1. Master Thesis Lefteris Benos
UNIVERSITY OF THESSALY
DEPARTMENT OF MECHANICAL ENGINEERING
LABORATORY OF FLUID MECHANICS & TURBOMACHINES
Master Thesis
Analytical and numerical study of the magnetohydrodynamic natural convection in an internally heated horizontal shallow cavity
Lefteris Benos
Physics Degree, Aristotelian Univ. of Thessaloniki
Advisor: Nicholas Vlachos
Professor emeritus of Univ. of Thessaly

2. Objectives of the present thesis
Αnalytically and numerically study of the 2-D MHD natural convection flow of an electrically conductive fluid in an internally heated horizontal shallow cavity in the presence of external uniform magnetic field in the vertical direction. All walls were electrically insulated, with the horizontal wall being adiabatic while the vertical was isothermal

3. OUTLOOK Controlled Thermonuclear Fusion
Main Characteristics of OpenFOAM
MHD natural convection in an internally heated horizontal shallow cavity using:
A low-Rm MHD numerical model developed in-house & validation of the numerical model
The method of the matched asymptotic expansions
Comparison of the analytical and numerical solutions

4. Controlled Thermonuclear Fusion
Nuclear fusion: The reaction in which two or more nuclei combine together in order to form a new element with higher atomic number

5. Fusion on Earth
A sufficiently high kinetic energy of the nuclei is needed to overcome the Coulomb force
Plasma temperature≈108 oC!
The density of the particles must be high
High Plasma confinement time
Lawson criterion

6. Tokamak Basic components Vacuum vessel Magnets Blanket Divertor
Cryostat
Diagnostics
Heating systems

7. OpenFOAM
A free-to-use Open Source numerical simulation software with extensive CFD and multi-physics capabilities, produced by OpenCFDLtd
Solvers are validated in detail and match the efficiency of commercial codes
It is an object-oriented package in C++
Interest greatly increased in the last six years, industry- sponsored PhD projects, study visits and funded projects

8. Low-Rm approximation : A common simplification in MHD studies
The induced magnetic field produced by the motion of the electrically conducting fluid is negligible compared to the applied magnetic field B0
Assuming negligible perturbations for the electric and magnetic fields:
Thus, in the case of 2-D enclosures with electrically insulating boundaries, the electric field vanishes everywhere.
The momentum equation takes the following form:
Boussinesq Approximation Lorentz Force

9. Numerical details
The system of the governing equations was solved with the Finite Volume Method
A non-uniform staggered grid with a finer distribution of nodes close to the walls
Numerical schemes:
(a) Transient terms: Crank-Nicolson
Diffusion terms: Central differencing
Convection terms: Hybrid differencing

10. Validation of the low-Rm MHD model applied in OpenFOAM
Dimensionless quantities
Governing Equations
Dimensionless numbers:
Prandtl
Hartmann
Flow configuration of Al-Najem et al (1998)
Grashof

11. Comparison between (a) the midsection velocities at Ha=10 and Gr=104
Results
OpenFoam (low-Rm)
Al-Najem et al. (1998)
Ha=0
Ha=15
Ha=50
(b)
(a)
Comparison between (a) the midsection velocities at Ha=10 and Gr=104
and (b) temperatures isolines for various values of Ha and Gr=106 derived by Al-Najem et al (1998) and OpenFOAM

12. Flow configuration and boundary conditions
Magnetohydrodynamic natural convection in an internally heated horizontal shallow cavity
Flow configuration and boundary conditions

13. a. Numerical Simulations
12/23
a. Numerical Simulations
Dimensionless governing equations
Dimensionless quantities
Dimensionless numbers
Prandtl
Hartmann
Rayleigh

14. b. Matched asymptotic expansions method
Governing equations:
Mass continuity:
x-momentum:
z-momentum:
energy balance:
Governing dimensionless equations
Dimensionless quantities

15. b. Matched asymptotic expansions method
The stream function and the temperature fields can be expanded as L→∞ with respect to ξ, z in the form:
Walls
Boundary conditions
Horizontal
ψ=∂ψ/∂z=∂T/∂z=0
Vertical
ψ=∂ψ/∂x=T=0
Symmetric nature of the flow
ψ(x,z)=-ψ(L-x,z)
T(x,z)=T(L-x,z)
Core solutions for the flow and temperature fields are :
Rs=Ra∙L: Scaled Rayleigh number

16. Analytical Solutions Core temperature: Core streamfunction:
Core vertical velocity:

17. Analytical core temperature profiles at the mid-cavity height
16/23
(a) Ha=5
(b) Ha=50
(c) Rs=200
(d) Rs=5000

18. Comparison of the analytical and numerical temperatures at ξ=0.5, z=0
17/23
The numerical results showed that the approaching value of C in the following equation:
depends on L, Rs and Ha, as follows:

19. Distribution of the vertical velocity at the mid-cavity height
18/23
(a) Ha=5
(b) Ha=50
(c) Rs=200
(d) Rs=5000

20. Distribution of the streamfunction at mid-cavity height
19/23
(a) Ha=5
(b) Ha=50
(c) Rs=200
(d) Rs=5000

21. Variation of the average Nusselt number
20/23
Variation of the average Nusselt number
According to Daniels and Jones (1998):

22. Temperature and streamlines contours for Ha=5
21/23

23. Temperature and streamlines contours for Rs=3000
22/23
Temperature and streamlines contours for Rs=3000

24. Discussion
23/23
Both the analytical and numerical results demonstrate that the fluid is decelerated by the external magnetic field leading to the dominance of conduction over convection and, therefore, reducing the heat transfer. As a consequence, the temperature rises and, thus, the vertical walls lose their ability to cool the enclosed fluid
The comparison between the numerical and analytical results showed that the latter are not accurate for convective flows. However, they are in good agreement for combinations of low Rayleigh numbers and large Hartmann numbers.
When Ha→0 MHD results give the hydrodynamic results of Daniels and Jones (1998)
Although the present analytical study is limited to two-dimensional flows and cannot handle the downward fluid motion near the isothermal walls, it permits a detailed assessment of the effect of Rayleigh and Hartmann numbers of the flow field

25. THANKS FOR YOUR ATTENTION
This work is financially supported by the European Commission within the Association EURATOM-Hellenic Republic