Simulation of incompressible multiphase flows with complex geometry using etching multiblock method

The incompressible two-phase flows are simulated using combination of an etching multiblock method and a diffuse interface (DI) model, particularly in the complex domain that can be decomposed into multiple rectangular subdomains. The etching multiblock method allows natural communications between the connected subdomains and the efficient parallel computation. The DI model can consider two-phase flows with a large density ratio, and simulate the flows with the moving contact line (MCL) when a geometric formulation of the MCL model is included. Therefore, combination of the etching method and the DI model has potential to deal with a variety of two-phase flows in industrial applications. The performance is examined through a series of numerical experiments. The convergence of the etching method is firstly tested by simulating single-phase flows past a square cylinder, and the method for the multiphase flow simulation is validated by investing drops dripping from a pore. The numerical results are compared with either those from other researchers or experimental data. Good agreement is achieved. The method is also used to investigate the impact of a droplet on a grooved substrate and droplet generation in flow focusing devices.     

A perturbation solution for non-Newtonian fluid two-phase flow through Porous media

In this paper, the mathematical problem of weak non-Newtonian fluid two-phase flow through porous media, including the effect of capillary pressure, is solved by singular perturbation method in combination with regular perturbation method. The asymptotic analytical solutions of the fractional flow function and the wetting phase saturation are obtained. The results are verified by numerical calculations and by classical solutions for corresponding Newtonian case. The influences of the non-Newtonian exponent and capillary pressure are discussed.     

Unified one‐fluid formulation for incompressible flexible solids and multiphase flows: Application to hydrodynamics using the immersed structural potential method (ISPM)

In this paper, we present a two‐dimensional computational framework for the simulation of fluid‐structure interaction problems involving incompressible flexible solids and multiphase flows, further extending the application range of classical immersed computational approaches to the context of hydrodynamics. The proposed method aims to overcome shortcomings such as the restriction of having to deal with similar density ratios among different phases or the restriction to solve single‐phase flows. First, a variation of classical immersed techniques, pioneered with the immersed boundary method (IBM), is presented by rearranging the governing equations, which define the behaviour of the multiple physics involved. The formulation is compatible with the “one‐fluid” formulation for two‐phase flows and can deal with large density ratios with the help of an anisotropic Poisson solver. Second, immersed deformable structures and fluid phases are modelled in an identical manner except for the computation of the deviatoric stresses. The numerical technique followed in this paper builds upon the immersed structural potential method developed by the authors, by adding a level set–based method for the capturing of the fluid‐fluid interfaces and an interface Lagrangian‐based meshless technique for the tracking of the fluid‐structure interface. The spatial discretisation is based on the standard marker‐and‐cell method used in conjunction with a fractional step approach for the pressure/velocity decoupling, a second‐order time integrator, and a fixed‐point iterative scheme. The paper presents a wide d range of two‐dimensional applications involving multiphase flows interacting with immersed deformable solids, including benchmarking against both experimental and alternative numerical schemes.     

Non-conservative pressure-based compressible formulation for multiphase flows with heat and mass transfer

A pressure-based compressible multiphase flow solver has been developed based on non-conservative discretization of the mixture continuity equation. The formulation is an extension of the single phase incompressible pressure-correction approach, such that it can be applied to both two-phase flows using interface resolving methods and general n-phase ensemble-averaged mixture flows. The formulation is currently presented with the single pressure and single temperature assumption, but extension to multiple temperatures is straightforward. A robust treatment of phase change allows the method to model conditions with rapid phase change such as expansion through nozzles and valves. The method has been validated thoroughly using canonical single phase problems such as the shock tube, tank filling and sudden valve closure problems. Multiphase flow validation has been carried out for sound propagation in mixtures using the ensemble-averaged model and pressure wave transmission and reflection across an air-water interface, using the level set interface tracking method. The method has been used to study sound propagation in saturated steam-water systems under thermodynamic non-equilibrium, where the expected drastic reduction in the speed of sound is reproduced. Finally the method is applied to the problem of critical (choked) flow in a nozzle for a saturated steam-water system.     

The numerical simulation of turbulent flows of Newtonian and a sort of non-Newtonian fluid in 180-degree square sectioned bend

Using k-εmodel of turbulence and measured wall functions.turbulent flows ofNewtonian(pure water)and a sort of non-Newtonian fluid(dilute,drag-reduction solutionof polymer in a180-degree curved bend were simulated numerically.The calculated resultsagreed well with the measured velocity profiles.On the basis of calculation andmeasurement,appropriateness of turbulence model to complicated flow in which the large-scale vortex exists was analyzed and discussed.     

