http://orcid.org/0000-0002-5148-5648
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ABSTRACT
In the present study, a novel FD_LBM hybrid numerical method with total variation diminishing (TVD) characteristics is proposed to solve the famous problems of MagnetoHydroDynamics (MHD) and natural convection in a closed cavity. Here, the momentum and energy equations are carried out via lattice Boltzmann method (LBM) and FD techniques, respectively. To enhance the stability and performance of the finite difference scheme for higher Ra numbers, two renowned Superbee and Minmod flux limiter functions were used. The results for heat transfer and entropy generation features for a wide range of Rayleigh and Hartman numbers of and
are presented. In addition for comparison purposes, two multiple relaxation time (MRT) and single relaxation time (SRT) algorithms presented in the open literature, were applied in the same case definitions. Not only the tests revealed an excellent agreement between the TVD method results and the published data, but they also proved that this new technique is numerically much more efficient and stable than the SRT and MRT methods, and hence one can assume the present method as a fantastic tool in the numerical solution of laminar convection problems.
Nomenclature
| B | = | magnetic field |
| c | = | lattice speed |
| ci | = | discrete particle speeds |
| Cp | = | specific heat at constant pressure |
| F | = | external forces |
| f | = | density distribution functions |
| feq | = | equilibrium density distribution functions |
| g | = | internal energy distribution functions |
| geq | = | equilibrium internal energy distribution functions |
| gy | = | gravity |
| Ha | = | Hartman number |
| M | = | lattice number in y-direction |
| Ma | = | Mach number |
| Nu | = | Nusselt number |
| Pr | = | Prandtl number |
| Ra | = | Rayleigh number |
| Sx | = | entropy generation rate due x |
| T | = | temperature |
| TVD_S | = | hybrid TVD scheme with Superbee limiter |
| TVD_M | = | hybrid TVD scheme with Minmod limiter |
| u, v | = | velocity components in x- and y-directions |
| x, y | = | Cartesian coordinates |
| Greek letters | = | |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| = | magnetic field inclination angle | |
| μ | = | dynamic viscosity |
| ρ | = | density |
| σ | = | electrical conductivity |
| τα | = | relaxation time for temperature |
| τυ | = | relaxation time for flow |
| υ | = | kinematic viscosity |
| ωi | = | weighted factor in i-direction |
| = | limiter function | |
| Subscripts | = | |
| c | = | cold |
| f | = | fluid |
| h | = | hot |
Nomenclature
| B | = | magnetic field |
| c | = | lattice speed |
| ci | = | discrete particle speeds |
| Cp | = | specific heat at constant pressure |
| F | = | external forces |
| f | = | density distribution functions |
| feq | = | equilibrium density distribution functions |
| g | = | internal energy distribution functions |
| geq | = | equilibrium internal energy distribution functions |
| gy | = | gravity |
| Ha | = | Hartman number |
| M | = | lattice number in y-direction |
| Ma | = | Mach number |
| Nu | = | Nusselt number |
| Pr | = | Prandtl number |
| Ra | = | Rayleigh number |
| Sx | = | entropy generation rate due x |
| T | = | temperature |
| TVD_S | = | hybrid TVD scheme with Superbee limiter |
| TVD_M | = | hybrid TVD scheme with Minmod limiter |
| u, v | = | velocity components in x- and y-directions |
| x, y | = | Cartesian coordinates |
| Greek letters | = | |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| = | magnetic field inclination angle | |
| μ | = | dynamic viscosity |
| ρ | = | density |
| σ | = | electrical conductivity |
| τα | = | relaxation time for temperature |
| τυ | = | relaxation time for flow |
| υ | = | kinematic viscosity |
| ωi | = | weighted factor in i-direction |
| = | limiter function | |
| Subscripts | = | |
| c | = | cold |
| f | = | fluid |
| h | = | hot |
Notes
1FHP.
2HPP.
3The model was also independently proposed by other authors.