Multiphase Flow Computations Research Articles

Development of higher order deterministic approaches of forward and inverse uncertainty quantification for nonlinear engineering systems

This paper introduces an innovative numerical scheme that may accurately quantify the parameter and response distributions with minimal computational costs, with specific applications to the numerical computation of multiphase flows under various fluid conditions. The probabilistic approach of uncertainty quantification, e.g., the Monte Carlo simulation, although known to be effective in estimating the parameter/response distribution for highly nonlinear problems, is usually not applicable to multi-physics and multi-scale integrated systems due to its high computational cost. The objective of this study is first, to reduce the computing demand for complex physical system calculations and, at the same time, to accurately predict actual probability distributions by developing a higher order deterministic approach to uncertainty quantification. To achieve this, instead of executing simulation codes multiple times to estimate the probability distribution, the first to fourth moments of the probability density function are derived directly by employing a second order Taylor expansion form of the responses along with Bayesian statistics. This characterizes the parameter and response distributions computed by inverse and forward uncertainty quantification based on their mean value, mode, covariance, skewness, and kurtosis. The parameter distribution is estimated by the numerically calculated higher moments of the parameters with a posteriori probability density function being derived by Bayesian inference. The parameter distribution is then mapped into the second order approximated response function to compute the characteristics of the response distribution through both analytical calculation and numerical approximation of the moments of the responses. The proposed approach is numerically demonstrated through simulations of two benchmark problems: MIT pressurizer and FEBA tests. The prediction of the responses is improved by inverse uncertainty quantification as numerical calculations with the a posteriori model parameters are in better agreement with the measured data over a wide range of transient periods. Additionally, the response distribution estimated by the proposed approach of forward uncertainty quantification accurately predicts the distribution computed by the Monte Carlo simulation.

Revisiting the Becker-Kistiakowsky-Wilson equation of state

The Jones-Wilkins-Lee (JWL) equation of state (EOS) is the most popular equation of state used in hydrocodes to model detonation products thermodynamics resulting from high explosives. However, the JWL EOS presents several major difficulties. Namely, its range of validity and convexity is limited as its parameters are adjusted on a given reference curve. When used in conditions where the thermodynamic state varies significantly, computational failure may happen. It occurs frequently in multiphase flow computations when heat and mass transfers with other phases are present, as the resulting thermodynamic state is far from the reference curve used for adjustment. Moreover, the detonation products composition is absent in the JWL formulation. This is another important limitation when additional physics are involved, such as post-combustion, phase transition, or when dealing with non-ideal explosives. In this work, the Becker-Kistiakowsky-Wilson (BKW) EOS is considered instead. The BKW EOS is widely used in thermochemical codes, as parent EOS to provide the reference data used to fit JWL EOS parameters. The BKW EOS present several advantages. Its formulation considers variable gas species in the detonation products. No specific adjustment is needed for a given explosive, and only the detonation products composition requires computational efforts. Its range of validity is large, as the various parameters have been adjusted for each gas species separately. However, the BKW EOS also present difficulties which limit its application in hydrocodes. These difficulties are the aim of the present work. The BKW EOS is currently restricted to thermochemical codes as its mathematical formulation is computationally expensive. It also requires the gas chemical composition to be computed via an appropriate chemical equilibrium solver. In the present work, the computational cost of the BKW EOS is reduced thanks to the thermodynamic relaxation method of Neron et al. (2023). The variable composition of the detonation products is accounted for through specific relaxation terms computed once, with the help of conventional thermochemical codes. The condensed species, such as solid carbon, are modelled with the Cochran-Chan (CC) EOS, having a simpler thermodynamic formulation and being thermodynamically consistent compared to usual EOSs used for condensed species. This multiphase formulation made of BKW gas mixture with variable composition and condensed phases is considered in temperature and pressure equilibrium. It is embedded in a multiphase flow formulation, with the help of the thermodynamic relaxation method mentioned above. It shows comparable computational cost as the JWL formulation, with significantly enhanced physics capabilities and improved robustness.

Smoothed particle hydrodynamics modelling of multiphase flows: an overview

AbstractSmoothed particle hydrodynamics (SPH) is a meshless, particle-based approach that has been increasingly applied for modelling of various fluid-flow phenomena. Concerning multiphase flow computations, an advantage of the Lagrangian SPH over Eulerian approaches is that the advection step is straightforward. Consequently, the interphasial surface can be explicitly determined from the positions of particles representing different phases; therefore, there is no need for the interface reconstruction step. In this review paper, we briefly recall the basics of the SPH approach, and in particular the physical modelling and numerical implementation issues. We also mention the weaknesses of the approach and some remedies to overcome them. Then, we demonstrate the applicability of SPH to selected interfacial flow cases, including the liquid column break-up, gas–liquid flow regimes in a channel capturing the transitions between them and the wetting phenomena. Concerning the two-fluid modelling, it is illustrated with sediment transport in the presence of surface waves. Various other applications are briefly recalled from the rich and growing literature on the subject, followed by a tentative list of challenges in multiphase SPH.

On the computation of compressible multiphase flows with heat and mass transfer in elastic pipelines

The present work is motivated by the engineering need to simulate the multiphase flows in the course of the start-up of a long-distance crude oil pipeline. We propose a model for compressible multiphase flows with heat and mass transfer and diffusion processes in elastic pipelines. The model is derived with two approaches, i.e., (a) the spatial averaging procedure of a single phase model and (b) the Arbitrary Lagrangian Eulerian (ALE) formulation of the Baer-Nunziato model with quasi-1D approximation. The diffusion processes include the viscous dissipation, wall friction, heat conduction, and heat exchange with external environment. In particular, the wall friction consists of steady friction term calculated by Darcy-Weisbach formula, and the unsteady friction determined by the instantaneous-acceleration-based (IAB) model. The unsteady friction modifies the characteristic structure of the hyperbolic part of the model. For the solution of the hyperbolic part, we propose a three-wave approximate Riemann solver incorporating the unsteady friction term. Mass and heat transfer are realized via instantaneous relaxations of the chemical potential and temperature, respectively. Efficient iterative relaxation procedures for N-phase flows have been proposed. We have validated the proposed model and numerical methods against some benchmark multiphase problems and applied the model to calculate the start-up of a realistic liquid pipeline with intermediate pump station boundary condition.

An appropriate numerical dissipation for SLAU2 towards shock-stable compressible multiphase flow simulations

This paper proposes a novel method for the computation of compressible multiphase flows under the assumption of pressure equilibrium based on a 6-equation model and the AUSM (Advection Upstream Splitting Method) family. In this study, we introduce a new numerical pressure flux dissipation term based on the relative velocities in the gas and liquid phases to develop an analogous carbuncle-suppression mechanism that is applicable to gas dynamics. We also propose a mass flux dissipation term based on the pressure ratio at the gas-liquid interface and incorporate both terms into SLAU2, an AUSM-family scheme, to achieve robustness against shock anomalies.

Application of a high-order MP scheme to computation of multi-phase flows with heat and mass transfer

Reliable prediction of unsteady multiphase flows with mass changes, large density ratios, and complex flow features is very challenging due to the need to accurately resolve both complex interface deformations and steep gradients associated with material interfaces. This paper describes an application of our recently developed high-order monotonicity preserving (MP) scheme – a reliable and robust scheme for long-term computations of flows contain complex smooth structures interspersed with discontinuities (Ha and Lee, 2020) − to the solution of unsteady incompressible two-phase flows in the presence of heat and mass changes. The flow is governed by the multiphase Navier–Stokes equations, which comprises of the mixture continuity equation, the mixture momentum equations, the mixture energy equation, and the continuity equation for the individual phase. The numerical discretization is implemented on generalized curvilinear grids for facilitating flow simulations of the complex geometries. Our high-order MP scheme is applied to the calculations of the convective and surface tension terms. To obtain an effectively simultaneous solution of all equations, a novel dual-time stepping approach in conjunction with an implicit Runge–Kutta three-stage scheme is proposed. The reliability of the proposed numerical algorithm is then examined for a 1D Stefan condensation problem and two condensing flows including single bubble and steam-jet condensation problems. Particularly, for the single bubble condensation test, the bubble characteristics at different time instants are evaluated both qualitatively and quantitatively. For the steam-jet condensation test, the capability for capturing the evolution of interface deformations under different thermal and hydrodynamic flow conditions is highlighted. Finally, the effects of subcooling degree and steam mass flow rate on heat and mass transfer characteristics are investigated and discussed.

