Numerical Assessment and Data-Driven Reduced Order Model for Natural Convection of Water-Copper Nanofluid in Porous Media

Document Type : Full Lenght Research Article


1 CFD, Turbulence, and Combustion Research Laboratory, Department of Mechanical Engineering, University of Qom, Iran.

2 Institute of Aerospace Studies, University of Qom, Iran


In this article, two computational frameworks are presented for the numerical simulation of flow and heat transfer under the effects of natural convection phenomena in a field containing water-copper Nano-fluid and including porous media. The first is a CFD model which is built based on accurate algorithms for spatial derivatives and time integration. The spatial derivatives have been calculated using first-order upwind and second-order central differencing approaches. Also, time integration is performed using the fourth-order Runge-Kutta method. In the second, a parametric reduced order model is developed to compute the whole flow field under the effects of some important parameters such as Darcy number and Rayleigh number. This model is constructed based on POD-snapshots method. The POD modes are calculated by the solution of an eigenvalues problem. The calculated eigenfunctions are POD modes which are ranked using energy-based criteria based on the total kinetic energy of the flow field. This approach leads to the development of a reduced-order model that can be used as a surrogate model of the CFD high-order approach. The results obtained from the reduced order model show relatively good agreements under variations of some important parameters such as Darcy and Rayleigh numbers and nanoparticles density on the flow and thermal fields with the benchmark DNS data. Also, from the results, it is concluded that the surrogate model has very small values of errors (order of 10-4 ~ 10-6) and the time spent on calculations is less than 10% of the time required for direct numerical simulation.


