High-order accurate partitioned schemes for conjugate heat transfer with advection-diffusion equations

Huang, Sijia
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Banks, Jeffrey
Henshaw, William D.
Kapila, Ashwani K.
Sahni, Onkar
Schwendeman, Donald W.
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This thesis presents a high-order accurate partitioned scheme for the solution of conjugate heat transfer (CHT) problems. The scheme is based on special second-order accurate CHAMP (Conjugate Heat transfer Advanced Multi-domain Partitioned) interface conditions and extended here to high-order accuracy and to the advection-diffusion equations. The solutions in each material domain are advanced independently with an implicit method, and domains are coupled at the interface using the new CHAMP interface conditions. These conditions are based on usual interface matching conditions involving continuity of temperature and heat flux, together with additional compatibility conditions derived from the governing equations. The new CHAMP conditions are implemented numerically using an optimized Schwarz approach, with a Taylor expansion leading to an effective domain overlap, which significantly improves the convergence rate. The partitioned time-stepping schemes are found to be stable with none or just a few sub-time-step iterations for a wide class of CHT problems. The first part of this thesis focuses on the conjugate heat transfer problem for the diffusion equations and extends the current second-order accurate CHAMP schemes to higher-order accuracy. A detailed fourth-order accurate derivation is given to demonstrate the approach, with a general discussion on deriving a pth-order accurate method. The scheme is then analyzed to determine the optimal coupling coefficients in the CHAMP condition based on solving an optimization problem. The CHAMP iteration is studied in isolation while keeping the time-step fixed, and the iteration amplification factor of the CHAMP scheme is compared to the optimized Schwarz scheme with different overlap widths. The CHAMP time-stepping scheme is then studied for the case where no sub-iteration is taken. The un-iterated CHAMP time-stepping scheme is analyzed to show the overall fourth-order accuracy when a fourth-order accurate CHAMP interface condition is applied. The CHAMP condition is also derived for general curvilinear grids. Numerical results using manufactured solutions on curvilinear grids are given showing the accuracy and stability of the un-iterated CHAMP schemes. To solve problems with large time steps, an adaptive variable sub-iteration CHAMP algorithm is proposed. The new algorithm chooses the number of sub-iterations adaptively based on a measure of the residual for the CHAMP conditions at each time step. A large time step study using the new adaptive algorithm is given to show the robustness of the method. The second part of this thesis generalizes the approach to solving the conjugate heat transfer problems with advection-diffusion equations, also to higher-order. Involving the advective terms in the governing equations and interface conditions complicates the analysis. The pth-order accurate method is derived first, followed by a presentation of the complete CHAMP time-stepping algorithm. A detailed second-order accurate analysis is presented to describe the approach, with a general discussion on analyzing a pth-order accurate method. The CHAMP iteration amplification factor for the advection-diffusion equations is computed numerically using some numerical software packages. The coupling parameters in the CHAMP condition are calculated by solving an optimization problem based on the convergence factor of the sub-iterations. The iteration amplification factors for the CHAMP schemes using the optimal coupling parameters with different orders of accuracy are presented and compared. The convergence factor of a pth-order accurate un-iterated CHAMP time-stepping scheme is also derived and analyzed. Finally, the accuracy of the scheme is verified using several numerical examples.
July 2022
School of Science
Dept. of Mathematical Sciences
Rensselaer Polytechnic Institute, Troy, NY
Rensselaer Theses and Dissertations Online Collection
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