Simplified lattice Boltzmann method for non-Newtonian power-law fluid flows

In this paper, we present a simplified lattice Boltzmann method for non-Newtonian power-law fluid flows. The new method adopts the predictor-corrector scheme and reconstructs solutions to the macroscopic equations recovered from the lattice Boltzmann equation through Chapman-Enskog expansion analysis. The truncated power-law model is incorporated into this method to locally adjust the physical viscosity and the associated relaxation parameter, which recovers the non-Newtonian behaviors. Compared with existing non-Newtonian lattice Boltzmann models, the proposed method directly evolves the macroscopic variables instead of the distribution functions, which eliminates the intrinsic drawbacks like high cost in virtual memory and inconvenient implementation of physical boundary conditions. The validity of the method is demonstrated by benchmark tests and comparisons with analytical solution or numerical results in the literature. Benchmark solutions to the three-dimensional lid-driven cavity flow of non-Newtonian power-law fluid are also provided for future reference.     

Code verification for multiphase flows using the method of manufactured solutions

Code verification is the process of ensuring, to the extent possible, that there are no algorithm deficiencies and coding mistakes (bugs) in a scientific computing simulation. Order of accuracy testing using the Method of Manufactured Solutions (MMS) is a rigorous technique that is employed here for code verification of the main components of an open-source, multiphase flow code – MFIX. Code verification is performed here on 2D and 3D, uniform and stretched meshes for incompressible, steady and unsteady, single-phase and two-phase flows using the two-fluid model of MFIX. Currently, the algebraic gas-solid exchange terms are neglected as these can be verified via techniques such as unit-testing. The no-slip wall, free-slip wall, and pressure outflow boundary conditions are verified. Temporal orders of accuracy for first-order and second-order time-marching schemes during unsteady simulations are also assessed. The presence of a modified SIMPLE-based algorithm in the code requires the velocity field to be divergence free in case of the single-phase incompressible model. Similarly, the volume fraction weighted velocity field must be divergence-free for the two-phase incompressible model. A newly-developed curl-based manufactured solution is used to generate manufactured solutions that satisfy the divergence-free constraint during the verification of the single-phase and two-phase incompressible governing equations. Manufactured solutions with constraints due to boundary conditions as well as due to divergence-free flow are derived in order to verify the boundary conditions.     

A generalized Brinkman number for non-Newtonian duct flows

When viscous dissipation effects are important in duct flows the Brinkman number is widely used to quantify the relationship between the heat generated by dissipation and the heat exchanged at the wall. For Newtonian laminar fully developed pipe flow the use of the classical expression for this dimensionless group is appropriate, but under different conditions it can lead to misleading conclusions, such as when comparing flows through different cross-section ducts, flow regimes and mainly non-Newtonian flows. In this work a generalized Brinkman number is proposed, based on an energy balance for the power dissipated by friction, that allows proper quantification of viscous heating effects and reduces to the classical definition in laminar Newtonian pipe flow. The advantages of the new definition are shown and expressions are given for generalized Brinkman numbers in the most common cases.     

Finite element approximation for single-phase multicomponent flows

With the objective of performing computational simulations of mixture problems, this work employed a mechanical modeling of single-phase incompressible multicomponent flows able to deal with geometric and material non-linearities. This model is based on conservative laws of mass and momentum in continuum mechanics, assuming the mixture as a superposition of a number of continuous media, each one with a variable volume fraction field. Using appropriate hypothesis, the model was formulated as a set of non-linear partial differential equations and associated boundary conditions. This set was approximated by a stabilized finite element method, based on a Galerkin/least-squares scheme, in order to circumvent Babu ka–Brezzi condition and to generate stable approximations even in highly advective situations. Some preliminary numerical results have been obtained for non-Newtonian axial injections in backward facing step incompressible flows.     

Direct numerical simulation of particulate flows with a fictitious domain method

Our works on the fictitious domain method for the direct numerical simulation of particulate flows are reviewed, and particularly our recent progresses in the simulations of the motion of particles in Poiseuille flow at moderately high Reynolds numbers are reported. The method is briefly described, and its capability to simulate the motion of spherical and non-spherical particles in Newtonian, non-Newtonian and non-isothermal fluids is demonstrated. In addition, the applications of the fictitious domain method reported in the literature are also reviewed, and some comments on the features of the fictitious domain method and the immersed boundary method are given.     

