# DEVELOPMENT OF A NON-OVERLAPPING DOMAIN DECOMPOSITION METHOD FOR PROBLEMS WITH BOUNDARY LAYER

• Hongru Li

Student thesis: Phd

### Abstract

Near-wall turbulence modelling has been a popular research object in computational fluid dynamics over the past few decades. Apart from low Reynolds number models and high Reynolds number models, the method of near-wall domain decomposition has been developed to be an effective approach for solving this issue. The basic idea of near-wall domain decomposition is to transfer the boundary condition from the wall to an interface boundary with no/little knowledge of the off-wall solution. It is important to note that unlike conventional methods, near-wall domain decomposition could be applied to both low Reynolds number models and high Reynolds number models, which makes the approach capable of overcoming the limitations of the two models by maintaining a trade-off between accuracy and efficiency. The approach allows for the distance between the interface boundary and the wall to be varied so as to improve either efficiency and accuracy. Previous research has proved that if the governing equation in the near-wall region is of boundary-layer type and locally one-dimensional, the transfer of the boundary condition could be exact. However, in more generalised cases the local interface boundary condition is found to be deficient in capturing the essential non-local feature of turbulent flows in both time and space. The challenge posed by this problem motivates the development of the near-wall domain decomposition approach with non-local interface boundary conditions. In this thesis an effective and novel near-wall domain decomposition approach with non-local interface boundary conditions is developed and tested in application to model equations that simulate high-Reynolds-number flow with a boundary layer. The approach is also compared with existing non-local near-wall domain decomposition approaches in the literature and is proven to be more efficient. The supreme convergence of the approach is analysed theoretically in Poisson's equation and the result is further confirmed to be valid in application to the model equations. Both theoretical and numerical analysis shows that the approach has great potential to be applied to near-wall turbulence modelling. The non-local interface boundary condition of the approach is obtained by approximating the non-local Steklov-Poincar\'{e} operators, which are decomposed into a few basic units under local Taylor expansions. The approximation is proved to be effective in retaining the non-local nature of the operators and is easy to implement. The convergence analysis is performed in two steps: first ignore the boundary effect when analysing the product of Steklov-Poincar\'{e} operators applied to given functions (standard analysis); next restore the ignored boundary effect. These two steps reflect the influence of the governing equation and boundary conditions respectively. Together they describe the complete non-local nature of Steklov-Poincar\'{e} operators in application to a given problem. As an advanced numerical method to solve large linear systems: the generalised minimum residual method is widely used in the project and its convergence is studied. The convergence of the generalised minimum residual method is found to be determined by the eigenvalue distribution of the coefficient matrix of the linear system. The result is intended to serve as a theoretical foundation to evaluate the comparative computing cost of applying the near-wall domain decomposition approach as opposed to the one-block approach.
Date of Award 31 Dec 2023 English The University of Manchester Dominique Laurence (Supervisor) & Sergey Utyuzhnikov (Supervisor)

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