Recently it has become increasingly clear that the role of numerical dissipation, originating from the discretization of governing equations of fluid dynamics, rarely can be ignored regardless of the formal order of accuracy of a numerical scheme used in either explicit or implicit Large Eddy Simulations (LES). The numerical dissipation inhibits the predictive capabilities of LES whenever it is of the same order of magnitude or larger than the subgrid-scale (SGS) dissipation. The need to estimate the numerical dissipation is most pressing for low-order methods employed by commercial CFD codes. Following the recent work of Schranner et al. (2015) the equations and procedure for estimating the numerical dissipation rate and the numerical viscosity
At high Reynolds numbers the use of explicit in time compressible flow simulations with spectral/hp element discretization can become significantly limited by time step. To alleviate this limitation we extend the capability of the spectral/hp element open-source software framework, Nektar++, to include an implicit discontinuous Galerkin compressible flow solver. The integration in time is carried out by a singly diagonally implicit Runge-Kutta method. The non-linear system arising from the implicit time integration is iteratively solved by the Jacobian-free Newton Krylov (JFNK) method. A favorable feature of the JFNK approach is its extensive use of the explicit operators available from the previous explicit in time implementation. The functionalities of different building blocks of the implicit solver are analyzed from the point of view of software design and placed in appropriate hierarchical levels in the C++ libraries. In the detailed implementation, the contributions of different parts of the solver to computational cost, memory consumption and programming complexity are also analyzed. A combination of analytical and numerical methods is adopted to simplify the programming complexity in forming the preconditioning matrix. The solver is verified and tested using cases such as manufactured compressible tex, turbulent flow over a circular cylinder at Re = 3900 and shock wave boundary-layer interaction. The results show that the implicit solver can speed-up the simulations while maintaining good simulation accuracy.
The thermodynamic regime of a complex mixture depends on the composition, the pressure and the temperature; the spinodal locus separates the regime of thermodynamic instability from the remainder of the phase space. Since diffusion is one of the phenomena affecting the local chemical composition, the first focus is here on evaluating diffusion models in the context of high-pressure (high-$p$) multispecies mixing and combustion. It is shown that the diffusion model equations previously used to create two high-$p$ direct numerical simulation (DNS) databases can reproduce classical experimental observations of uphill diffusion in an accurate spatiotemporal manner, whereas the popular model which has a diagonal diffusion matrix and uses a velocity correction lacks spatiotemporal accuracy. Further, a mathematical formalism is used to compute the spinodal locus for mixtures for which either experimental data or previous computations from the literature are available, and it is shown that the agreement of the present calculations with that previously existing information is excellent. Using the spinodal-calculation mathematical formalism, the aforementioned DNS databases are then examined to determine the thermodynamic regime of the mixture at important stages of the simulations. In the first subset of the DNS databases that portrays mixing of five species under high-$p$ conditions, this stage is that of the transitional state representing the individual time station at which each simulation, having been initiated in a laminar state, transitions to a state having turbulent characteristics. In the second subset of the DNS databases that portrays high-$p$ turbulent combustion, this stage represents the individual time station at the peak $p$ achieved during the calculations. In both databases, the influence of the initial Reynolds number, the free-stream composition and the free-stream $p$ is studied. The results show that in all cases the mixture is in the single-phase regime. The present DNS databases have only five species, but it is shown that the methodology for computing the spinodal locus can be applied to very complex mixtures, with examples given for a twelve-species mixture and surrogate diesel fuels, thereby boding well for determining the thermodynamic regime of practical mixtures in high-$p$ turbulent flow simulations for engineering applications. According to these calculations, diesel-fuel surrogates are always in the single-phase regime at injection-conditions $p$ and temperatures existing in diesel-engine combustion chambers.
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