Dilute Two-phase Flow Research Articles

An optimisation method for the cold-spray process: On the nozzle geometry

Currently, the cold-spray process, or simply cold spray, is an extensively used technique in coating applications. The low temperature of the deposition process is the distinctive feature that makes it suitable for many additive manufacturing activities such as repair and restoration of damaged components. The reliability of the coatings is strongly dependent on the velocity of the powder during its impact on the target surface. Spraying conditions such as the pressure and temperature of the carrier gas and the geometry of the nozzle control the acceleration of the powder particles. Consequently, there is an increasing interest in the optimisation of nozzle geometry so as to maximise the acceleration of the particles through the nozzle path that they follow. In contrast with various extant approaches to achieve this aim (finite element modelling, experimental approach, and analytical methods), an alternative model based on the one-dimensional isentropic theory that accounts for the dynamics of the dilute two-phase flow was developed in this study. First, an analysis of the common hypotheses used to obtain the equation of motion of the particle was carried out. Subsequently, with the new insights revealed from the previous analysis, a new theoretical model for the optimisation of the divergent part of the nozzle was performed considering a geometric angle restriction. This model is based on the numerical integration of the equation of motion of the particle, ensuring the maximisation of the particle drag force by means of the Lagrange multiplier method. Once the analytical model is formulated, a set of curves describing the optimal geometric parameters for different conditions is obtained. Moreover, some optimal geometries are presented demonstrating the low influence of the angle restriction. Additionally, the inversely proportional relationship between stagnation pressure and temperature is revealed.

Two-phase Interactions between Propagating Shock Waves and Evaporating Water Droplets

One-dimensional numerical simulations based on hybrid Eulerian-Lagrangian method are performed to study the interactions between propagating shocks and dispersed evaporating water droplets. Two-way coupling for exchanges of mass, momentum, energy and vapour species is adopted for the dilute two-phase gas-droplet flows. Interphase interactions and droplet breakup dynamics are investigated with initial droplet diameters of 30, 50, 70 and 90 {\mu}m under an incident shock wave Mach number of 1.3. Novel two-phase flow phenomena are observed when droplet breakup occurs. First, droplets near the two-phase contact surface show obvious dispersed distribution because of the reflected pressure wave that propagates in the reverse direction of the leading shock. The reflected pressure wave grows stronger for larger droplets. Second, spatial oscillations of the gas phase pressure, droplet quantities (e.g., diameter and net force) and two-phase interactions (e.g., mass, momentum, and energy exchange), are observed in the post-shock region when droplet breakup occurs, which are caused by shock / droplet interactions. Third, the spatial distribution of droplets (i.e., number density, volume fraction) also shows strong oscillation in the post-shock region when droplet breakup occurs, which is caused by the oscillating force exerted on the droplets.

Modelling compressible dense and dilute two-phase flows

Many two-phase flow situations, from engineering science to astrophysics, deal with transition from dense (high concentration of the condensed phase) to dilute concentration (low concentration of the same phase), covering the entire range of volume fractions. Some models are now well accepted at the two limits, but none are able to cover accurately the entire range, in particular regarding waves propagation. In the present work, an alternative to the Baer and Nunziato (BN) model [Baer, M. R. and Nunziato, J. W., “A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials,” Int. J. Multiphase Flow 12(6), 861 (1986)], initially designed for dense flows, is built. The corresponding model is hyperbolic and thermodynamically consistent. Contrarily to the BN model that involves 6 wave speeds, the new formulation involves 4 waves only, in agreement with the Marble model [Marble, F. E., “Dynamics of a gas containing small solid particles,” Combustion and Propulsion (5th AGARD Colloquium) (Pergamon Press, 1963), Vol. 175] based on pressureless Euler equations for the dispersed phase, a well-accepted model for low particle volume concentrations. In the new model, the presence of pressure in the momentum equation of the particles and consideration of volume fractions in the two phases render the model valid for large particle concentrations. A symmetric version of the new model is derived as well for liquids containing gas bubbles. This model version involves 4 characteristic wave speeds as well, but with different velocities. Last, the two sub-models with 4 waves are combined in a unique formulation, valid for the full range of volume fractions. It involves the same 6 wave speeds as the BN model, but at a given point of space, 4 waves only emerge, depending on the local volume fractions. The non-linear pressure waves propagate only in the phase with dominant volume fraction. The new model is tested numerically on various test problems ranging from separated phases in a shock tube to shock–particle cloud interaction. Its predictions are compared to BN and Marble models as well as against experimental data showing clear improvements.

