A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations - Nchupang, Mojalefa (Department of Mechanical Engineering, 2025)
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.