Main Subjects

[1]    Eiyad Abu-Nada, Ziyad Masoud, Hakan F. Oztop, Antonio Campo, 2010, Effect of Nanofluid variable properties on natural convection in enclosures, Int. Journal of Thermal Sciences, 49, pp.479–491.
[2]    Mefteh Bouhalleb, Hassan Abbassi, 2014, Natural convection of nanofluids in enclosures with low aspect ratios, Int. Journal of hydrogen energy, 39, pp.15275-15286.
[3]    Bishwajit Sharma, Basant Kumar, Rabindra Nath Barman, 2018, Numerical investigation of cu-water nanofluid in a differentially heated square cavity with conducting solid square cylinder at center, Int. Journal of Heat and Technology, 36(2), pp.714-722.
[4]    M. Izadi, R. Mohebbi, D. Karimi, M. A. Sheremet, 2018, Numerical simulation of natural convection heat transfer inside a ┴ shaped cavity filled by a MWCNT-Fe3O4/water hybrid nanofluids using LBM, Chemical Engineering and Processing - Process Intensification, 125, pp. 56-66.
[5]    M. Izadi, A. Behzadmehr & M. M. Shahmardan, 2015, Effects of Inclination Angle on Mixed Convection Heat Transfer of a Nanofluid in a Square Cavity, International Journal for Computational Methods in Engineering Science and Mechanics, 16(1), pp.11-21.
[6]    M. Izadi, M.M. Shahmardan, A. Behzadmehr, A.M. Rashidi, and A. Amrollahi, 2015, Modeling of Effective Thermal Conductivity and Viscosity of Carbon Structured Nanofluid, Challenges in Nano and Micro Scale Science and Technology, 3(1), pp.1-13.
[7]    M. Izadi, M.M. Shahmardan, and A.M. Rashidi, 2013, Study on Thermal and Hydrodynamic Indexes of a Nanofluid Flow in a Micro Heat Sink”, Challenges in Nano and Micro Scale Science and Technology, 1(1), pp. 53-63.
[8]    M. Izadi, H. M. Alshehri, F. Hosseinzadeh, M. Shokri Rad, M. Bechir Ben Hamida, 2023, Numerical study on forced convection heat transfer of TiO2/water nanofluid flow inside a double-pipe heat exchanger with spindle-shaped turbulators”, Engineering Analysis with Boundary Elements, 150, pp. 612-623.
[9]    M. Izadi, T. Tayebi, H.M. Alshehri, 2023, Transient magneto-buoyant convection of a magnetizable nanofluid inside a circle sensible storage subjected to double time-dependent thermal sources, Journal of Thermal Analysis and Calorimetry, 148, pp. 8511–8531.
[10] Nazia, S, Seshaiah, B, Sudarsana Reddy, P, Sreedevi, P., 2023, Silver–ethylene glycol and copper–ethylene glycol based thermally radiative nanofluid characteristics between two rotating stretchable disks with modified Fourier heat flux, Heat Transfer, 52, pp.289- 316.
[11] P Sudarsana Reddy, and P Sreedevi, 2021, Flow and heat transfer analysis of carbon nanotubes based nanofluid flow inside a cavity with modified Fourier heat flux, Phys. Scr, 96 055215.
[12] M.R. Aminian, A.R. Miroliaei, and B. Mirzaei Ziapour, 2019, Numerical study of flow and heat transfer characteristics of CuO/H2O nanofluid within a mini tube, Journal of Heat and Mass Transfer Research, 6(1), pp.11-20.
[13] S. Kaviany, 1995, Principles of Heat Transfer in Porous Media, Springer-Verlag, NY.
[14] Ling, X.; Yan, Z.; Liu, Y.; Lu, G. 2021, Transport of nanoparticles in porous media and its effects on the co-existing pollutants, Environmental Pollutants, 283, 117098.
[15] G.C. Bourantas, E.D. Skouras, V.C. Loukopoulos, V.N. Burganos, 2014, Heat transfer and natural convection of nanofluids in porous media, European Journal of Mechanics - B/Fluids, 43, pp. 45-56.
[16] Prabir Barman and PS Rao, 2022, Natural convection of nanofluids in a wavy porous cavity, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236(7), pp. 3847-3863.
[17] Basil Mahdi Al-Srayyih, Shian Gao, and Salam Hadi Hussain , 2019, Natural convection flow of a hybrid nanofluid in a square enclosure partially filled with a porous medium using a thermal non-equilibrium model, Physics of Fluids, 31, 043609.
[18] Z. A. S. Raizah and A. M. Aly, 2021, Natural convection in an h-shaped porous enclosure filled with a nanofluid, Computers, Materials & Continua, 66(3), pp. 3233–3251.
[19] S.A.M. Mehryan, M. Ghalambaz, A. J. Chamkha, M. Izadi, 2020, Numerical study on natural convection of Ag–MgO hybrid/water nanofluid inside a porous enclosure: A local thermal non-equilibrium model, Powder Technology, 367, pp. 443-455.
[21] M. Izadi, B. Bastani and M. A. Sheremet, 2020, Numerical simulation of thermogravitational energy transport of a hybrid nanoliquid within a porous triangular chamber using the two-phase mixture approach, Advanced Powder Technology, Vol. 31(6), pp. 2493-2504.
