Comparison of turbulence models in simulating swirling pipe flows

Swirling flow is a common phenomenon in engineering applications. A numerical study of the swirling flow inside a straight pipe was carried out in the present work with the aid of the commercial CFD code fluent. Two-dimensional simulations were performed, and two turbulence models were used, namely, the RNG k–ε model and the Reynolds stress model. Results at various swirl numbers were obtained and compared with available experimental data to determine if the numerical method is valid when modeling swirling flows. It has been shown that the RNG k–ε model is in better agreement with experimental velocity profiles for low swirl, while the Reynolds stress model becomes more appropriate as the swirl is increased. However, both turbulence models predict an unrealistic decay of the turbulence quantities for the flows considered here, indicating the inadequacy of such models in simulating developing pipe flows with swirl.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

基于微观方法对液态铝泡沫宏观析液过程的数值模拟研究

通过微元管内流动模型,研究了液态金属熔体泡沫体内单条Plateau边界内析液过程中的速度场.分析了不同Newton表面粘度,即不同的气液界面运动能力(无量纲参数M)下,Plateau边界内速度的分布.结果显示:在相同的气液界面运动能力和曲率半径条件下,泡沫体内固壁处Plateau边界内速度约是内部Plateau边界内速度的6～8倍,从而解释了不同容器内泡沫体析液速率的差异现象;发现M存在1个临界值,在此值的两边,液膜厚度与曲率半径的比值对Plateau边界内速度的影响呈现出相反的趋势.结合多尺度方法,进而利用微观计算结果建立了泡沫体的整体宏观析液模型,将模型计算结果和经典析液方程计算结果及实验值作了比较,结果表明:该文模型计算结果与实验值在泡沫层上部、中部吻合较好,M值和气泡大小对析液过程有显著影响.     

Numerical analysis of flow-induced gas entrainment in roll coating

Gas entrainment by plane liquid jets which plunge into a liquid pool is analyzed by numerical simulations. The numerical model is based on the equations of incompressible newtonian fluids flow. The two-phase flow problem is described with the volume-of-fluid method. The dynamic behaviour of the interface is characterized by two similarity parameters, the capillary number Ca = uη/σ and the property number Γ = σ(ρ/η⁴g)^1/3 where u is the velocity, η the dynamic viscosity, σ the interface tension, ρ the density and g the gravitational constant. Numerical simulations are performed with the open source CFD code OpenFOAM. In the simulations the stability of the gas–liquid meniscus is tested for different sets of Ca and Γ. Critical values of Ca which indicate the beginning gas entrainment are deduced from the inspection of the simulation results. The findings of the numerical investigations agree well with corresponding experimental results.     

Computational fluid dynamics modelling of gas jets impinging onto liquid pools

Gas jets impinging onto a gas–liquid interface of a liquid pool are studied using computational fluid dynamics modelling, which aims to obtain a better understanding of the behaviour of the gas jets used metallurgical engineering industry. The gas and liquid flows are modelled using the volume of fluid technique. The governing equations are formulated using the density and viscosity of the “gas–liquid mixture”, which are described in terms of the phase volume fraction. Reynolds averaging is applied to yield a set of Reynolds-averaged conservation equations for the mass and momentum, and the k–ε turbulence model. The deformation of the gas–liquid interface is modelled by the pressure jump across the interface via the Young–Laplace equation. The governing equations in the axisymmetric cylindrical coordinates are solved using the commercial CFD code, FLUENT. The computed results are compared with experimental and theoretical data reported in the literature. The CFD modelling allows the simultaneous evaluation of the gas flow field, the free liquid surface and the bulk liquid flow, and provides useful insight to the highly complex, and industrially significant flows in the jetting system.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

Theoretical model for swirling thin film flows inside nozzles with converging-diverging shapes

A quasi-one-dimensional model was developed to describe a swirling, thin, liquid film inside nozzles with different wall profiles. The model quantifies the effects of swirl strength, initial film thickness, and Reynolds and Weber numbers on the film thickness along the nozzle surface. Moreover, the model allows for a rapid (at least, qualitative) evaluation of different effects, e.g. of the swirl strength and nozzle geometry, and can serve as a benchmark case for the subsequent more involved numerical simulations. Steady-state solutions are presented as a function of various parameters. The effect of the nozzle geometry on film thickness is explored. As swirling flow entered the expanding (diverging) section of the nozzle, film thickness decreased to satisfy continuity (to conserve mass). Conversely, film thickness increased upon entering the contracting (converging) region of the nozzle. Geometric effects controlled film thicknesses much more than other flow parameters. This quasi-one-dimensional model for a swirling thin film can be useful for designing a swirl jet used in various industrial applications.     

