Abstract
An analysis is presented for the transient thermal response of a laminar boundary layer in the vicinity of an axisymmetric stagnation flow on an infinite circular cylinder. The final approach to steady state temperature field is shown to have exponential decay with time. The characteristic factors appearing in the exponents result in the solution of an eigenvalue problem in ordinary linear differential equations. Numerical results are presented for a range of values of the Reynolds number and Prandtl number.
Zusammenfassung
Es wird eine Analyse für das transiente Temperaturverhalten einer laminaren Grenzschicht in der Nähe einer achsensymmetrischen Staupunktströmung an einem unendlich ausgedehnten, kreisförmigen Zylinder vorgestellt. Die Approximation an das stationäre Temperaturfeld hat —wie gezeigt wird-einen exponentiellen zeitlichen Abfall. Die charakteristischen Faktoren, die in den Exponenten erscheinen, führen zur Lösung eines Eigenwertproblems in gewöhnlichen linearen Differentialgleichungen. Es werden Zahlenwerte angegeben für verschiedene Reynolds-und Prandtl-Zahlen.
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Abbreviations
- A :
-
Proportionality constant [A=(U∞/a) for a cylinder]
- a :
-
radius of cylinder
- f :
-
stream function
- Pr :
-
Prandtl number
- P :
-
pressure
- qw :
-
wall heat flux
- Re :
-
Reynolds number (Aa2/2v)
- r :
-
coordinate normal to the cylindrical surface
- T :
-
temperature
- t :
-
time
- U∞ :
-
free stream velocity
- u, w :
-
velocity components
- r, z :
-
coordinate directions
- \(\tilde \alpha \) :
-
thermal diffusivity
- α :
-
constant in Eq. (18)
- α1 :
-
first eigenvalue
- α2 :
-
second eigenvalue
- \(\bar \theta (\zeta ,p)\) :
-
Laplace transform ofθ (ζ,τ)
- Φ :
-
eigenfunction defined by Eq. (17)
- μ :
-
dynamic viscosity
- v :
-
kinematic viscosity
- ϱ :
-
fluid density
- τ :
-
dimensionless time
- 1(t):
-
Heaviside unit operator=0 fort <0 and=1 fort ≧0
- s :
-
steady state
- w :
-
conditions at the wall
- ∞ :
-
conditions far away from wall
References
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Gorla, R.S.R. The final approach to steady state in a nonsteady axisymmetric stagnation point heat transfer. Wärme- und Stoffübertragung 22, 37–44 (1988). https://doi.org/10.1007/BF01001570
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DOI: https://doi.org/10.1007/BF01001570