Abstracts - KTH Mechanics

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16 4 3.5 3 2.5 2 1.5 1 0.5 0 -1 -0.5 0 0.5 1 V z 100 1 0.01 0.0001 1e-06 1e-08 1e-10 ω 1 ω 0 /2 ω 2 ω 0 ω 0 +ω 1 3ω 0 /2 ε=0.09 ε=0.05 ε=0.01 ω 0 +ω 2 2ω 0 0 0.5 1 1.5 2 ω/ω0
The influence of centrifugal buoyancy in rotating convection. F. Marques ∗ ,J.M. Lopez † , O. Batiste ∗ and I. Mercader ∗ Rotating thermal convection is a paradigm problem that incorporates fundamental processes of great importance to atmospheric and oceanic circulations, as well as of astrophysical importance. We analyze rotating convection in a simple geometrical setting, consisting of an enclosed circular cylinder of depth-to-radius ratio Γ=d/R = 1.0, rotating at an angular rate ω, with the bottom endwall at temperature T0 +∆T and the top endwall at T0. The nondimensional parameters are the Coriolis parameter Ω=ωd 2 /ν, theRayleigh number Ra = αg∆Td 3 /νκ, the Prandtl number σ = ν/κ and the Froude number Fr = ω 2 R/g. Three-dimensional numerical simulations using a spectral code have been made in this geometry. Traditionally, density variation was only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. Inthislimit, corresponding to Fr → 0, the governing equation admit a trivial conduction solution, resulting in important simplifications of the problem. Linear stability analysis neglecting centrifugal buoyancy in an enclosed rotating cylinder finds that typically the onset of thermal convection from the conduction state is to a so-called wall mode which consists of alternating hot and cold thermals rising and decending in the cylinder boundary layer 1 . Experiments to test this linear theory have needed to be carefully designed in order to minimize the effects of the neglected centrifugal buoyancy, and have found good agreement with the theory for the onset of convection 2 . The presence of centrifugal buoyancy changes the problem in a fundamental manner. The buoyancy force has a radial component that destroys the horizontal translation invariance assumed in the unbounded theoretical treatments of the problem. Furthermore, the Z2 reflection symmetry about the cylinder mid-height is also destroyed. The centrifugal buoyancy drives a large scale circulation in which the cool denser fluid is centrifuged radially outward and the hot less dense fluid is centrifuged radially inward 3 . This large scale circulation exists for any ∆T = 0,andsothereis no trivial conduction state when the centrifugal buoyancy is incorporated. For small Froude numbers the transition from the base axisymetric state to 3D flow happens around Ra ≈ 7.500. We have found that for moderate Froude numbers (Fr ≥ 4) the centrifugal buoyance delays this transition to much larger Rayleigh numbers, Ra ≈ 50.000. At intermediate Fr the transition to 3D flow happens via four different Hopf bifurcations, with complex interactions between the bifurcated states,resulting in different coexisting branches of 3D solutions. How the centrifugal buoyancy and the gravitational buoyancy interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The main conclusion is that centrifugal buoyancy changes quantitatively and qualitatively the flow dynamics, and should not be neglected in rotating convection problems. ∗Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain. † Department of Mathematics and Statistics, Arizona State University, Tempe AZ 85287, USA. 1Goldstein et. al., J. Fluid Mech. 248, 583–604 (1993). 2Zhong et. al., Phys. Rev. Lett. 67, 2473–2476 (1991) and J. Fluid Mech. 249, 135–159 (1993). 3Barcilon &Pedlosky, J. Fluid Mech. 29, 673–690 (1967); Homsy & Hudson, J. Fluid Mech. 35, 33–52 (1969). 17
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Abstracts - KTH Mechanics

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