Application of MUSTA Scheme for Simulation of Dilute Particle-Gas Two-Phase Flow

The Multi-Stage (MUSTA) scheme was used to construct the numerical fluxes when the hyperbolically degenerate governing equations, in which the independent eigenvectors can not be found, are numerically solved for dilute particle-gas two-phase flows. The abilities of MUSTA scheme used in various problems were analyzed. It was found that the MUSTA scheme is effective even for very complicated problem although appropriate total number of stages should be carefully selected. When the total number of stages equals 3 or 4, the MUSTA scheme is as accurate as Riemann solver in most cases.

Modeling of Interior Ballistic Gas-Solid Flow Using a Coupled Computational Fluid Dynamics-Discrete Element Method.

In conventional models for two-phase reactive flow of interior ballistic, the dynamic collision phenomenon of particles is neglected or empirically simplified. However, the particle collision between particles may play an important role in dilute two-phase flow because the distribution of particles is extremely nonuniform. The collision force may be one of the key factors to influence the particle movement. This paper presents the CFD-DEM approach for simulation of interior ballistic two-phase flow considering the dynamic collision process. The gas phase is treated as a Eulerian continuum and described by a computational fluid dynamic method (CFD). The solid phase is modeled by discrete element method (DEM) using a soft sphere approach for the particle collision dynamic. The model takes into account grain combustion, particle-particle collisions, particle-wall collisions, interphase drag and heat transfer between gas and solid phases. The continuous gas phase equations are discretized in finite volume form and solved by the AUSM+-up scheme with the higher order accurate reconstruction method. Translational and rotational motions of discrete particles are solved by explicit time integrations. The direct mapping contact detection algorithm is used. The multigrid method is applied in the void fraction calculation, the contact detection procedure, and CFD solving procedure. Several verification tests demonstrate the accuracy and reliability of this approach. The simulation of an experimental igniter device in open air shows good agreement between the model and experimental measurements. This paper has implications for improving the ability to capture the complex physics phenomena of two-phase flow during the interior ballistic cycle and to predict dynamic collision phenomena at the individual particle scale.

Modeling heat transfer in dilute two-phase flows using the Mesoscopic Eulerian Formalism

In dilute two-phase flows, the accurate prediction of the temperature of the dispersed phase can be of paramount importance. Indeed, processes such as evaporation or chemical reactions are strongly non-linear functions of heat transfer between the carrier and dispersed phases. This study is devoted to the validation of an Eulerian description of the dispersed phase – the Mesoscopic Eulerian Formalism (MEF) – in the case of non-isothermal flows. Direct Numerical Simulations using the MEF are compared to a reference Lagrangian simulation for a two-dimensional non-isothermal turbulent jet laden with solid particles. The objectives of this paper are (1) to study the influence of the thermal inertia of particles on their temperature distribution and (2) conduct an a posteriori validation of the MEF, which was recently extended to non-isothermal flows. The focus is on the influence of additional terms in the MEF governing equations, namely heat fluxes arising from the Random Uncorrelated Motion (RUM). Results show that mean and rms of particle temperature are strongly dependent of the thermal Stokes number. The mean temperature is satisfactorily predicted by the MEF, comparing to the Lagrangian reference. Under the conditions of the present study, the RUM heat fluxes have a marginal influence on the mean particle temperature. However, a significant impact was observed on the magnitude of particle temperature fluctuations. Neglecting the RUM heat fluxes leads to erroneous results while the Lagrangian statistics are recovered when it is accounted for in the regimes of low to moderate thermal Stokes number.

