[Paper Review] Modified steady discrete unified gas kinetic scheme for multiscale radiative heat transfer

Xinliang Song, Yue Zhang+3 more|arXiv (Cornell University)|Nov 19, 2022

Thermal Radiation and Cooling Technologies4 citations

TL;DR

This paper proposes a modified steady discrete unified gas kinetic scheme (T-SDUGKS) for multiscale radiative heat transfer by replacing the rectangular rule with a trapezoidal rule in interface flux reconstruction. This change links the characteristic line length to the Courant-Friedrichs-Lewy (CFL) number, removing dependence on the extinction coefficient and enhancing accuracy and robustness, especially for coarse meshes and optical thick regimes, while maintaining efficiency over transient DUGKS for steady problems.

ABSTRACT

In this work, a steady discrete unified gas kinetic scheme (SDUGKS) is proposed to solve the steady radiative transfer equation (RTE), which is an improvement of the original SDUGKS [X. F. Zhou et al., J. Comput. Phys. 423, 109767 (2020)]. The trapezoidal rule other than the rectangular rule used in the original SDUGKS is adopted in the proposed method in the reconstruction of energy flux across cell interface, just as the unsteady DUGKS. By this way, the characteristic line length of the modified SDUGKS establishes a relationship with the Courant-Friedrichs-Lewy (CFL) number in the DUGKS, which guarantees the accuracy of the modified SDUGKS. Furthermore, the characteristic line length is no longer limited by the extinction coefficient like in original SDUGKS. As a result, the modified SDUGKS is more accurate and robust than original SDUGKS, and more efficient than the DUGKS for steady radiation problems. Furthermore, the smooth linear interpolation and the van Leer limiter are used for problems with smooth and discontinuous optical thicknesses, respectively. Several numerical tests with optical thickness varying from optical thin to thick are conducted to validate the present scheme. Numerical results demonstrate that the modified SDUGKS can serve as an effective tool in the study of multiscale steady radiative heat transfer in participating media.

Motivation & Objective

• Address the inaccuracy of the original steady DUGKS (R-SDUGKS) in optical thick regions with coarse meshes due to rectangular rule-based flux reconstruction.
• Overcome the limitation in the original R-SDUGKS where the characteristic line length was strictly constrained by the extinction coefficient.
• Develop a more accurate and robust scheme for steady multiscale radiative transfer that maintains efficiency compared to transient DUGKS.
• Ensure mesh size independence from the photon mean free path (MFP), enabling coarse-mesh simulations without loss of accuracy.
• Validate the method across a range of optical thicknesses, including both smooth and discontinuous optical thickness profiles.

Proposed method

• Replace the rectangular rule in interface flux reconstruction with a trapezoidal rule, aligning the characteristic line length with the CFL number in the DUGKS framework.
• Introduce a modified interface intensity reconstruction that couples absorption, emission, and scattering along the characteristic line of the radiative transfer equation (RTE).
• Use the trapezoidal rule to compute the energy flux across cell interfaces, improving accuracy in both smooth and discontinuous optical thickness scenarios.
• Apply smooth linear interpolation for smooth optical thickness distributions and the van Leer limiter for discontinuous cases to maintain numerical stability.
• Implement the scheme within a finite-volume framework using Gauss-Legendre quadrature for angular discretization and uniform spatial meshes.
• Ensure the method remains consistent with the underlying physics by preserving the balance between flux and source terms across cell interfaces.

Experimental results

Research questions

• RQ1 Can replacing the rectangular rule with a trapezoidal rule in the steady DUGKS improve accuracy for multiscale radiative transfer problems?
• RQ2 Does the modified scheme eliminate the dependence of the characteristic line length on the extinction coefficient, enabling larger mesh sizes?
• RQ3 How does the T-SDUGKS perform in comparison to the original R-SDUGKS and the diamond difference (DD) method in terms of accuracy and robustness for optical thick and thin regimes?
• RQ4 Can the T-SDUGKS maintain accuracy and avoid nonphysical oscillations in problems with large temperature gradients and high optical thickness?
• RQ5 Is the mesh size in the T-SDUGKS independent of the photon mean free path, as claimed for the DUGKS framework?

Key findings

• The T-SDUGKS achieves higher accuracy than the original R-SDUGKS, especially on coarse meshes, as demonstrated by the slab problem with $N_x = 40$ where R-SDUGKS produces incorrect results.
• With $N_x = 40$, the T-SDUGKS produces results closer to the reference solution than R-SDUGKS, confirming improved accuracy and robustness.
• For the large temperature gradient problem at $ au_L = 100$, the T-SDUGKS produces accurate results even at $N_x = 10$, while the DD method exhibits nonphysical oscillations.
• At $ au_L = 500$, the DD method requires finer meshes to suppress oscillations, whereas the T-SDUGKS remains stable and accurate at $N_x = 10$, demonstrating mesh size independence from the MFP.
• The characteristic line length in T-SDUGKS is no longer restricted by the extinction coefficient, enabling larger time steps and improved efficiency.
• The T-SDUGKS converges faster than transient DUGKS and maintains high accuracy across optical thin to thick regimes, confirming its suitability for steady multiscale radiative transfer.