Incompressible Two-fluid Model Research Articles

Remedy for ill-posedness and mass conservation error of 1D incompressible two-fluid model with artificial viscosities

The two-fluid model is widely used to describe two-phase flows in complex systems such as nuclear reactors. Although the two-phase flow was successfully simulated, the standard two-fluid model suffers from an ill-posed nature. There are several remedies for the ill-posedness of the one-dimensional (1D) two-fluid model; among those, artificial viscosity is the focus of this study. Some previous works added artificial diffusion terms to both mass and momentum equations to render the two-fluid model well-posed and demonstrated that this method provided a numerically converging model. However, they did not consider mass conservation, which is crucial for analyzing a closed reactor system. In fact, the total mass is not conserved in the previous models. This study improves the artificial viscosity model such that the 1D incompressible two-fluid model is well-posed, and the total mass is conserved. The water faucet and Kelvin-Helmholtz instability flows were simulated to test the effect of the proposed artificial viscosity model. The results indicate that the proposed artificial viscosity model effectively remedies the ill-posedness of the two-fluid model while maintaining a negligible total mass error.

On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

Abstract The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.

Simplified 1D Incompressible Two-Fluid Model with Artificial Diffusion for Slug Flow Capturing in Horizontal and Nearly Horizontal Pipes

This article proposes a numerical resolution of a one-dimensional (1D), transient, simplified two-fluid model regularized with an artificial diffusion term for modeling stratified, wavy and slug flow in horizontal and nearly horizontal pipes. Artificial diffusion is introduced to prevent the unbounded growth of instabilities where the 1D two-fluid model is ill-posed. We propose a method to set the artificial diffusion case by case to obtain the desired cut-off at short wavelengths by combining the choice of the spatial discretisation and the amplification factors obtained by the linear stability analysis of the model. A proper criterion to simulate two-phase to single-phase flow transition, which occurs during slug formation, is also developed. Flow pattern transitions have been numerically computed and compared against theoretical transition boundaries and experimental observations. Moreover, we showed that the developed code computes slug initiation and slug characteristics, in a reasonably accurate way considering the simplicity of the model, comparing numerical results with well-known empirical correlations and experimental data. Furthermore, the model simplicity leads to a computationally-inexpensive numerical resolution; this can be useful in engineering applications where obtaining fast numerical results is fundamental, such as applications involving automated control for two-phase flows.

A dual grid method for the compressible two-fluid model which combines robust flux splitting methodology with high-resolution capturing of incompressible dynamics

The speed of sound in two-phase pipe flow systems is often several orders of magnitude greater than the travelling speed of hydraulic information (volume fractions). Dynamically simulating such flows requires resolution of acoustic and hydraulic waves existing at vastly different spatial and temporal scales. If simulated on the same numerical grid, the need for accuracy in hydraulic waves will necessitate an exaggerated resolution of acoustic waves. Likewise, time steps restricted by the speed of sound are small compared to the time scales active in hydraulic waves. This constitutes a waste of computational potential. The method proposed herein decouples the hydraulic and acoustic scales, greatly improving computational efficiency.The proposed dual grid method solves a four-equation compressible two-fluid model on a principal grid which robustly accounts for the pressure evolution and conserves mass and momentum. An incompressible two-equation model is at the same time solved on a finer grid, resolving the details of the hydraulic evolution. Information from both model formulations is coupled through the terms of the governing transport equations, providing consistency between the grids. Accurate and finely resolved schemes can then be employed for the incompressible two-fluid model without suffering from the time and stability restrictions otherwise enforced by acoustic waves. At the same time, the Hybrid Central-Upwind flux splitting scheme of Evje and Flåtten (2005a) allows for an explicit and numerically robust treatment of the acoustic waves without losing hydraulic accuracy.The dual grid method is tested against four dissimilar problems: A shock tube problem, the water faucet problem, a surge wave and pressure wave problem and a roll-wave case. In all problems, the proposed scheme provided significant increases in computational efficiency and accuracy as compared with a single grid arrangement.

Characteristic methods and Roe’s method for the incompressible two-fluid model for stratified pipe flow

