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Application of Rayleigh-Ritz method to forced convection turbulent heat transfer

Anwendung der Rayleigh-Ritz-Methode auf die Wärmeübertragung bei erzwungener turbulenter Strömung

Wärme - und Stoffübertragung volume 15, pages 67–77 (1981)Cite this article

Abstract

An analytical model to predict heat transfer rates to an incompressible fluid in turbulent flow, with fully developed velocity profile, between a heated plate and a parallel, insulated plate is developed. The model employs van Driest's mixing length expression near the wall, a constant eddy diffusivitiy in the core and a constant turbulent Prandtl number. An approximate solution obtained by employing Rayleigh-Ritz method is shown to compare well with the ‘exact’ solution obtained by numerical integration of the differential equations. The results are compared with the available experimental data and analytical solutions.

Zusammenfassung

Es wird ein analytisches Modell zur Berechnung der Wärmeübertragung an ein inkompressibles Fluid in turbulenter Strömung mit voll ausgebildetem Geschwindigkeitsprofil zwischen einer beheizten Platte und einer dazu parallelen isolierten Platte angegeben. Das Modell verwendet van Driest's Ausdruck für die wandnahe Mischungslänge, eine konstante Wirbeldiffusivität im Kern und eine konstante turbulente PrandtlZahl. Eine Näherungslösung nach der Rayleigh-Ritz-Methode läßt sich gut mit der “exakten” Lösung vergleichen, die durch numerische Integration der Differentialgleichungen erhalten wurde. Die Ergebnisse werden mit verfügbaren Versuchswerten und analytischen Lösungen verglichen.

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Abbreviations

A+ :

dimensionless constant in van Driest formula

a+ :

dimensionless distance from the wall after which the eddy diffusivity of momentum is constant

b:

half-gap of passage

b+ :

dimensionless half-gap=bu*/ν

Cf :

skin friction coefficient

Cp :

constant pressure specific heat

d:

hydraulic mean diameter defined as 4xarea/perimeter=4b

h:

convective heat transfer coefficient

K+ :

dimensionless constant in van Driest formula

k:

fluid thermal conductivity

m:

mass flow rate of fluid

Nu:

Nusselt number hd/k

P:

pressure

Pr:

Prandtl number=ν/α

Prt :

turbulent Prandtl number=ɛm/ɛν

qw :

heat flux at wall

Re:

Reynolds number=vmd/ν

T:

Temperature

u+ :

dimensionless velocity=Vx/u*

u* :

friction velocity=\(\sqrt {^\tau _w /^\rho } \)

Vx :

axial velocity

x:

axial distance from the entrance

x+ :

dimensionless distance=x/d

y:

distance from the heated wall

y+ :

dimensionless distance=yu*/ν

α :

thermal molecular diffusivity

γ:

function equal to (ɛH+α)/ν

δ:

boundary layer thickness

ɛH :

eddy diffusivity of heat

ɛm :

eddy diffusivity of momentum

ɛm0 :

uniform eddy diffusivity of momentum in the core

θ:

dimensionless temperature

T-Ti/qwd/k:

uniform heat flux

T-Tw/Ti-Tw :

uniform temperature

ν:

fluid kinematic viscosity

ρ:

fluid density

τ:

fluid shearing stress

Φ:

bulk mean temperature—fully developed region

γ:

fully developed transverse temperature profile

1:

fully developed

2:

in the entrance region

i:

at the inlet

m:

bulk mean value

w:

at the heated wall

References

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Authors and Affiliations

  1. Department of Mechanical Engineering, Michigan Technological University, 49931, Houghton, Michigan, USA

    L. A. Diaz (Graduate Student) &  N. V. Suryanarayana

Authors
  1. L. A. Diaz
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  2. N. V. Suryanarayana
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Diaz, L.A., Suryanarayana, N.V. Application of Rayleigh-Ritz method to forced convection turbulent heat transfer. Wärme- und Stoffübertragung 15, 67–77 (1981). https://doi.org/10.1007/BF01002404

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  • DOI: https://doi.org/10.1007/BF01002404

Keywords

  • Heat Transfer
  • Prandtl Number
  • Heat Transfer Rate
  • Incompressible Fluid
  • Forced Convection

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