Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page

Modeling, Numerical Analysis, and Predictions for the Detonation of Multi-Component Energetic Solids

Metal powders are often used as an additive to conventional high explosives to enhance the post-detonation blast wave. Piston-impact simulations are commonly utilized to predict performance metrics such as detonation speed and strength, as well as assessing the impact and shock sensitivity of these materials. The system response is strongly influenced by the initial particle size distribution and material composition. Multiphase continuum models have been routinely applied at the macroscale to characterize the detonation of solid high explosives over engineering length scales. Current models lack a description of the physically permissible constitutive relations for mass transfer due to general chemical reactions between multiple components. The model developed in this study is a major extension of one formulated for an inert mixture to include these reactions, which features a rigorous analysis of the energetic processes that identically satisfy the Second Law of Thermodynamics. Additional features of the model include evolutionary equations which predict phase temperature changes due to individual dissipative heating processes. Macroscale models often include nonconservative source terms that prevent the system of evolutionary equations from being posed in divergence form. A significant challenge in the development of numerical methods to solve these model equations is the proper inclusion of discretizations for the nonconservative sources. In the present work a novel modification of a centered finite-volume scheme is formulated, which is a rigorous extension of a conservative method to include nonconservative sources. This numerical scheme was used to perform a parametric study of metalized explosives containing the high explosive HMX (C4H8N8O8), with both inert and reactive aluminum. Wave speeds, structures, and energetics were shown to exhibit a strong dependence on metal grain size, with reactive aluminum significantly accelerating the detonation speed for the mixture above that of pure HMX for d_m \u3c 500 nm

Microscopically implicit-macroscopically explicit schemes for the BGK equation

In this work a new class of numerical methods for the BGK model of kinetic equations is introduced. The schemes proposed are implicit with respect to the distribution function, while the macroscopic moments are evolved explicitly. In this fashion, the stability condi- tion on the time step coincides with a macroscopic CFL, evaluated using estimated values for the macroscopic velocity and sound speed. Thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of ener- getic particles either. With the technique proposed here, the updating of the distribution function requires the solution of a linear system of equations, even though the BGK model is highly non linear. Thus the proposed schemes are particularly effective for high or moderate Mach numbers, where the macroscopic CFL condition is comparable to accuracy requirements. We show results for schemes of order 1 and 2, and the generalization to higher order is sketched

Analysis of the oscillations induced by a supersonic jet applied to produce nanofibers

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGHigh-performance fibers are key components for enhancing the mechanical properties of composite materials. The development of high strength nanofibers augurs the production of new nano-composites with outstanding features. However, the robust production of continuous glass nanofibers that can be feasible processed for efficiently manufacturing nanocomposites is still challenging. Recently, Cofiblas (Continuous Fiberizing by Laser melting and Supersonic dragging) was demonstrated as a technique capable of producing continuous glass nanofibers with unlimited length. Cofiblas process has some similarities with the widely known melt blowing technique for the production of polymeric fibers. In both techniques, the design of the gas nozzle is key to ensure the feasibility of the process since the turbulences of the gas jet may induce strong whipping of the filament. This paper gives novel experimental evidences on the correlation of the supersonic gas jet instabilities with the oscillation of the filament in the melt-blowing and Cofiblas processes, relating these oscillations with the presence of shock waves and unsteadiness in the flow, and gives valuable insight into the use of supersonic jets in the melt blowing process as an effective approach for the formation of nanofibers. A thin 3D-axisymmetric model in OpenFOAM® was put to test by comparing the performance of different solvers which were validated by flow visualization of the exit jet using digital holography (DH). In order to perform a realistic and thorough validation, we simulated the optical measurements of the flow from the CFD simulations of the mass density by Abel transform and numerical differentiation. The application of digital holography as the flow visualization technique makes possible both a precise validation of the density maps obtained from the Abel transformation of the 2D-alike results, and the analysis of the shockwave pattern in the air jet. Conversely, the numerical reconstruction of time-averaged holograms is employed to detect unsteadiness in the flow and to analyze the fiber oscillation, which is essential to assess the stability of the process. Lastly, the analysis and comparison of the vibration of the filament using the basic design and the optimized nozzle demonstrates a clear influence of the shock waves and flow unsteadiness in the stability of the filament.Agencia Estatal de Investigación | Ref. PGC2018-094900-B-I00Xunta de Galicia | Ref. ED431C 2019/23Ministerio de Universidades | Ref. FPU20/0311

