Finite-amplitude convection in the presence of finitely conducting boundaries

Westerburg, M.; Busse, Friedrich H.

doi:10.1017/s002211200000327x

Cited by 12 publications

(9 citation statements)

References 14 publications

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“…The balloons are shifted towards smaller wavenumbers with decreased conductivity of the slabs. As was found for cases of infinitely thick slabs by Westerburg & Busse (2001), the balloons shrink towards the critical point and vanish for sufficiently small K. The rolls are replaced by squares at K = 0.2356 (calculated using the weakly nonlinear theory). The results with H = 0.1 suggest that cases of thin poorly conducting slabs might remain stable at even larger Rayleigh numbers.…”

Section: Stability Regionsmentioning

confidence: 64%

“…The Nusselt number decreases significantly with increased thickness of the slabs. It follows from the discussion above that the heat transfer in the case of infinitely thick slabs by Westerburg & Busse (2001) is not at all influenced by the convection in the fluid layer. From (3.3), it is realized that the change of their Nusselt number is related entirely to the change of their temperature scale, namely the difference between the mean temperatures at the fluid boundaries, which depends not only on the prescribed outer temperature difference, T , but also on the steady solution itself.…”

Section: Steady Solutions and Their Heat Transfermentioning

confidence: 94%

“…The weakly nonlinear perturbation techniques are restricted to small Rayleigh numbers and degenerates for slab conductivities corresponding to the border between stable rolls and squares. For cases of infinitely thick heat-conducting slabs, fully nonlinear solutions and their heat transfer and stability regions were reported by Westerburg & Busse (2001). They found that the stability region shrinks onto the critical point for onset of convection as the slab the conductivity decreases.…”

Section: Introductionmentioning

confidence: 99%

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Stability of squares and rolls in Rayleigh–Bénard convection in an infinite-Prandtl-number fluid between slabs

Holmedal

2005

J. Fluid Mech.

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Nonlinear solutions in the form of squares and rolls are investigated for Rayleigh–Bénard convection in an infinite-Prandtl-number fluid enclosed between two symmetric slabs. It is found that the heat transfer depends strongly on the thickness and thermal conductivity of the slabs, but hardly on the planform of convection. Examples of stability regions of rolls are calculated, showing that for certain slab selections, rolls remain stable at even larger Rayleigh numbers than with fixed temperatures at the boundaries. The region of stable squares is restricted by a zigzag and a long-wavelength cross-roll instability in addition to a new three-dimensional instability. As the slab conductivity is increased, the stability region of the squares shrinks onto a point located well above the critical point for the onset of convection. For a small range of slab conductivities, stability regions for squares and rolls both exist for the same set-up. In the present calculations, the regions never overlap. An example, where both patterns are stable at the same Rayleigh number, provides an explanation for the co-existence of rolls and squares where transparent slabs with a low thermal conductivity were applied.

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Section: Stability Regionsmentioning

confidence: 64%

Section: Steady Solutions and Their Heat Transfermentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Stability of squares and rolls in Rayleigh–Bénard convection in an infinite-Prandtl-number fluid between slabs

Holmedal

2005

J. Fluid Mech.

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“…A basic consideration in investigating the influence of poorly conducting boundaries on convection is the choice of thermal boundary conditions (BCs). Numerous researchers have concentrated solely on idealized fixed flux conditions corresponding to perfectly insulating boundaries (for instance ; Otero et al (2002); Verzicco & Sreenivasan (2008); Johnston & Doering (2009)), while other studies (including Sparrow et al (1964); Gertsberg & Sivashinsky (1981); Westerburg & Busse (2001)) have imposed more general mixed conditions of fixed Biot number η at the fluid boundaries. Note, though, that the Biot number in general depends on the horizontal "disturbance" wave number in the plates (Normand et al 1977, Section V.C.1).…”