Analysis of Inertial Particle Drift Dispersion by Direct Numerical Simulation of Two-Phase Wall-Bounded Turbulent Flows

This paper analyzes inertial particle drift dispersion based on the numerical database established by direct numerical simulation of a dilute particle-laden two-phase turbulent channel flow, in which the released particles are tracked by a Lagrangian trajectory method. Low-inertia particles tend to travel toward the walls at negative velocities. High-inertia particles however display very low drift in the near-wall regions. Effects of particle size and material density on particle drifts are also jointly studied. The statistical results show that particles drift dispersion depends significantly on particle Stokes number. Low-inertia particle drift dispersion causes a direct low-inertia particle deposition on the wall, and hence increases the particle concentration close to the walls. The results are valuable for understanding the mechanisms of particle dispersion by turbulence.

FENSAP-ICE-Unsteady: Unified In-Flight Icing Simulation Methodology for Aircraft, Rotorcraft, and Jet Engines

In-flight ice accretion, even though driven by a steady flow airstream, is an inherently unsteady phenomenon. It is, however, completely ignored in icing simulation codes (one-shot) or, at best approximated via quasi-steady modeling (multishot). The final ice shapes thus depend on the length of the total accretion time (one-shot), or of the arbitrarily prescribed time intervals (multishot), during which the impact of ice growth on both airflow and water impingement is ignored. Such a longstanding heuristic approximation is removed in this paper by coupling in time the dilute two-phase flow (air and water droplets flow) with ice accretion, and is implemented in a new code, FENSAP-ICE-Unsteady. The two-phase flow is solved using the coupled Navier–Stokes and water concentration equations, and the water film characteristics and ice shapes are obtained from laws of conservation of mass and energy within the thin film layer. To continually update the geometry of the iced surface in time, arbitrary Lagrangian–Eulerian terms are added to all governing equations to account for mesh movement in the case of stationary components. In the case of rotating/stationary interacting components, a dynamically stitched grid is used. The numerical results clearly show that unsteady modeling improves the accuracy of both rime and glaze ice shape prediction, compared with the traditional quasi-steady approach with frozen solutions. The unsteady model is shown to open the door for a unified approach to icing on fixed wings, on helicopters with blades/rotors/fuselage systems. Problems of current concern in the icing community such as the ingestion of ice crystals at high altitude become tractable with the new formulation.

Kinetic modeling of dilute solid-liquid two-phase flows with inelastic collisions

A kinetic model was presented for dilute solid-liquid two-phase flows with inelastic collisions by modeling the inelastic collisions with the revised BGK model (Santos and Astillero, 2005) and particle turbulence interactions with the Fokker-Planck operator. The formulated model kinetic equation was solved with the Chapman-Enskog method and the obtained approximate solution was further adopted to derive the constitutive relationships for the conservation equations of the particle phase. The new constitutive relationships would be suitable for the study on dilute solid-liquid two-phase flows such as sediment-laden flows in open channels or rivers.

Optimization of Venturi tube design for pipeline pulverized coal flow measurements

A Venturi flow meter was designed to accurately measure the mass flow rate of pulverized coal in power plant pipelines. Numerical simulations of the dilute turbulent gas-solid two-phase flow in a horizontal Venturi tube were used to study the effects of Venturi tube geometry on the pressure distribution in the mixture. The results show that Venturi tube geometry strongly influences the metering of the dilute gas-particle two-phase flow. The geometry can be optimized to improve the precision of the measurement and ensure stable measurements. Furthermore, the geometries of three types of Venturi tubes were optimized for an experimental study of pulverized coal mass flow rate measurements.

Computation of dilute two-phase flow in a pump

This paper is a report on a joint project between academia and industry which is concerned with computation of dilute two-phase flow through a pump in turbulent condition. The flow field for the continuous phase is computed using the Reynolds averaged Navier–Stokes equations together with mixing length turbulence modeling. The dispersed phase is treated using the Lagrangian approach by tracking it's trajectory along which the information is passed. It is found that the bubbles and small solid particles flow out of the chamber (between the rotating impeller and the casing wall) with the conveying fluid. The solid particles of relatively bigger sizes accumulate at the low pressure zones near the cashing wall or the rotating shaft.