Numerical simulation of a highly underexpanded carbon dioxide jet

The underexpanded jets are present in many processes such as rocket propulsion, mass spectrometry, fuel injection, as well as in the process called rapid expansion of supercritical solutions (RESS). In the RESS process a supercritical solution flows through a capillary nozzle until an expansion chamber where the strong changes in the thermodynamic properties of the solvent are used to encapsulate the solute in very fine particles. The research project was focused on the hydrodynamic modeling of an hypersonic carbon dioxide jet produced in the context of the RESS process. The mathematical modeling of the jet was developed using the set of the compressible Navier-Stokes equations along with the generalized Bender equation of state. This set of PDE was solved using an adaptive discontinuous Galerkin discretization for space and the exponential Rosenbrock-Euler method for the time integration. The numerical solver was implemented in C++ using several libraries such as deal.ii and Sacado-Trilinos

On the numerical simulation of compressible flows

In this thesis, numerical tools to simulate compressible flows in a wide range of situations are presented. It is intended to represent a step forward in the scientific research of the numerical simulation of compressible flows, with special emphasis on turbulent flows with shock wave-boundary-layer and vortex interactions. From an academic point of view, this thesis represents years of study and research by the author. It is intended to reflect the knowledge and skills acquired throughout the years that at the end demonstrate the author’s capability of conducting a scientific research, from the beginning to the end, present valuable genuine results, and potentially explore the possibility of real world applications with tangible social and economic benefits. Some of the applications that can take advantage of this thesis are: marine and offshore engineering, combustion in engines or weather forecast, aerodynamics (automotive and aerospace industry), biomedical applications and many others. Nevertheless, the present work is framed in the field of compressible aerodynamics and gas combustion with a clear target: aerial transportation and engine technology. The presented tools allow for studies on sonic boom, drag, noise and emissions reduction by means of geometrical design and flow control techniques on subsonic, transonic and supersonic aerodynamic elements such as wings, airframes or engines. Results of such studies can derive in new and ecologically more respectful, quieter vehicles with less fuel consumption and structural weight reduction. We start discussing the motivation for this thesis in chapter one, which is placed into the upcoming second generation of supersonic aircraft that surely will be flying the skies in no more than 20 years. Then, compressible flows are defined and the equations of motion and their mathematical properties are presented. Navier Stokes equations arise from conservation laws, and the hyperbolic properties of the Euler equations will be used to develop numerical schemes. Chapter two is focused on the numerical simulation with Finite Volumes techniques of the compressible Navier-Stokes equations. Numerical schemes commonly found in the literature are presented, and a unique hybrid-scheme is developed that is able to accurately predict turbulent flows in all the compressible regimens (subsonic, transonic and supersonic). The scheme is applied on the flow around a NACA0012 airfoil at several Mach numbers, showing its ability to be used as a design tool in order to reduce drag or sonic boom, for example. At subsonic regimens, results show excellent agreement with reference data, which allowed the study of the same case at transonic conditions. We were able to observe the buffet phenomenon on the airfoil, which consists of shock-waves forming and disappearing, causing a dramatic loss of aerodynamic performance in a highly unsteady process. To perform a numerical simulation, however, boundary conditions are also required in addition to numerical schemes. A new set of boundary conditions is introduced in chapter three. They are developed for three-dimensional turbulent flows with or without shocks. They are tested in order to assess its suitability. Results show good performance for three-dimensional turbulent flows with additional advantages with respect traditional boundary conditions formulations. Unfortunately, compressible flows usually require high amounts of computational power to its simulation. High speeds and low viscosity result in very thin boundary layers and small turbulent structures. The grid required in order to capture this flow structures accurately often results in unfeasible simulations. This fact motivates the use of turbulent models and wall models in order to overcome this restriction. Turbulent models are discussed in chapter four. The Reynolds-Averaged Navier Stokes (RANS) approach is compared with Large-Eddy Simulation (LES) with and without wall modeling (WMLES). A transonic diffuser is simulated in order to evaluate its performance. Results showed the ability of RANS methods to capture shock-wave positions accurately, but failing in the detached part of the flow. LES, on the other hand, was not able to reproduce shock-waves positions accurately due to the lack of precision on the shock wave-boundary-layer interaction (SBLI). The use of a wall model, nevertheless, allowed to overcome this issue, resulting in an accurate method to capture shock-waves and also flow separation. More research on WMLES is encouraged for future studies on SBLIs, since they allow three-dimensional unsteady studies with feasible levels of computational requirements. With all these tools, we are able to solve at this point any problem concerned with the aerodynamic design of high-speed vehicles which were identified in previous paragraphs. Finally, multi-component flows are discussed in chapter five. Our hybrid scheme is upgraded to deal with multi-component gases and tested in several cases. We demonstrate that with a redefinition of the discontinuity sensor multi-components flows can be solved with low levels of diffusion while being stable in the presence of high scalar gradients. Because of the work of this thesis, a complete numerical approach to the numerical simulation of compressible turbulent multi-component flows with or without discontinuities in a wide range of Reynolds and Mach numbers is proposed and validated. Direct applications can be found in civil aviation (subsonic and supersonic) and engine operation.En aquesta tesis es presenten tècniques numèriques per a la simulació de compressibles en una gran varietat de situacions. L’objectiu és el de donar un pas endavant en la investigació científica de la simulació numèrica de fluids compressibles, amb especial èmfasi en fluxos turbulents amb interaccions entre ones de xoc, capa límit y vòrtex. Algunes de les aplicacions que es poden beneficiar d’aquesta investigació són: enginyeria marítima, combustió en motors, predicció meteorològica, aerodinàmica en la industria automotriu y aeronàutica, aplicacions biomèdiques y moltes altres. Tot i així, aquest treball s’emmarca en el camp de l’aerodinàmica compressible y la combustió de gasos amb un clar objectiu: el transport aeri i la tecnologia de motors. Les ferramentes presentades permeten l’estudi del sònic boom, resistència aerodinàmica, soroll y reducció d’emissions mitjançant el disseny geomètric i tècniques de control de flux en elements aerodinàmics tals com ales o motors en règims subsònics, transsònics i supersònics. Els resultats de tals estudis poden donar lloc a nous vehicles més ecològics, respectuosos amb el medi ambient, més silenciosos, amb menor peso estructural i menys consum de combustible.Postprint (published version

Investigation of a Jacobian-free Newton-Krylov solution to multiphase flows

The current study is focused on investigating a Jacobian-Free Newton-Krylov (JFNK) method to obtain a fully-implicit solution for two phase flows. In the JFNK formulation, the Jacobian matrix is not directly determined potentially leading to major computational savings compared to a simple Newton's solver. Prior to the implementation of JFNK to solve two-phase flow problem, it is utilized to solve the governing equations corresponding to single phase flow. The objectives of the present study are (i) Application of the JFNK method to two-fluid models, (ii) Investigation of the advantages and disadvantages of the method compared to commonly used explicit methods, and (iii) Comparison of the numerical predictions with those obtained by the current version of the Network thermalhydraulics code, CATHENA. The background information required is presented and the numerical setup for each test case is discussed in detail. Three well-known benchmarks are considered, the 1D dam break problem, the water faucet and the oscillating manometer. For single phase flow simulations, the Shallow Water Wave Equations is selected to model the motion of the fluid and a backward Euler scheme is utilized for the temporal discretization along with a central-upwind Godonuv scheme for the spatial discretization. For the two-phase simulations, an isentropic (four equation) two fluid model is chosen. Time discretization is performed by a backward Euler scheme and the AUSM+ scheme is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. For comparison, one explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented for each set of governing equations. A detailed grid and model parameter sensitivity analysis is performed to identify the advantages and disadvantages of JFNK for each case. For all three benchmarks, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL (Courant–Friedrichs–Lewy ) numbers up to 100 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared to the calculations requiring the determination of the Jacobian matrix. This reduction is in the order of 80%

Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE