Fluctuating hydrodynamics of multi-species, non-reactive mixtures

In this paper we discuss the formulation of the fuctuating Navier-Stokes (FNS) equations for multi-species, non-reactive fluids. In particular, we establish a form suitable for numerical solution of the resulting stochastic partial differential equations. An accurate and efficient numerical scheme, based on our previous methods for single species and binary mixtures, is presented and tested at equilibrium as well as for a variety of non-equilibrium problems. These include the study of giant nonequilibrium concentration fluctuations in a ternary mixture in the presence of a diffusion barrier, the triggering of a Rayleigh-Taylor instability by diffusion in a four-species mixture, as well as reverse diffusion in a ternary mixture. Good agreement with theory and experiment demonstrates that the formulation is robust and can serve as a useful tool in the study of thermal fluctuations for multi-species fluids. The extension to include chemical reactions will be treated in a sequel paper

Modeling of magnetized expanding plasmas

In fusion reactors, the walls are exposed to very high particle and energy fluxes. To study the problem of wall erosion and hydrogen retention in these conditions, the Magnum-PSI experiment at the FOM Institute of Plasma Physics is set up. The plasma source for Magnum-PSI is a cascaded arc, where a strong magnetic field is applied to obtain the desired conditions. The focus of this thesis is the development of a numerical model for studying the plasma creation in the source and the consecutive magnetized expansion. To describe the behavior of the different species in the range of conditions in the plasma – from gas to fully ionized, from non-magnetized to strongly magnetized – a multicomponent diffusion description is needed. Numerical techniques are developed to successfully apply multicomponent diffusion to magnetized expanding plasmas. Multi-component diffusion results in a system of coupled continuity equations for all species. In addition this coupled system is subject to mass and charge conservation constraints. To deal with the coupling between the species a new finite volume discretization method is introduced to discretize the system of continuity equations. For numerical stability, mass and charge constraints are not explicitly applied. Instead, all species mass fractions are treated as independent unknowns and mass and charge constraints are a result of the continuity equations, the boundary conditions, the diffusion algorithm and the new discretization scheme. With this method, mass and charge constraints can be satisfied exactly, although they are not explicitly applied. To verify the suitability of the method, simulations of both magnetized and nonmagnetized expansions have been performed. The simulations are able to reproduce important characteristics of magnetic confinement. Results show that in the magnetized case, the plasma production cannot be modeled by considering the source alone, since plasma production extends into the expansion region

ASHEE: a compressible, equilibrium-Eulerian model for volcanic ash plumes

A new fluid-dynamic model is developed to numerically simulate the non-equilibrium dynamics of polydisperse gas-particle mixtures forming volcanic plumes. Starting from the three-dimensional N-phase Eulerian transport equations for a mixture of gases and solid particles, we adopt an asymptotic expansion strategy to derive a compressible version of the first-order non-equilibrium model, valid for low concentration regimes and small particles Stokes $St<0.2$. When $St < 0.001$ the model reduces to the dusty-gas one. The new model is significantly faster than the Eulerian model while retaining the capability to describe gas-particle non-equilibrium. Direct numerical simulation accurately reproduce the dynamics of isotropic turbulence in subsonic regime. For gas-particle mixtures, it describes the main features of density fluctuations and the preferential concentration of particles by turbulence, verifying the model reliability and suitability for the simulation of high-Reynolds number and high-temperature regimes. On the other hand, Large-Eddy Numerical Simulations of forced plumes are able to reproduce their observed averaged and instantaneous properties. The self-similar radial profile and the development of large-scale structures are reproduced, including the rate of entrainment of atmospheric air. Application to the Large-Eddy Simulation of the injection of the eruptive mixture in a stratified atmosphere describes some of important features of turbulent volcanic plumes, including air entrainment, buoyancy reversal, and maximum plume height. Coarse particles partially decouple from the gas within eddies, modifying the turbulent structure, and preferentially concentrate at the eddy periphery, eventually being lost from the plume margins due to the gravity. By these mechanisms, gas-particle non-equilibrium is able to influence the large-scale behavior of volcanic plumes.Comment: 29 pages, 22 figure

Mathematical Modeling of Mixtures and Numerical Solution with Applications to Polymer Physics