This article examines the use of characteristic methods in stratified two-phase pipe flow simulations for obtaining non-dissipative flow predictions. A Roe scheme and several methods based on the principle of characteristics are presented for the two-fluid model. The main focus is finding numerically efficient ways of capturing wave dynamics and flow regime transitions through direct simulation. Capturing flow regime transitions both effciently and precisely necessitates schemes which remain stable in a numerical sense, yet predicts wave growth once the flow conditions turn unstable. Characteristic methods offer the possibility of simulating hyperbolic systems without numerical dissipation. These methods do however lack certain fundamental conservation properties. Challenges related to information scattering and clustering in space and time, particularly around shocks, are also an issue in some method variants. Hybridisations with the finite volume method are proposed which overcome these shortcomings. All methods are compared, evaluating predictions on the onset of linear wave growth and simulations of non-linear, discontinuous roll-waves. The following observations are made: 1. Characteristic methods are excellent at predicting the onset of linear hydrodynamic instability, even with a small number of computational nodes. 2. Dissipative errors in finite volume methods and characteristic hybrids will be closely linked to the CFL number. The Roe scheme and the characteristic hybrids give very little dissipation error as the CFL number approaches unity. This then becomes a question of numerical stability. 3. Adapting dynamic grid cells, moving along with the characteristics information drift, greatly improves simulation efficiency by allowing for longer time steps. Dynamic grid cells are also useful for alleviating the need for interpolation in characteristic methods.On the whole, the performance of the characteristic hybridisations are similar to that of the Roe scheme at large CFL numbers. Characteristic hybridisations perform somewhat better in predicting linear instability while the Roe scheme is better suited for the shock fronts found in the roll-wave flow regime. These method easily out-perform the more basic upwind and Lax–Friedrich schemes.

An Artificial Viscosity for the III-Posed One-Dimensional Incompressible Two-Fluid Model

Methods to remedy the ill-posedness of the basic one-dimensional two-fluid model, which is widely used in nuclear reactor safety codes, have been the subject of considerable study. Both of the two prevalent methods have drawbacks. Unconditional hyperbolization uses nonphysical constitutive relations to create a well-posed two-fluid model that is hyperbolic over all flow conditions. However, when the model is hyperbolized, it is also stabilized, which is not a universal property of two-phase flows. The second method, the preferred method of the U.S. Nuclear Regulatory Commission safety codes, is to simply use a first-order upwind numerical method that relies on numerical viscosity to regularize the ill-posedness of the model by damping the short-wavelength instabilities. Unfortunately, the scale of the “short wavelength” is related to a particular numerical grid or discretization. Because of the consistency of the numerical method, in the limit of an infinitely resolved grid, i.e., Δx, Δt→0, the numerical viscosity vanishes, as does its regularization effect. This results in a somewhat heuristic user guideline that suggests a lower limit on the grid size based on a cross-sectional dimension that is a combination of the long-wavelength assumption and experience. However, a cutoff wavelength achieved by numerical viscosity is not set by the grid size alone but also depends on the time step, the material, and the flow properties, as demonstrated with a von Neumann stability analysis. This can create poor resolution in areas where numerical stability may not be a substantial problem, unless the guideline is intentionally violated. Additionally, strict observance of this limit makes verification by convergence difficult or impossible. Therefore, it is proposed that an artificial viscosity be prescribed explicitly, i.e., independently of any particular numerical method or grid. An artificial viscosity model is derived that prescribes exactly a cutoff in the linear stability growth rate at a specified wavelength, e.g., λ0 = 2DH consistent with the aforementioned user guideline. It is shown, using the water faucet problem, that the proposed artificial viscosity model can be used to remove the high-frequency component of the solution without limiting the resolution of the grid. Furthermore, the solution also converges, which was not the case without the artificial viscosity.

Streamwise oscillations of a cylinder beneath a free surface: Free surface effects on vortex formation modes

A computational study of a viscous incompressible two-fluid model with an oscillating cylinder is investigated at a Reynolds number of 200 and at a dimensionless displacement amplitude of A=0.13 and for the dimensionless forcing cylinder oscillation frequency-to-natural vortex shedding frequency ratios, f/f0=1.5,2.5,3.5. Specifically, two-dimensional flow past a circular cylinder subject to forced in-line oscillations beneath a free surface is considered. The method is based on a finite volume discretization of the two-dimensional continuity and unsteady Navier–Stokes equations (when a solid body is present) on a fixed Cartesian grid. Two-fluid model based on improved volume-of-fluid method is used to discretize the free surface interface. The study focuses on the laminar asymmetric flow structure in the near wake region and lock-on phenomena at a Froude number of 0.2 and for the dimensionless cylinder submergence depths, h=0.25, 0.5 and 0.75. The equivorticity patterns and pressure distribution contours are used for the numerical flow visualization. The code validations in special cases show good comparisons with previous numerical results.

Numerical simulations of basic interfacial instabilities with incompressible two-fluid model

The incompressible two-fluid model for stratified flow was improved. The interface of the stratified two-phase flow was successfully recognized and sharpened within the two-fluid model. After the advection step of volume fraction the numerical diffusion of the interface was reduced in such a way that the thickness of the interface is kept constant during the simulation. The surface tension force was implemented in the system of the two-fluid model equations. The two basic instabilities of stratified flows: Rayleigh–Taylor and Kelvin–Helmholtz instability were used to validate the proposed two-fluid model. The proposed two-fluid model with interface sharpening presents a step towards the simulations of mixed flows, where locally dispersed flow or stratified flow will be simulated with appropriated submodels within the two-fluid model.

