Numerical assessments of high-order accurate shock capturing schemes: Kelvin–Helmholtz type vortical structures in high-resolutions
Abstract
This paper investigates the performance of extensions of the state-of-the-art high-resolution shock-capturing schemes by solving hyperbolic conservation laws in gas dynamics. Such numerical methods used to integrate compressible flow simulations should provide accurate solutions for these flows’ long-time integrations. To this end, several joint solvers are developed within the framework of the reconstruction and flux-splitting approaches using the underlying MUSCL and WENO frameworks. The numerical assessments include testing and evaluation of various interpolation procedures, flux-limiters, Riemann solvers, flux-splitting schemes, and their formal order of accuracy. For temporal integration, a three-stage optimal TVD Runge–Kutta time stepping is employed. The modular development of these joint solvers provides ease in characterizing the solution procedures. The performances of these high-resolution solvers are compared for several carefully selected two-dimensional Riemann problems, including shock and rarefaction waves and joint discontinuities. Based on solutions obtained by all five-point stencil schemes, we demonstrate that the reconstruction-based WENO scheme with the Roe solver is more accurate than all the versions of the flux-splitting WENO solvers tested in this study. We also show that results are highly dependent on the choice of the flux limiter. It is shown that the Euler equations discretized by the fifth-order WENO scheme produce solutions that convect vorticity and create small-scale vortical flow structures, which are usually associated with the high Reynolds number viscous flows. Surprisingly, it is found that these Kelvin–Helmholtz instability-like vortical structures are not captured in any form of the third-order five-point stencil schemes.
Kursat Kara
Associate Professor, Mechanical and Aerospace Engineering
Kursat Kara is an Associate Professor of Mechanical and Aerospace Engineering at Oklahoma State University and principal investigator of the Kara Aerodynamics Research Laboratory. His research spans hypersonic boundary-layer physics, unsteady aerodynamics, and the emerging interface of quantum computing and fluid dynamics. A dedicated educator and mentor, he teaches core and advanced courses—including Fundamentals of Aerodynamics, Computational Fluid Dynamics, Boundary-Layer Theory and Transition, and Quantum Computing—and supervises graduate and undergraduate projects in high-fidelity simulation and data-driven modeling. His work has been funded by NASA, NSF, Oklahoma NASA-EPSCoR, NAVAIR, ANSYS, and IBM Quantum. In 2025, he received the CEAT Excellent Faculty Award and was nominated for both the 2024 Excellent Teacher Award and the 2025 Excellent Faculty Award by OSU’s School of Mechanical and Aerospace Engineering. Dr. Kara earned his Ph.D. from Old Dominion University with a dissertation on hypersonic boundary layer receptivity to acoustic disturbances. He began his career as a research engineer at New England Analytics (supporting Sikorsky Aircraft), then completed a post-doctoral appointment at Penn State in hot jet simulations for aeroacoustics. In 2010, he helped establish the Aerospace Engineering Department at Khalifa University—where he won the President’s Faculty Excellence Award for Outstanding Teaching in 2015—before joining OSU. An active member of AIAA and APS, he served on the AIAA Applied Aerodynamics Technical Committee (2012–2021) and chaired/co-chaired multiple AIAA conferences. He also sits on the editorial board of Nature Scientific Reports and guest-edits its Quantum Computing collection.