Free convection due to buoyancy force from temperature gradients is important in a wide range of scientific scenarios. For example, heat transfer in ocean ridge hydrothermal systems 1 and oceanic crust convection 2 3, food & beverage processing 4 5 6 etc. In many of these scenarios, free convection is coupled with phase change phenomena. In these cases, free convection analysis should include the moving interfaces among difference phases 7 8. Other examples include passive cooling of electronic devices 9, controlling thermal conditions on melt convection during crystal growth 10.

To better analysis free convections, important quantities such as temperature, pressure, density and the flow filed need to be plotted over the entire domain. In order to determine these physical quantities, free convection coefficient derived from Newton’s law of cooling as a function of time and space can be used to predict/control the heat transfer process in free convection scenarios. A great effort have been devoted for decades to analysis the mass and energy transfer in free convection with traditional Euler CFD methods such as finite volume and finite element methods. Even with a great success in single phase natural convection 11 12 13, these Euler CFD methods cannot efficiently handle the natural convection in multiphase conditions when solid-liquid and/or liquid-vapor interfaces variating with time and space occurs 14, also considerable accuracy can be lost when handling large deformation 15. However, multiphase or free surface free convection is important in fluid-structure interactions 16, heart simulation 17, bubble and droplet analysis 18 19, and naval engineering 20.

The moving interfaces occurring in multiphase and free surface free convection require extra numerical schemes such as front tracking 21 22 23, adaptive mesh generation method 24 25, level set method 26 27, and phase field approach 28 for Euler CFD. Although with some success in simulating multiphase free convection, these additional numerical schemes can increase the computational loads as well as inducing numerical instability especially with complex geometry 29. Compared with these Euler method, Lagrangian methods moving along with the controlled mass of the simulated domain can better handle the moving boundaries/interfaces 30. Smoothed Particle Hydrodynamics (SPH) is one of the meshless Lagrangian CFD method proposed by Lucy 31 and Gingold and Monaghan in 1977 32. As a meshless method, there is no need for meshing process or less reliance on the quality of mesh, so meshless method is the best choice for fast dynamic, multiphase, and highly nonlinear problems 33. These advantages make SPH popular in astrophysics and geographic such as motion of glaciers 32, rapid landslide34 35, ship flooding scenarios in naval engineering 36, bubble and foam formation/rising simulation 37 38 etc..

Paul W. Cleary 39 simulated single phase natural convection and two phases Rayleigh-Benard convection with SPH recently. In this study, liquid was simulated as weakly compressible water which satisfied Tait water equation of state. Buoyancy force was approximated as an additional body force term following Boussinesq approximation. M.E Danis et al. 40 and A. M. Aly et al. 41 applied SPH to model the transient natural/mixed convection of incompressible fluid with Boussinesq approximation to facilitate the simulation time in Eulerian form discretised using SPH operators. Since Boussinesq approximation was applied in all previous cases, they cannot properly handle for the situations with large temperature difference and highly transient dynamics 42 43. Boussinesq model conserve volume without mass conservation which is not negligible under lots of real phenomena, Szewc et al. 44 extended the Boussinesq formulation by introducing Gay-Lussac number to better handle the free convection situations with larger temperature gradient. Until now, no direct SPH simulation with real equation of states of fluids to handle the free convection without Boussinesq type assumptions has been demonstrated. This is because of the difficulty to represent real equation of state of water as solvable math functions which can be used in calculation. To prevent the limitations from these Boussinusq like approximation, we approach the multiphase natural convection with compressible SPH based on real equation of states of the simulation fluids. To Such approach was prohibitive in existing SPH algorithms since the large sound speed of real fluid in its liquid phase can cause numerical instability 45 46.

We combine real physics properties of fluid with SPH methods to intrinsically solve the free convection problems with less assumption. To handle the numerical instability caused by large compressibility, high order Runge-Kutta method is used the numerical integration to diminish the noises in compressible SPH 47 48. And, we used tabulated data of real fluid’s physical properties instead of math functions which can better predict real phenomenon with our modified multiphase (liquid-gas, liquid-solid boundaries, and gas-solid boundaries) interaction formulas.

Heat Conduction of multiphase system

This is a multiphase system with high density ratio (Top:Bottom=1:1000), high heat conductivity ratio (Top:Bottom=1:25).

Temperature contour plot – Natural Convection inside Enclosure

SPH simulation of natural convection of water inside enclosure with left handside heating and right handside cooling using: 1) Real water EoS 2) Non-boussinesq approximation 3) Laminar Viscosity (non-Artificial Viscosity)

Rayleigh–Bénard convection with air presented

Temperature Plot, Particle animation, and interface plot of SPH simulation of Rayleigh–Bénard convection of water inside enclosure with air presented on the top using: 1) Real water EoS 2) Non-boussinesq approximation 3) Laminar Viscosity (non-Artificial Viscosity)

Bubble Formation due to local heating

  1. Coumou, D., et al., High-resolution three-dimensional simulations of mid-ocean ridge hydrothermal systems. Journal of Geophysical Research-Solid Earth, 2009. 114. ↩

  2. Williams, C.F., et al., Convection in the Oceanic-Crust - Simulation of Observations from Deep-Sea Drilling Project Hole 504b, Costa-Rica Rift. Journal of Geophysical Research-Solid Earth and Planets, 1986. 91(B5): p. 4877-4889. ↩

  3. Geiger, S., et al., Multiphase thermohaline convection in the earth’s crust: I. A new finite element - Finite volume solution technique combined with a new equation of state for NaCl-H2O. Transport in Porous Media, 2006. 63(3): p. 399-434. ↩

  4. Ghani, A.G.A., et al., Thermal sterilization of canned food in a 3-D pouch using computational fluid dynamics. Journal of Food Engineering, 2001. 48(2): p. 147-156. ↩

  5. Singh, A., A.P. Singh, and H.S. Ramaswamy, Computational techniques used in heat transfer studies on canned liquid-particulate mixtures. Trends in Food Science & Technology, 2015. 43(1): p. 83-103. ↩

  6. Huang, E. and G.S. Mittal, Meatball Cooking - Modeling and Simulation. Journal of Food Engineering, 1995. 24(1): p. 87-100. ↩

  7. Huber, C., et al., Lattice Boltzmann model for melting with natural convection. International Journal of Heat and Fluid Flow, 2008. 29(5): p. 1469-1480. ↩

  8. Lacroix, M. and M. Benmadda, Numerical simulation of natural convection-dominated melting and solidification from a finned vertical wall. Numerical Heat Transfer Part a-Applications, 1997. 31(1): p. 71-86. ↩

  9. Sheikholeslami, M., et al., MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFEM. Neural Computing & Applications, 2014. 24(3-4): p. 873-882. ↩

  10. Son, Y.G., et al., Numerical study of three-dimensional convection due to buoyancy force in an aluminum oxide melt for Kyropoulos growth. Journal of Ceramic Processing Research, 2015. 16(1): p. 68-73. ↩

  11. Danaila, I., et al., A Newton method with adaptive finite elements for solving phase-change problems with natural convection. Journal of Computational Physics, 2014. 274: p. 826-840. ↩

  12. Abderrahman Nait Alla, M.F., Hicham Meftah,, Numerical study of the evaporation by mixed convection of ethanol in partially heated plate along a vertical channel. International Journal of Heat and Mass Transfer, 2015. 89: p. 206-215. ↩

  13. Xia, J., K. Luo, and J. Fan, Simulating heat transfer from moving rigid bodies using high-order ghost-cell based immersed-boundary method. International Journal of Heat and Mass Transfer, 2015. 89(0): p. 856-865. ↩

  14. Gurau, V. and J.A. Mann, A Critical Overview of Computational Fluid Dynamics Multiphase Models for Proton Exchange Membrane Fuel Cells. Siam Journal on Applied Mathematics, 2009. 70(2): p. 410-454. ↩

  15. Liu, G.R., Meshfree Methods: Moving beyond the Finite Element Method, 2009, CRC Press ↩

  16. Grandmont, C. and Y. Maday, Existence for an unsteady fluid-structure interaction problem. Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique, 2000. 34(3): p. 609-636. ↩

  17. Peskin, C.S., Numerical-Analysis of Blood-Flow in Heart. Journal of Computational Physics, 1977. 25(3): p. 220-252. ↩

  18. Boissonneau, P. and P. Byrne, An experimental investigation of bubble-induced free convection in a small electrochemical cell. Journal of Applied Electrochemistry, 2000. 30(7): p. 767-775. ↩

  19. Sato, J., et al., Effects of Natural-Convection on High-Pressure Droplet Combustion. Combustion and Flame, 1990. 82(2): p. 142-150. ↩

  20. Oro, J.M.F., et al., Numerical simulation of the fuel oil cooling process in a wrecked ship. Journal of Fluids Engineering-Transactions of the Asme, 2006. 128(6): p. 1390-1393. ↩

  21. Gan, H., et al., Direct numerical simulation of the sedimentation of solid particles with thermal convection. Journal of Fluid Mechanics, 2003. 481: p. 385-411. ↩

  22. Zarzalejo, L.J., K.S. Schmaltz, and C.H. Amon, Molten droplet solidification and substrate remelting in microcasting - Part I: numerical modeling and experimental verification. Heat and Mass Transfer, 1999. 34(6): p. 477-485. ↩

  23. Schmaltz, K.S., L.J. Zarzalejo, and C.H. Amon, Molten droplet solidification and substrate remelting in microcasting - Part II: Parametric study and effect of dissimilar materials. Heat and Mass Transfer, 1999. 35(1): p. 17-23. ↩

  24. Lebeau, M. and J.M. Konrad, Natural convection of compressible and incompressible gases in undeformable porous media under cold climate conditions. Computers and Geotechnics, 2009. 36(3): p. 435-445. ↩

