Eric Daniel

Two-phase flows: Second-order schemes and boundary conditions

A second-order numerical method suitable to unstructured grids is developed for the solution of two-phase flow equations. A number of assumptions are introduced, justified on physical grounds, when the dispersed phase concentration is low; their effect is to change the mathematical nature of the system of equations. The new numerical method put forward builds on Van Leers original ideas. The solution of two Riemann problems is involved: one for the gas phase and a second for the dispersed phase. The method is first tested on a one-dimensional case, with excellent results; it is then generalized to two dimensions

A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations

A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures.

A multiphase formulation for two phase flows

This paper investigates the multi‐phase behaviour of droplets injected into a nozzle at two separate wall locations. The physical features of the droplets (rate of mass, density and radius) at each injector location are identical. This system can be described by a two‐phase Eulerian—Eulerian approach that yields classical systems of equations: three for the gaseous phase and three for the dispersed droplet phase. An underlying assumption in the two phase model is that no interaction occurs between droplets. The numerical solution of the model (using the MacCormack scheme) indicates however that the opposite jets do interact to form one jet. This inconsistency is overcome in the current paper by associating the droplets from a given injection location with a separate phase and subsequently solving equations describing a multiphase system (here, three‐phase system). Comparison of numerical predications between the two‐phase and the multiphase model shows significantly different results. In particular the mu...

Droplet break-up through an oblique shock wave

The interaction of a two-phase flow with a wedge where a stationary shock wave is initially settled is studied in a two-dimensional configuration. Before the introduction of the dispersed phase, the flow around the wedge is a supersonic one phase flow such as an attached stationary shock wave is present. Then, the dispersed phase is introduced upstream the initial position of the stationary shock wave. The purpose of this study is to point out two-phase and droplets break-up effects on the oblique shock wave. The two-dimensional equations are solved by a TVD scheme where fluxes are computed by using Riemann solver for the gas phase equations and also for the dispersed phase equations wich is an original approach due to the authors (Saurel et al. 1994). In addition to drag forces and heat and mass transfers, the process of droplets fragmentation based on the particle oscillation is considered.

Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

In nuclear reactor safety and optimization there are key issues that rely on in-depth understanding of basic two-phase flow phenomena with heat and mass transfer. Within the context of multiphase flows, two bubble-dynamic phenomena – boiling (heterogeneous) and flashing or cavitation (homogeneous boiling), with bubble collapse, are technologically very important to nuclear reactor systems. The main difference between boiling and flashing is that bubble growth (and collapse) in boiling is inhibited by limitations on the heat transfer at the interface, whereas bubble growth (and collapse) in flashing is limited primarily by inertial effects in the surrounding liquid. The flashing process tends to be far more explosive (and implosive), and is more violent and damaging (at least in the near term) than the bubble dynamics of boiling. However, other problematic phenomena, such as crud deposition, appear to be intimately connecting with the boiling process. In reality, these two processes share many details.

Stability of Acoustic Wave in Two-Phase Dilute Flow with Mass Transfer

The behavior of an acoustic wave propagating in a two-phase dilute flow is analytically and numerically investigated. The focus is on the effects of a mass transfer modeled by the so-called rapid-mixing model. An analytical solution is carried out that shows a possible unstable flow regime, which means that the magnitude of a pressure wave may be amplified under particular conditions. The neutral stability condition is mainly driven by a mass transfer number, which links the heat of phase change and the equilibrium temperature. Even the mass transfer is a simplified one and far from the actual combustion of metal particles, when the analysis is applied to aluminum particles in solid rocket motor environment, unstable flow behavior is seen at low frequencies. One-dimensional simulations of the propagation of an acoustic wave are performed, and the results recovered the theoretical ones. A simulation in a two-dimensional motor leads to an oscillatory flow, which is sustained, and the amplitude of the pressure oscillation reaches an asymptotic value. This result, obtained by solving the nonlinear coupled two-phase flow equations shows that the mass transfer might be a driven mechanism for instabilities in solid rocket motor two-phase flows.

Numerical simulation of a two‐phase dilute flow in a diffuser pipe

A numerical simulation of a two‐phase dilute flow (droplet‐gas mixture) is carried out by using a finite volume method based on Riemann solvers. The computational domain represents a one‐ended pipe with holes at its upper wall which lead into an enclosure. The aim of this study is to determine the parameters of such a flow. More specially, an analytical solution is compared with numerical results to assess the mass flow rates through the vents in the pipe. Inertia effects dominate the dynamic behaviour of droplets, which causes a non‐homogeneous flow in the cavity. The unsteady effects are also important, which makes isentropical calculation irrelevant and shows the necessity of the use of CFD tools to predict such flows. No relation can be extracted from the numerical results between the gas and the dispersed mass flow rates across the holes. But a linear variation law for the droplet mass flow versus the position of the holes is pointed out, which is independent of the incoming flow when the evaporating effects are quite low.

Analytical and numerical results on the attenuation and dispersion of an acoustic wave through a two‐phase flow

Presents validation of the Eulerian approach for unsteady two‐phase flows, whose behaviour depends on the coupling between the two phases, on the basis of the study of attentuation and dispersion of an acoustic wave propagating into a one dimensional two‐phase flow. This approach and the corresponding numerical aspects are accurate enough for later applications in more complex geometries, where “vortex shedding” phenomena take place. Attenuation and dispersion of a pressure wave in a two‐phase medium of rest was previously studied by Temkin and Dobbins. Present work is an extension of this theory to the case of a two‐phase flow. This theoretical approach leads to a numerical solution of the problem. Compares the derived results with those obtained from a direct numerical simulation based on MacCormack scheme in a finite volume formulation. Verifies that analytical and numerical approaches are in good agreement.

Effects of the Injection of Droplets on a Stationary Shock Wave in a Nozzle

A numerical simulation of the flow of gases through a converging-diverging nozzle, where droplets are injected in a given section of the divergent, is presented. The two-dimensional equations are solved by a TVD scheme where fluxes are computed by using a new Riemann solver for the dispersed phase, and an exact Riemann solver for the gas phase equations. The behaviour of the initial shock wave as a function of the particles injection location is examined in this paper.

Nonlinear Development of Particle-Laden Mixing Layers at Low Mach Number

The two-phase coupling effects on the particle-laden compressible mixing layer are studied for low convective Mach numbers. A full Eulerian approach is used to obtain a set of conservative partial differential equations. The initial condition is provided by the solution of the solution of the linear problem. The numerical solutions show that the presence of particles in the flow does not alter the typical development of the mixing layer observed for one-phase flows: the amplification of linear instability waves, the formation of two-dimensional large-scale vortices, and the pairing of vortices. The shape of the vortical structures is modified compared to single-phase calculations. The particle density distribution is greatly dependant on their inertia that can lead to a particle-free zone in the vortex. The compressibility effects are clearly seen on the modal kinetic energy, which decreases when the convective Mach number increases