Purdue University Graduate School
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posted on 2022-07-22, 19:15 authored by Victor de Carvalho Britto SousaVictor de Carvalho Britto Sousa

The present dissertation focuses on the numerical solution of compressible flows with an emphasis on simulations of transitional hypersonic boundary layers. Initially, general concepts such as the governing equations, numerical approximations and theoretical modeling strategies are addressed. These are used as a basis to introduce two innovative techniques, the Quasi-Spectral Viscosity (QSV) method, applied to high-order finite difference settings and the Legendre Spectral Viscosity (LSV) approach, used in high-order flux reconstruction schemes. Such techniques are derived based on the mathematical formalism of the filtered compressible Navier-Stokes equations. While the latter perspective is only typically used for turbulence modeling in the context of Large-Eddy Simulations (LES), both the QSV and LSV subfilter scale (SFS) closure models are capable of performing simulations in the presence of shock-discontinuities. On top of that, the QSV approach is also shown to support dynamic subfilter turbulence modeling capabilities.

QSV’s innovation lies in the introduction of a physical-space implementation of a spectral-like subfilter scale (SFS) dissipation term by leveraging residuals of filter operations, achiev- ing two goals: (1) estimating the energy of the resolved solution near the grid cutoff; (2) imposing a plateau-cusp shape to the spectral distribution of the added dissipation. The QSV approach was tested in a variety of flows to showcase its capability to act interchangeably as a shock capturing method or as a SFS turbulence closure. QSV performs well compared to previous eddy-viscosity closures and shock capturing methods. In a supersonic TGV flow, a case which exhibits shock/turbulence interactions, QSV alone outperforms the simple super- position of separate numerical treatments for SFS turbulence and shocks. QSV’s combined capability of simulating shocks and turbulence independently, as well as simultaneously, effectively achieves the unification of shock capturing and Large-Eddy Simulation.

The LSV method extends the QSV idea to discontinuous numerical schemes making it suitable for unstructured solvers. LSV exploits the set of hierarchical basis functions formed by the Legendre polynomials to extract the information on the energy content near the resolution limit and estimate the overall magnitude of the required SFS dissipative terms, resulting in a scheme that dynamically activates only in cells where nonlinear behavior is important. Additionally, the modulation of such terms in the Legendre spectral space allows for the concentration of the dissipative action at small scales. The proposed method is tested in canonical shock-dominated flow setups in both one and two dimensions. These include the 1D Burgers’ problem, a 1D shock tube, a 1D shock-entropy wave interaction, a 2D inviscid shock-vortex interaction and a 2D double Mach reflection. Results showcase a high-degree of resolution power, achieving accurate results with a small number of degrees of freedom, and robustness, being able to capture shocks associated with the Burgers’ equation and the 1D shock tube within a single cell with discretization orders 120 and higher.

After the introduction of these methods, the QSV-LES approach is leveraged to perform numerical simulations of hypersonic boundary layer transition delay on a 7-half-angle cone for both sharp and 2.5 mm-nose tip radii due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) in the Reynolds number range Rem = 2.43 · 106 – 6.40 · 106 m−1 at the freestream Mach number of M = 7.4. A low-order impedance model was fitted through experimental measurements of acoustic absorption taken at discrete frequencies yielding a continuous representation in the frequency domain that was imposed in the simulations via a broadband time domain impedance boundary condition (TDIBC). The stability of the base flow is studied over impermeable and porous walls via pulse-perturbed axisymmetric simulations with second-mode spatial growth rates matching linear predictions. This shows that the QSV-LES approach is able to dynamically deactivate its dissipative action in laminar portions of the flow making it possible to accurately capture the boundary layer’s instability dynamics. Three-dimensional transitional LES were then performed with the introduction of grid independent pseudorandom pressure perturbations. Comparison against previous experiments were made regarding the frequency content of the disturbances in the transitional region with fairly good agreement capturing the shift to lower frequencies. Such shift is caused by the formation of near-wall low-temperature streaks that concentrate the pressure disturbances at locations with locally thicker boundary layers forming trapped wavetrains that can persist into the turbulent region. Additionally, it is shown that the presence of a porous wall representative of a C/C material does not affect turbulence significantly and simply shifts its onset downstream.




Reynolds-Averaged Modeling and Real-Gas Effects in Hypersonic Turbulent Boundary Layers

United States Department of the Navy

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Degree Type

  • Doctor of Philosophy


  • Mechanical Engineering

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Carlo Scalo

Additional Committee Member 2

Gregory Blaisdell

Additional Committee Member 3

Joe Jewell

Additional Committee Member 4

Alexander Wagner