  25. Meng, X.Y. and Y. Li, Numerical study of natural convection in a horizontal cylinder filled with water-based alumina nanofluid. Nanoscale Research Letters, 2015. 10. ↩

  26. Enright, D., et al., A hybrid particle level set method for improved interface capturing. Journal of Computational Physics, 2002. 183(1): p. 83-116. ↩

  27. Osher, S. and R.P. Fedkiw, Level set methods: An overview and some recent results. Journal of Computational Physics, 2001. 169(2): p. 463-502. ↩

  28. Beckermann, C., et al., Modeling melt convection in phase-field simulations of solidification. Journal of Computational Physics, 1999. 154(2): p. 468-496. ↩

  29. Magoulès, F., Computational Fluid Dynamics, 2011, Chapman and Hall/CRC 2011. ↩

  30. BENNETT, A., LAGRANGIAN FLUID DYNAMICS, 2006, Cambridge University Press. ↩

  31. Lucy, L.B., Numerical Approach to Testing of Fission Hypothesis. Astronomical Journal, 1977. 82(12): p. 1013-1024. ↩

  32. Gingold, R.A. and J.J. Monaghan, Smoothed Particle Hydrodynamics - Theory and Application to Non-Spherical Stars. Monthly Notices of the Royal Astronomical Society, 1977. 181(2): p. 375-389. ↩ ↩2

  33. Magoulès, F., Computational Fluid Dynamics, 2011, CRC Press. ↩

  34. Huang, Y., et al., Run-out analysis of flow-like landslides triggered by the Ms 8.0 2008 Wenchuan earthquake using smoothed particle hydrodynamics. Landslides, 2012. 9(2): p. 275-283. ↩

  35. Ataie-Ashtiani, B. and G. Shobeyri, Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics. International Journal for Numerical Methods in Fluids, 2008. 56(2): p. 209-232. ↩

  36. Veen, D. and T. Gourlay, A combined strip theory and Smoothed Particle Hydrodynamics approach for estimating slamming loads on a ship in head seas. Ocean Engineering, 2012. 43: p. 64-71. ↩

  37. Das, A.K. and P.K. Das, Bubble evolution through submerged orifice using smoothed particle hydrodynamics: Basic formulation and model validation. Chemical Engineering Science, 2009. 64(10): p. 2281-2290. ↩

  38. Das, A.K. and P.K. Das, Bubble Evolution through a Submerged Orifice Using Smoothed Particle Hydrodynamics: Effect of Different Thermophysical Properties. Industrial & Engineering Chemistry Research, 2009. 48(18): p. 8726-8735. ↩

  39. Cleary, P.W., Modelling confined multi-material heat and mass flows using SPH. Applied Mathematical Modelling, 1998. 22(12): p. 981-993. ↩

  40. Danis, M.E., M. Orhan, and A. Ecder, ISPH modelling of transient natural convection. International Journal of Computational Fluid Dynamics, 2013. 27(1): p. 15-31. ↩

  41. Aly, A.M. and S.E. Ahmed, An incompressible smoothed particle hydrodynamics method for natural/mixed convection in a non-Darcy anisotropic porous medium. International Journal of Heat and Mass Transfer, 2014. 77: p. 1155-1168. ↩

  42. Johannsen, K., On the validity of the Boussinesq approximation for the Elder problem. Computational Geosciences, 2003. 7(3): p. 169-182. ↩

  43. Boos, W.R., J.R. Scott, and K.A. Emanuel, Transient diapycnal mixing and the meridional overturning circulation. Journal of Physical Oceanography, 2004. 34(1): p. 334-341. ↩

  44. Szewc, K., J. Pozorski, and A. Taniere, Modeling of natural convection with Smoothed Particle Hydrodynamics: Non-Boussinesq formulation. International Journal of Heat and Mass Transfer, 2011. 54(23-24): p. 4807-4816. ↩

  45. Balsara, D.S., Von-Neumann Stability Analysis of Smoothed Particle Hydrodynamics - Suggestions for Optimal-Algorithms. Journal of Computational Physics, 1995. 121(2): p. 357-372. ↩

  46. Swegle, J.W., D.L. Hicks, and S.W. Attaway, Smoothed Particle Hydrodynamics Stability Analysis. Journal of Computational Physics, 1995. 116(1): p. 123-134. ↩

  47. Blanc, T. and M. Pastor, A stabilized Runge-Kutta, Taylor smoothed particle hydrodynamics algorithm for large deformation problems in dynamics. International Journal for Numerical Methods in Engineering, 2012. 91(13): p. 1427-1458. ↩

  48. Mabssout, M. and M.I. Herreros, Runge-Kutta vs Taylor-SPH: Two time integration schemes for SPH with application to Soil Dynamics. Applied Mathematical Modelling, 2013. 37(5): p. 3541-3563. ↩