[22] H. Sajjadi,  A. A. Delouei,  R. Mohebbi, M. Izadi, and  S. Succi, 2020, Natural convection heat transfer in a porous cavity with sinusoidal temperature distribution using Cu/Water nanofluid: Double MRT Lattice Boltzmann Method, Communications in Computational Physics, 29(1), pp.292-318.
[23] M. Izadi, R. Mohebbi, A. A. Delouei, H. Sajjadi, 2019, Natural convection of a magnetizable hybrid nanofluid inside a porous enclosure subjected to two variable magnetic fields, International Journal of Mechanical Sciences, 151, pp.154-169.
[24] M. Izadi, B. Alshuraiaan, A. Hajjar, M. A. Sheremet, M. B. Ben Hamida, 2023, Free convection of nanofluids in a porous sensible heat storage unit: Combined effect of time periodic heating and external magnetic field, International Journal of Thermal Sciences, 192 part A.
[25] P. Holmes, J. L. Lumley, G. Berkooz, 1996, Turbulence coherent structures, dynamical systems and symmetry, Cambridge, UK, Cambridge University Press.
[26] Rathinam, Muruhan and Petzold, Linda., 2023, A New Look at Proper Orthogonal Decomposition, SIAM J. Numerical Analysis, 41, pp.1893-1925.
[27] Feng, J. W., Cen, S., Li, C. F., and Owen, D. R. J., 2015, Statistical reconstruction and Karhunen–Loève expansion for multiphase random media, Int. Journal of Numerical Methods in Engineering, 105, pp.3– 32.
[28] Holmes P.J., Lumley J.L., Berkooz G., Mattingly J.C., Wittenberg R. W., 1997, Low-dimensional models of coherent structures in turbulence, Physics Reports, 287(4), pp. 337-384.
[28] Holmes P.J., Lumley J.L., Berkooz G., Mattingly J.C., Wittenberg R. W., 1997, Low-dimensional models of coherent structures in turbulence, Physics Reports, 287(4), pp. 337-384.
[29] Smith, T. R., Moehlis, J., and Holmes, P., 2004, Low-Dimensional Modeling of Turbulence Using the Proper Orthogonal Decomposition: A tutorial”, Kluwer Academic, Boston.
[30] L. Sirovich, M. Kirby, 1987, Low-dimensional procedure for the characterization of human faces, J. Optical Society America, 4(3), 519–24.
[31] T. Lieu, C. Farhat, 2005, Adaptation of POD-based aeroelastic ROMs for varying Mach number and angle of attack: Application to a complete F16 configuration, AIAA Journal.
[32] Li, J., Zhang, T., Sun, S. and Yu, B., 2019, Numerical investigation of the POD reduced-order model for fast predictions of two-phase flows in porous media”, International Journal of Numerical Methods for Heat & Fluid Flow, 29(11), pp. 4167-4204.
[33] Thomas A. Brenner, Raymond L. Fontenot, Paul G.A. Cizmas, Thomas J. O’Brien, Ronald W. Breault, 2012, A reduced-order model for heat transfer in multiphase flow and practical aspects of the proper orthogonal decomposition, Computers & Chemical Engineering, 43, pp. 68-80.
[34] Alexandra Tallet, Cyrille Allery & Cédric Leblond, 2016, Optimal flow control using a POD-based reduced-order model, Numerical Heat Transfer, Part B: Fundamentals, Vol.70(1), pp.1-24.
[35] Li, S., Li, W., & Noack, B., 2022, Machine-learned control-oriented flow estimation for multi-actuator multi-sensor systems exemplified for the fluidic pinball, Journal of Fluid Mechanics, 952, A36.
[36] J. R. Connell, D. Kulasiri, 2007, Computational modeling of turbulent velocity structures for an open channel flow using Karhunen-Loeve expansion, Lincoln University.
[37] T. Bui-Thanh, M. Damodaran, K. E. Willcox, 2004, Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition, AIAA Journal.
[38] R. Bourguet, M. Braza, A. Dervieux, 2011, Reduced-Order modeling of transonic flows around an airfoil submitted to small deformations, Journal of Computational Physics, 230, 159–184.
[39] M.K. Moayyedi, M. Najafbeygi, 2018, A high fidelity cost efficient tensorial method based on combined POD-HOSVD reduced order model of flow field, European Journal of Computational Mechanics, 27(4), 342–366.
[40] Zhendong Luo, 2015, Proper Orthogonal Decomposition-based Reduced-order Stabilized Mixed Finite Volume Element Extrapolating Model for the Non-stationary Incompressible Boussinesq equations, Journal of Mathematical Analysis and Applications, 425(1), pp. 259-280.
[41] M.K. Moayyedi, 2017, Numerical simulation and reduced order modeling of mass transfer due to natural convection based on coupling between temperature and contaminant, Sharif Journal of Mechanical Engineering, 33.3(2), pp.43-52.
[42] M. K. Moayyedi, F. Sabaghzadeghan, 2021, Development of parametric and time dependent reduced order model for diffusion and convection-diffusion problems based on proper orthogonal decomposition method”, Amirkabir Journal of Mechanical Engineering, 53(7), pp. 4241-4260.