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A higher-order analysis of natural convection in a corner

Untersuchung freier Konvektion höherer Ordnung in einer Ecke

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Abstract

A study of natural convection flow in a right angled corner formed by a semi-infinite vertical plate, which is maintained at the ambient temperature, and a semi-infinite horizontal plate, which is prescribed with a uniform heat flux, is carried out for moderately large values of the Grashof number by the method of matched asymptotic expansions. Higher-order corrections are found for the velocity and temperature fields as well as for the heat transfer and skin friction coefficients. The interaction between the two boundary-layers, which form on the vertical and horizontal plates, takes place through an isothermal outer flow. Eigenvalues and their corresponding eigenfunctions which are associated with the inner expansions have been sought. We are able to continue the solution up to the contribution played by the first eigenvalue and to uniquely find the first eigensolution. Numerical results have been obtained for a wide range of values of the Prandtl number, σ, but the results are only presented for σ=0.72 (air) and 6.7 (water). It is found that higher-order corrections to the classical boundary-layer theory are quite significant even for Grashof numbers of order 109.

Zusammenfassung

Eine Untersuchung über freie Konvektionsströmung in einer rechtwinkligen Ecke, die aus einer horizontalen und vertikalen, halbunendlichen Platte gebildet wird, wurde für ziemlich hohe Werte der Grashofzahl mit der asymptotisch, erwartungstreuen Schätzung durchgeführt. Bei der vertikalen Platte wird die Umgebungstemperatur konstant gehalten und die Horizontale Platte steht unter gleichmäßigem Wärmefluß. Für die Geschwindigkeits-und Temperaturfelder sind gleich gute Verbesserungen höherer Ordnung gefunden worden, wie für die Wärmeübertragung und den Oberflächenreibungskoeffizient. Die gegenseitige Beeinflussung der beiden Grenzschichten, die sich auf der horizontalen und vertikalen Platte bilden, entstand durch eine isotherme Außenströmung. Eigenwerte und ihre darauf bezogenen Eigenfunktionen, die mit der Expansion verbunden sind, wurden ermittelt. Wir sind in der Lage, die Lösung durch eine Approximation hoher Ordnung zu ersetzen und dadurch eine einzige Eigenlösung zu finden. Die numerischen Ergebnisse sind für weite Bereiche der Prandtlzahl, σ erhalten worden. Dargestellt wurden jedoch nur die für σ=0,72 (Luft) und 6,7 (Wasser). Es wurde herausgefunden, daß die Verbesserungen höherer Ordnung für die klassische Grenzschichttheorie sehr bedeutend sind, besonders für die Grashofzahlen der Ordnung 109.

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Abbreviations

A i, Āi :

constants

C 1 :

constant defined in Eq. (34)

C I f ,C II f :

skin friction coefficients

c p :

specific heat of fluid

Gr :

Grashof number

Gr x, Gry :

local Grashof numbers

g :

acceleration due to gravity

h w :

heat transfer at the horizontal plate

k :

thermal conductivity

L :

characteristic length of the plates

Nu :

local Nusselt number defined in Eq. (39)

q w :

heat flux at the horizontal plate

Q :

total heat transfer from the horizontal plate defined in Eq. (32)

r :

polar radial coordinate

T :

temperature

T w :

plate temperature

T ∞ :

ambient temperature

u, v :

velocity components alongx andy axes, respectively

x, y :

Cartesian coordinates

X, Y :

inner variables

α i,\(\bar \alpha _i \) :

constant

β :

coefficient of thermal expansion

γ :

constant

η,\(\bar \eta \) :

similarity variables

θ :

vorticity

ε x, εy :

perturbation parameters

ϑ :

angular coordinate

μ :

dynamic viscosity

ν :

kinematic viscosity

ϱ :

density

φ :

non-dimensional temperature

ψ :

stream function

σ :

Prandtl number

τ w :

skin friction

∧:

quantities associated with dimensional variables

∼:

quantities associated with outer flow

-:

quantities associated with vertical boundary-layer

\:

differentiation with respect to η or\(\bar \eta \)

BL:

boundary-layer

o:

classical boundary-layer

w:

conditions at the wall

∞:

ambient condition

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Authors and Affiliations

  1. Department of Applied Mathematical Studies, The University of Leeds, LS2 9JT, Leeds, West Yorkshire, England

    D. B. Ingham

  2. Faculty of Mathematics, University of Cluj, R-3400, Cluj, CP 253, Romania

    I. Pop

Authors
  1. D. B. Ingham
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  2. I. Pop
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Ingham, D.B., Pop, I. A higher-order analysis of natural convection in a corner. Wärme- und Stoffübertragung 26, 289–298 (1991). https://doi.org/10.1007/BF01590001

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  • DOI: https://doi.org/10.1007/BF01590001

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