Drukcorrectiealgoritmen voor willekeurige fluida bij lage snelheden, toegepast op simulaties van niet-voorgemengde vlammen

Dit doctoraatsonderzoek situeert zich binnen het domein van de numerieke stromingsmechanica. Deze wetenschap, die tot doel heeft de stroming van een vloeistof of een gas te simuleren aan de hand van computerberekeningen, wordt alsmaar belangrijker in de ontwerpfase van hedendaagse systemen waar reagerende stromingen deel van uitmaken. Numerieke simulaties worden dan ook veelvuldig gebruikt bij ontwerp en optimalisatie van bijvoorbeeld industriële branders. Een goede voorspelling van de ingewikkelde processen die zich voordoen in dergelijke systemen is mogelijk indien men beschikt over nauwkeurige modellen en geavanceerde numerieke methoden. Helaas zijn deze technieken enkel in staat kwantitatieve voorspellingen te maken als de onderliggende algoritmen geschikt zijn voor tijdsnauwkeurige simulaties van reagerende stromingen. Frequent gebruikte algoritmen blijken aanleiding te geven tot onstabiele oplossingen als deze toegepast worden op stromingen met sterk variabele dichtheid, zoals in verbrandingprocessen. Andere gangbare algoritmen blijken wel stabiel, maar voorspellen oplossingen die fysisch niet mogelijk zijn. Omwille van deze tekortkomingen, wordt in dit doctoraat een algoritme ontwikkeld, die de goede eigenschappen van beide klassen bundelt: het is stabiel en voorspelt fysisch correcte oplossingen. Uiteindelijk draagt dit doctoraatswerk bij tot betere numerieke simulaties, en dus tot de ontwikkeling van branders met een hoger rendement en verminderde uitstoot van schadelijke stoffen

Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations eliminate the fast isentropic fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatio-temporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions. We construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplications neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing

An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

We present an Asymptotic-Preserving 'all-speed' scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and Two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties

Numerical simulations of the Euler system with congestion constraint

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [1, 3] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [2,4] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes

A Second Order Godunov Method for Multidimensional Relativistic Magnetohydrodynamics

We describe a new Godunov algorithm for relativistic magnetohydrodynamics (RMHD) that combines a simple, unsplit second order accurate integrator with the constrained transport (CT) method for enforcing the solenoidal constraint on the magnetic field. A variety of approximate Riemann solvers are implemented to compute the fluxes of the conserved variables. The methods are tested with a comprehensive suite of multidimensional problems. These tests have helped us develop a hierarchy of correction steps that are applied when the integration algorithm predicts unphysical states due to errors in the fluxes, or errors in the inversion between conserved and primitive variables. Although used exceedingly rarely, these corrections dramatically improve the stability of the algorithm. We present preliminary results from the application of these algorithms to two problems in RMHD: the propagation of supersonic magnetized jets, and the amplification of magnetic field by turbulence driven by the relativistic Kelvin-Helmholtz instability (KHI). Both of these applications reveal important differences between the results computed with Riemann solvers that adopt different approximations for the fluxes. For example, we show that use of Riemann solvers which include both contact and rotational discontinuities can increase the strength of the magnetic field within the cocoon by a factor of ten in simulations of RMHD jets, and can increase the spectral resolution of three-dimensional RMHD turbulence driven by the KHI by a factor of 2. This increase in accuracy far outweighs the associated increase in computational cost. Our RMHD scheme is publicly available as part of the Athena code.Comment: 75 pages, 28 figures, accepted for publication in ApJS. Version with high resolution figures available from http://jila.colorado.edu/~krb3u/Athena_SR/rmhd_method_paper.pd