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Vector eigenfunction expansion in the integral transform solution of transient natural convection

Kleber Marques Lisboa (Department of Mechanical Engineering, POLI/COPPE/UFRJ, Rio de Janeiro, Brazil)
Jian Su (Department of Nuclear Engineering, POLI/COPPE/UFRJ, Rio de Janeiro, Brazil)
Renato M. Cotta (Nuclear and Technological Development, DGDNTM, Brazilian Navy and Department of Mechanical Engineering, POLI/COPPE/UFRJ, Rio de Janeiro, Brazil)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 15 August 2019

Issue publication date: 11 September 2019

Abstract

Purpose

The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation.

Design/methodology/approach

The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme.

Findings

A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification.

Originality/value

A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

Keywords

Acknowledgements

The authors are grateful for the financial support offered by the Brazilian Government agencies CNPq (projects no. 401237/2014-1 and no. 207750/2015-7), CAPES-INMETRO, and FAPERJ. This work was partially sponsored by ETN - Eletrobrás Termonucleares S.A., Project no. PEM-17909 with the COPPETEC Foundation. RMC is also grateful to the Leverhulme Trust for the Visiting Professorship (VP1-2017-028) and to the kind hospitality of the Department of Mechanical Engineering, University College London (UCL), UK.

Citation

Lisboa, K.M., Su, J. and Cotta, R.M. (2019), "Vector eigenfunction expansion in the integral transform solution of transient natural convection", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 8, pp. 2684-2708. https://doi.org/10.1108/HFF-10-2018-0543

Publisher

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Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited