On a simple and effective thermal open boundary condition for convective heat transfer problems

Abstract

We present an effective thermal open boundary condition for convective heat transfer problems on domains involving outflow/open boundaries. This boundary condition is energy-stable, and it ensures that the contribution of the open boundary will not cause an "energy-like" temperature functional to increase over time, irrespective of the state of flow on the open boundary. It is effective in coping with thermal open boundaries even in flow regimes where strong vortices or backflows are prevalent on such boundaries, and it is straightforward to implement. Extensive numerical simulations are presented to demonstrate the stability and effectiveness of our method for heat transfer problems with strong vortices and backflows occurring on the open boundaries. Simulation results are compared with previous works to demonstrate the accuracy of the presented method.

An unconditionally energy-stable scheme for the convective heat transfer equation

“…The spatial and temporal convergence rates of the method are first demonstrated and then the effects of the algorithmic parameters on the simulation results will be studied, especially the stability and the accuracy at large time step sizes will be demonstrated. A survey of literature indicates the semi-implicit schemes based on the BDF-2 and based on the Crank–Nicolson/Adams–Bashforth (CNAB-2) scheme are the most commonly used methods for heat transfer problems; see Liu et al (2020), Zheng et al (2015), Rakotondrandisa et al (2020), Pan et al (2021) and Qaddah et al (2022) for BDF and Yoon et al (2020), Son and Park (2021) and Seo et al (2020) for CNAB. Therefore, we also provide a comparison of the current scheme with the semi-implicit BDF and CNAB schemes in the following tests.…”

“…Within a time step, the semi-implicit scheme only requires solving linear algebraic systems with a constant and time-independent coefficient matrix that can be precomputed. Thanks to the low computational cost, the semi-implicit type schemes have been widely used in the simulations of convective heat transfer in fluid flow (Woodruff, 2022; Bhinder et al , 2012; Chandra and Chhabra, 2011; Wang and Pepper, 2009; Feldman, 2018; Soo et al , 2017; Liu et al , 2020). A downside of the schemes is their conditional stability.…”

“…The two semi-implicit methods used are second-order backward differentiation formula (BDF-2) and Crank–Nicolson Adam–Bashforth (CNAB-2) methods. Semi-implicit BDF-2 algorithm for solving the convective heat transfer equation Here, we use the same notation as in the main text. Given T n and u n+1 , we can compute T n +1 based on the BDF-2 scheme (Liu et al , 2020; Qaddah et al , 2022), which reads as follows: In the above equations, the detailed definition of all the related variables can be found in Section 2.3. Furthermore, we can use the C 0 -continuous high-order spectral elements for spatial discretizations.…”

“…Here, we use the same notation as in the main text. Given T n and u n+1 , we can compute T n +1 based on the BDF-2 scheme (Liu et al , 2020; Qaddah et al , 2022), which reads as follows: …”

Numerical considerations for advection‐diffusion problems in cardiovascular hemodynamics

“…Similar scenarios of numerical instability can arise in scalar advection‐diffusion systems 19 . Despite the numerous reports on backflow stabilization for flow problems 4,5,7‐16,20 and 2D heat mass transfer, 21‐25 these strategies have not been adopted for 3D cardiovascular scalar advection‐diffusion systems. Instead, to circumvent the numerical instability issues in the presence of backflow, mass transport models have resorted to unphysical approaches such as the imposition of arbitrary Dirichlet boundary conditions at the outlet faces, 26,27 artificial extensions of the computational domain 28 that seek to regularize the flow profile, or an artificial increase in the diffusivity of the scalar 29,30 .…”