A novel PIV technique for measurements in multiphase flows and its application to two-phase bubbly flows

A new experimental procedure for performing simultaneous, phase-separated velocity measurements in two-phase flows is introduced. Basically, the novel particle image velocimetry (PIV) technique is a combination of the three most often used PIV techniques in multiphase flows: PIV with fluorescent tracer particles, shadowgraphy, and the digital phase separation with a masking technique. The combination of these three independent measurement techniques is achieved by shifting the background intensity of a PIV recording to a higher, but uniform gray value level. In order to combine the advantages of these multiphase-PIV methods, a new PIV set-up was developed. With this set-up the velocity distributions of the two phases are measured simultaneously with only one b/w camera. This experimental set-up is aimed at providing a means for characterizing the modification of turbulence in the liquid phase by bubbles. This phenomenon is often called "pseudo-turbulence".     

A modified SPH method to model the coalescence of colliding non-Newtonian liquid droplets

This paper develops a modified smoothed particle hydrodynamics (SPH) method to model the coalescence of colliding non-Newtonian liquid droplets. In the present SPH, a van der Waals (vdW) equation of state is particularly used to represent the gas-to-liquid phase transition similar to that of a real fluid. To remove the unphysical behavior of the particle clustering, also known as tensile instability, an optimized particle shifting technique is implemented in the simulations. To validate the numerical method, the formation of a Newtonian vdW droplet is first tested, and it clearly demonstrates that the tensile instability can be effectively removed. The method is then extended to simulate the head-on binary collision of vdW liquid droplets. Both Newtonian and non-Newtonian fluid flows are considered. The effect of Reynolds number on the coalescence process of droplets is analyzed. It is observed that the time up to the completion of the first oscillation period does not always increase as the Reynolds number increases. Results for the off-center binary collision of non-Newtonian vdW liquid droplets are lastly presented. All the results enrich the simulations of the droplet dynamics and deepen understandings of flow physics. Also, the present SPH is able to model the coalescence of colliding non-Newtonian liquid droplets without tensile instability.     

带有障碍物溃坝流动问题的有限元数值模拟研究

针对下游带有障碍物的溃坝流动问题,本文基于两相流动模型,在有限元算法框架下对其进行数值模拟研究。依据水平集(Level Set)方法追踪运动界面,并引入了一个简单的修正技术,保证较好的质量守恒性。为了精确表示运动界面,采用稳定和有效的间断有限元方法求解双曲型Level Set及其重新初始化方程。对于两相统一Navier-Stokes方程,首先利用分裂格式对其解耦,然后通过SUPG(Streamline Upwind Petrov Galerkin)方法进行数值求解。模拟研究了下游带有障碍物的牛顿流体溃坝流动问题,得到的数值结果与文献已有模拟结果及实验结果均吻合较好。此外,还考虑了幂律型非牛顿流体,并分析了不同特性非牛顿流体对于溃坝流动过程和界面形态等的影响。     

Derivation of a spectral pressureless formulation for direct numerical simulation of incompressible channel flows based on a functional formalism

In this article, a new computational spectral algorithm is developed for simulation of general three-dimensional, time-dependent, incompressible channel flow. The development is based on a general functional formalism of non-equilibrium thermodynamics, and, although it is illustrated here for a Newtonian fluid, it is easily adapted to non-Newtonian fluids. The advantage of this algorithm is that the scalar pressure is eliminated from the discrete spectral analog to the equations of motion, which are expressed solely in terms of the spectral coefficients of the velocity vector field. This alleviates the need for the application of boundary conditions on the pressure, the specification of which can be a major source of difficulty in direct numerical simulations. At the same time, the velocity spectrum is quite general, and not subject to any a priori constraints. Thus, it is anticipated that the ideas exposed in the present algorithm can lead to the development of better numerical simulation techniques for complicated three-dimensional and turbulent flows.     