Section: Effect Of Imperfectly Conducting Plates Bounding the Fluidmentioning

confidence: 99%

Bounds on Rayleigh–Bénard convection with imperfectly conducting plates

Wittenberg

2010

J. Fluid Mech.

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We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Bénard convection on analytical upper bounds on convective heat transport. We model imperfectly conducting bounding plates in two ways: using idealized mixed thermal boundary conditions of constant Biot number η, continuously interpolating between the previously studied fixed temperature (η = 0) and fixed flux (η = ∞) cases; and by explicitly coupling the evolution equations in the fluid in the Boussinesq approximation through temperature and flux continuity to identical upper and lower conducting plates. In both cases, we systematically formulate a bounding principle and obtain explicit upper bounds on the Nusselt number Nu in terms of the usual Rayleigh number Ra measuring the average temperature drop across the fluid layer, using the "background method" developed by Doering and Constantin. In the presence of plates, we find that the bounds depend on σ = d/λ, where d is the ratio of plate to fluid thickness and λ is the conductivity ratio, and that the bounding problem may be mapped onto that for Biot number η = σ. In particular, for each σ > 0, for sufficiently large Ra (depending on σ) we show that Nu ≤ c(σ)R 1/3 ≤ CRa 1/2 , where C is a σ-independent constant, and where the control parameter R is a Rayleigh number defined in terms of the full temperature drop across the entire plate-fluid-plate system. In the Ra → ∞ limit, the usual fixed temperature assumption is a singular limit of the general bounding problem, while fixed flux conditions appear most relevant to the asymptotic Nu-Ra scaling even for highly conducting plates.

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“…When the convection flow starts, it distorts the originally horizontal isotherms. Since, this deviation from the basic state occurs in the fluid layer as well as in the wall boundary, the wavelength of the convection pattern becomes larger with decreasing ξ[36]. From mathematical point of view, the decrease of Ra c is caused by the weakening of the thermal boundary condition(29) for θ as ξ decreases from 10 3 to 10 −3 .…”

mentioning

confidence: 96%

Weakly nonlinear analysis of Rayleigh-Bénard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects

Bouteraa

Nouar

2015

Phys. Rev. E

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Finite-amplitude thermal convection in a shear-thinning fluid layer between two horizontal plates of finite thermal conductivity is considered. Weakly nonlinear analysis is adopted as a first approach to investigate nonlinear effects. The rheological behavior of the fluid is described by the Carreau model. As a first step, the critical conditions for the onset of convection are computed as a function of the ratio ξ of the thermal conductivity of the plates to the thermal conductivity of the fluid. In agreement with the literature, the critical Rayleigh number Ra(c) and the critical wave number k(c) decrease from 1708 to 720 and from 3.11 to 0, when ξ decreases from infinity to zero. In the second step, the critical value α(c) of the shear-thinning degree above which the bifurcation becomes subcritical is determined. It is shown that α(c) increases with decreasing ξ. The stability of rolls and squares is then investigated as a function of ξ and the rheological parameters. The limit value ξ(c), below which squares are stable, decreases with increasing shear-thinning effects. This is related to the fact that shear-thinning effects increase the nonlinear interactions between sets of rolls that constitute the square patterns [M. Bouteraa et al., J. Fluid Mech. 767, 696 (2015)]. For a significant deviation from the critical conditions, nonlinear convection terms and nonlinear viscous terms become stronger, leading to a further diminution of ξ(c). The dependency of the heat transfer on ξ and the rheological parameters is reported. It is consistent with the maximum heat transfer principle. Finally, the flow structure and the viscosity field are represented for weakly and highly conducting plates.

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Finite-amplitude convection in the presence of finitely conducting boundaries

Cited by 12 publications

References 14 publications

Stability of squares and rolls in Rayleigh–Bénard convection in an infinite-Prandtl-number fluid between slabs

Stability of squares and rolls in Rayleigh–Bénard convection in an infinite-Prandtl-number fluid between slabs

Bounds on Rayleigh–Bénard convection with imperfectly conducting plates

Weakly nonlinear analysis of Rayleigh-Bénard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects

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