Nonlinear Stability of two-layer flows

Milewski, Paul A.; Tabak, Esteban G.; Turner, Cristina; Rosales, Ruben; Menzaque, Fernando

doi:10.4310/cms.2004.v2.n3.a5

Cited by 37 publications

(11 citation statements)

References 3 publications

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“…This is the value K should be set to in order to achieve global energy conservation, for periodic boundary conditions, when Q is unknown. We note that a similar expression was derived by Milewski et al, 25 for a two-layer shallow water model with a rigid lid, but the approximation K(t) = 0 was made for the analysis and computations. We will use the precise expression (25), meaning that K is set as a function of the global solution at time t. With this choice, (23) reduces to (18).…”

Section: Determining the Pftfm Free Parameter Based On Energy Conserv...mentioning

confidence: 86%

“…We note that a similar expression was derived by Milewski et al, 25 for a two-layer shallow water model with a rigid lid, but the approximation K(t) = 0 was made for the analysis and computations. We will use the precise expression (25), meaning that K is set as a function of the global solution at time t. With this choice, (23) reduces to (18). Using the proof that with this choice the PFTFM is equivalent to the TFM (see subsection 3.3), ( 24) can be written as [Qp]…”

Section: Determining the Pftfm Free Parameter Based On Energy Conserv...mentioning

confidence: 86%

“…22 In addition, the volumetric flow rate is often assumed to be constant in time, which excludes unsteady inflow boundary conditions, and which is incorrect in case of periodic boundary conditions. Similar two-equation models have been given by Keyfitz et al 23 and Etrati et al, 24 and a comparable approach has been taken for a two-layer shallow water model with rigid-lid assumption, 25 which describes flow in a closed channel. A different two-equation two-fluid model was derived by Jones and Prosperetti, 26 in which the volumetric flow constraint is used to eliminate both the pressure and the mass equations.…”

Section: Introductionmentioning

confidence: 94%

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Energy‐consistent formulation of the pressure‐free two‐fluid model

Buist

Sanderse

Dubinkina

et al. 2023

Numerical Methods in Fluids

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The pressure‐free two‐fluid model (PFTFM) is a recent reformulation of the one‐dimensional two‐fluid model (TFM) for stratified incompressible flow in ducts (including pipes and channels), in which the pressure is eliminated through intricate use of the volume constraint. The disadvantage of the PFTFM was that the volumetric flow rate had to be specified as an input, even though it is an unknown quantity in case of periodic boundary conditions. In this work, we derive an expression for the volumetric flow rate that is based on the demand for energy (and momentum) conservation. This leads to PFTFM solutions that match those of the TFM, justifying the validity and necessity of the derived choice of volumetric flow rate. Furthermore, we extend an energy‐conserving spatial discretization of the TFM, in the form of a finite volume scheme, to the PFTFM. We propose a discretization of the volumetric flow rate that yields discrete momentum and energy conservation. The discretization is extended with an energy‐conserving discretization of the source terms related to gravity acting in the streamwise direction. Our numerical experiments confirm that the discrete energy is conserved for different problem settings, including sloshing in an inclined closed tank, and a traveling wave in a periodic domain. The PFTFM solutions and the volumetric flow rates match the TFM solutions, with reduced computation time, and with exact momentum and energy conservation.

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Section: Determining the Pftfm Free Parameter Based On Energy Conserv...mentioning

confidence: 86%

Section: Determining the Pftfm Free Parameter Based On Energy Conserv...mentioning

confidence: 86%

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

Energy‐consistent formulation of the pressure‐free two‐fluid model

Buist

Sanderse

Dubinkina

et al. 2023

Numerical Methods in Fluids

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“…In such a mixed-type model, a crucial mathematical question is whether initially wave-like (hyperbolic) solutions remain inside the hyperbolic domain. It is known that this is always the case for two-layer flows with a rigid lid in the Boussinesq approximation [11], but that it fails in the non-Boussinesq case [12], where more restricted initial data needs to be imposed to ensure hyperbolicity.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear stability of two-layer shallow water flows with a free surface

Viríssimo

Milewski

2020

Proc. R. Soc. A.

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The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.

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“…The study of two-layer flows in open systems is a vast topic with a substantial literature, including works by [19,20,21,22,23,24] to name a few. However of interest here are two-layer closed systems, where the literature is more limited.…”

Section: Introductionmentioning

confidence: 99%