Numerical Linear Stability Analysis for Compressible Fluids

Bormann, Andreas S.

doi:10.1007/3-540-27907-5_5

Cited by 6 publications

(9 citation statements)

References 7 publications

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“…In view of the hypotheses (2.10), (2.12), p(̺, ϑ) ≈ ̺ 5 3 + ϑ 4 . As already pointed out, the exponent γ = 5 3 is critical in the simplified isentropic case. To handle this problem we use the fact that (i) the gravitational force acting on the fluid is of potential type, and (ii) the entropy satisfies the Third law of thermodynamics, notably (2.15).…”

Section: Resultsmentioning

confidence: 66%

The Rayleigh Benard problem for compressible fluid flows

Feireisl¹,

Świerczewska-Gwiazda²

2021

Preprint

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We consider the physically relevant fully compressible setting of the Rayleigh-Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to the gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor A. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure -a stationary statistical solution sitting on A. In addition, the Birkhoff-Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to A a.s. with respect to the invariant measure.

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Section: Resultsmentioning

confidence: 66%

The Rayleigh Benard problem for compressible fluid flows

Feireisl¹,

Świerczewska-Gwiazda²

2021

Preprint

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“…Moreover our critical values increase with g while the ones from the Rayleigh parameter decrease with g. One explanation to this discrepancy is that the Rayleigh critical value is obtained in the Boussinesq approximation that is assuming that the fluid is non-compressible. Some numerical analysis of the linearized Navier Stokes equations found that the critical temperatures for the onset of convection, increase with g [4]. In fact, the increment is argumented to be given by gαT L 4 y /c p .…”

Section: Computer Simulation Results: Global Magnitudesmentioning

confidence: 99%

“…177 for g = 0 5, 10 and 15 (from left to right). At each figure we plot the points with T0 = 1, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 from bottom to top. Lines are phenomenological fits of sixth order polynomials: w = a0 + a1y + .…”

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confidence: 99%

Rayleigh-Benard convection in a hard disk system

Garrido

2018

Preprint

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We do a generic study of the behavior of a hard disk system under the action of a thermal gradient in presence of an uniform gravity field. We observe the conduction-convection transition and measure the main system observables and fields as the thermal current, global pressure, velocity field, temperature field,... We can highlight two of the main results of this overall work: (1) for large enough thermal gradients and a given gravity, we show that the hydrodynamic fields (density, temperature and velocity) have a natural scaling form with the gradient. And (2) we show that local equilibrium holds if the mechanical pressure and the thermodynamic one are not equal, that is, the Stoke's assumption does not hold in this case. Moreover we observe that the best fit to the data is obtained when the bulk viscosity depends on the mechanical pressure.

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“…As pointed out, the key point of the analysis is the Levinson dissipativity or the existence of a universal bounded absorbing set for the "monoatomic" equation of state introduced in [16, Chapters 1,2]. This is rather surprising as this constitutive equation can be seen as a temperature dependent counterpart of the isentropic pressure law p( ) ≈ γ , with γ = 5 3 . Note that for the isentropic model, the existence of a bounded absorbing set is known only if γ > 5 3 , see [19], whereas the limit case γ = 5 3 requires smallness of the total mass of the fluid, see Wang and Wang [37].…”

Section: Introductionmentioning

confidence: 99%

“…This is rather surprising as this constitutive equation can be seen as a temperature dependent counterpart of the isentropic pressure law p( ) ≈ γ , with γ = 5 3 . Note that for the isentropic model, the existence of a bounded absorbing set is known only if γ > 5 3 , see [19], whereas the limit case γ = 5 3 requires smallness of the total mass of the fluid, see Wang and Wang [37]. Moreover, uniform boundedness of global trajectories for the Navier-Stokes-Fourier system is a delicate issue.…”

Section: Introductionmentioning

confidence: 99%

The Rayleigh–Bénard Problem for Compressible Fluid Flows

Feireisl

Świerczewska-Gwiazda

2023

Arch Rational Mech Anal

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We consider the physically relevant fully compressible setting of the Rayleigh–Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor $$\mathcal {A}$$ A . The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure—a stationary statistical solution sitting on $$\mathcal {A}$$ A . In addition, the Birkhoff–Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to $$\mathcal {A}$$ A a.s. with respect to the invariant measure.

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Numerical Linear Stability Analysis for Compressible Fluids

Cited by 6 publications

References 7 publications

The Rayleigh Benard problem for compressible fluid flows

The Rayleigh Benard problem for compressible fluid flows

Rayleigh-Benard convection in a hard disk system

The Rayleigh–Bénard Problem for Compressible Fluid Flows

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