Direct discontinuous Galerkin methods and hybrid turbulence modeling for high-speed compressible flows

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2023-08
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Danis, Mustafa Engin
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Yan, Jue
Durbin, Paul
Liu, Hailang
Rossmanith, James
GS, Sidharth
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Mathematics
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The present study considers two aspects of high-speed compressible flow simulations: numerical algorithms for high-order viscous flux construction, and turbulence modeling for high-speed and hypersonic flow. In the first project, a generalized direct discontinuous Galerkin method (DDG) is developed for the simulation of compressible Navier-Stokes (NS) equations. The original version of the DDG method is unable to define a proper numerical viscous flux for the total energy equation of NS, mainly due to the requirement for computing an anti-derivative of the diffusion matrix. This requirement also negatively affects the applicability of the original DDG method to important equations frequently encountered in fluid dynamics, such as turbulence model equations and transport equations of chemically reactive species. In the present study, the numerical viscous flux is calculated only through the numerical flux of the gradient of the conserved variable along a direction vector. The anti-derivative of the diffusion matrix is also no longer needed. This technique is then extended to NS by expressing the viscous flux as a combination of multiple individual diffusion processes. For nonlinear diffusion equations, the nonlinear stability of the new DDG methods is proven, and the high-order accuracy of the proposed method is shown through a wide range of numerical examples, from the heat equation to supersonic flows. In the second project, a hybrid $\ell^2-\omega$ model is developed for hypersonic turbulent boundary layers (TBLs) with cold walls. The first part of this project presents improvements to the Reynolds Averaged Navier-Stokes (RANS) component ($k-\omega$ model) of the hybrid model, which becomes inaccurate at high Mach numbers under intense cooling effects. It is shown that the eddy viscosity prediction of the standard $k-\omega$ models violates the compressible scaling hypotheses. To improve their accuracy, a compressibility correction that modifies the production and destruction rates of the $\omega$-equation is devised. Upon applying the compressibility scaling relations, it is demonstrated that the predicted eddy viscosity profiles of the corrected models agree well with the incompressible DNS data in the near-wall region. This, in turn, improves the accuracy of flow and heat transfer predictions of the $k-\omega$ models. The second part of this project is devoted to the development of a unified $\ell^2-\omega$ framework to compute turbulent stress and heat flux models. By introducing a turbulent thermal length scale, an eddy diffusivity model is defined, which is analogous to the eddy viscosity model. Various strategies are also explored to model subgrid-scale (SGS) model coefficients dynamically. Compressibility corrections to existing models are proposed. It is found that existing models are very sensitive to the grid resolution, especially in hypersonic TBLs with cold walls. It is shown that the grid sensitivity is significantly reduced by reformulating these models and also revising the hybrid blending function.
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