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Improved perturbation solutions for laminar natural convection on a vertical cylinder

Verbesserte Störungslösungen für laminare freie Konvektion am senkrechten Zylinder

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Abstract

The method of extended perturbation series is applied to solve for laminar natural convection from an isothermal, thin vertical cylinder. The series in terms of the transverse curvature parameterξ extended to five terms and is subsequently improved by applying the Shanks transformation twice. The validity of the solution is extended up toξ=10 and possibly even beyond. Up toξ=10, the results for wall shear as well as the local and average Nusselt numbers agree very closely with those of local nonsimilarity and finite difference solutions. The ease of computation coupled with high accuracy makes the present approach far more attractive than the currently popular local nonsimilarity and finite difference methods. Its success with the present problem should motivate applications to a host of nonsimilar boundary layer flows.

Zusammenfassung

Die Methode der erweiterten Störungsserien wird auf die laminare freie Konvektion am isothermen senkrechten dünnen Zylinder angewendet. Die Serien in Ausdrücken des Krümmungsparametersξ werden auf 5 Terme ausgedehnt und weiter durch doppelte Auswertung der Shank-Transformation verbessert. Die Lösung gilt mindestens bisξ=10, vielleicht sogar weiter. Bisξ=10 stimmen die Lösungen für die Wandschubspannung und die örtliche und mittlere Nußelt-Zahl gut überein mit jenen, die auf der örtlichen Nicht-Ähnlichkeit und finiten Differenzen beruhen. Die leichte Berechenbarkeit und die hohe Genauigkeit machen diesen Lösungsweg attraktiver als die heute populären Verfahren der örtlichen Nicht-Ähnlichkeit und der finiten Differenzen. Der hier aufgezeigte Erfolg sollte zur Anwendung auf nicht-ähnliche Grenzschichtströmungen motivieren.

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Abbreviations

e :

operator defined by Eq. (31)

F :

dimensionless stream function

F n :

nth order approximation for dimensionless stream function

g :

acceleration of gravity

h :

local heat transfer coefficient

¯h :

average heat transfer coefficient

k :

thermal conductivity

Nu x :

local Nusselt number,h x/k

\(\begin{array}{*{20}c} {\_\_} \\ {Nu_x } \\ \end{array} \) :

average Nusselt number,¯h x/k

Pr:

Prandtl number,v/α

r :

radial coordinate

r0 :

radius of cylinder

T :

temperature

T w :

wall temperature of cylinder

T ∞ :

ambient temperature

u,ν :

velocity components

x :

axial coordinate

α :

thermal diffusivity

β :

coefficient of thermal expansion

η :

similarity variable, Eq. (7)

θ :

dimensionless temperature, (T−T ∞)/(T w —T ∞ )

θ n :

nth order approximation for dimensionless temperature

ν :

kinematic viscosity

ξ :

transverse curvature parameter, Eq. (8)

ψ :

stream function

fp:

flat plate

cyl:

cylinder

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Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering, P. O. Box 800, Riyadh, Saudi Arabia

    A. Aziz

  2. Department of Mechanical Engineering, University of Michigan-Dearborn, Dearborn, Michigan, USA

    T. Y. Na

Authors
  1. A. Aziz
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  2. T. Y. Na
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Aziz, A., Na, T.Y. Improved perturbation solutions for laminar natural convection on a vertical cylinder. Wärme- und Stoffubertragung 16, 83–87 (1982). https://doi.org/10.1007/BF01459764

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  • DOI: https://doi.org/10.1007/BF01459764

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