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A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations
A Provably Stable and High-Order Accurate Finite Difference Approximation for the Incompressible Boundary Layer Equations Mojalefa Prince Nchupang In recent years, there has been considerable interest in numerical simulations of incompress-ible flows due to their numerous industrial applications. Thes...
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| Format: | Thesis |
| Language: | English English |
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Department of Mechanical Engineering
2025
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| _version_ | 1867613237309079552 |
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| access_status_str | Open Access |
| author | Nchupang, Mojalefa |
| author2 | Malan, Arnaud |
| author_browse | Malan, Arnaud Nchupang, Mojalefa |
| author_facet | Malan, Arnaud Nchupang, Mojalefa |
| author_sort | Nchupang, Mojalefa |
| collection | Thesis |
| description | A Provably Stable and High-Order Accurate Finite Difference Approximation for the Incompressible Boundary Layer Equations Mojalefa Prince Nchupang In recent years, there has been considerable interest in numerical simulations of incompress-ible flows due to their numerous industrial applications. These include weather forecasting, modeling blood circulation, and analysing airflow around vehicles. Traditional second order nu-merical schemes have been widely used to analyse and predict flow parameters such as velocities and pressure. However, these second order accurate approaches numerically damp flow vortexes while requiring excessive element numbers in the boundary layers. Further, mainstream incom- pressible flow solution schemes augment the incompressible mass conservation equation to avoid the resulting singular coefficient matrix. The two main augmentation approaches are the so-called pressure-based (projection scheme) and density-based (artificial compressibility) methods. These approaches introduce the need for more boundary conditions which place additional constraints on pressure gradients at bound- aries. Finally, the ubiquitous practice of upwinding convective terms when solving incompress- ible flows adds both complexity and non-physical dissipation to the flow solution. The key contributions of this study address these concerns. For this purpose we employ the celebrated incompressible boundary layer equations as a model problem and endeavour to prove the exis- tence of a stable and high order accurate solution without any need for additional augmented pressure/density based equations and without the use of upwinding. We develop a high order accurate method to solve the incompressible boundary layer equa- tions in a provably stable manner. We will derive continuous energy estimates, and then we will proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the con- tinuous counterpart is obtained thus proving stability. We also show that these newly derived boundary conditions remove the singularities associated with the nullspace of the nonlinear dis-crete spatial operator. Numerical experiments that verify the high-order accuracy of the scheme and coincide with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution, as well as that resulting from the solution of the incompressible Navier-Stokes equations. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/41837 |
| institution | University of Cape Town (South Africa) |
| language | English eng |
| last_indexed | 2026-06-10T12:32:57.328Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Department of Mechanical Engineering |
| publisherStr | Department of Mechanical Engineering |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/41837 A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations Nchupang, Mojalefa Malan, Arnaud Nordstrom, Jan Layer equations A Provably Stable and High-Order Accurate Finite Difference Approximation for the Incompressible Boundary Layer Equations Mojalefa Prince Nchupang In recent years, there has been considerable interest in numerical simulations of incompress-ible flows due to their numerous industrial applications. These include weather forecasting, modeling blood circulation, and analysing airflow around vehicles. Traditional second order nu-merical schemes have been widely used to analyse and predict flow parameters such as velocities and pressure. However, these second order accurate approaches numerically damp flow vortexes while requiring excessive element numbers in the boundary layers. Further, mainstream incom- pressible flow solution schemes augment the incompressible mass conservation equation to avoid the resulting singular coefficient matrix. The two main augmentation approaches are the so-called pressure-based (projection scheme) and density-based (artificial compressibility) methods. These approaches introduce the need for more boundary conditions which place additional constraints on pressure gradients at bound- aries. Finally, the ubiquitous practice of upwinding convective terms when solving incompress- ible flows adds both complexity and non-physical dissipation to the flow solution. The key contributions of this study address these concerns. For this purpose we employ the celebrated incompressible boundary layer equations as a model problem and endeavour to prove the exis- tence of a stable and high order accurate solution without any need for additional augmented pressure/density based equations and without the use of upwinding. We develop a high order accurate method to solve the incompressible boundary layer equa- tions in a provably stable manner. We will derive continuous energy estimates, and then we will proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the con- tinuous counterpart is obtained thus proving stability. We also show that these newly derived boundary conditions remove the singularities associated with the nullspace of the nonlinear dis-crete spatial operator. Numerical experiments that verify the high-order accuracy of the scheme and coincide with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution, as well as that resulting from the solution of the incompressible Navier-Stokes equations. 2025-09-18T07:27:31Z 2025-09-18T07:27:31Z 2025 2025-09-18T07:24:36Z Thesis / Dissertation Doctoral PhD http://hdl.handle.net/11427/41837 en eng application/pdf Department of Mechanical Engineering Faculty of Engineering and the Built Environment University of Cape Town |
| spellingShingle | Layer equations Nchupang, Mojalefa A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations |
| thesis_degree_str | Doctoral |
| title | A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations |
| title_full | A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations |
| title_fullStr | A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations |
| title_full_unstemmed | A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations |
| title_short | A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations |
| title_sort | provably stable and high order accurate finite difference approximation for the incompressible boundary layer equations |
| topic | Layer equations |
| url | http://hdl.handle.net/11427/41837 |
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