1. Laminar Natural Convection in 2D Glazing Cavities
MIE 605 Finite Element Analysis
Prof. Dr. Ian Grosse
Submitted by
Bhaskar Adusumalli

2. Project outline
Introduction
Problem description
Modelling asssumptions
Boundary conditions
Governing equations
Non dimensionalisation
Meshing
Solution procedure
Results and discussions

3. Problem description
Rectangular cavity enclosed by glazing units
Air in the cavity
Isothermal side walls of different temperatures
Adiabatic top and bottom walls
Gravity acting downwards
Bouyancy acting upwards

4. Geometry and Boundary conditions for the glazing cavity

5. Modelling Assumptions
An incompressible flow with negligible viscous dissipation
Constant fluid properties
No internal heat sources
Wall gradients in the actual enclosure are negligible compared to the fluid gradients
So neglect solid boundaries in the model

6. uu = -p + g(T-Tr) + 2u momentum cpuT = k2T energy
Governing Equations
u = continuity
uu = -p + g(T-Tr) + 2u momentum
cpuT = k2T energy

7. Non dimensional parameters
 defining a reference velocity by,
U = (α/L) (Ra*Pr)½ ,
the non dimensional variables are given by,
u* = u/U
T* = (T-Tr)/(T2-Tr)
x* = x/L
p* = pL/U

8. Non dimesional properties of air
cp = Pr
  = 1
 g = 1
  = 1
 k = 1
ρ = (Ra*Pr)½
Where,
Pr = (μcp/k)
Ra = ρβg(Th-Tc)L3/(μα)

9. Non dimensional form of governing equations
u = continuity
(Ra*Pr)½ (uu) = -p + 2u - (Ra*Pr)½ T ĵ momentum
(Ra*Pr)½ (uT) = 2T energy

10. Boundary Conditions Temperature boundary conditions T(x=0,y) = Tc
T(x=L,y) = Th
Adiabatic boundary conditions on top and bottom walls
q(x,y=0) = 0
q(x,y=L) = 0
Velocity boundary conditions on the walls
u(x=0,y) = v(x=0,y) = 0
u(x=L,y) = v(x=L,y) = 0
u(x,y=0) = v(x,y=0) = 0
u(x,y=L) = v(x,y=L) = 0

11. Meshing
Mesh generated by FIMESH
Mesh density to resolve velocity and thermal boundary layers
first element chosen such that it lies well within the boundary layer thickness, δ, given by
 = 5/Ra0.25

12. Mesh generated by FIMESH

13. Solution procedure
After non dimensionalisation, the temperature difference is applied in the form of Rayleigh number
A combination of Picard iteration and Newton Raphson method has been used by setting the commands SOLUTION (N.R. = 7) and STRATEGY(S.S. = 1)
Convergence achieved in just 4 to 8 iterations for the different Rayleigh numbers

14. Tabulation of average Nusselt number variation with rayleigh number
Elsherbiny
FIDAP
5000
1.77
1.18
6000
1.22
1.217
7000
1.27
1.28
8000
1.32
1.33
9000
1.36
1.43
10000
1.40
1.70

15. Variation average Nusselt number with rayleigh number

16. Streamline contours for different Rayleigh number

17. Results and Discussions
At low Rayleigh number the cavity has a uni-cellular flow.
At high rayleigh number secondary eddies begin to form
Flow pattern is multicellular at high Rayleigh number
Therefore increased mixing and increased convective heat transfer

18. Thank You