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Wärme - und Stoffübertragung

, Volume 22, Issue 1–2, pp 83–90 | Cite as

Unsteady mixed convection over two-dimensional and axisymmetric bodies

  • C. D. Surma Devi
  • G. Nath
Article

Abstract

The unsteady laminar incompressible mixed convection flow over a two-dimensional body (cylinder) and an axisymmetric body (sphere) has been studied when the buboyancy forces arise from both thermal and mass diffusion and the unsteadiness in the flow field is introduced by the time dependent free stream velocity. The nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The results indicate that for the thermally assisting flow the local skin friction, heat transfer and mass diffusion are enhanced when the buoyancy force from mass diffusion assists the thermal buoyancy force. But this trend is opposite for the thermally opposing flow. The point of zero skin friction moves upstream due to unsteadiness. No singularity is observed at the point of zero skin friction for unsteady flow unlike steady flow. The flow reversal is observed after a certain instant of time. The velocity overshoot occurs for assisting flows.

Keywords

Skin Friction Buoyancy Force Unsteady Flow Mixed Convection Mass Diffusion 
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.

Nomenclature

c

species, mass fraction or concentration

Cf

local skin-friction coefficient

Cp

specific heat at a constant pressure

D

binary diffusion coefficient

E

Eckert number (viscous dissipation parameter)

f

dimensionless stream function

F(ft)

dimensionless velocity

g

acceleration due to the gravity

G, H

dimensionless temperature and concentration, respectively

Gr1, Gr2

thermal Grashof number and Grashof number for mass diffusion, respectively

j

index

k

thermal conductivity of the fluid

L

characteristic length

mw

local mass flux of the diffusing species

Nu

local Nusselt number

P

function of ¯Pr Prandtl number

qw

local surface heat transfer rate

r

radius of revolution of an axisymmetric body

r1 function of ¯R

radius of a cylinder or a sphere

Rex

local Reynolds number defined with respect tox

ReL

Reynolds number defined with respect toL

Sc, Sh

Schmidt number and local Sherwood number, respectively

t, t*

dimensional and dimensionless times, respectively

T

temperature

u, v

velocity components in thex andy directions, respectively

x, y

distances along and perpendicular to the surfaces

¯x

dimensionless distance

¯ x0

the distance (point) where the skin friction parameter F′w vanishes

Greek Symbols

α1

function of ¯β pressure gradient parameter

β12

volumetric coefficients of thermal expansion and mass fraction expansion, respectively

ɛ, ɛ1

constants

η, ξ

transformed coordinates

θ

angle between the normal drawn outwards from the body and the downward vertical

λ1, λ2

buoyancy forces due to thermal and mass diffusion, respectively

μ,ν

viscosity and kinematic viscosity, respectively

ϱ

density

τw

local shear stress at the wall

ϕ

function of t*

ψ

dimensional stream function

ω*

frequency parameter

Superscript

'

prime denotes derivatives with respect toη

Subscripts

e, w,∞

conditions at the edge of the boundary layer, at the wall, and in the free stream, respectively

i

initial conditions

t, t*,x, y

derivatives with respect tot, t*,x, y, respectively

s

steady-state conditions

Mischkonvektion über zweidimensionale und über achsensymmetrische Körper

Zusammenfassung

Es wurde die instationäre, laminare, inkompressible Mischkonvektion über einen zweidimensionalen (Zylinder) und einen achsensymmetrischen Körper (Kugel) studiert, wobei die Auftriebskräfte sowohl thermisch bedingt sind als auch von der Massendiffusion herrühren und das Strömungsfeld durch einen zeitabhängigen Freistrahl eingeleitet wird. Die nichtlinearen, partiellen Differentialgleichungen mit drei unabhängigen Variablen, welche die Strömung beschreiben, wurden numerisch gelöst unter Anwendung eines impliziten Finite-Differenzen-Verfahrens, in Kombination mit einer Quasi-Linearisierungstechnik. Die Ergebnisse zeigen, daß für thermisch unterstützten Auftrieb die örtliche Wandreibung, der Wärmeübergang und die Stoffdiffusion verbessert werden, wenn die Auftriebskräfte von der Stoffdiffusion den thermischen Auftrieb unterstützen. Dieser Trend ist jedoch umgekehrt, wenn der thermische Auftrieb in entgegengesetzter Richtung wirkt. Die Stelle, wo die Wandschubspannung Null wird, bewegt sich stromaufwärts infolge von Unstetigkeiten. Für instationäre Strömung wurde keine Singularität an der Stelle, an der die Wandschubspannung Null ist, festgestellt, im Gegensatz zu stationärer Strömung. Die Strömungsumkehr wurde nach einer gewissen Verzögerungszeit beobachtet. Ein überschießen der Geschwindigkeit tritt für sich stützende Strömungen auf.

