A decoupled monolithic projection method for natural convection problems

doi:10.1016/j.jcp.2016.03.019

Journal of Computational Physics

Volume 314, 1 June 2016, Pages 160-166

https://doi.org/10.1016/j.jcp.2016.03.019 Get rights and content

Abstract

We propose an efficient monolithic numerical procedure based on a projection method for solving natural convection problems. In the present monolithic method, the buoyancy, linear diffusion, and nonlinear convection terms are implicitly advanced by applying the Crank–Nicolson scheme in time. To avoid an otherwise inevitable iterative procedure in solving the monolithic discretized system, we use a linearization of the nonlinear convection terms and approximate block lower–upper (LU) decompositions along with approximate factorization. Numerical simulations demonstrate that the proposed method is more stable and computationally efficient than other semi-implicit methods, preserving temporal second-order accuracy.

Introduction

Over the decades, the natural convection phenomenon has received considerable attention [1], [2], [3], [4], [5], [6], [7], [8], [9] because of a variety of real-world applications, such as mantle convection, processor cooling device, indoor ventilation, and solar convection zone. Since a typical Rayleigh number (Ra) is extremely large and ranges from 10⁷ to 10²⁴ depending on problem scales [1], it is important to investigate time-dependent dynamics in natural convection, relying on a strong coupling between incompressible flows and heat transfers. This coupling is related to the fact that the thermo-fluid flow is driven by a buoyancy force depending on temperature distribution while the temperature is convected by the background fluid flow. De Vhal Davis [2] numerically solved the natural convection problems based on stream function–vorticity formulations and provided numerical solutions for Ra ranging from 10³ to 10⁶ by using forward Euler discretization in time and second-order central difference in space. Quéré [4] obtained numerical solutions for Ra up to 10⁸ with a pseudo-spectral Chebyshev algorithm and a temporally second-order advancement approach that combines the second-order backward difference formula for the linear diffusion terms with the Adams–Bashforth scheme for the buoyancy and nonlinear convection terms. Armfield et al. [6] investigated natural convection problems using a projection method that has been proven to be effective and widely used for incompressible fluid problems. This projection method is based on a semi-implicit discretization in time, in which the buoyancy and nonlinear convection terms are explicitly treated by applying the Adams–Bashforth scheme, and the linear diffusion terms are implicitly treated by applying the Crank–Nicolson scheme. The explicit treatments of the buoyancy and nonlinear convection terms allow a numerical decoupling between the momentum and energy equations [2], [4], [6], which leads to a severe restriction on the computational time step due to numerical instability.

Recently, in order to mitigate the time step restriction of the stability requirement, several researchers developed more stable and robust methods based on iterative monolithic procedures [7], [8], [9] for solving time-dependent natural convection problems. Zhang et al. [8] established two monolithic nonlinear projection-based numerical schemes with the backward Euler and Crank–Nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization, respectively. Deteix et al. [9] proposed a coupled prediction scheme based on a fixed-point iterative procedure in accordance with a projection method, whereas the Crank–Nicolson scheme is used for all temporal discretizations. As expected, these monolithic methods [8], [9] admit a fairly large computational time step for obtaining stable numerical solutions. However, they require a time-consuming iterative procedure for solving coupled nonlinear discrete systems.

Inspired by the work of Kim et al. [10], who developed a non-iterative implicit projection scheme for solving incompressible Navier–Stokes equations, we extend this idea to natural convection problems. In this short note we propose a decoupled monolithic projection method (DMPM), preserving a temporal second-order accuracy. All terms in the momentum and energy equations are implicitly discretized, based on the Crank–Nicolson scheme along with linearization treatments for the nonlinear convection terms. We use approximate block LU decompositions of the coupled global operator matrix in the linear discretized system to obtain decoupled linear subsystems. All of the above treatments lead to a non-iterative monolithic procedure in the sense that momentum and energy equations are solved only once per time step, while other previously proposed monolithic schemes [7], [8], [9] for natural convection problems are mostly based on iterative approaches. The present DMPM, although it requires solving the Poisson equation twice per time step, allows a much larger computational time step than other semi-implicit methods, and thus saves significant computation time. Validation and performance of the proposed scheme are compared against the scheme that treats the nonlinear convection terms explicitly.

Section snippets

Construction of the DMPM

Under the Boussinesq approximation, governing equations for a three-dimensional (3D) natural convection flow can be written as $\frac{\partial u}{\partial t} + u \cdot \nabla u = - \nabla p + \Pr \nabla^{2} u + F θ,$ $\nabla \cdot u = 0,$ $\frac{\partial θ}{\partial t} + u \cdot \nabla θ = \nabla^{2} θ,$ where $u = {(u_{1}, u_{2}, u_{3})}^{T}$ , p, θ, and $F θ = {(0, 0, \Pr Ra θ)}^{T}$ are the velocity vector, pressure, temperature, and buoyancy force vector, respectively. In the equations above, the following non-dimensionalization is employed: $x_{i} = \frac{{\tilde{x}}_{i}}{L}, u_{i} = \frac{{\tilde{u}}_{i}}{α / L}, t = \frac{\tilde{t}}{L^{2} / α}, p = \frac{\tilde{p}}{ρ α^{2} / L^{2}}, θ = \frac{\tilde{θ} - θ_{c}}{θ_{h} - θ_{c}}, \Pr = \frac{ν}{α}, Ra = \frac{g β (θ_{h} - θ_{c}) L^{3}}{ν α},$ where L, $θ_{h}$ , $θ_{c}$ , α, β, ν, ρ, and

Numerical results

We consider a two-dimensional (2D) natural convection in a cavity to investigate the numerical accuracy, stability, and computational efficiency of the proposed DMPM. Numerical simulations based on a semi-implicit projection method (SIPM) are also performed for comparison purposes. In the SIPM, the buoyancy and linear diffusion terms are implicitly treated by applying the Crank–Nicolson scheme, while the nonlinear convection terms in momentum as well as energy equations are explicitly treated

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP) (NRF-2014R1A2A2A01006544, NRF-2014R1A2A1A11053140, and NRF-20151009350) and in part by the Yonsei University Future–leading Research Initiative of 2014.

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Short noteA decoupled monolithic projection method for natural convection problems

Abstract

Introduction

Section snippets

Construction of the DMPM

Numerical results

Acknowledgements

J. Comput. Phys.

Comput. Fluids

Comput. Math. Appl.

New perspectives in turbulent Rayleigh–Bénard convection

Eur. Phys. J. E

Natural convection of air in a square cavity: a benchmark numerical solution

Int. J. Numer. Methods Fluids

Short note
A decoupled monolithic projection method for natural convection problems