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Heat and Mass Transfer

, Volume 46, Issue 4, pp 421–429 | Cite as

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

  • Amitesh Kumar
  • Mitesh Vegad
  • Subhransu RoyEmail author
Original

Abstract

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.

Keywords

Aspect Ratio Prandtl Number Rayleigh Number Critical Rayleigh Number Sustained Oscillation 
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.

List of symbols

hc

Melt height

k

Thermal conductivity

\(\overline{KE}\)

Average kinetic energy

p

Dynamic pressure

Pr

Prandtl number

qw

Heat flux at side wall

rz

Radial, axial coordinates in cylindrical coordinate system

AR

Aspect ratio (melt height/crucible radius)

Ra

Rayleigh number

rc

Radius of crucible

rs

Crystal radius

T

Temperature

t

Time

\(\overline{TE}\)

Average thermal energy

Tm

Reference melt temperature

vr, vz

Radial and axial velocity

α

Thermal diffusivity

β

Coefficient of thermal expansion

ν

Kinematic viscosity

ψ

Stream function

ρ

Fluid density

ρm

Density at melt temperature

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of TechnologyKharagpurIndia

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