4 results
A playground for compressible natural convection with a nearly uniform density
- Thierry Alboussière, Jezabel Curbelo, Fabien Dubuffet, Stéphane Labrosse, Yanick Ricard
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- Journal:
- Journal of Fluid Mechanics / Volume 940 / 10 June 2022
- Published online by Cambridge University Press:
- 05 April 2022, A9
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In the quest to understand the basic universal features of compressible convection, one would like to disentangle genuine consequences of compression from spatial variations of transport properties. For instance, one may choose to consider a fluid with uniform dynamic viscosity, but, then, compressible effects will generate a density gradient and consequently the kinematic viscosity will not be uniform. In the present work, we consider a very peculiar equation of state, whereby entropy is solely dependent on density, so that a nearly isentropic fluid domain is nearly isochoric. Within this class of equations of state, there is a thermal adiabatic gradient and a key property of compressible convection is still present, namely its capacity to viscously dissipate a large fraction of the thermal energy involved, of the order of the well-named dissipation number. In the anelastic approximation, under the assumption of an infinite Prandtl number, the number of governing parameters can be brought down to two, the Rayleigh number and the dissipation number. This framework is proposed as a playground for compressible convection, an opportunity to extend the vast corpus of theoretical analyses on the Oberbeck–Boussinesq equations regarding stability, bifurcations or the determination of upper bounds for the turbulent heat transfer. Here, in a two-dimensional geometry, we concentrate on the structure of numerical solutions. For all Rayleigh numbers, a change in the vertical temperature profile is observed in the range of dissipation number between $0$ and less than $0.4$, associated with the weakening of ascending plumes. For larger dissipation numbers, the heat flux dependence on this number is found to be well predicted by Malkus's model of critical layers. For dissipation numbers of order unity, and large Rayleigh numbers, dissipation becomes related to the entropy heat flux at each depth, so that the vertical dissipation profile can be predicted, and so does the total ratio of dissipation to convective heat flux.
The effects of a Robin boundary condition on thermal convection in a rotating spherical shell
- Thibaut T. Clarté, Nathanaël Schaeffer, Stéphane Labrosse, Jérémie Vidal
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- Journal:
- Journal of Fluid Mechanics / Volume 918 / 10 July 2021
- Published online by Cambridge University Press:
- 17 May 2021, A36
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Convection in a spherical shell is widely used to model fluid layers of planets and stars. The choice of thermal boundary conditions in such models is not always straightforward. To understand the implications of this choice, we report on the effects of the thermal boundary condition on thermal convection, in terms of instability onset, fully developed transport properties and flow structure. We impose a Robin boundary condition, enforcing linear coupling between the temperature anomaly and its radial derivative, with the Biot number ${{\textit {Bi}}}$ as a proportionality factor in non-dimensional form. Varying ${{\textit {Bi}}}$ allows us to transition from fixed temperature for ${{\textit {Bi}}}=+\infty$, to imposed heat flux for ${{\textit {Bi}}}=0$. We find that the onset of convection is only affected by ${{\textit {Bi}}}$ in the non-rotating case. Far from onset, considering an effective Rayleigh number and a generalized Nusselt number, we show that the Nusselt and Péclet numbers follow standard universal scaling laws, independent of ${{\textit {Bi}}}$ in all cases considered. However, for the non-rotating limit, the large-scale flow structure keeps the signature of the boundary condition with more vigorous large scales for smaller ${{\textit {Bi}}}$, even though the global heat transfer and kinetic energy are the same. For all practical purposes, the Robin condition can be safely replaced by a fixed flux when ${{\textit {Bi}}} \lesssim 0.03$ and by a fixed temperature for ${{\textit {Bi}}}\gtrsim 30$. For turbulent rapidly rotating convection, the thermal boundary condition does not seem to have any impact, once the effective numbers are considered and a reference temperature profile has been chosen.
Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations
- Jezabel Curbelo, Lucia Duarte, Thierry Alboussière, Fabien Dubuffet, Stéphane Labrosse, Yanick Ricard
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- Journal:
- Journal of Fluid Mechanics / Volume 873 / 25 August 2019
- Published online by Cambridge University Press:
- 26 June 2019, pp. 646-687
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We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.
Rayleigh–Bénard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries
- Stéphane Labrosse, Adrien Morison, Renaud Deguen, Thierry Alboussière
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- Journal:
- Journal of Fluid Mechanics / Volume 846 / 10 July 2018
- Published online by Cambridge University Press:
- 03 May 2018, pp. 5-36
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Solid-state convection can take place in the rocky or icy mantles of planetary objects, and these mantles can be surrounded above or below or both by molten layers of similar composition. A flow towards the interface can proceed through it by changing phase. This behaviour is modelled by a boundary condition taking into account the competition between viscous stress in the solid, which builds topography of the interface with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , and convective transfer of the latent heat in the liquid from places of the boundary where freezing occurs to places of melting, which acts to erase topography, with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}$ . The ratio $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ controls whether the boundary condition is the classical non-penetrative one ( $\unicode[STIX]{x1D6F7}\rightarrow \infty$ ) or allows for a finite flow through the boundary (small $\unicode[STIX]{x1D6F7}$ ). We study Rayleigh–Bénard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly nonlinear analyses. When both boundaries are phase-change interfaces with equal values of $\unicode[STIX]{x1D6F7}$ , a non-deforming translation mode is possible with a critical Rayleigh number equal to $24\unicode[STIX]{x1D6F7}$ . At small values of $\unicode[STIX]{x1D6F7}$ , this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, $\unicode[STIX]{x1D706}_{c}\sim 8\sqrt{2}\unicode[STIX]{x03C0}/3\sqrt{\unicode[STIX]{x1D6F7}}$ . Both modes lead to very efficient heat transfer, as expressed by the relationship between the Nusselt and Rayleigh numbers. When only one boundary is subject to a phase-change condition, the critical Rayleigh number is $\mathit{Ra}_{c}=153$ and the critical wavelength is $\unicode[STIX]{x1D706}_{c}=5$ . The Nusselt number increases approximately two times faster with the Rayleigh number than in the classical case with non-penetrative conditions, and the average temperature diverges from $1/2$ when the Rayleigh number is increased, towards larger values when the bottom boundary is a phase-change interface.