Shock capturing has been a challenge for computational fluid dynamicists over the years. This article deals with discontinuous Galerkin method to solve the hyperbolic equations in which solutions may develop discontinuities in finite time. The high order discontinuous Galerkin method combining the basis of finite volume and finite element methods has shown a lot of attractive features for a wide range of applications. Various techniques proposed in the literature to deal with discontinuities basically reduce the order of interpolation in the region around these discontinuities. The accuracy of the scheme therefore may be degraded in the vicinity of the shock. The proposed method resolves the discontinuities presented in the solution by applying viscosity into the shock-containing elements. The discontinuity is spread over a distance and is well approximated in the space of interpolation functions. The technique of adding viscosity to the system and the indicator based on the expansion coefficients of the solution are presented. A number of numerical examples in one and two dimensions is carried out to show the capability of the scheme for shock capturing. / Singapore-MIT Alliance (SMA)

Aquesta tesi doctoral proposa formulacions de Galerkin Discontinu (DG) d’alt ordre per la captura de shocks, obtenint alhora solucions altament precises per problemes de flux compressible. En les últimes dècades, la investigació en els mètodes de DG ha estat en constant creixement. L'èxit dels mètodes DG en problemes hiperbòlics ha conduit el seu desenvolupament en lleis de conservació no lineals i problemes de convecció dominant. Entre els avantatges dels mètodes DG, destaquen la seva estabilitat inherent i les propietats locals de conservació. D'altra banda, els mètodes DG estan especialment dissenyats per l’ús aproximacions d'ordre superior. De fet, en els últims anys s'ha demostrat que la resolució de problemes de convecció dominant ja no es restringeix només a elements d'ordre inferior. De fet, es necessiten models numèrics d'alta precisió per aconseguir prediccions altament fiables dins la dinàmica de fluids computacional (CFD). En aquest context es presenten i discuteixen dos tècniques de captura de shocks. En primer lloc, es presenta una tècnica novedosa i senzilla basada en la introducció d'una nova base de funcions de forma. Aquesta base té la capacitat de canviar a nivell local entre una interpolació contínua o discontínua, depenent de la suavitat de la funció que es vol aproximar. En presència de xocs, les discontinuïtats introduïdes dins l’element permeten incloure l'estabilització necessària gràcies a l’ús dels fluxos numèrics, i alhora exploten les propietats intrínsiques del mètodes DG. En conseqüència, es poden utilitzar malles grolleres amb elements d’ordre superior. Amb aquestes discretitzacions i, utilitzant el mètode proposats, els xocs queden continguts a l’interior de l’element i per tant, és possible evitar l’ús de tècniques de refinament adaptatiu de la malla, alhora que es manté la localitat i compacitat dels esquemes DG. En segon lloc, es proposa una tècnica clàssica i, aparentment simple: la introducció de la viscositat artificial. Primerament es realitza un estudi detallat per al cas unidimensional. S’obté una viscositat d’alta precisió que escala segons el valor hk amb 1 ≤ k ≤ p i essent h la mida de l’element. En conseqüència, s’obté un xoc amb amplitud del mateix ordre. Seguidament, l'estudi de la viscositat unidimensional obtenida s'extén al cas multidimensional per a malles triangulars. L'extensió es basa en la projecció de la viscositat unidimensional en unes determinades direccions espacials dins l’element. Es demostra de manera consistent que la viscositat introduïda és, com a molt, del mateix ordre que la resolució donada per la discretització espacial, és a dir, h/p. El mètode és especialment eficient per aproximacions de Galerkin discontinu d’alt ordre, per exemple p≥ 3. Les dues metodologies es validen mitjançant una àmplia selecció d’exemples numèrics. En alguns exemples, els mètodes proposats permeten una reducció en el nombre de graus de llibertat necessaris per capturar xocs acuradament de fins i tot un ordre de magnitud, en comparació amb mètodes estàndar de refinament adaptatiu amb aproximacions de baix ordre. / This thesis proposes shock-capturing methods for high-order Discontinuous Galerkin (DG) formulations providing highly accurate solutions for compressible flows. In the last decades, research in DG methods has been very active. The success of DG in hyperbolic problems has driven many studies for nonlinear conservation laws and convection-dominated problems. Among all the advantages of DG, their inherent stability and local conservation properties are relevant. Moreover, DG methods are naturally suited for high-order approximations. Actually, in recent years it has been shown that convection-dominated problems are no longer restricted to low-order elements. In fact, highly accurate numerical models for High-Fidelity predictions in CFD are necessary. Under this rationale, two shock-capturing techniques are presented and discussed. First, a novel and simple technique based on on the introduction of a new basis of shape functions is presented. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization thanks to the numerical fluxes, thus exploiting DG inherent properties. Large high-order elements can therefore be used and shocks are captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Second, a classical and, apparently simple, technique is advocated: the introduction of artificial viscosity. First, a one-dimensional study is perfomed. Viscosity of the order O(hk) with 1≤ k≤ p is obtained, hence inducing a shock width of the same order. Second, the study extends the accurate one-dimensional viscosity to triangular multidimensional meshes. The extension is based on the projection of the one-dimensional viscosity into some characteristic spatial directions within the elements. It is consistently shown that the introduced viscosity scales, at most, withthe DG resolutions length scales, h/p. The method is especially reliable for highorder DG approximations, say p≥3. A wide range of different numerical tests validate both methodologies. In some examples the proposed methods allow to reduce by an order of magnitude the number of degrees of freedom necessary to accurately capture the shocks, compared to standard low order h-adaptive approaches.

Supersonic flows over both simple and complex geometries involve features over a wide spectrum of spatial and temporal scales, whose resolution in a numerical solution is of significant importance for accurate predictions in engineering applications. While CFD has been greatly developed in the last 30 years, the desire and necessity to perform more complex, high fidelity simulations still remains. The present thesis has introduced two major innovations regarding the fidelity of numerical solutions of the compressible \ns equations. The first one is the development of new a priori mesh quality measures for the Finite Volume (FV) method on mixed-type (quadrilateral/triangular) element meshes. Elementary types of mesh distortion were identified expressing grid distortion in terms of stretching, skewness, shearing and non-alignment of the mesh. Through a rigorous truncation error analysis, novel grid quality measures were derived by emphasizing on the direct relation between mesh distortion and the quality indicators. They were applied over several meshes and their ability was observed to identify faithfully irregularly-shaped small or large distortions in any direction. It was concluded that accuracy degradation occurs even for small mesh distortions and especially at mixed-type element mesh interfaces the formal order of the FV method is degraded no matter of the mesh geometry and local mesh size. Therefore, in the present work, the high-order Discontinuous Galerkin (DG) discretization of the compressible flow equations was adopted as a means of achieving and attaining high resolution of flow features on irregular mixed-type meshes for flows with strong moving shocks. During the course of the thesis a code was developed and named HoAc (standing for High Order Accuracy), which can perform via the domain decomposition method parallel $p$-adaptive computations for flows with strong shocks on mixed-type element meshes over arbitrary geometries at a predefined arbitrary order of accuracy. In HoAc in contrast to other DG developments, all the numerical operations are performed in the computational space, for all element types. This choice constitutes the key element for the ability to perform $p$-adaptive computations along with modal hierarchical basis for the solution expansion. The time marching of the DG discretized Navier-Stokes system is performed with the aid of explicit Runge-Kutta methods or with a matrix-free implicit approach. The second innovation of the present thesis, which is also based on the choice of implementing the DG method on the regular computational space, is the development of a new $p$-adaptive limiting procedure for shock capturing of the implemented DG discretization. The new limiting approach along with positivity preserving limiters is suitable for computations of high speed flows with strong shocks around complex geometries. The unified approach for $p$-adaptive limiting on mixed-type meshes is achieved by applying the limiters on the transformed canonical elements, and it is fully automated without the need of ad hoc specification of parameters as it has been done with standard limiting approaches and in the artificial dissipation method for shock capturing. Verification and validation studies have been performed, which prove the correctness of the implemented discretization method in cases where the linear elements are adequate for the tessellation of the computational domain both for subsonic and supersonic flows. At present HoAc can handle only linear elements since most grid generators do not provide meshes with curved elements. Furthermore, p-adaptive computations with the implemented DG method were performed for a number of standard test cases for shock capturing schemes to illustrate the outstanding performance of the proposed $p$-adaptive limiting approach. The obtained results are in excellent agreement with analytical solutions and with experimental data, proving the excellent efficiency of the developed shock capturing method for the DG discretization of the equations of gas dynamics. / -

Turbulent combustion has a profound effect on the way we live our lives; homes and businesses predominantly rely on power generated by burning some form of fuel, and the vast majority of transport of passengers and cargo are driven by combustion. Fossil fuels remain readily available and relatively cheap, and so will continue to power the modern world for the foreseeable future. Combustion of fossil fuels produces emissions that detrimentally affect air quality, particularly in highly-populated cities, and are also widely believed to be contributing to global climate change. Consequently, increasing attention is being focused on alternative fuels, increased efficiency and reduced emissions. One alternative fuel is hydrogen, which introduces challenges in end-usage, storage and safety that are not encountered with more conventional fuels. Advances in computational power and software technology means that numerical simulation has a growing role in the development of combustors and safety evaluation. Despite these advances, many challenges remain; the broad range of time and length scales involved are coupled with complex thermodynamics and chemistry on top of turbulent fluid mechanics, which means that detailed simulations of even relatively-simple burners are still prohibitively expensive. Engineering turbulent flame models are required to reduce computational expense, and the challenge is to retain as much of the flow physics as possible. Furthermore, the choice of numerical approach has a significant effect on the quality of simulation, and different target applications place different demands on the numerical scheme. In the case of hydrogen explosion, the approach needs to be able to capture a range of physical behaviours including turbulence, low-speed deflagration, high-speed shock waves and potentially detonations. One such numerical approach that has enjoyed widespread success is finite volumes schemes based on the Godunov method. These methods perform well at all speeds, and have positive shock-capturing capability, but recent studies have demonstrated difficulties with numerical stability for more complex thermodynamics, specifically in the case of fully-conservative methods for multi-component fluids with varying thermodynamic properties. A recent development is the so-called double-flux method, which retains many of the positive properties of the fully-conservative approaches and does not suffer from the same numerical instabilities, but is quasi-conservative and involves additional computational expense. The present work consolidates the state-of-the-art in the literature, and considers two equation sets, based on mass fraction and volume fraction, respectively, along with fully-conservative and quasiconservative schemes. Comprehensive validation and evaluation of the different approaches is presented. It was found that both quasi-conservative approaches performed well, with a better conservative behaviour for the quasi-conservative volume fraction, but a better stability for the quasi-conservative mass fraction. Finally, the numerical tool developed is applied to turbulent combustion of premixed hydrogen in the context of the semi-confined experiments from the University of Sydney. The LES results showed an good overall agreement with the experimental data, and the critical parameters such as overpressure and flame speed where globally well captured, highlighting the large potential of LES for safety analysis.

Truncation errors and computational cost are obstacles that still hinder large-scale applications of the Computational Fluid Dynamics method. The discontinuous Galerkin method is one of the high-order schemes utilized extensively in recent years, which is locally conservative, stable, and high-order accurate. Besides that, it can handle complex geometries and irregular meshes with hanging nodes. In this document, the nondimensional compressible Euler equations and Reynolds- Averaged Navier-Stokes equations are discretized by discontinuous Galerkin methods with a two-equations turbulence model on both structured and unstructured meshes. The traditional equation of state for an ideal gas model is substituted by a multispecies thermodynamics model in order to complete the governing equations. An approximate Riemann solver is used for computing the convective flux, and the diffusive flux is approximated with some internal penalty based schemes. The temporal discretization of the partial differential equations is either performed explicitly with the aid of Rung-Kutta methods or with semi-implicit methods. Inspired by the artificial viscosity diffusion based limiter for shock-capturing method, which has been extensively studied, a novel and robust technique based on the introduction of mass diffusion to the species governing equations to guarantee that the species mass fractions remain positive has been thoroughly investigated. This contact-surface-capturing method is conservative and a high order of accuracy can be maintained for the discontinuous Galerkin method. For each time step of the algorithm, any trouble cell is first caught by the contact-surface discontinuity detector. Then some amount of mass diffusions are added to the governing equations to change the gas mixtures and arrive at an equilibrium point satisfying some conditions. The species properties are reasonable without any oscillations. Computations are performed for many steady and unsteady flow problems. For general non-mixing fluid flows, the classical air-helium shock bubble interaction problem is the central test case for the high-order discontinuous Galerkin method with a mass diffusion based limiter chosen. The computed results are compared with experimental, exact, and empirical data to validate the fluid dynamic solver.

This thesis consists of an introduction and five papers concerning different numerical and mathematical aspects of viscous shocks. Hyperbolic conservation laws are used to model wave motion and advect- ive transport in a variety of physical applications. Solutions of hyperbolic conservation laws may become discontinuous, even in cases where initial and boundary data are smooth. Shock waves is one important type of discontinu- ity. It is also interesting to study the corresponding slightly viscous system, i.e., the system obtained when a small viscous term is added to the hyper- bolic system of equations. By a viscous shock we denote a thin transition layer which appears in the solution of the slightly viscous system instead of a shock in the corresponding purely hyperbolic problem. A slightly viscous system, a so called modified equation, is often used to model numerical solutions of hyperbolic conservation laws and their beha- vior in the vicinity of shocks. Computations presented elsewhere show that numerical solutions of hyperbolic conservation laws obtained by higher order accurate shock capturing methods in many cases are only first order accurate downstream of shocks. We use a modified equation to model numerical solu- tions obtained by a generic second order shock capturing scheme for a time dependent system in one space dimension. We present analysis that show how the first order error term is related to the viscous terms and show that it is possible to eliminate the first order downstream error by choosing a special viscosity term. This is verified in computations. We also extend the analysis to a stationary problem in two space dimensions. Though the technique of modified equation is widely used, rather little is known about when (for what methods etc.) it is applicable. The use of a modified equation as a model for a numerical solution is only relevant if the numerical solution behaves as a continuous function. We have experimentally investigated a range of high resolution shock capturing methods. Our experiments indicate that for many of the methods there is a continuous shock profile. For some of the methods, however, this not the case. In general the behavior in the shock region is very complicated. Systems of hyperbolic conservation laws with solutions containing shock waves, and corresponding slightly viscous equations, are examples where the available theoretical results on existence and uniqueness of solutions are very limited, though it is often straightforward to find approximate numerical solu- tions. We present a computer-assisted technique to prove existence of solu- tions of non-linear boundary value ODEs, which is based on using an approx- imate, numerical solution. The technique is applied to stationary solutions of the viscous Burgers' equation.We also study a corresponding method suggested by Yamamoto in SIAM J. Numer. Anal. 35(5)1998, and apply also this method to the viscous Burgers' equation.

<p>This thesis consists of an introduction and five papers concerning different numerical and mathematical aspects of viscous shocks. </p><p>Hyperbolic conservation laws are used to model wave motion and advect- ive transport in a variety of physical applications. Solutions of hyperbolic conservation laws may become discontinuous, even in cases where initial and boundary data are smooth. Shock waves is one important type of discontinu- ity. It is also interesting to study the corresponding slightly viscous system, i.e., the system obtained when a small viscous term is added to the hyper- bolic system of equations. By a viscous shock we denote a thin transition layer which appears in the solution of the slightly viscous system instead of a shock in the corresponding purely hyperbolic problem. </p><p>A slightly viscous system, a so called modified equation, is often used to model numerical solutions of hyperbolic conservation laws and their beha- vior in the vicinity of shocks. Computations presented elsewhere show that numerical solutions of hyperbolic conservation laws obtained by higher order accurate shock capturing methods in many cases are only first order accurate downstream of shocks. We use a modified equation to model numerical solu- tions obtained by a generic second order shock capturing scheme for a time dependent system in one space dimension. We present analysis that show how the first order error term is related to the viscous terms and show that it is possible to eliminate the first order downstream error by choosing a special viscosity term. This is verified in computations. We also extend the analysis to a stationary problem in two space dimensions. </p><p>Though the technique of modified equation is widely used, rather little is known about when (for what methods etc.) it is applicable. The use of a modified equation as a model for a numerical solution is only relevant if the numerical solution behaves as a continuous function. We have experimentally investigated a range of high resolution shock capturing methods. Our experiments indicate that for many of the methods there is a continuous shock profile. For some of the methods, however, this not the case. In general the behavior in the shock region is very complicated.</p><p>Systems of hyperbolic conservation laws with solutions containing shock waves, and corresponding slightly viscous equations, are examples where the available theoretical results on existence and uniqueness of solutions are very limited, though it is often straightforward to find approximate numerical solu- tions. We present a computer-assisted technique to prove existence of solu- tions of non-linear boundary value ODEs, which is based on using an approx- imate, numerical solution. The technique is applied to stationary solutions of the viscous Burgers' equation.We also study a corresponding method suggested by Yamamoto in SIAM J. Numer. Anal. 35(5)1998, and apply also this method to the viscous Burgers' equation.</p>

The main objective of this dissertation is to develop large-eddy simulation (LES) based computational tools for supersonic inlet and combustor design. In the recent past, LES methodology has emerged as a viable tool for modeling turbulent combustion. LES computes the large scale mixing process accurately, thereby providing a better starting point for small-scale models that describe the combustion process. In fact, combustion models developed in the context of Reynolds-averaged Navier Stokes (RANS) equations exhibit better predictive capability when used in the LES framework. The development of a predictive computational tool based on LES will provide a significant boost to the design of scramjet engines. Although LES has been used widely in the simulation of subsonic turbulent flows, its application to high-speed flows has been hampered by a variety of modeling and numerical issues. In this work, we develop a comprehensive LES methodology for supersonic flows, focusing on the simulation of scramjet engine components. This work is divided into three sections. First, a robust compressible flow solver for a generalized high-speed flow configuration is developed. By using carefully designed numerical schemes, dissipative errors associated with discretization methods for high-speed flows are minimized. Multiblock and immersed boundary method are used to handle scramjet-specific geometries. Second, a new combustion model for compressible reactive flows is developed. Subsonic combustion models are not directly applicable in high-speed flows due to the coupling between the energy and velocity fields. Here, a probability density function (PDF) approach is developed for high-speed combustion. This method requires solution to a high dimensional PDF transport equation, which is achieved through a novel direct quadrature method of moments (DQMOM). The combustion model is validated using experiments on supersonic reacting flows. Finally, the LES methodology is used to study the inlet-isolator component of a dual-mode scramjet. The isolator is a critical component that maintains the compression shock structures required for stable combustor operation in ramjet mode. We simulate unsteady dynamics inside an experimental isolator, including the propagation of an unstart event that leads to loss of compression. Using a suite of simulations, the sensitivity of the results to LES models and numerical implementation is studied. / text

<p>The present dissertation focuses on the numerical solution of compressible flows with an emphasis on simulations of transitional hypersonic boundary layers. Initially, general concepts such as the governing equations, numerical approximations and theoretical modeling strategies are addressed. These are used as a basis to introduce two innovative techniques, the Quasi-Spectral Viscosity (QSV) method, applied to high-order finite difference settings and the Legendre Spectral Viscosity (LSV) approach, used in high-order flux reconstruction schemes. Such techniques are derived based on the mathematical formalism of the filtered compressible Navier-Stokes equations. While the latter perspective is only typically used for turbulence modeling in the context of Large-Eddy Simulations (LES), both the QSV and LSV subfilter scale (SFS) closure models are capable of performing simulations in the presence of shock-discontinuities. On top of that, the QSV approach is also shown to support dynamic subfilter turbulence modeling capabilities.</p> <p>QSV’s innovation lies in the introduction of a physical-space implementation of a spectral-like subfilter scale (SFS) dissipation term by leveraging residuals of filter operations, achiev- ing two goals: (1) estimating the energy of the resolved solution near the grid cutoff; (2) imposing a plateau-cusp shape to the spectral distribution of the added dissipation. The QSV approach was tested in a variety of flows to showcase its capability to act interchangeably as a shock capturing method or as a SFS turbulence closure. QSV performs well compared to previous eddy-viscosity closures and shock capturing methods. In a supersonic TGV flow, a case which exhibits shock/turbulence interactions, QSV alone outperforms the simple super- position of separate numerical treatments for SFS turbulence and shocks. QSV’s combined capability of simulating shocks and turbulence independently, as well as simultaneously, effectively achieves the unification of shock capturing and Large-Eddy Simulation.</p> <p>The LSV method extends the QSV idea to discontinuous numerical schemes making it suitable for unstructured solvers. LSV exploits the set of hierarchical basis functions formed by the Legendre polynomials to extract the information on the energy content near the resolution limit and estimate the overall magnitude of the required SFS dissipative terms, resulting in a scheme that dynamically activates only in cells where nonlinear behavior is important. Additionally, the modulation of such terms in the Legendre spectral space allows for the concentration of the dissipative action at small scales. The proposed method is tested in canonical shock-dominated flow setups in both one and two dimensions. These include the 1D Burgers’ problem, a 1D shock tube, a 1D shock-entropy wave interaction, a 2D inviscid shock-vortex interaction and a 2D double Mach reflection. Results showcase a high-degree of resolution power, achieving accurate results with a small number of degrees of freedom, and robustness, being able to capture shocks associated with the Burgers’ equation and the 1D shock tube within a single cell with discretization orders 120 and higher.</p> <p>After the introduction of these methods, the QSV-LES approach is leveraged to perform numerical simulations of hypersonic boundary layer transition delay on a 7^◦-half-angle cone for both sharp and 2.5 mm-nose tip radii due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) in the Reynolds number range Re_m = 2.43 · 106 – 6.40 · 10⁶ m⁻¹ at the freestream Mach number of M_∞ = 7.4. A low-order impedance model was fitted through experimental measurements of acoustic absorption taken at discrete frequencies yielding a continuous representation in the frequency domain that was imposed in the simulations via a broadband time domain impedance boundary condition (TDIBC). The stability of the base flow is studied over impermeable and porous walls via pulse-perturbed axisymmetric simulations with second-mode spatial growth rates matching linear predictions. This shows that the QSV-LES approach is able to dynamically deactivate its dissipative action in laminar portions of the flow making it possible to accurately capture the boundary layer’s instability dynamics. Three-dimensional transitional LES were then performed with the introduction of grid independent pseudorandom pressure perturbations. Comparison against previous experiments were made regarding the frequency content of the disturbances in the transitional region with fairly good agreement capturing the shift to lower frequencies. Such shift is caused by the formation of near-wall low-temperature streaks that concentrate the pressure disturbances at locations with locally thicker boundary layers forming trapped wavetrains that can persist into the turbulent region. Additionally, it is shown that the presence of a porous wall representative of a C/C material does not affect turbulence significantly and simply shifts its onset downstream.</p>

With the advances in the computing world, computational fluid dynamics (CFD) is becoming more and more critical tool in the field of fluid dynamics. In the past few decades, a huge number of CFD models have been developed with ever improved performance. In this research a robust CFD model, called Riemann2D, is extended to model flow over a mobile bed and applied to a full scale dam break problem. Riemann2D, an object oriented hyperbolic solver that solves shallow water equations with an unstructured triangular mesh and using high resolution shock capturing methods, provides a generic framework for the solution of hyperbolic problems. The object-oriented design of Riemann2D has the flexibility to apply the model to any type of hyperbolic problem with the addition of new information and inheriting the common components from the generic part of the model. In a part of this work, this feature of Riemann2D is exploited to enhance the model capabilities to compute flow over mobile beds. This is achieved by incorporating the two dimensional version of the one dimensional non-capacity model for erodible bed hydraulics by Cao et al. (2004). A few novel and simple algorithms are included, to track the wet/dry and dry/wet fronts over abruptly varying topography and stabilize the solution while using high resolution shock capturing methods. The negative depths computed from the surface gradient by the limiters are algebraically adjusted to ensure depth positivity. The friction term contribution in the source term, that creates unphysical values near the wet/dry fronts, are resolved by the introduction of a limiting value for the friction term. The model is validated using an extensive variety of tests both on fixed and mobile beds. The results are compared with the analytical, numerical and experimental results available in the literature. The model is also tested against the actual field data of 1957 Malpasset dam break. Finally, the model is applied to simulate dam break flow of Warsak Dam in Pakistan. Remotely sensed topographic data of Warsak dam is used to improve the accuracy of the solution. The study reveals from the thorough testing and application of the model that the simulated results are in close agreement with the available analytical, numerical and experimental results. The high resolution shock capturing methods give far better results than the traditional numerical schemes. It is also concluded that the object oriented CFD model is very easy to adapt and extend without changing the generic part of the model.