finite prandtl number convection catherine a. hier-majumder department of terrestrial magnetism...
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Finite Prandtl Number Convection
Catherine A. Hier-MajumderDepartment of Terrestrial Magnetism
Carnegie Institution of Washington
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Collaborators
• University of Minnesota, Minneapolis:David A. Yuen
• Université de Montréal:Alain P. Vincent
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Purpose of Study
• Many fluids with large, but finite Pr numbers (Pr~104) in geophysical systems (mushy ice, Mg-magmas)
• Usually approximated as infinite Pr, same as mantle (Pr~1025)
• Pr~104 fluids still have significant inertial behaviors missed by infinite Pr approximation
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Outline
• Equations of Finite Prandtl Convection
• 2-D Finite Prandtl Plumes
• Infinite vs. finite Prandtl plumes
• 3-D Finite Prandtl Plumes
• Conclusions
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Finite Prandtl Convection Equations
0=•∇ v
vev 21
∇++−∇= zRaTpDtD
Pr
TDt
DT 2∇=
Nondimensionalized by
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Finite Prandtl Convection Equations
0=•∇ v
vev 2∇++−∇=
RaPr
TpDtD
z
TRaPrDt
DT 2∇=
zgLTU )(by onalizedNondimensi Δ≡ α
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Ra = 107,Pr = 7Ra = 107,Pr = 7
Nondimensional Temperature
Red is hot T > 1Blue is cold T ~ 0
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Ra = 107,Pr = 7
Nondimensional Temperature
Red is hot T > 1Blue is cold T ~ 0
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Ra = 108,Pr = 2 x 104
Nondimensional Temperature
Red is hot T > 1Blue is cold T ~ 0
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Ra = 108,Pr = ∞
Nondimensional Temperature
Red is hot T > 1Blue is cold T ~ 0
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Ra = 108,Pr = 2 x 104
t = t1
Ra = 108,Pr = infinityt = 2t1
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Implications for Plume Heat Flux
• Heat flux = vz * T • Real heat fluxes are likely to
be at least 2X larger than those calculated using infinite Pr approximation
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Conclusions
• Prandtl number 104 plumes have significantly different behaviors from infinite Prandtl number plumes:
o Hotter
o Grow faster
• Differences increase with Rayleigh number
• This may have significant effects on heat fluxes in icy satellites, very hot bodies like Io, and magma ocean phases of the early terrestrial planets and moons
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Ra = 6.57 x 105, Pr = 10
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Questions?