Mathematical and physical simulation of the interaction between a gas jet and a liquid free surface

In this work a two phase 3D mathematical model was developed using the volume of fluid (VOF) algorithm, which is able to accurately describe the cavity geometry and size as well as the liquid flow patterns created when a gas jet that impinges on a liquid free surface. These phenomena are commonly found in steelmaking operations such as in the Electric Arc Furnace (EAF) and the Basic Oxygen Furnace (BOF) where oxygen jets impinge on a steel bath and they control heat, momentum and mass transfer. The model was successfully validated with measurements made on a physical model through velocity fields obtained by Particle Image Velocimetry (PIV) and high speed camera images of the cavity. Agreement between model predictions and experimental measurements is excellent in both x-velocity component of the liquid and cavity sizes. The cavity formed in the liquid by the impinging jet depends on a force balance at the free surface where the inertial force of the jet governs this phenomena, while the liquid circulation depends on also the jet inertial force of the jet, but its angle plays an important role, being the lowest angle the best choice to shear the bath and promote stronger circulation and better mixing in the liquid.     

A compressible two-phase flow framework for Large Eddy Simulations of liquid-propellant rocket engines

Atomization of liquid jets is a key feature of many propulsion systems, such as jet engines, internal combustion engines or liquid-propellant rocket engines (LRE). As it controls the characteristics of the spray, atomization has a great influence on the complex interaction between phenomena such as evaporation, turbulence, acoustics and combustion. In this context, Computational Fluid Dynamics is a promising way to bring better understanding of dynamic phenomena involving atomization, such as e.g. high-frequency combustion instabilities in LRE. However the unsteady simulation of primary atomization in reactive compressible two-phase flows is very challenging, due to the variety of the spatial and temporal scales, as well as to the high density, velocity and temperature gradients which require robust and efficient numerical methods. To address this issue, a numerical strategy is proposed in this paper, which is able to describe the dynamics of the whole chain of mechanisms from the liquid injection to its atomization and combustion. Primary atomization is modeled by a coupling between a homogeneous diffuse interface model and a kinetic-based Eulerian model for the spray. This strategy is successfully applied to the unsteady simulation of an operating point of the Onera’s Mascotte test bench, representative of one coaxial injector of LRE operating under subcritical conditions. The dynamics of the liquid core is retrieved and the flame shape as well as Sauter mean diameters are in good agreement with experimental results. These results demonstrate the ability of the strategy to deal with the harsh conditions of cryogenic combustion, and provide a promising framework for future studies of combustion instabilities in LRE.     

Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model

In this paper simulation of cavitating flow over the Clark-Y hydrofoil is reported using the large eddy simulation (LES) turbulence model and volume of fluid (VOF) technique. We applied an incompressible LES modelling approach based on an implicit method for the subgrid terms. To apply the cavitation model, the flow has been considered as a single fluid, two-phase mixture. A transport equation model for the local volume fraction of vapour is solved and a finite rate mass transfer model is used for the vapourization and condensation processes. A compressive volume of fluid (VOF) method is applied to track the interface of liquid and vapour phases. This simulation is performed using a finite volume, two phase solver available in the framework of the OpenFOAM (Open Field Operation and Manipulation) software package. Simulation is performed for the cloud and super-cavitation regimes, i.e., σ = 0.8, 0.4, 0.28. We compared the results of two different mass transfer models, namely Kunz and Sauer models. The results of our simulation are compared for cavitation dynamics, starting point of cavitation, cavity’s diameter and force coefficients with the experimental data, where available. For both of steady state and transient conditions, suitable accuracy has been observed for cavitation dynamics and force coefficients.     

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare-Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1−O(h^1/4) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.     

Stokes flow of an assemblage of porous particles: stress jump condition

The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (??), volume fraction (??), and effective viscosity.     

Numerical investigation of impinging gas jets onto deformable liquid layers

Impinging jets over liquid surfaces are a common practice in the metallurgy and chemical industries. This paper presents a numerical study of the fluid dynamics involved in this kind of processes. URANS simulations are performed using the volume of fluid (VOF) method to deal with the multiphase physics. This unsteady approach with the appropriate computational domain allows resolution of the big eddies responsible for the low frequency phenomena. The solver we used is based on the finite volume method and turbulence is modelled with the realisable k-? model. Two different configurations belonging to the dimpling and splashing modes are under consideration. The results are compared with PIV and LeDaR experimental data previously obtained by the authors. Attention is focused on the surroundings of the impingement, where the interaction between jet and liquid film is much stronger. Finally, frequency analysis is carried out to study the flapping motion of the jet and cavity oscillations.     

