Publications

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The use of p-, and ℎp-nonconformal interfaces enables greater geometric flexibility in performing computational science simulations, especially when relying on efficient tensor-product-based high-order Summation-by-Parts element schemes. For high-speed compressible computational fluid dynamics, the underlying numerical method must be conservative such that the discretization of the governing equations satisfies the Rankine Hugoniot relations. This paper extends the conservative nonconformal interface method of [1] to Summation-by-Parts elements with face quadratures of degree less than 2p, specifically allowing which allows the use of tensor-product elements on the Legendre-Gauss-Lobatto nodes, which are accurate up to degree 2p − 1. This formulation does not satisfy the inner-product preservation property of [1], but nonetheless remains conservative, entropy stable, and free-stream preserving. Mathematical theory is developed to determine the required accuracy of the mortar grid quadrature rule, and numerical results verify the mathematical results.

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Entropy stable numerical methods for compressible flow have been demonstrated to exhibit better robustness than purely linearly stable methods and need less overall artificial dissipation for long simulations in subsonic and transonic flows. In this work we seek to extend these benefits to multicomponent, multitemperature flows in thermochemical nonequilibrium such as combustion and hypersonic flight. We first derive entropy functions that symmetrize the governing equations and allow stability proofs for such systems. The impact of diffusion model selection on provable entropy stability is considered in detail, including both rigorous models of irreversible thermodynamics and simplified models of greater practical interest. Based on the proven entropy functions we develop affordable, entropy conservative two-point flux functions for solution in conservation form. We derive entropy conservative fluxes for calorically and thermally perfect mixtures, with heat capacities described by either polynomials of the temperature or formulas from statistical thermodynamics.

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A new cell-centered third-order entropy stable Weighted Essentially Non-Oscillatory (SS-WENO) finite difference scheme in multi-block domains is developed for compressible flows. This new scheme overcomes shortcomings of the conventional SSWENO finite difference scheme in multi-domain problems by incorporating non-dissipative Simultaneous Approximation Term (SAT) penalties into the construction of a dual flux. The stencil of the generalized dual flux allows for full stencil biasing across the interface while maintaining the nonlinear stability estimate. We demonstrate the shock capturing improvement across multi-block domain interfaces using the generalized SSWENO in comparison to the conventional entropy stable high-order finite difference with interface penalty in shock problems. Furthermore, we test the new scheme in multi-dimensional turbulent flow problems to assess the accuracy and stability of the multi-block domain formulation.

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A new cell-centered third-order entropy stable Weighted Essentially Non-Oscillatory (SS-WENO) finite difference scheme in multi-block domains is developed for compressible flows. This new scheme overcomes shortcomings of the conventional SSWENO finite difference scheme in multi-domain problems by incorporating non-dissipative Simultaneous Approximation Term (SAT) penalties into the construction of a dual flux. The stencil of the generalized dual flux allows for full stencil biasing across the interface while maintaining the nonlinear stability estimate. We demonstrate the shock capturing improvement across multi-block domain interfaces using the generalized SSWENO in comparison to the conventional entropy stable high-order finite difference with interface penalty in shock problems. Furthermore, we test the new scheme in multi-dimensional turbulent flow problems to assess the accuracy and stability of the multi-block domain formulation.