Numerical simulation of transient roll-waves in two-phase pipe flow

This paper presents a transient roll-wave simulator based on a one-dimensional incompressible two-fluid model. Using an efficient numerical method and a grid cell length of one pipe diameter, the simulator is able to accurately predict experimental data in the roll-wave regime. Essential to obtaining the stable roll-wave solutions is the use of a modified version of the Biberg friction model. Based on an algebraic eddy viscosity model, the Biberg model yields mechanistic expressions for the wall and interfacial shear stresses in stratified two-phase flow. Input to the model is closure relations for the turbulence levels at either side of the interface. In order to model the increased dissipation of energy in breaking wave fronts, we propose to modify the original closures by adding a term proportional to the negative gradient of the liquid height. As the waves grow and come close to breaking, this term redistributes the shear stresses such that the waves are stabilized. The numerical method used in the simulator is an extension of a previously presented pseudospectral Fourier method. In particular, spectral vanishing viscosity terms are introduced to stabilize short wavelengths inconsistent with the long wavelength approximation of the two-fluid model. The result is a simulator that provides reliable and convergent numerical solutions for all stratified flow conditions. In the last part of the paper, the simulator is tested against roll-wave experiments with horizontal and upward inclined flows of water and a dense gas in a 10 cm pipe. Wave heights and speeds as well as mean pressure drops and holdups are predicted within 20% accuracy when compared with 474 experiments. Furthermore, the transient simulations enable us to study the dynamics and interactions between waves. In agreement with experimental observations, two different wave growth mechanisms are identified, and their influence on the evolution of the roll-wave flow regime is analyzed.

Analysis of a 1D incompressible two-fluid model including artificial diffusion

This article examines a 1D incompressible two-fluid model including artificial tensor diffusion. The aim is to obtain a formulation that provides convergent numerical solutions for all flow conditions within the stratified and the stratified wavy flow regime. With appropriate simplifications, the two-fluid model reduces to one momentum balance, one mass conservation and two algebraic equations. It has previously been established that a formulation that is well posed in possessing exclusively real characteristics can be obtained by including an axial diffusion term in the momentum balance. In this article, however, we demonstrate that this is not sufficient to obtain a system suitable for numerical simulations. Although the unbounded growth rates of the standard two-fluid model are eliminated, linear stability theory predicts that infinitesimal wavelengths still experience finite growth. This entails that grid refinement always will result in new unstable wavelengths being resolved. On the other hand, if artificial axial diffusion is added to both the mass and the momentum equations as suggested here, a cut-off wavelength is established below which all wavelengths are stable. Thus, a numerically converging model is formed, which retains the long-wavelength properties of the standard two-fluid model. The conclusions of the mathematical analysis are substantiated by numerical simulations of ID gravity waves.

Numerical study on the bubbly flow around a hydrofoil in pitching and heaving motions

The air-water bubbly flow around a hydrofoil of NACA65-010, undergoing the heaving and pitching motions in the uniform flow on a horizontal plane, is simulated by an incompressible two-fluid model. The finite element method proposed in a prior paper is applied to solve the model. The Reynolds number defined by the volumetric velocity of the water is 10000 and the volumetric fraction of the air upstream of the hydrofoil, αg0, ranges from 0 to 0.06. The simulation reveals the effects of the αg0 value, the phase difference between the heaving and pitching motions, and the oscillating frequency. The propulsive performance is also discussed in relation to the time variation of the flow properties around the hydrofoil.

Numerical prediction of added mass and damping for a cylinder oscillating in confined incompressible gas–liquid two-phase mixture

This paper is concerned with the numerical analysis of the added mass and damping of a circular cylinder, which oscillates in an air–water bubbly mixture enclosed by a concentric shell. The mixture is assumed to be incompressible. This is because the oscillation frequency of the cylinder is low in this study, and accordingly the pressure change around the cylinder is not so large. An incompressible two-fluid model is solved by the finite element method, proposed by the author in a prior paper, to calculate the bubbly flow around the oscillating cylinder. The analysis reveals the effects of the diameter ratio of the cylinder to the shell, the air volumetric fraction and the bubble diameter. It also clarifies that the increase of damping ratio in the bubbly mixture is attributable to the phase lag of the drag force acting on the cylinder behind the cylinder displacement.

Numerical study on the propulsive performance of a wiggling blade in bubbly flow

In order to search for an efficient propulsion mechanism in an air-water bubbly flow, the propulsive performance of a blade wiggling in the bubbly flow is analysed by a two-dimensional numerical method. The blade, whose geometry is similar to an NACA65–010 hydrofoil, is set in a straight conduit, in which the bubbly mixture flows. The wiggling motion is expressed by a progressive wave with reference to the swimming motions of fish. The bubbly flow is calculated by an incompressible two-fluid model in conjunction with the finite element method proposed by the author in an earlier paper. The calculations reveal the effects of a progressive waveform and volumetric fraction of air upstream of the blade on the propulsive performance of the blade. The time variations of the flow properties around the blade are also discussed in relation to the blade motion and propulsive performance.

Numerical Prediction of Bubbly Flow Around an Oscillating Hydrofoil by Incompressible Two-Fluid Model

Numerical Prediction of Bubbly Flow Around an Oscillating Hydrofoil by Incompressible Two-Fluid Model

ALE finite element method for gas—liquid two-phase flow including moving boundary based on an incompressible two-fluid model

ALE finite element method for gas—liquid two-phase flow including moving boundary based on an incompressible two-fluid model

Numerical Prediction of Bubbly Flow Around an Oscillating Hydrofoil by Incompressible Two‐Fluid Model

This paper is concerned with the calculation of bubbly flow around a hydrofoil (NACA65-010), which is forced to oscillate normal to the uniform flow. An incompressible two-fluid model, expressed in the ALE (Arbitrary Lagrangian-Eulerian) form, is solved by the finite clement method using an upstream scheme. The flow field around the hydrofoil changes in accordance with the oscillation of the hydrofoil, and vortices are periodically shed from the trailing edge. The unsteady change in the forces acting on the hydrofoil is clarified, and the relation between the forces and the periodical flow around the hydrofoil is discussed.

Finite element analysis of bubbly flow around an oscillating square-section cylinder

The bubbly flow around a square-section cylinder, which is forced to oscillate normal to a uniform flow, is simulated by an incompressible two-fluid model. The model is solved by the Arbitrary Lagrangian–Eulerian (ALE) finite element method. The width of the cylinder is 30 mm . A bubbly flow is assumed, and the bubble diameter is set as 1 mm . The volumetric fraction of the gas-phase upstream of the cylinder is 0.05, where the Reynolds number defined by the volumetric velocity of the liquid-phase is 200. With an increment in the cylinder oscillation frequency Sc, the vortex shedding frequency St approaches the Sc value. When the Sc is increased beyond a critical value, the lock-in occurs, in which the vortex shedding is synchronized with the cylinder oscillation. The time-averaged drag coefficient and the root-mean-square of the lift coefficient for the cylinder take their maximum values at the lock-in state. The volumetric fraction of the gas-phase α g is higher in the wake of the cylinder, irrespective of the Sc value. The α g value is lower below the lock-in state and higher in the lock-in state than the value for the stationary cylinder.

ALE finite element method for gas–liquid two-phase flow including moving boundary based on an incompressible two-fluid model

This paper proposes an ALE (Arbitrary Lagrangian–Eulerian) finite element method for gas–liquid two-phase flow, based on an incompressible two-fluid model, to analyze the two-phase flow including moving boundaries. The basic equations are derived by describing the two-fluid model in the ALE form. The solution algorithm is parallel to a fractional step method, and the Galerkin method is employed for the formulation. A quadrilateral element with four nodes is used for the discretization of the computational domain. The present method is also applied to calculate the flow around a circular cylinder, which is forced to oscillate in a quiescent air–water two-phase mixture. The drag coefficients of the cylinder exhibit periodical change in accordance with the variation of the flow around the cylinder. The time variations of the flow field and drag coefficients are discussed in relation to the oscillation of the cylinder.

Numerical Analysis of Air-Water Two-Phase Flow across a Staggered Tube Bundle Using an Incompressible Two-Fluid Model

The air-water two-phase flow across a staggered tube bundle at a pitch-to-diameter ratio of 1.4 is analyzed by an incompressible two-fluid model using the upstream finite element method proposed in a prior study. The Reynolds number, based on the tube diameter and the volumetric velocity of the liquid phase at the tube gap, is 41 000, and the volumetric fraction of the gas phase upstream of the bundle αg0 ranges from 0 to 0.15. The calculated flows exhibit unsteady and complicated behavior irrespective of αg0. The change in the drag coefficient of a tube in the bundle due to αg0 agrees with the experimental result. The distribution of the volumetric fraction of the gas phase around the tube is also in good agreement with the measurement trend. These results indicate that the finite element method is usefully applicable to the two-phase-flow analysis in staggered tube bundles. It is also clarified that the unsteady flows are attributable to the occurrence and movement of vortices of both phases around the tubes.

ALE Finite Element Method Based on an Incompressible Two-Fluid Model for Gas-Liquid Two-Phase Flow Including Moving Boundary.

ALE Finite Element Method Based on an Incompressible Two-Fluid Model for Gas-Liquid Two-Phase Flow Including Moving Boundary.