A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations
| dc.contributor.author | Ranjan, Rakesh | |
| dc.contributor.author | Catabriga, Lucia | |
| dc.contributor.author | Araya, Guillermo | |
| dc.date.accessioned | 2024-06-28T15:07:35Z | |
| dc.date.available | 2024-06-28T15:07:35Z | |
| dc.date.issued | 2024-01-08 | |
| dc.date.updated | 2024-06-28T15:07:35Z | |
| dc.description.abstract | The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>p</mi></mrow></semantics></math></inline-formula> stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>p</mi></mrow></semantics></math></inline-formula> framework. We introduce revised stabilization parameters, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>τ</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi><mi>g</mi></mrow></msub></semantics></math></inline-formula>, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>: the first is described with entropy variables, and the other is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mi>Z</mi><mi>β</mi></mrow></semantics></math></inline-formula> parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3.0</mn></mrow></semantics></math></inline-formula> and numerically validate the shock standoff distance with results obtained from NASA Overflow <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.2</mn></mrow></semantics></math></inline-formula> code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries. | |
| dc.identifier | doi: 10.3390/fluids9010018 | |
| dc.identifier.citation | Fluids 9 (1): 18 (2024) | |
| dc.identifier.uri | https://hdl.handle.net/20.500.12588/6456 | |
| dc.title | A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations |