Steady Euler Flows with Large Vorticity and Characteristic Discontinuities in Arbitrary Infinitely Long Nozzles

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by nonlinear arguments. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic-sonic limit of the solutions is also shown. Finally, through the incompressible limit, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.Comment: 43 pages; 2 figures; To be published in Advances in Mathematics (2019

Regularization of the Shock Wave Solution to the Riemann Problem for the Relativistic Burgers Equation

The regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation is considered. For Riemann initial data consisting of a single decreasing jump, we find that the regularization of nonlinear convective term cannot capture the correct shock wave solution. In order to overcome it, we consider a new regularization technique called the observable divergence method introduced by Mohseni and discover that it can capture the correct shock wave solution. In addition, we take the Helmholtz filter for the fully explicit computation

Application and assessment of time-domain DGM for intake acoustics using 3D linearized Euler equations

Fan noise is one of the major sources of aircraft noise. This can be modelled by means of frequency and time domain CAA methods. Frequency domain methods based on the convected Helmholtz equation are widely used for noise propagation and radiation from turbofan intakes. However, these methods are unsuited to deal easily with turbofan exhaust noise and presently unable to solve large 3D (three-dimensional) problems at high frequencies. In this thesis the application of time-domain Discontinuous Galerkin Methods (DGM) for solving linearized Euler equations is investigated. The research is focused on large 3D problems with arbitrary mean flows. A commercially available DGM code, Actran DGM, is used.An automatic procedure has been developed to perform the DGM simulations for axisymmetric and 3D intake problems by providing simple control of all the parameters (flow, geometry, liners). Moreover, a new method for integrating source predictions obtained from CFD calculations for the fan stage of a turbofan engine with the DGM code to predict tonal noise radiation in the far field has been proposed, implemented and validated.The DGM is validated and benchmarked for intake and exhaust problems against analytical solutions and other numerical methods. The principal properties of the DGM are assessed, best practice is defined, and important issues which relate to the accuracy and stability of the liner model are identified. The accuracy and efficiency of the CFD/CAA coupling are investigated and results obtained are compared to rig test data.The influence of the 3D intake shapes and the mean flow distortion on the sound field is investigated for static rig and flight conditions by using the DGM approach. Moreover, it is shown that the mean flow distortion can have a significant effect on the sound attenuation by a liner

Hamiltonian regularisation of the unidimensional barotropic Euler equations

Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh. This system is Galilean invariant, linearly non-dispersive and conserves formally an $H^1$-like energy. In this paper, we generalise this regularisation for the barotropic Euler system preserving the same properties. We prove the local (in time) well-posedness of the regularised barotropic Euler system and a periodic generalised two-component Hunter-Saxton system. We also show for both systems that if singularities appear in finite time, they are necessary in the first derivatives

Asymptotics, structure, and integration of sound-proof atmospheric flow equations

Relative to the full compressible flow equations, sound-proof models filter acoustic waves while maintaining advection and internal waves. Two well-known sound-proof models, an anelastic model by Bannon and Durran's pseudo-incompressible model, are shown here to be structurally very close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing @t in the mass continuity equation and slightly modifying the gravity term, whereas the pseudoincompressible model results from dropping @tp from the pressure equation. For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudo-incompressible model approaches the zero Mach number, variable density flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible flow equations, yet the pseudo-incompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphere-ocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dörnbrack (2007). Despite these useful properties, neither model can be derived by a low-Mach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura & Phillips model, which assumes weak stratication of the order of the Mach number squared, can be obtained as a leading-order model from systematic low Mach number asymptotic analysis. Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudo-incompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper we propose a second-order finite-volume projection method for the anelastic and pseudo-incompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the descent of a cold air bubble in the small-scale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work

Methods for higher order numerical simulations of complex inviscid fluids with immersed boundaries

Within this thesis, we study inviscid compressible flows of fluids modelled by several equations of state. Namely, these are the ideal gas law, the stiffened gas law, Tait's law and the covolume gas law. In their entirety, these equations of state can be used as models for the behaviour of many gases and liquids. After deriving new exact solutions for the corresponding variants of the Euler equations, we use the results as a tool for the verification of a higher-order accurate numerical scheme that has been implemented during the course of this thesis. The scheme is based on a Runge-Kutta Discontinuous Galerkin Method and the presented verification results show that we are able to obtain the expected rates of convergence in both, space and time. In the main part of this thesis, we consider an important building block for the extension of this conventional discretization by means of a treatment for generic immersed boundaries, namely the numerical integration of general functions over domains that are at least partly defined by the zero iso-contour of a level set function defining the domain boundary. Here, we study two new, generally applicable approaches in terms of their robustness and convergence behaviour. The first approach is based on a classical adaptive strategy, while the second approach is based on a hierarchical moment-fitting strategy with variable Ansatz order P. Both methods have been designed such that they are applicable on general element types. Most notably, the results of our numerical experiments suggest that the moment-fitting procedure leads to integration errors that decrease with a rate of O(h^(P+1)), thus allowing for a severe increase of integration accuracy at constant computational effort