC. [http://dx.

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Section: A Taylor's Configurationmentioning

confidence: 88%

Section: A Stability Analysis For a Bounded Domainmentioning

confidence: 99%

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Elementary stratified flows with stability at low Richardson number

Barros

Choi

2014

Physics of Fluids

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“…In this case, internal solitary waves with multi-humped profiles have been observed by Barros and Gavrilyuk [8]. Also, on the hyperbolicity of the two-layer shallow water equations, distinct features between the two configurations can be found [9]. These indicate that the free-surface effects could be worth to explore.…”

Section: Introductionmentioning

confidence: 55%

“…As a result, we can reduce the set of conditions in Theorem 1 to 0. The case of = 0 separating the two scenarios (two or four real solutions) corresponds to the special case for which the straight line described by (7) becomes tangent to the curve (5) (see [9]).…”

Section: A Geometrical Formulation Of the Problemmentioning

confidence: 99%

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

Barros

Choi

2009

Stud Appl Math

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We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin-Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.

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Air entrainment in transient flows in closed water pipes : A two-layer approach

Bourdarias¹,

Ersoy²,

Gerbi³

2013

ESAIM: M2AN

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In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme in which a special treatment for the "missing" boundary condition is performed. Several numerical tests on closed water pipes are performed and the impact of the loss of hyperbolicity is discussed and illustrated. Finally, we make a numerical study of the order of the kinetic method in the case where the system is mainly non hyperbolic. This provides a useful stability result when the spatial mesh size goes to zero.

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On the Hyperbolicity of Two-Layer Flows

Cited by 11 publications

References 8 publications

Elementary stratified flows with stability at low Richardson number

Elementary stratified flows with stability at low Richardson number

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

Air entrainment in transient flows in closed water pipes : A two-layer approach

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