Numerical simulation of binary convection within the Soret regime in a tilted cylinder

Arantxa Alonso; Isabel Mercader; Oriol Batiste; Alvaro Meseguer

doi:10.1515/jnet-2024-0064

Article

Numerical simulation of binary convection within the Soret regime in a tilted cylinder

Arantxa Alonso , Isabel Mercader , Oriol Batiste and Alvaro Meseguer

Published/Copyright: February 11, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Journal of Non-Equilibrium Thermodynamics Volume 50 Issue 2

Abstract

This study computationally investigates the time-dependent patterns emerging in the Soret regime for binary fluid convection in slightly inclined cylinders heated from below, with a particular focus on positive Soret coefficient thermophobic mixtures (S _T > 0) and aspect ratios Γ = 5.2, Γ = 5.3, and Γ = 5.4. By varying the Rayleigh number (Ra) and smoothly adjusting its increments, we capture a range of spatio-temporal behaviours, revealing the coexistence of large-scale shear flows (LSF) and superhighway convection (SHC) patterns. SHC-like structures, characterised by a high base frequency, involve oscillating plumes arranged in adjacent lanes, moving in opposite directions along the inclination. Remarkably, this frequency remains nearly constant across different Ra values. Some of the observed coherent structures, such as periodic and modulated solutions, exhibit equivariance with respect to some elements of the D 2 symmetry group inherent to the physical system. In the case of Γ = 5.4, we identify three-frequency orbits, with modulations up to two orders of magnitude smaller than the base frequency. The observed dynamics is highly sensitive to small variations of Γ, with different patterns being stabilized depending on the aspect ratio of the cell. The bifurcation scenarios are complex and case-specific, and their precise determination is computationally demanding.

Keywords: direct numerical simulation; inclined cylindrical cells; positive Soret coefficient mixtures; time-dependent binary fluid convection; thermodiffusion

Corresponding author: Arantxa Alonso, Department of Physics, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain, E-mail: arantxa.alonso@upc.edu

Funding source: Ministerio de Ciencia, Innovación y Universidades (Agencia Estatal de Investigación)

Award Identifier / Grant number: PID 2020-114043 GB-I 00 (MCIN/AEI/10.13039/501100011033)

Funding source: Ministerio de Ciencia, Innovación y Universidades (Agencia Estatal de Investigación)

Award Identifier / Grant number: PID2023-150029NB-I00 (MCIN/AEI/10.13039/501100011033 /FEDER, UE)

Acknowledgement

This research is supported by the Ministerio de Ciencia, Innovación y Universidades (Agencia Estatal de Investigación, project nos. PID2020-114043GB-I00 (MCIN/AEI/10.13039/501100011033) and PID2023-150029NB-I00 (MCIN/AEI/10.13039/501100011033/feder, ue).

Research ethics: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Informed consent: Not applicable.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflict of interest.
Research funding: This work was supported by the Ministerio de Ciencia, Innovación y Universidades (Agencia Estatal de Investigación, project nos. PID2020-114043GB-I00 (MCIN/AEI/10.13039/501100011033) and PID2023-150029NB-I00 (MCIN/AEI/10.13039/501100011033/FEDER, UE).
Data availability: Not applicable.

References

[1] J. K. Platten, “The Soret effect: a review of recent experimental results,” ASME J. Appl. Mech., vol. 73, no. 1, pp. 5–15, 2006. https://doi.org/10.1115/1.1992517.Search in Google Scholar

[2] W. Köhler and K. I. Morozov, “The Soret effect in liquid mixtures – a review,” J. Non-Equilib. Thermodyn., vol. 41, no. 3, pp. 151–197, 2016. https://doi.org/10.1515/jnet-2016-0024.Search in Google Scholar

[3] W. Köhler, A. Mialdun, M. M. Bou-Ali, and V. Shevtsova, “The measurement of Soret and thermodiffusion coefficients in binary and ternary liquid mixtures,” Int. J. Thermophys., vol. 44, no. 140, p. 140, 2023. https://doi.org/10.1007/s10765-023-03242-x.Search in Google Scholar

[4] P. Kolodner, J. A. Glazier, and H. Williams, “Dispersive chaos in one-dimensional traveling-wave convection,” Phys. Rev. Lett., vol. 65, no. 13, pp. 1579–1582, 1990. https://doi.org/10.1103/physrevlett.65.1579.Search in Google Scholar

[5] P. Kolodner, S. Slimani, N. Aubry, and R. Lima, “Characterization of dispersive chaos and related states of binary-fluid convection,” Physica D, vol. 85, nos. 1–2, pp. 165–224, 1995, https://doi.org/10.1016/0167-2789(95)00061-8.Search in Google Scholar

[6] O. Batiste, E. Knobloch, A. Alonso, and I. Mercader, “Spatially localized binary-fluid convection,” J. Fluid Mech., vol. 560, pp. 149–158, 2006, https://doi.org/10.1017/s0022112006000759.Search in Google Scholar

[7] A. Alonso, O. Batiste, A. Meseguer, and I. Mercader, “Complex dynamical states in binary mixture convection with weak negative Soret coupling,” Phys. Rev. E, vol. 75, no. 2, p. 2007, 2007. https://doi.org/10.1103/physreve.75.026310.Search in Google Scholar

[8] P. Le Gal, A. Pocheau, and V. Croquette, “Square versus roll pattern at convective threshold,” Phys. Rev. Lett., vol. 54, no. 23, pp. 2501–2504, 1985. https://doi.org/10.1103/physrevlett.54.2501.Search in Google Scholar

[9] E. Moses and V. Steinberg, “Convective patterns in a convective binary mixture,” Phys. Rev. Lett., vol. 57, no. 16, pp. 2018–2021, 1986, https://doi.org/10.1103/physrevlett.57.2018.Search in Google Scholar PubMed

[10] Ch. Jung, B. Hucke, and M. Lücke, “Subharmonic bifurcation cascade of pattern oscillations caused by winding number increasing entrainment,” Phys. Rev. Lett., vol. 81, no. 17, pp. 3651–3654, 1998. https://doi.org/10.1103/physrevlett.81.3651.Search in Google Scholar

[11] S. Weggler, B. Huke, and M. Lücke, “Roll and square convection in binary liquids: a few-mode Galerkin model,” Phys. Rev. E, vol. 81, no. 1, p. 016309, 2010. https://doi.org/10.1103/physreve.81.016309.Search in Google Scholar PubMed

[12] Z. C. Hu and X. R. Zhang, “A new oscillatory instability in Rayleigh-Bénard convection of a binary mixture with positive separation ratio,” Phys. Fluids, vol. 33, no. 5, p. 054113, 2021. https://doi.org/10.1063/5.0049247.Search in Google Scholar

[13] J. F. Torres, D. Henry, A. Komiya, and S. Maruyama, “Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls,” J. Fluid Mech., vol. 756, pp. 650–688, 2014, https://doi.org/10.1017/jfm.2014.448.Search in Google Scholar

[14] P. Subramanian, O. Brausch, K. E. Daniels, E. Bodenschatz, T. M. Schneider, and W. Pesch, “Spatio-temporal patterns in inclined layer convection,” J. Fluid Mech., vol. 794, pp. 719–745, 2016, https://doi.org/10.1017/jfm.2016.186.Search in Google Scholar

[15] F. Reetz and T. M. Schneider, “Invariant states in inclined layer convection. part 1. temporal transitions along dynamical connections between invariant states,” J. Fluid Mech., vol. 898, p. A22, 2020, https://doi.org/10.1017/jfm.2020.317.Search in Google Scholar

[16] F. Reetz and T. M. Schneider, “Invariant states in inclined layer convection. part 2. bifurcations and connections between branches of invariant states,” J. Fluid Mech., vol. 898, p. A23, 2020, https://doi.org/10.1017/jfm.2020.318.Search in Google Scholar

[17] O. Shishkina and S. Horn, “Thermal convection in inclined cylindrical containers,” J. Fluid Mech., vol. 790, no. R3, pp. 1–12, 2016, https://doi.org/10.1017/jfm.2016.55.Search in Google Scholar

[18] I. Mercader, O. Batiste, A. Alonso, and E. Knobloch, “Effect of small inclination on binary convection in elongated rectangular cells,” Phys. Rev. E, vol. 99, no. 2, p. 023113, 2019. https://doi.org/10.1103/physreve.99.023113.Search in Google Scholar PubMed

[19] A. Alonso, O. Batiste, and I. Mercader, “Stationary localized solutions in binary convection in slightly inclined rectangular cells,” Phys. Rev. E, vol. 106, no. 5, 2022. https://doi.org/10.1103/physreve.106.055106.Search in Google Scholar PubMed

[20] A. Zebib and M. M. Bou-Ali, “Inclined layer Soret instabilities,” Phys. Rev. E, vol. 79, no. 5, p. 056305, 2009. https://doi.org/10.1103/physreve.79.056305.Search in Google Scholar

[21] A. Alonso, I. Mercader, and O. Batiste, “Time-dependent patterns in quasivertical cylindrical binary convection,” Phys. Rev. E, vol. 97, no. 2, p. 023108, 2018. https://doi.org/10.1103/physreve.97.023108.Search in Google Scholar

[22] A. Alonso, I. Mercader, O. Batiste, and J. M. Vega, “Analyzing slightly inclined cylindrical binary fluid convection via higher order dynamic mode decomposition,” SIAM J. Appl. Dyn. Sys., vol. 21, no. 3, pp. 2148–2186, 2022, https://doi.org/10.1137/21m1447416.Search in Google Scholar

