Browsing by Author "Nordstrom, Jan"

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    A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations
    (2025) Nchupang, Mojalefa; Malan, Arnaud; Nordstrom, Jan
    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.
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    Stable and high order accurate finite difference method for the incompressible laminar boundary layer equations
    (2020) Nchupang, Mojalefa Prince; Malan, Arnaud G; Nordstrom, Jan
    Numerical simulations of incompressible flows are unequivocally important due to their numerous industrial applications. These applications ranges from the large-scale fluid's flow modelling such as aerodynamics [1], atmospheric-ocean modelling [2] to a simple pipe flows in the petroleum industry [3]. This study is devoted to develop a provably stable and high order approximation for the incompressible laminar boundary layer equations. A new set of energystable boundary conditions are derived using the energy method. It is shown that both the weak and strong implementation of these boundary conditions yields an energy estimate. The semidiscrete problem is formulated by discretizing the continuous spatial derivatives using high order finite difference approximations on summation-by-parts form. The boundary conditions are implemented weakly using the simultaneous approximation terms methods. The discrete energy estimate is derived by mimicking the continuous analysis and hence, the numerical approximation is proved to be stable. The accuracy and linear stability of the developed scheme is also validated by solving the celebrated laminar flat plate flow problem. This is done by injecting the Blasius solution into the coefficient matrix as well as weak boundary conditions