Computational interfacial rheology

Fluid–fluid interfaces, laden with polymers, particles or other surface-active moieties, often show a rheologically complex response to deformations, in particular when strong lateral interactions are present between these moieties. The response of the interface can then no longer be described by an isotropic surface tension alone. These “structured” soft-matter interfaces are found in many industrial applications, ranging from foods, cosmetics and pharmaceuticals, to oil recovery. Also many biomedical applications involve such interfaces, including those involving lung surfactants and biofilms. In order to understand, design and optimize processes in which structured interfaces are present, flow predictions of how such multiphase systems deform are of the utmost importance, which is the goal of “computational interfacial rheology”, the main topic of this review. We start by rigorously establishing the stress boundary condition used in the computation of multi-phase flows, and show how this changes when the interface is rheologically complex. Then, constitutive models for the extra stress in interfaces, ranging from 2D generalized Newtonian to hyperelastic and viscoelastic, are reviewed extensively, including common pitfalls when applying these models. This is followed by an overview of different approaches to measure interfacial rheological properties, and a discussion of advanced numerical implementations for deforming interfaces. We conclude with an outlook for this relatively young and exciting field.

An improvement of interface computation of incompressible two-phase flows based on coupling volume of fluid with level-set methods

ABSTRACTA coupling technique for interface simulations applied to the fields of ship hydrodynamics and multiphase flow computations is presented. The technique takes advantages of mass conservation property of the volume-of-fluid method and sharpened interface computation of the Level-Set approach. An improved scheme for the gradient computation of the Level-Set function during re-initialisation is also introduced. A solver with the present coupling is built into the interFoam solver of the OpenFOAM platform. Three test cases, i.e. a free rising air bubble in still water, dam-break simulation with a hump in dry bed channel and free surface flow around a container ship are conducted to demonstrate the capability of the new solver. A good agreement between the simulated results and the benchmark and/or measured data shows the significant potential of the present technique in both academic research and practical applications.

A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows

The monotonicity preserving (MP) polynomial-based reconstruction scheme with Runge-Kutta time stepping by Suresh and Huynh (1997) has been applied extensively in computational fluid dynamics for solving hyperbolic partial differential equations and other convection-dominated problems. However, despite of its algorithmic simplicity in achieving a desired high-order of accuracy, the MP scheme is shown to lose the accuracy and robustness for a long-term computation of flows containing complex smooth structures interspersed with discontinuities. In this paper, the original MP scheme is first reviewed with a focus on the potential causes that lead to its failure. We then propose an improved MP scheme that can overcome the weakness in the MP scheme. The proposed MP scheme is constructed so that it can bypass smooth extrema without loss of accuracy while preserving monotonicity and properly reducing the order of accuracy near discontinuities. Various challenging scalar advection test cases are considered to demonstrate the performance of the presented method. Numerical results reveal that it achieves the highly accurate solutions in smooth parts of the flow and yields a significant improvement in accuracy and shape-preserving property. Moreover, the successful application to several classes of two- and three-dimensional unsteady multi-phase flows highlights the potential of the present method for resolving both sharp and complex interface deformations.

An energy-preserving level set method for multiphase flows

The computation of multiphase flows presents a subtle energetic equilibrium between potential (i.e., surface) and kinetic energies. The use of traditional interface-capturing schemes provides no control over such a dynamic balance. In the spirit of the well-known symmetry-preserving and mimetic schemes, whose physics-compatible discretizations rely upon preserving the underlying mathematical structures of the space, we identify the corresponding structure and propose a new discretization strategy for curvature. The new scheme ensures conservation of mechanical energy (i.e., surface plus kinetic) up to temporal integration. Inviscid numerical simulations are performed to show the robustness of such a method.

A high-order finite volume method with improved isotherms reconstruction for the computation of multiphase flows using the Navier–Stokes–Korteweg equations

In this work we solve the Navier–Stokes–Korteweg (NSK) equations to simulate a two-phase fluid with phase change. We use these equations on a diffuse interface approach, where the properties of the fluid vary continuously across the interface that separates the different phases. The model is able to describe the behavior of both phases with the same set of equations, and it is also able to handle problems with great changes in the topology of the problem. However, high-order derivatives are present in NSK equations, which is a difficulty for the design of a numerical method to solve the problem. Here, we propose the use of a high-order Finite Volume method with Moving Least Squares approximations to handle high-order derivatives and solve the NSK equations. Moreover, a new methodology to obtain accurate equations of state is presented. In this method, we use any accurate equation of state for the pure phases. Under the saturation curve, a B-spline reconstruction fulfilling a given set of thermodynamic criteria is performed. The new EOS can be used for computations using diffuse interface modeling. Several numerical examples to show the accuracy of the new approach are presented.

