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

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.

Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

“…Recently, hybrid analytical-numerical approaches, the finite integral transform method and the generalized integral transform technique (GITT) (Cotta and Mikhailov, 1997;Cotta, 1998;An and Su, 2014), has been developed to solve the structural mechanics problems (Li et al, 2019;Zhang et al, 2019a;Ullah et al, 2019;An et al, 2020;Li et al, 2020;He et al, 2021), and heat and fluid problems (An et al, 2013;Fu et al, 2018;Lisboa et al, 2018Lisboa et al, , 2019Machado dos Santos et al, 2022). The GITT method essentially preserves the properties of partial differential equations, which is different from the physics-preserving schemes of the finite difference method and the finite element method (Zhao et al, 2017;Shen et al, 2018;Jiang et al, 2021Jiang et al, , 2022.…”