MSAT: Matrix stability analysis tool for shock-capturing schemes

The simulation of supersonic or hypersonic flows often suffers from numerical shock instabilities if the flow field contains strong shocks, limiting the further application of shock-capturing schemes. In this paper, we develop the unified matrix stability analysis method for schemes with three-point stencils and present MSAT, an open-source tool to quantitatively analyze the shock instability problem. Based on the finite-volume approach on the structured grid, MSAT can be employed to investigate the mechanism of the shock instability problem, evaluate the robustness of numerical schemes, and then help to develop robust schemes. Also, MSAT has the ability to analyze the practical simulation of supersonic or hypersonic flows, evaluate whether it will suffer from shock instabilities, and then assist in selecting appropriate numerical schemes accordingly. As a result, MSAT is a helpful tool that can investigate the shock instability problem and help to cure it.Comment: 18 pages, 6 figure

Contributions to the Development of Entropy-Stable Schemes for Compressible Flows

Entropy-Stable (ES) schemes have gathered considerable attention over the last decade, especially in the context of under-resolved simulations of compressible turbulent flows, where achieving both high-order accuracy and robustness is difficult. ES schemes provide stability in a nonlinear and integral sense: the total entropy of the discrete solution can be made non-decreasing, in agreement with the second principle of thermodynamics. Additionally, the amount of entropy produced by the scheme is known and can be modified, making room for analysis and improvements. This thesis delves into some of the challenges currently limiting their use in practice. The current state of the art solves the compressible Navier-Stokes equations for a single-component perfect gas in chemical and thermal equilibrium. This model is inappropriate in aerospace engineering applications such as hypersonics and combustion, which typically involve chemically reacting gas mixtures far from equilibrium. As a first step towards enabling their use for these applications, we formulated ES schemes for the multicomponent compressible Euler equations. Special care had to be taken as we found out that the theoretical foundations of ES schemes begin to crumble in the limit of vanishing partial densities. The realization that ES schemes can only go as far as their theory led us to review some of it. A fundamental result supporting the development of limiting strategies for high-order methods is the minimum entropy principle for the compressible Euler equations. It states that the specific entropy of the physically relevant weak solution does not decrease. We proved that the same result holds for the specific entropy of the gas mixture in the multicomponent case. While entropy-stability is a valuable property, it does not imply a well-behaved solution. One must recall that the second principle is a prescription on the correct behavior of a system at the global level only. To better understand how ES schemes may or may not improve the quality of the numerical solution, we revisited two classical problems encountered in the development of shock-capturing techniques. First, we studied the receding flow problem, which is a simple setup used to study the anomalous temperature rise, termed "overheating", typically observed in shock reflection and shock interaction calculations. Previous studies showed that the anomaly can be cured if conservation of entropy is enforced, but at the considerable price of total energy conservation. Entropy-Conservative (EC) schemes, a particular instance of ES schemes, can achieve both simultaneously and therefore appeared as a potential solution. We showed that while the overheating is correlated to entropy production, entropy conservation does not necessarily prevent it. Second, we studied the behavior of ES schemes in the low Mach number regime, where shock-capturing schemes are known to suffer from severe accuracy degradation issues. A classic remedy to this problem is the flux-preconditioning technique, which consists in modifying artificial dissipation terms to enforce consistent low Mach behavior. We showed that ES schemes suffer from the same issues and that the flux-preconditioning technique can improve their behavior without interfering with entropy-stability. Furthermore, we demonstrated analytically that these issues stem from an acoustic entropy production field which scales improperly with the Mach number, generating spatial fluctuations that are inconsistent with the equations. An important outgrowth of this effort is the discovery that skew-symmetric dissipation operators can alter the way entropy is produced or conserved locally.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155304/1/gouasmia_1.pd

Development of an Unstructured CFD Solver for External Aerothermodynamics and Nano/Micro Flows