Investigation of internal flow pattern of a multiphase axial pump

Due to recent advances in computational techniques, single phase performances of pumps can be easily estimated using CFD. However, when the working fluid is a mixture of liquid and gas (multi-phase flow) computations may struggle to capture the more complex physical phenomena that occurs, and the precision of the prediction worsen as the gas volume fraction at pump inlet is increased. To provide reliable benchmark for the improvement of simulations, in this study we experimentally investigated the internal flow pattern of a multiphase pump using high speed camera and high frequency pressure measurements, so as to obtain the data of pressure distribution inside of impeller, or to obtain blade loading, for several gas concentration conditions. Unsteady phenomena were identified at low flow rate conditions, and an explanation of their nature has been attempted.

A Study of Pressure Gradient in Multiphase Flow in Vertical Pipes

The study of multiphase flow in vertical pipes is aimed at effective and accurate design of tubing, surface facilities and well performance optimization for the production of oil and gas in the petroleum industry by developing a better approach for predicting pressure gradient. In this study, field data was analyzed using mathematical model, multiphase flow correlations, statistical model, and computer programming to predict accurately the flow regime, liquid holdup and pressure drop gradient which are important in the optimization of well. A Computer programme was used to prediction pressure drop gradient. Four dimensionless parameters liquid velocity number (Nlv), gas velocity number (Ngv), pipe diameter number (Nd), liquid viscosity number (Nl), were chosen because they represent an integration of the two dominant components that influence pressure drop in pipes. These dominant component are flow channel/media and the flowing fluid. The model was found to give a fit of 100% to the selected data points. Hagedorn & Brown, Griffith &Wallis correlations and model were compared with field data and the overall pressure gradient for a total depth of 10000ft was predicted. The predicted pressure gradient measured was found to be 0.320778psi/ft, Graffith& Wallis gave 0.382649Psi/ft, Hagedorn & Brown gave 0.382649Psi/ft; whereas generated model gave 0.271514Psi/ft. These results indicate that the model equation generated is better and leads to a reasonably accurate prediction of pressure drop gradient according to measured pressure gradient.

Study of Pressure Gradient in Multiphase Flow in Vertical Pipes

The study of multiphase flow in vertical pipes is aimed at effective and accurate design of tubing, surface facilities and well performance optimization for the production of oil and gas in the petroleum industry by developing a better approach for predicting pressure gradient. In this study, field data was analyzed using mathematical model, multiphase flow correlations, statistical model, and computer programming to predict accurately the flow regime, liquid holdup and pressure drop gradient which are important in the optimization of well. A Computer programme was used to prediction pressure drop gradient. Four dimensionless parameters liquid velocity number (Nlv), gas velocity number (Ngv), pipe diameter number (Nd), liquid viscosity number (Nl), were chosen because they represent an integration of the two dominant components that influence pressure drop in pipes. These dominant component are flow channel/media and the flowing fluid. The model was found to give a fit of 100% to the selected data points. Hagedorn & Brown, Griffith &Wallis correlations and model were compared with field data and the overall pressure gradient for a total depth of 10000ft was predicted. The predicted pressure gradient measured was found to be 0.320778psi/ft, Graffith& Wallis gave 0.382649Psi/ft, Hagedorn & Brown gave 0.382649Psi/ft; whereas generated model gave 0.271514Psi/ft. These results indicate that the model equation generated is better and leads to a reasonably accurate prediction of pressure drop gradient according to measured pressure gradient.