[23] F. Croccolo, F. Scheffold, and A. Vailati, “Effect of a marginal inclination on pattern formation in a binary liquid mixture under thermal stress,” Phys. Rev. Lett., vol. 111, no. 1, p. 014502, 2013, https://doi.org/10.1103/physrevlett.111.014502.Search in Google Scholar

[24] J. M. Mihaljan, “A rigorous expositon of the Boussinesq approximation applicable to a thin fluid layer,” Astrophys. J., vol. 136, pp. 1126–1133, 1962. https://doi.org/10.1086/147463.Search in Google Scholar

[25] J. K. Platten and J. C. Legros, Convection in Liquids, 1st ed Heidelberg, Springer-Verlag, 1984.10.1007/978-3-642-82095-3Search in Google Scholar

[26] I. Mercader, O. Batiste, and A. Alonso, “An efficient spectral code for incompressible flows in cylindrical geometries,” Comput. Fluids, vol. 39, no. 2, pp. 215–224, 2010. https://doi.org/10.1016/j.compfluid.2009.08.003.Search in Google Scholar

[27] S. Hughes and A. Randriamampianina, “An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations,” Intnl J. Num. Meth. Fluids, vol. 28, no. 3, pp. 501–521, 1998. Available at: https://doi.org/10.1002/(sici)1097-0363(19980915)28:3<501::aid-fld730>3.0.co;2-s.10.1002/(SICI)1097-0363(19980915)28:3<501::AID-FLD730>3.0.CO;2-SSearch in Google Scholar

[28] C. G. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Berlin, Heidelberg, Springer, 2007.10.1007/978-3-540-30728-0Search in Google Scholar

[29] C. G. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, 2nd printing. 3rd printing. edition Berlin, Heidelberg, Springer, 2010.Search in Google Scholar

[30] O. Sánchez, X. Ruíz, M. Pujalte, I. Mercader, O. Batiste, and J. Gavaldà, “On the determination of diffusion coefficients in two-component alloys and doped semiconductors. Several implications concerning the International Space Station,” Intnl J. Heat Mass Transfer, vol. 88, pp. 508–518, 2015. https://doi.org/10.1016/j.ijheatmasstransfer.2015.02.061.Search in Google Scholar

[31] O. Sánchez, I. Mercader, O. Batiste, and A. Alonso, “Natural convection in a horizontal cylinder with axial rotation,” Phys. Rev. E, vol. 93, no. 6, pp. 2470–0045, 2016. https://doi.org/10.1103/physreve.93.063113.Search in Google Scholar

[32] J. K. Platten, M. M. Bou Ali, and J. K. Dutrieux, “Enhanced molecular separation in inclined thermogravitational columns,” J. Phys. Chem. B, vol. 107, no. 42, pp. 11763–11767, 2003, https://doi.org/10.1021/jp034780k.Search in Google Scholar

[33] I. Ryzhkov and M. Shevtsova, “On thermal diffusion and convection in multicomponent mixtures with application to the thermogravitational column,” Phys. Fluids, vol. 19, no. 2, 2007. https://doi.org/10.1063/1.2435619.Search in Google Scholar

[34] F. Croccolo, S. Castellini, F. Scheffold, and A. Vailati, “Nonequilibrium solid-solid phase transition in a lattice of liquid jets,” Phys. Rev. E, vol. 98, no. 6, p. 063104, 2018. https://doi.org/10.1103/physreve.98.063104.Search in Google Scholar

[35] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Cham, Springer-Verlag, 2004.10.1007/978-1-4757-3978-7Search in Google Scholar

[36] I. Mercader, O. Batiste, and X. Ruíz, “Quasi-periodicity and chaos in a differentially heated cavity,” Theoret. Comput. Fluid Dyn., vol. 18, no. 2–4, pp. 221–229, 2004. https://doi.org/10.1007/s00162-004-0128-2.Search in Google Scholar

[37] J. Jiménez and P. Moin, “The minimal flow unit in near-wall turbulence,” J. Fluid Mech., vol. 225, pp. 213–240, 1991, https://doi.org/10.1017/s0022112091002033.Search in Google Scholar

[38] B. Wang, R. Ayats, K. Deguchi, F. Mellibovsky, and A. Meseguer, “Self-sustainment of coherent structures in counter-rotating Taylor–Couette flow,” J. Fluid Mech., vol. 951, 2022. https://doi.org/10.1017/jfm.2022.828.Search in Google Scholar

Received: 2024-07-25

Accepted: 2025-01-17

Published Online: 2025-02-11

Published in Print: 2025-04-28

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/jnet-2024-0064

Keywords for this article

direct numerical simulation; inclined cylindrical cells; positive Soret coefficient mixtures; time-dependent binary fluid convection; thermodiffusion