Computational aerothermodynamics is the branch of science which focuses on the computation of the effect of thermodynamic and transport models on aerodynamics and heating. They are widely used for external ow cases. On the other hand, the computation of heat and stress in the design of Nano/Micro Electronic Mechanical Systems from the point of view of a Fluid mechanics engineer is also an important area of study. A generalized computational tool which can simulate the low and high speed flows at both the macro and micro levels is desirable from the perspective of industry, academics and research. For a developing nation, it is extremely important to have such a solver developed indigenously to create self-sufficiency and self-reliance. In this work, a robust three-dimensional density-based general purpose computational fluid dynamics solver was developed in house by our research group. The cell centred finite volume discretization method is used on an unstructured grid, which is more desirable for computation on a complex geometry from the perspective of pre-processing (meshing). Compressible ow solutions obtained from density-based solvers usually do not work well at low speeds where the ow is close to incompressible, unless special schemes and/or special treatments are used. An all-speed algorithm was incorporated using two different methods: (a) preconditioning of the governing equations or (b) through the use of the recently developed SLAU2 all speed convective scheme. The time-stepping discretization is done implicitly, using the lower-upper symmetric-Gauss-Seidel method, which allows us to take a high CFL number during computations. Throughout this work, we have used a second-order accurate reconstruction with limiters to accurately capture the shocks without dispersive error. Turbulence modelling is done using Favre- and Reynolds- Averaged Navier-Stokes equations using the Spalart Allmaras turbulence model. The developed solver is used to solve external ow problems at low and high speeds (hypersonic regimes). In these problems, the thesis focus is on the implementation and testing of an automatic wall function treatment for the Spalart-Allmaras turbulence model. The applicability of the solver is extended to rarefied gas ow regimes in the following manner. Thermal non-equilibrium which exists in the rarefied ow regime is tackled using non-equilibrium boundary conditions in the slip ow regime. The use of non-equilibrium boundary conditions allows the applicability of the Navier-Stokes equation to be extended beyond the continuum to the slip regime. This approach is used to solve problems of hypersonic rarefied flows and nano/micro flows; and for testing and validation of several recently proposed boundary conditions for several problems in the slip ow regime. The main focus of this work is in developing newer numerical methods and on testing and improving other recently proposed numerical techniques that are used for solving the problems covered in this thesis. In the following paragraphs we present the major outcomes of the thesis. The Spalart-Allmaras (SA) is one of the most popular turbulence models in the aerospace CFD community. In its original (low-Reynolds number) formulation it requires a very tight grid (with y+ ' 1) spacing near the wall to resolve the high ow gradients. The use of _ne grids increases the computational cost of the solutions. However, the use of wall functions with an automatic feature of switching from the wall function to the low-Reynolds number approach is an effective solution to this problem. We have extended Menter's automatic wall treatment (AWT), devised for the

A hybrid compressible-incompressible computational fluid dynamics method for richtmyer-meshkov mixing

This paper presents a hybrid compressible–incompressible approach for simulating the Richtmyer–Meshkov instability (RMI) and associated mixing. The proposed numerical approach aims to circumvent the numerical deficiencies of compressible methods at low Mach (LM) numbers, when the flow has become essentially incompressible. A compressible flow solver is used at the initial stage of the interaction of the shock wave with the fluids interface and the development of the RMI. When the flow becomes sufficiently incompressible, based on a Mach number prescribed threshold, the simulation is carried out using an incompressible flow solver. Both the compressible and incompressible solvers use Godunov-type methods and high-resolution numerical reconstruction schemes for computing the fluxes at the cell interfaces. The accuracy of the model is assessed by using results for a two-dimensional (2D) single-mode RMI

Neutrino Physics

The fundamental properties of neutrinos are reviewed in these lectures. The first part is focused on the basic characteristics of neutrinos in the Standard Model and how neutrinos are detected. Neutrino masses and oscillations are introduced and a summary of the most important experimental results on neutrino oscillations to date is provided. Then, present and future experimental proposals are discussed, including new precision reactor and accelerator experiments. Finally, different approaches for measuring the neutrino mass and the nature (Majorana or Dirac) of neutrinos are reviewed. The detection of neutrinos from supernovae explosions and the information that this measurement can provide are also summarized at the end.Comment: 50 pages, contribution to the 2011 CERN-Latin-American School of High-Energy Physics, Natal, Brazil, 23 March-5 April 2011, edited by C. Grojean, M. Mulders and M. Spiropulu. arXiv admin note: text overlap with arXiv:1010.5112, arXiv:1010.4131, arXiv:0704.1800 by other author