Abstract
This paper presents a two-dimensional numerical procedure which employs finite-difference techniques to solve the partial-differential equations governing the steady flow and heat transfer of a surface thermal plume above a crossflowing ambient fluid in the near field. An earlier model developed for predicting thermal plumes in stagnant surroundings is extended by the inclusion of curvature terms in the momentum equations. An integral equation is also solved to determine the trajectory of the thermal plume. Comparisons of present model calculations with two sets of experimental data show that the centre-line trajectories as well as the velocity and temperature decays are well predicted.
Zusammenfassung
Dieser Bericht beschreibt ein zweidimensionales Rechenverfahren, das mit Hilfe Finiter-Differenzen-Techniken die partiellen Differentialgleichungen löst, welche das Geschwindigkeits- und Temperaturfeld stationärer Oberflächenauftriebsstrahlen über einer querströmenden umgebenden Flüssigkeit beschreiben. Ein früheres Modell, entwickelt für die Berechnung von Auftriebsstrahlen in stehenden Umgebungen, wurde durch Einfügung von Krümmungseffekten in den Impulsgleichungen erweitert. Eine Integralgleichung wurde ebenfalls gelöst, um den Bahnverlauf der Auftriebsstrahlen zu bestimmen. Vergleiche der vorgestellten Modellrechnungen mit zwei experimentell bestimmten Datensätzen zeigen, daß sowohl die Mittellinienbahnverläufe als auch die Geschwindigkeits- und Temperaturverläufe gut berechnet wurden.
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Abbreviations
- AR :
-
(b/z)IN, initial aspect ratio
- b :
-
initial layer width
- C :
-
specific heat of water
- D :
-
(initial area)1/2, characteristic length scale
- E :
-
external layer boundary
- E t :
-
entrainment coefficient
- F :
-
\(\frac{U}{{\left( {\frac{{\Delta \varrho }}{\varrho }gz} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}\), densimetric Froude number
- g :
-
acceleration due to gravity
- I :
-
internal layer boundary
- \(\dot m_h^{''} \) :
-
lateral entrainment rate per unit area
- \(\dot m_v^{''} \) :
-
vertical entrainment rate per unit area
- n :
-
lateral distance normal tos
- P :
-
pressure
- R :
-
radius of curvature of centre-line trajectory
- Ri :
-
\(\frac{1}{{F^2 }}\), overall Richardson number
- r :
-
radius of curvature of other longitudinal grid lines
- s :
-
distance along centre-line trajectory
- T :
-
temperature
- ΔT :
-
To-T
- U :
-
longitudinal velocity tangential to trajectory
- V :
-
lateral velocity normal to trajectory
- x :
-
longitudinal distance in Cartesian coordinates
- y :
-
lateral distance in Cartesian coordinates
- z :
-
layer depth
- α :
-
angle between crossflow direction and the normal to the trajectory
- Γ T :
-
thermal diffusion coefficient
- μ :
-
effective viscosity
- ϱ :
-
density
- ψ :
-
stream function
- σ :
-
Prandtl number
- θ :
-
inclination of a longitudinal grid line to the centre-line trajectory
- ω :
-
\(\frac{{\psi - \psi _I }}{{\psi _E - \psi _I }}\), normalized stream function
- C L :
-
centre-line condition
- e q :
-
equilibrium condition
- IN :
-
initial condition
- I, E :
-
conditions along theI andE boundaries
- o :
-
ambient fluid property
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Demuren, A.O. Prediction of the steady surface thermal plume in crossflow. Warme- und Stoffubertragung 19, 41–46 (1985). https://doi.org/10.1007/BF01682545
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DOI: https://doi.org/10.1007/BF01682545