Rigorous Bounds on Scaling Laws in Fluid Dynamics

Otto, Félix; Pottel, Steffen; Nobili, Camilla

doi:10.1007/978-3-319-67600-5_3

Cited by 7 publications

(10 citation statements)

References 8 publications

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“…Interestingly, rigorous upper bounds on heat transport are sensitive to the boundary conditions, being bounded by 1/2 in the no-slip case [23] and by 5/12 in the 2D free-slip case [43,44]. We also mention that substantial effort has been devoted to determining the effect of P r on the heat transport [45][46][47][48][49][50]. Finally, the Grossmann-Lohse (GL) framework [19,20] introduces nonlinear relationships for N u (Ra, P r) and Re (Ra, P r) with six free parameters that have been fit to experimental data [42].…”

Section: Introductionmentioning

confidence: 99%

Coherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-Bénard Convection

Kooloth,

Sondak,

Smith

2020

Preprint

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For two-dimensional (2D) Rayleigh-Bénard convection, classes of unstable, steady solutions were previously computed using numerical continuation [1,2]. The 'primary' steady solution bifurcates from the conduction state at Ra ≈ 1708, and has a characteristic aspect ratio (length/height) of approximately 2. The primary solution corresponds to one pair of counterclockwise-clockwise convection rolls with a temperature updraft in between and an adjacent downdraft on the sides. By adjusting the horizontal length of the domain, [1, 2] also found steady, maximal heat transport solutions, with characteristic aspect ratio less than 2 and decreasing with increasing Ra. Compared to the primary solutions, optimal heat transport solutions have modifications to boundary layer thickness, the horizontal length scale of the plume, and the structure of the downdrafts. The current study establishes a direct link between these (unstable) steady solutions and transition to turbulence for P r = 7 and P r = 100. For transitional values of Ra, the primary and optimal-heattransport solutions both appear prominently in appropriately-sized sub-fields of the time-evolving temperature fields. For Ra beyond transitional, our data analysis shows persistence of the primary solution for P r = 7, while the optimal heat transport solutions are more easily detectable for P r = 100. In both cases P r = 7 and P r = 100, the relative prevalence of primary and optimal solutions is consistent with the N u vs. Ra scalings for the numerical data and the steady solutions.

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Section: Introductionmentioning

confidence: 99%

Coherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-Bénard Convection

Kooloth,

Sondak,

Smith

2020

Preprint

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“…, leading to Nu Ra 1 2 . Therefore the localization method is equivalent to the background field method with the choice (3.8).…”

Section: Story Of the Background Field Methods In Five Actsmentioning

confidence: 99%

“…On the other hand, for the choice L s ∼ Ra α and Pr ≥ Ra if 0 ≤ α ≤ 1 24 . If we compare this bound for small slip-lengths with the upper bound for no-slip boundary the existence of a flow which is not buoyancy driven but nevertheless realizes Nu ∼ Ra 1 2 . Another important problem to study is the role of rough boundaries in the bounds.…”

Section: Boundary Conditionsmentioning

confidence: 98%

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The role of boundary conditions in scaling laws for turbulent heat transport

Nobili¹

2021

Preprint

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In most results concerning bounds on the heat transport in the Rayleigh-Bénard convection problem no-slip boundary conditions for the velocity field are assumed. Nevertheless it is debatable, whether these boundary conditions reflect the behavior of the fluid at the boundary. This problem is important in theoretical fluid mechanics as well as in industrial applications, as the choice of boundary conditions has effects in the description of the boundary layers and its properties. In fact, different boundary conditions may inhibit or enhance heat transport. This review presents a selection of contributions in the theory of rigorous bounds on Nusselt number, distinguishing and comparing the results for no-slip, free-slip and Navier-slip boundary conditions.

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“…There is a large amount of mathematical literature devoted to the asymptotic behaviour of solutions to the Rayleigh-Bénard problem in the simplified incompressible framework, where the system (1.1)-(1.3) is replaced by the Oberbeck-Boussinesq approximation, see Constantin et al [11], Foias, Manley and Temam [22], Cao et al [8] and the references therein. Recently, the problem motivated a series of studies by Otto et al [10], [30], [34] concerning the associated scaling laws.…”

Section: Introductionmentioning

confidence: 99%

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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Rigorous Bounds on Scaling Laws in Fluid Dynamics

Cited by 7 publications

References 8 publications

Coherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-Bénard Convection

Coherent Solutions and Transition to Turbulence in Two-Dimensional Rayleigh-Bénard Convection

The role of boundary conditions in scaling laws for turbulent heat transport

The Rayleigh Benard problem for compressible fluid flows

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