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References

  1. 1.
    Oosthuizen, P. H.; Hart, A.: A numerical study of laminar combined convective flow over a flat plate. J. Heat Transfer 95 (1973) 60–63Google Scholar
  2. 2.
    Gryzagoridis, J.: Combined free and forced convection from an isothermal vertical plate. Int. J. Heat Mass Transfer 18 (1975) 911–916Google Scholar
  3. 3.
    Dey, J.; Nath, G.: Mixed convection flow on vertical surface. Wärme-Stoffübertrag. 15 (1981) 279–283Google Scholar
  4. 4.
    Mucoglu, A.; Chen, T. S.: Analysis of combined forced and free convection across a horizontal cylinder. The Canad. J. Chem. Eng. 55 (1977) 265–271Google Scholar
  5. 5.
    Chen, T. S.; Mucoglu, A.: Analysis of mixed forced and free convection about a sphere. Int. J. Heat Mass Transfer 20 (1977) 867–876Google Scholar
  6. 6.
    Katagiri, M.; Pop, I.: Unsteady combined convection from an isothermal circular cylinder. Z.A.M.M. 59 (1979) 51–60Google Scholar
  7. 7.
    Ingham, D. B.; Merkin, J. H.: Unsteady mixed convection from an isothermal circular cylinder. Acta Mechanica 38 (1981) 55–69Google Scholar
  8. 8.
    Chen, T. S.; Yuh, C. F.; Moutsoglou, A.: Combined heat and mass transfer in mixed convection along vertical and inclined plates. Int. J. Heat Mass Transfer 23 (1980) 527–537Google Scholar
  9. 9.
    Surma Devi, C. D.; Nath, G.: Unsteady nonsimilar laminar boundary-layer flows with heat and mass transfer. Acta Technica Csav 28 (1983) 225–239Google Scholar
  10. 10.
    Inouge, K.; Tate, A.: Finite-difference version of quasilinearization applied to boundary-layer equations. A.I.A.A. J. 12 (1974) 558–560Google Scholar
  11. 11.
    Sears, V. R.; Telionis, D. P.: Boundary-layer separation in unsteady flow. SIAM J. Appl. Math. 28 (1975) 215–235Google Scholar
  12. 12.
    Williams, J. C. III: Incompressible laminar boundary layer separation. Ann. Rev. Fluid Mech. 9 (1977) 113–144Google Scholar
  13. 13.
    Telionis, D. P.: Unsteady Viscous Flows. New York Berlin Heidelberg: Springer-Verlag 1981Google Scholar
  14. 14.
    Van Dommelen, L. L.; Shen, S. F.: An unsteady iterative separation process. A.I.A.A. J. 21 (1983) 358–362Google Scholar
  15. 15.
    Smith, F. T.: Steady and unsteady boundary-layer separation. Ann. Rev. Fluid Mech. 18 (1986) 197–220Google Scholar
  16. 16.
    Williams, J. G. III; Johnson, W. D.: Semi-similar solutions of unsteady boundary layer flows including separation. A.I.A.A. j.12 (1974) 1388–1393Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • C. D. Surma Devi
    • 1
  • G. Nath
    • 2
  1. 1.Department of MathematicsBangalore Institute of TechnologyBangaloreIndia
  2. 2.Department of Applied MathematicsIndian Institute of ScienceBangaloreIndia

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