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Development of a turbulent near-wall temperature model and its application to channel flow

Entwicklung eines turbulenten, wandnahen Temperaturmodells und seine Anwendung auf Kanalströmung

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Abstract

A numerical study of fluid flow and heat transfer in a two-dimensional channel under fully developed turbulent conditions is reported. A computer program which is capable of treating both forced and natural convection problems under turbulent conditions has been developed. The code uses the high-Reynolds-number form of the two equation turbulent model(k-ɛ) in which a turbulent kinetic energy near-wall model is incorporated in order to accurately represent the behavior of the flow near the wall, particularly in the viscous sublayer where the turbulent Reynolds number is small. A near-wall temperature model has been developed and incorporated into the energy equation to allow accurate prediction of the temperature distribution near the wall and, therefore, accurate calculation of heat transfer coefficients.

The sensitivity of the prediction of flow and heat transfer to variations in the coefficients used in the turbulence model is investigated. The predictions of the model are compared to available experimental and theoretical results; good agreement is obtained. The inclusion of the near-wall temperature model has further improved the predictions of the temperature profile and heat transfer coefficient. The results indicate that the turbulent kinetic energy Prandtl number should be a function of Reynolds number.

Zusammenfassung

Es wird über eine numerische Studie der Strömung und des Wärmetransportes in einem zweidimensionalen Kanal mit voll entwickelter Turbulenz berichtet. Ein Computer-programm wurde entwickelt, das in der Lage ist, sowohl Zwangsais auch Naturkonvektion unter turbulenten Bedingungen zu behandeln. Das Programm verwendet die „High-Reynolds-Number“-Form des turbulenten Zweigleichungsmodells(k-ɛ-Modell in das ein Ansatz für die wandnahe, turbulente, kinetische Energie eingearbeitet ist, um das Verhalten der Strömung nahe der Wand genau wiederzugeben, insbesondere in der viskosen Unterschicht, wo die turbulente Reynolds-Zahl klein ist. Es wurde ein wandnahes Temperaturmodell entwickelt und in die Energiegleichung eingearbeitet, um eine genaue Vorhersage der Temperaturverteilung nahe der Wand zu ermöglichen und damit die Wärmeübergangskoeffizienten genau zu berechnen.

Die Empfindlichkeit für die Vorhersage der Strömung und der Wärmeübertragung auf Veränderungen in den Koeffizienten des verwendeten Turbulenzmodells wird untersucht. Die Vorhersagen des Modells werden mit verfügbaren experimentellen Daten und theoretischen Ergebnissen verglichen. Es wurde gute Übereinstimmung erzielt. Die Einbeziehung des wandnahen Temperaturmodells brachte weitere Verbesserung der Vorhersage des Temperaturprofils und des Wärmeübergangskoeffizienten. Die Ergebnisse zeigen, daß die turbulente, kinetische Prandtl-Zahl eine Funktion der Reynolds-Zahl sein sollte.

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Abbreviations

A :

empirical coefficient, Eq. (27)

b :

channel width

cl :

constant of proportionality between length scalel and distance from wall (applied over near-wall cell)=2.55, Eq. (17)

C 1, C2,C μ∞ :

constants in thek-ɛ turbulent model, Eq. (8)

c p :

specific heat

E * :

constant in logarithmic velocity profile=5.0, Eq. (12)

k :

turbulent kinetic energy

K :

empirical coefficient, Eq. (27)

l m :

mixing length

Nu :

Nusselt number, Eq. (19)

p :

pressure

Pr :

Prandtl number=ν/α

Re :

Reynolds number=υmb/2ν

Re t :

turbulent Reynolds number, Eq. (10)

Re Γ :

viscous sublayer Reynolds number=y v k 1/2 v /v=20.0

¯Re :

average Reynolds number

t :

time

T :

temperature

u :

velocity inx-direction

ν :

velocity iny-direction

x :

horizontal coordinate; main flow direction

y :

vertical coordinate

α :

thermal diffusivity

ɛ :

dissipation rate of turbulence energy

χ :

von Karman constant=0.4187

χ * :

constant in logarithmic velocity profile=χ C 1/4μ∞ =0.23

λ :

thermal conductivity

μ :

laminar dynamic fluid viscosity

μ t :

turbulent dynamic fluid viscosity

ν :

laminar kinematic fluid viscosity

ϱ :

density

σ k :

turbulent kinetic energy Prandtl number

σ t :

turbulent temperature Prandtl number

σɛ :

turbulent dissipation Prandtl number

τ :

shear stress

b :

bulk

c :

cold

e :

edge of near-wall cell

E :

center of cell adjacent to the near-wall cell

h :

hot

m :

maximum of experimental value

P :

center of near-wall cell

t :

turbulent

υ :

edge of viscous sublayer

w :

wall

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Authors and Affiliations

  1. Passive Research and Development Group Lawrence Berkeley Laboratory, University of California, 94720, Berkeley, California, USA

    H. Akbari, A. Mertol, A. Gadgil, R. Kammerud & F. Bauman

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  1. H. Akbari
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  4. R. Kammerud
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Akbari, H., Mertol, A., Gadgil, A. et al. Development of a turbulent near-wall temperature model and its application to channel flow. Wärme- und Stoffübertragung 20, 189–201 (1986). https://doi.org/10.1007/BF01303450

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