Ranjan, RakeshCatabriga, LuciaAraya, Guillermo2024-06-282024-06-282024-01-08Fluids 9 (1): 18 (2024)https://hdl.handle.net/20.500.12588/6456The 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>&tau;</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>&delta;</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>&beta;</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&rsquo;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.2024-06-28