A time viscosity-splitting method for incompressible flows with temperature-dependent viscosity and thermal conductivity

El-Amrani, Mofdi; Obbadi, Anouar; Seaid, Mohammed; Yakoubi, Driss

Abstract

A fractional-step method is proposed and analyzed for solving the incompressible thermal Navier–Stokes equations coupled to the convection–conduction equation for heat transfer with a generalized source term for which the viscosity and thermal conductivity are temperature-dependent under the Boussinesq assumption. The proposed method consists of four steps all based on a viscosity-splitting algorithm where the convection and diffusion terms of both velocity and temperature solutions are separated while a viscosity term is kept in the correction step at all times. This procedure preserves the original boundary conditions on the corrected velocity and it removes any pressure inconsistencies. As a main feature, our method allows the temperature to be transported by a non-divergence-free velocity, in which case we show how to handle the subtle temperature convection term in the error analysis and establish full first-order error estimates for the velocity and the temperature solutions and 1/2-order estimates for the pressure solution in their appropriate norms. The theoretical results are examined by an accuracy test example with known analytical solution and using a benchmark problem of Rayleigh–Bénard convection with temperature-dependent viscosity and thermal conductivity. We also apply the method for solving a problem of unsteady flow over a heated airfoil. The obtained results demonstrate the convergence, accuracy and applicability of the proposed time viscosity-splitting method.

Citation

El-Amrani, M., Obbadi, A., Seaid, M., & Yakoubi, D. (2024). A time viscosity-splitting method for incompressible flows with temperature-dependent viscosity and thermal conductivity. Computer Methods in Applied Mechanics and Engineering, 429, Article 117103. https://doi.org/10.1016/j.cma.2024.117103

• Journal Article Type: Article
• Acceptance Date: May 27, 2024
• Online Publication Date: Jun 12, 2024
• Publication Date: Sep 1, 2024
• Deposit Date: Jul 12, 2024
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 429
• Article Number: 117103
• DOI: https://doi.org/10.1016/j.cma.2024.117103
• Public URL: https://durham-repository.worktribe.com/output/2493126