The simulation of micro droplet behavior of molten lead-free solder in inkjet printing process and its experimental validation

Micro droplets of molten lead-free solder were ejected at 230 °C using a piezoelectric inkjet printing process. The effect of the micro droplet formation of molten lead-free solder was investigated on the pulse time of the waveform. In this study, a numerical system for simulating the shape evolution of micro droplet of molten lead-free solder in the inkjet printing process was developed based on a solution algorithm (SOLA) scheme for the solution of velocity and pressure fields. It is coupled with the volume-of-fluid (VOF) and piecewise-linear interface construction (PLIC) techniques for the transport of mass and construction of the interface. For the treatment of surface tension effects, a CSF (continuum surface force) model is employed. The simulation results were validated with experimental observations. The numerical result was used to understand the mechanisms of the extrusion of the liquid column, the contraction of the liquid thread, and the pinch-off of the liquid thread at the nozzle exit.     

CFD simulation of precession in sudden pipe expansion flows with low inlet swirl

In this paper, three-dimensional, time-dependent calculations are carried out using the finite volume CFD code CFX4 and the VLES approach with standard k–ε model to simulate the turbulent swirl flow in an axisymmetric sudden expansion with an expansion ratio of 5.0 for a Reynolds number of 10⁵. This flow is unstable over the entire swirl number range considered between 0 and 0.48, and a large-scale coherent structure is found to precess about the centerline. Compared with the unswirled case, inclusion of a slight inlet swirl (swirl number below 0.23) can reduce the precession speed, cause the precession to be against the mean swirl and suppress the flapping motion. Several modes of precession are predicted as the swirl intensity increases, in which the precession, as well as the spiral structure, reverses direction. Accompanying the transition between different modes, abrupt changes in precession frequency are also experienced. Grid sensitivity and comparison with smaller expansion ratio data are also discussed.     

壁面效应对剪切稀化流体内气泡上浮特性的影响

数值研究了壁面效应对剪切稀化流体内气泡上浮运动特性的影响,气液两相的界面捕捉采用流体体积(VOF)法,剪切稀化流体流变特性和气液相间表面张力的计算分别采用Carreau模型和连续表面张力模型.详细研究了不同流变指数下,壁面效应对气泡形状、液相流场和气泡终端速度的影响.结果表明,强的壁面效应或弱的剪切稀化程度会限制气泡的变形和尾涡的形成,使气泡的终端速度减小;气泡终端速度最易受壁面效应的影响;强的壁面效应和强的剪切稀化程度会导致高剪切速率区域出现在壁面附近,引起壁面附近液相表观黏度大幅度的下降.     

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H¹ bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.     

Numerical study of sediment transport in turbulent two-phase flows around an obstacle

This paper numerically investigates the transport of dissolved and particulate pollutants in turbulent channel flows. We present a predictive hydrodynamic model in order to explore the dispersion phenomenon of a pollutant injected at a free surface around an obstacle. The air/water interface was modeled using the volume of fluid method (VOF). Numerical results agree well with experimental data and the penetration of pollutant released at different inlet positions of the channel is studied. The Lagrangian tracking of individual particles was performed, and the transport and deposition of various particle size, density and velocity in the channel were analyzed. The standard k–ε turbulence model was chosen for this simulation.We found that large particles with a density of 1600 kg/m³, a velocity of 2 m/s and a diameter higher than 70 mm are deposited around the obstacle and near the end sill of the channel, while particles of very small size (lower than 5 mm) remain suspended in the flow and arrive at the outlet of the channel without any deposition rate. This factor must be taken into account during the discharge of effluents and pollutants in coastal water.     

射流速度剖面形状变化及其对界面稳定性的影响

用线性稳定性理论研究了内部为轴对称液体射流、外部为同轴气体射流的粘性双流体模型．在液体射流和气体射流分别处于不同压力梯度作用的条件下,讨论了数学模型中射流速度剖面形状变化,并用Chebyshev配点法研究了剪切层厚度变化及其对界面稳定性产生的影响．     

On two-dimensional gas expansion for pressure-gradient equations of Euler system

In this paper we establish the existence of global continuous solutions of gas expansion into a vacuum for the two-dimensional pressure-gradient equations in gas dynamics. Under irrotational condition, By hodograph transformation, the flow is governed by the equation (p−p²_v)p_uu+2p_up_vp_uv+(p−p²_u)p_vv=0, which can be further reduced to a inhomogeneous linearly degenerate system of three equations. Then the problem of the expansion of a wedge of gas into a vacuum is solved in the same way.