Theoretical approach on the turbulence intensity of the carrier fluid in dilute two-phase flows

The purpose of this research is to investigate the dominating factors of carrier-phase turbulence modulation, i.e. drag, wake, and particle concentration, and to develop a model for the turbulence modulation of the carrier-phase due to the presence of the dispersed phase in various types of dilute two-phase flows. Available experimental data were surveyed and the dependency of turbulence characteristics for the carrier-phase on the characteristic length scales, drag coefficient, relative velocities, size of dispersed phase, and the concentration of particles were examined. This study takes into account the effect of the wakes induced by the laden phase, the drag between the phases and the velocity gradient in the wake to estimate the production of the turbulence. The model of turbulence modulation using the mixing length theory under the assumption of equilibrium flow is proposed. Numerical results show that the turbulence modulation model is successful in predicting the turbulence attenuation of a relatively small particle size and low loading ratio and the turbulence increase of a large particle size and high loading ratio both qualitatively and quantitatively.

Improved subgrid scale model for dense turbulent solid-liquid two-phase flows

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules. Assuming that the solid-phase velocity distributions obey the Maxwell equations, the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow, the solid-particle's governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations. Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale (SGS) model, in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor, is proposed to model the two-phase governing equations by applying dimension analyses. Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls, the velocity and pressure fields, and the volumetric concentration are calculated. The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.

Effect of Clustering on Gas−Solid Drag in Dilute Two-Phase Flow

When coarse-grid calculations in two-phase flow are performed, mesoscale fluctuations, such as clusters, cannot be explicitly captured. Their impact on macroscale fluctuations, however, has to be taken into account by the introduction of an appropriate closure model. A new closure model to describe gas−solid drag is introduced. The effect of particle clustering on the interphase momentum-transfer coefficient is taken into account by introducing the concept of effective drag. Clustering results in a decreasing value of the interphase momentum-transfer coefficient because the contribution of particles in the cluster is considered to be negligible in dilute phase flow. For solids fractions greater than 1%, clustering phenomena become increasingly important, resulting in an important decrease of the interphase momentum-transfer coefficient. Calculation results for a riser reactor, both neglecting and including the effects of clustering on the gas−solid drag, are presented and validated with experimental data....

Analysis of the impact velocity of powder particles in the cold-gas dynamic-spray process

Abstract While built on a sound physical foundation, isentropic, one-dimensional models generally used to analyze the dynamics of dilute two-phase (feed-powder particles suspended in a carrier gas) flow during the cold-gas dynamic-spray process, require the use of numerical procedures to obtain solutions for the governing equations. Numerical solutions, unfortunately, do not enable an easy establishment of the relationships between the gas, process and feed-powder parameters on one side and the gas and the particle velocities at the nozzle exit and the particle impact velocity, on the other. Analytical solutions for the governing equations in the limits of small and large relative particle/gas velocities and a multiple non-linear regression analysis are used, in the present work, to develop analytical functions which can be used to compute the gas and the particle exit velocities and the particle impact velocity for a given set of the gas, process, and feed-powder parameters. The results obtained using the analytical functions are found to be in a very good agreement with their numerical and experimental counterparts.

Flow analysis and nozzle-shape optimization for the cold-gas dynamic-spray process

An isentropic, one-dimensional model is used to analyse the dynamics of dilute two-phase (feed powder particles plus the carrier gas) flow during the cold-spray process. While the physical foundation of the model is quite straightforward, the solution for the model can be obtained only numerically. The results obtained show that there is a particle-velocity-dependent, carrier-gas-invariant optimal value of the relative gas/particle Mach number that maximizes the drag force acting on feed powder particles and, hence, maximizes the acceleration of the particles. Furthermore, it is found that if the cold-spray nozzle is designed in such a way that at each axial location the acceleration of the particles is maximized, a significant increase in the average velocity of the particles at the nozzle exit can be obtained. For the optimum design of the nozzle, helium as the carrier gas is found to give rise to a substantially higher exit velocity of the particles than air. All these findings are in good agreement with experimental observations.