We consider in this dissertation the mathematical modeling and simulation of a general diffuse interface mixture model based on the principles of energy dissipation. The model developed allows for a thermodynamically consistent description of systems with an arbitrary number of different components, each of which having perhaps differing densities. We also provide a mathematical description of processes which may allow components to source or sink into other components in a mass conserving, energy dissipating way, with the motivation of applying this model to phase transformation. Also included in the modeling is a unique set of thermodynamically consistent boundary conditions which allows flow across the boundary of a select number of components. The result of this modeling is a unique Cahn-Hilliard, Allen-Cahn-like system of equations. For numerical solution of this model, we use cell-centered finite difference methods for discretization and Full Approximation Storage (FAS) multigrid methods to solve the resulting system of equations via use of the BSAM (Block- Structured Adaptive Multigrid) libraries. Upon development of the mathematical model, we consider two applications. The primary application of this mathematical modeling is the time evolution of a quaternary mixture consisting of a volatile solvent in the liquid phase, solvent in the vapor phase, and two polymers. This modeling is motivated by the need to better understand the active layer in Organic Photovoltaics (OPVs). In this mixture, the volatile solvent is evaporating into the its vapor phase, and upon fully evaporating the polymer mixture which results is the active layer of the OPV device. Simulations are provided which demonstrate the solvent evaporation phenomenon and the resulting microstructure of the active layer. As a future application, we consider a mixture of a charged polymer and its counterion. We provide a description of the system based on the dissipation of the electrochemical free energy which allows for the permittivity to be dependent on the volume fractions. Simulations are provided which vary the gradient energies and polymer chain length and demonstrate the different steady-state microstructures which can result

Finite-volume WENO scheme for viscous compressible multicomponent flows

We develop a shock- and interface-capturing numerical method that is suitable for the simulation of multicomponent flows governed by the compressible Navier–Stokes equations. The numerical method is high-order accurate in smooth regions of the flow, discretely conserves the mass of each component, as well as the total momentum and energy, and is oscillation-free, i.e. it does not introduce spurious oscillations at the locations of shockwaves and/or material interfaces. The method is of Godunov-type and utilizes a fifth-order, finite-volume, weighted essentially non-oscillatory (WENO) scheme for the spatial reconstruction and a Harten–Lax–van Leer contact (HLLC) approximate Riemann solver to upwind the fluxes. A third-order total variation diminishing (TVD) Runge–Kutta (RK) algorithm is employed to march the solution in time. The derivation is generalized to three dimensions and nonuniform Cartesian grids. A two-point, fourth-order, Gaussian quadrature rule is utilized to build the spatial averages of the reconstructed variables inside the cells, as well as at cell boundaries. The algorithm is therefore fourth-order accurate in space and third-order accurate in time in smooth regions of the flow. We corroborate the properties of our numerical method by considering several challenging one-, two- and three-dimensional test cases, the most complex of which is the asymmetric collapse of an air bubble submerged in a cylindrical water cavity that is embedded in 10% gelatin

Kinetic-energy- and pressure-equilibrium-preserving schemes for real-gas turbulence in the transcritical regime

Numerical simulations of compressible turbulent flows governed by real-gas equations of state, such as high-pressure transcritical flows, are strongly susceptible to instabilities. In addition to the inherent multi-scale nature of the flow, the presence of a pseudo-interface can generate spurious pressure oscillations when conventional schemes are utilized. This study proposes a general framework to derive and analyze discretization methods that are able to preserve kinetic energy by convection, and simultaneously maintain pressure equilibrium in discontinuity-free compressible real-gas flows. The formal analysis reveals that the discrete pressure-equilibrium condition can be fulfilled at most to second-order accuracy, as it requires the spatial differential operator to satisfy a discrete chain rule when total, or internal energy, are directly discretized. A novel class of schemes based on the solution of a pressure equation is thus proposed, which preserves mass, momentum, kinetic energy and pressure equilibrium, but not total energy. Extensive numerical tests of increasing complexity confirm the theoretical predictions, and show that the proposed scheme is capable of providing non-dissipative, stable and oscillation-free simulations, unlike existing methods tailored for the transcritical regime.This work is funded by the European Union (ERC, SCRAMBLE, 101040379).Peer ReviewedPostprint (author's final draft

A fast, low-memory, and stable algorithm for implementing multicomponent transport in direct numerical simulations

Implementing multicomponent diffusion models in reacting-flow simulations is computationally expensive due to the challenges involved in calculating diffusion coefficients. Instead, mixture-averaged diffusion treatments are typically used to avoid these costs. However, to our knowledge, the accuracy and appropriateness of the mixture-averaged diffusion models has not been verified for three-dimensional turbulent premixed flames. In this study we propose a fast,efficient, low-memory algorithm and use that to evaluate the role of multicomponent mass diffusion in reacting-flow simulations. Direct numerical simulation of these flames is performed by implementing the Stefan-Maxwell equations in NGA. A semi-implicit algorithm decreases the computational expense of inverting the full multicomponent ordinary diffusion array while maintaining accuracy and fidelity. We first verify the method by performing one-dimensional simulations of premixed hydrogen flames and compare with matching cases in Cantera. We demonstrate the algorithm to be stable, and its performance scales approximately with the number of species squared. Then, as an initial study of multicomponent diffusion, we simulate premixed, three-dimensional turbulent hydrogen flames, neglecting secondary Soret and Dufour effects. Simulation conditions are carefully selected to match previously published results and ensure valid comparison. Our results show that using the mixture-averaged diffusion assumption leads to a 15% under-prediction of the normalized turbulent flame speed for a premixed hydrogen-air flame. This difference in the turbulent flame speed motivates further study into using the mixture-averaged diffusion assumption for DNS of moderate-to-high Karlovitz number flames.Comment: 36 pages, 14 figure

Mixing and Phase Separation of Fluid Mixtures

During the three years of the PhD project we extended the di®use interface (DI) method and apply it to engineering related problems, particularly re- lated to mixing and demixing of two °uids. To do that, ¯rst the DI model itself was validated, showing that, in agreement with its predictions, a single drop immersed in a continuum phase moves whenever its composition and that of the continuum phase are not at mutual equilibrium [D. Molin, R. Mauri, and V. Tricoli, "Experimental Evidence of the Motion of a Single Out-of-Equilibrium Drop," Langmuir 23, 7459-7461 (2007)]. Then, we de- veloped a computer code and validated it, comparing its results on phase separation and mixing with those obtained previously. At this point, the DI model was extended to include heat transport e®ects in regular mixtures In fact, in the DI approach, convection and di®usion are coupled via a nonequi- librium, reversible body force that is associated with the Kortweg stresses. This, in turn, induces a material °ux, which enhances both heat and mass transfer. Accordingly, the equation of energy conservation was developed in detail, showing that the in°uence of temperature is two-folded: on one hand, it determine phase transition directly, as the system is brought from the single-phase to the two-phase region of its phase diagram. On the other hand, temperature can also change surface tension, that is the excess free en- ergy stored within the interface at equilibrium. These e®ects were described using the temperature dependence of the Margules parameter. In addition, the heat of mixing was also taken into account, being equal to the excess free energy. [D. Molin and R. Mauri, "Di®use Interface Model of Multiphase Fluids," Int. J. Heat Mass Tranf., submitted]. The new model was applied to study the phase separation of a binary mixture due to the temperature quench of its two con¯ning walls. The results of our simulations showed that, as heat is drawn from the bulk to the walls, the mixture phase tends to phase separate ¯rst in vicinity of the walls, and then, deeper and deeper within the bulk. During this process, convection may arise, due to the above mentioned non equilibrium reversible body force, thus enhancing heat transport and, in particular increasing the heat °ux at the walls [D. Molin, and R. Mauri, "Enhanced Heat Transport during Phase Separation of Liquid Binary Mix- tures," Phys. Fluids 19, 074102-1-10 (2007)]. The model has been extended then and applied to the case where the two phases have di®erent heat con- 3 ductivities. We saw that heat transport depends on two parameters, the Lewis number and the heat conductivity ratio. In particular, varying these parameters can a®ect the orientation of the domains that form during phase separation. Domain orientation has been parameterized using an isotropy coe±cient », varying from -1 to 1, with » = 0 when the morphology is isotropic, » = +1 when it is composed of straight lines along the transversal (i.e. perpendicular to the walls) direction, and » = ¡1 when it is composed of straight lines along the longitudinal (i.e. parallel to the walls) direction [D. Molin, and R. Mauri, "Spinodal Decomposition of Binary Mixtures with Composition-Dependent Heat Conductivities," Int. J. Engng. Sci., in press (2007)]. In order to further extend the model, we removed the constraint of a constant viscosity, and simulated a well known problem of drops in shear °ows. There we found that, predictably, below a certain threshold value of the capillary number, the drop will ¯rst stretch and then snap back. At lager capillary numbers, though, we predict that the drop will stretch and then, eventually, break in two or more satellite drops. On the other hand, applying traditional °uid mechanics (i.e. with in¯nitesimal interface thick- ness) such stretching would continue inde¯nitely [D. Molin and R. Mauri, " Drop Coalescence and Breakup under Shear using the Di®use Interface Model," in preparation]. Finally, during a period of three months at the Eindhoven University, we extended the DI model to a three component °uid mixture, using a di®erent form of the free energy, as derived by Lowengrub and Coworkers.. With this extension, we simulated two simple problems: ¯rst, the coalescence/repulsion of two-component drops immersed in a third component continuum phase; second, the e®ect of adding a third component to a separated two phase system. Both simulations seem to capture physical behaviors that were observed experimentally [D. Molin, R. Mauri and P. Anderson, " Phase Separation and Mixing of Three Component Mixtures," in preparation]