Wärme - und Stoffübertragung

, Volume 22, Issue 3–4, pp 159–168 | Cite as

Mixed convection on vertical plate for fluids of any prandtl number

  • H. -T. Lin
  • C. -C. Chen


A mixed convection parameterζ=(σRa)1/4/(ωRe)1/2, withσ=Pr/(1+Pr) andω=Pr/(1 +Pr)1/2, is proposed to replace the conventional Richardson number, Gr/Re2, for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra1/4/(Re1/2Pr1/3) forPr ≫ 1, and (RaPr)1/2/(RePr1/2 forPr ≪ 1. Furthermore, new coordinates and dependent variables are properly defined in terms ofζ, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.


Convection Prandtl Number Heat Transfer Rate Free Convection Force Convection 
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.



local friction coefficient


specific heat capacity


reduced stream function


gravitational acceleration


local Grashoff number,g βTw−T)x3/v2


local Nusselt number


Prandtl number,v/α


local Rayleigh number,g βTw−Tx3/(α v)


local Reynolds number,ux/v


Richardson number,Gr/Re2


fluid temperature


wall temperature


free stream temperature


velocity component in thex direction


free stream velocity


velocity component in they direction


vertical coordinate measuring from the leading edge


horizontal coordinate

Greek symbols


thermal diffusivity


thermal expansion coefficient


mixed convection parameter (σRa)1/4/ωRe)1/2


pseudo-similarity variable,(y/x)λ


conventional similarity variable,(y/x)Re1/2


dimensionless temperature, (T−TTW−T


unified mixed-flow parameter, [(ωRe)1/2 + (σRa)1/4]


dynamic viscosity


kinematic viscosity


stretched streamwise coordinate or mixed convection parameter, [1 + (ωRe)1/2/(σRa)1/4]−1=ζ/(1 +ζ)



σ Pr/(1 + Pr)τw

wall shear stress


stream function



Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl


Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameterζ=(σ Ra)1/4 (ωRe)1/2, mitσ=Pr/(1 +Pr) undω=Pr/(1 +Pr)1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re2, ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra1/4/(Re1/2Pr1/3) fürPr ≫ 1 und (RaPr)1/4/(RePr)1/2 fürPr ≪ 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion vonζ definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.


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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • H. -T. Lin
    • 1
  • C. -C. Chen
    • 1
  1. 1.Dept. of Chemical EngineeringNational Central UniversityChung-liTaiwan 32054, ROC

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