Grid-Averaged Lagrangian Equations of Dispersed Phase in Dilute Two-Phase Flow

k0 pi i is derived and demonstrated through a well-dee ned test problem of droplet-loading mixing layer. It is shown that accurate predictions of the turbulence kinetic energy of the droplets can be achieved by solving from this set of grid-averaged Lagrangian equations with signie cantly less computational droplets than those of a purely stochasticmodel. The information of the turbulence modulation quantity, which is solved from the grid-averaged Lagrangian transport equation of hu 0u k 0 pii, can also be used to evaluate accurately the source term accounting for the two-phase interaction in the turbulence kinetic energy k transport equation of the carrier e uid, which is important for determining the turbulence characteristics of the carrier e uid. Nomenclature CD = drag coefe cient C3 = turbulence model constant; see Eq. (10) dp = droplet diameter g = gravity K = history-force kernel k = turbulence kinetic energy N = number e ow rate P, Q = values of probability density function Rep = particle Reynolds number Sp, sp = instantaneous and mean time source terms due to turbulence- particle interactions, respectively T = time U, u = instantaneous and mean velocities, respectively hu 0 i u

Simulation of Particle Impact with a Wedge in Dilute Two-phase Flow

Dilute solid-fluid flow over a wedge in a stationary channel is numerically solved using one-way coupling between fluid and solid particles. The two-dimensional, steady, laminar carrierphaseflow is determined by Galerkin finite-element method using Newton's iteration for primitive variables, pressure, and velocity. Velocity is interpolated biquadratically and pressure is interpolated linearly. Parameter continuation is used to compute solutions for relatively large values of flow Reynolds number. Individual particles are tracked from specified inlet positionsby a fourth-order Runge-Kutta method applied to the equations of motion ofthe particle. Forces acting on the particle include drag, pressure, and inertia. Forces due to particle-particle interaction and Basset forces are neglected. Collisions with the wedge and the walls of the channel are modelled via assumed coefficients ofrestitution in both the normal and the tangential directions. The point of actual impact is determined by interpolation. Results are presented for various parameters, such as particle diameter, wedge angle, Reynolds number, particle density, etc.

Modelling of micro-particle agglomeration in turbulent flows

The present study introduces a stochastic Lagrangian agglomeration model to be applied within the framework of the classical Euler/Lagrange approach. The main subject of this contribution will concentrate on the modelling of the turbulent particle transport and a comprehensive modelling of particle collision and agglomeration. In the Lagrangian approach, the dynamic equation of motion for each individual particle is solved, and the properties of the particulate phase are given by the ensemble average over a large number of particles. The description of particle agglomeration in turbulent flow is governed by three consecutive and interrelated processes: The particle collisions in turbulent flow are calculated on the basis of the stochastic collision model (Sommerfeld, Int. J. Multiphase Flow 27 (2001) 1828), considering the collision efficiency due to geometrical size difference as a function of relative Stokes number, and modelling the agglomeration efficiency determined by the sticking potential due to the van der Waals forces. To analyse the agglomeration model, a homogeneous isotropic turbulent field is considered. The particle volume fraction is chosen to be small, so that the concentration condition for dilute two-phase flow is fulfilled and the modulation of turbulence due to the particle phase may be neglected. Under different boundary conditions, such as turbulence intensity and particle relaxation times, the time-dependent change of particle size distribution and number concentration is calculated. These numerical results allow one to understand the effect of the considered parameters on the agglomeration process.

Linear instability of dilute particle-laden compressible mixing layers

The effects of fluid–particle interaction on the temporal development of the compressible mixing layer is investigated using a linearization of the set of equations used to describe dilute two-phase flows with an Eulerian approach. The evolution of the characteristics of the most unstable perturbation as a function of the convective Mach number is studied, including the variation of the wave number, the maximum growth rate, and the obliquity of the disturbance. A parametric investigation is conducted on the influence of the particle inertia, mass loading, and specific heat. The results show the important influence of the particles on the three-dimensional perturbation amplification and the growth rate attenuation when compressibility effects increase.