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Wärme - und Stoffübertragung

, Volume 16, Issue 2, pp 83–87 | Cite as

Improved perturbation solutions for laminar natural convection on a vertical cylinder

  • A. Aziz
  • T. Y. Na
Article

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.

Keywords

Boundary Layer Finite Difference Nusselt Number Finite Difference Method Present Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

e

operator defined by Eq. (31)

F

dimensionless stream function

Fn

nth order approximation for dimensionless stream function

g

acceleration of gravity

h

local heat transfer coefficient

¯h

average heat transfer coefficient

k

thermal conductivity

Nux

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

Tw

wall temperature of cylinder

T

ambient temperature

u,ν

velocity components

x

axial coordinate

Greek symbols

α

thermal diffusivity

β

coefficient of thermal expansion

η

similarity variable, Eq. (7)

θ

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

θn

nth order approximation for dimensionless temperature

ν

kinematic viscosity

ξ

transverse curvature parameter, Eq. (8)

ψ

stream function

Subscripts

fp

flat plate

cyl

cylinder

Verbesserte Störungslösungen für laminare freie Konvektion am senkrechten Zylinder

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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References

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • A. Aziz
    • 1
  • T. Y. Na
    • 2
  1. 1.Department of Mechanical EngineeringCollege of EngineeringRiyadhSaudi Arabia
  2. 2.Department of Mechanical EngineeringUniversity of Michigan-DearbornDearbornUSA

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