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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

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

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

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

Temperature and streamlines contours for Ha=5 21/23

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

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

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