A Hybrid Eulerian–Eulerian Discrete-Phase Model of Turbulent Bubbly Flow

The Eulerian–Eulerian two-fluid model (EE) is a powerful general model for multiphase flow computations. However, one limitation of the EE model is that it has no ability to estimate the local bubble sizes by itself. In this work, we have combined the discrete phase model (DPM) to estimate the evolution of bubble sizes with the EE model. In the DPM, the change of bubble size distribution is estimated by coalescence, breakup, and volumetric expansion modeling of the bubbles. The time-varying bubble distribution is used to compute the local interface area between gas and liquid phase, which is then used to estimate the momentum interactions such as drag, lift, wall lubrication, and turbulent dispersion forces for the EE model. In this work, this newly developed hybrid model Eulerian–Eulerian discrete-phase model (EEDPM) is applied to compute an upward flowing bubbly flow in a vertical pipe and the results are compared with previous experimental work of Hibiki et al. (2001, “Axial Interfacial Area Transport of Vertical Bubbly Flows,” Int. J. Heat Mass Transfer, 44(10), pp. 1869–1888). The EEDPM model is able to reasonably predict the locally different bubble size distributions and the velocity and gas fraction fields. On the other hand, the standard EE model without the DPM shows good comparison with measurements only when the prescribed constant initial bubble size is accurate and does not change much. Parametric studies are implemented to understand the contributions of bubble interactions and volumetric expansion on the size change of bubbles quantitatively. The results show that coalescence is larger than other effects, and naturally increases in importance with increasing gas fraction.

Computations of Homogeneous Multiphase Real Fluid Flows at All Speeds

Computations of all-speed multiphase real fluid flows for aerospace launch vehicles are known to be very demanding, owing to several issues that include robust capturing of two-phase shock/phase discontinuity and proper scaling of numerical flux at steady/unsteady all-speed flows. This paper deals with the development of computational components to overcome difficulties arising from these issues. First, the shock-discontinuity-sensing term used in the two-phase and RoeM schemes is modified because the existing shock-discontinuity-sensing term is not suitable for complex equation of state of real fluids. The accuracy of the two-phase and RoeM schemes for unsteady low-Mach-number flows is then improveed through separate scaling of the velocity- and pressure-difference terms. It is demonstrated that the resulting schemes are capable of robustly and accurately capturing phenomena involving shock and phase discontinuities and interactions between them for numerous two-phase problems. Finally, numerical simulations of cryogenic cavitation and three-dimensional cavitating flows around the Korea Aerospace Research Institute turbopump are presented to confirm accurate and robust behavior of the proposed approaches in computations of multiphase real fluid flows at all speeds.

A simple and fast phase transition relaxation solver for compressible multicomponent two-phase flows

Abstract The present paper aims at building a fast and accurate phase transition solver dedicated to unsteady multiphase flow computations. In a previous contribution (Chiapolino et al. 2017), such a solver was successfully developed to compute thermodynamic equilibrium between a liquid phase and its corresponding vapor phase. The present work extends the solver’s range of application by considering a multicomponent gas phase instead of pure vapor, a necessary improvement in most practical applications. The solver proves easy to implement compared to common iterative procedures, and allows systematic CPU savings over 50%, at no cost in terms of accuracy. It is validated against solutions based on an accurate but expensive iterative solver. Its capability to deal with cavitating, evaporating and condensing two-phase flows is highlighted on severe test problems both 1D and 2D.

Robust second-order scheme for multi-phase flow computations

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

A simple phase transition relaxation solver for liquid–vapor flows

SummaryDetermining liquid–vapor phase equilibrium is often required in multiphase flow computations. Existing equilibrium solvers are either accurate but computationally expensive or cheap but inaccurate. The present paper aims at building a fast and accurate specific phase equilibrium solver, specifically devoted to unsteady multiphase flow computations. Moreover, the solver is efficient at phase diagram bounds, where non‐equilibrium pure liquid and pure gas are present. It is systematically validated against solutions based on an accurate (but expensive) solver. Its capability to deal with cavitating, evaporating, and condensing two‐phase flows is highlighted on severe test problems both 1D and 2D. Copyright © 2016 John Wiley & Sons, Ltd.

Direct multiphase mesh generation from 3D images using anisotropic mesh adaptation and a redistancing equation

In this paper, a new methodology to build automatically 3D adapted meshes, ready for numerical simulations and directly from images, is proposed. It is based on the Immersed Image Method, which interpolates the image information on an initial mesh and combines it with parallel automatic anisotropic mesh adaptation with a control of the number of mesh nodes. Simultaneously, a smooth redistancing technique, based on the resolution of a Hamilton–Jacobi equation, is developed to produce phase functions for the objects to be detected in the image. The proposed methodology is applied on mesh generation and in multiphase flow computations based on 2D and 3D images.