Numerical simulations of non-isothermal three-dimensional flows in an extruder by a finite-volume method

This study considers numerical applications of a finite-volume method to steady non-isothermal flows in geometries close to a single-screw extruder. Two geometrical configurations of the channel, with gap and zero gap, are investigated. The simulations concern incompressible fluids obeying different constitutive equations: Newtonian, generalized Newtonian with shear-thinning properties (Carreau–Yasuda law), and two viscoelastic differential models, the upper convected maxwell (UCM) and the Phan–Thien/Tanner (PTT). The temperature dependence is described by a Williams–Landel–Ferry (WLF) equation. For discretizing the equations and unknowns, we use a staggered grid with a QUICK scheme for the convective-type terms and solve the set of governing equations by a decoupled algorithm, stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) technique, in the viscoelastic case for the UCM model. The numerical results enable us to state the influence of temperature and rheological properties on the flow characteristics in the geometries investigated and underline the complex behaviour of the materials in such configurations.     

An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics

We report on the development and applications of an interface-capturing method aimed at computing three-dimensional incompressible two-phase flows involving high density and viscosity ratios, together with capillary effects. The numerical approach borrows some features to the Volume of Fluid method (since it is essentially based on the transport of the local volume fraction of the liquid) as well as to the Level Set technique (as no explicit reconstruction of the interface is carried out). The transport of the volume fraction is achieved by using a flux-limiting Zalesak scheme and the fronts are prevented from spreading in time by a specific strategy in which the velocity at nodes crossed by the interface is modified to keep the thickness of the transition region constant. As shown on several test cases, this algorithm allows the interface to deform properly while maintaining the numerical thickness of the transition region within three computational cells whatever the structure of the local flow field. The full set of governing equations is then used to investigate some fundamental aspects of bubble dynamics. More precisely we focus on the evolution of shape and rise velocity of a single bubble over a wide range of physical parameters and on head-on and side-by-side interactions between two rising bubbles.     

Efficient simulation of particle-laden turbulent flows with high mass loadings using LES

The paper is concerned with the simulation of particle-laden two-phase flows based on the Euler–Lagrange approach. The methodology developed is driven by two major requirements: (i) the necessity to tackle complex turbulent flows by eddy-resolving schemes such as large-eddy simulation; (ii) the demand to predict dispersed multiphase flows at high mass loadings. First, a highly efficient particle tracking algorithm was developed working on curvilinear, block-structured grids. Second, to allow the prediction of dense two-phase flows, the fluid–particle interaction (two-way coupling) as well as particle–particle collisions (four-way coupling) had to be taken into account. For the latter instead of a stochastic collision model, in the present study a deterministic collision model is considered. Nevertheless, the computational burden is minor owing to the concept of virtual cells, where only adjacent particles are taken into account in the search for potential collision partners. The methodology is applied to different test cases (plane channel flow, combustion chamber flow). The computational results are compared with experimental measurements and good agreement is found.     

Hybrid algorithm for modeling of fluid-structure interaction in incompressible, viscous flows

The objective of this paper is to present and to validate a new hybrid coupling (HC) algorithm for modeling of fluid-structure interaction (FSI) in incompressible, viscous flows. The HC algorithm is able to avoid numerical instability issues associated with artificial added mass effects, which are often encountered by standard loosely coupled (LC) and tightly coupled (TC) algorithms, when modeling the FSI response of flexible structures in incompressible flow. The artificial added mass effect is caused by the lag in exchange of interfacial displacements and forces between the fluid and solid solvers in partitioned algorithms. The artificial added mass effect is much more prominent for light/flexible structures moving in water, because the fluid forces are in the same order of magnitude as the solid forces, and because the speed at which numerical errors propagate in an incompressible fluid. The new HC algorithm avoids numerical instability issues associated with artificial added mass effects by embedding Theodorsen’s analytical approximation of the hydroelastic forces in the solution process to obtain better initial estimates of the displacements. Details of the new HC algorithm are presented. Numerical validation studies are shown for the forced pitching response of a steel and a plastic hydrofoil. The results show that the HC algorithm is able to converge faster, and is able to avoid numerical instability issues, compared to standard LC and TC algorithms, when modeling the transient FSI response of a plastic hydrofoil. Although the HC algorithm is only demonstrated for a NACA0009 hydrofoil subject to pure pitching motion, the method can be easily extended to model general 3-D FSI response and stability of complex, flexible structures in turbulent, incompressible, multiphase flows.     

A consistent projection-based SUPG/PSPG XFEM for incompressible two-phase flows

In this paper, a consistent projection-based streamline upwind/pressure stabilizing Petrov-Galerkin (SUPG/PSPG) extended finite element method (XFEM) is presented to model incompressible immiscible two-phase flows. As the application of linear elements in SUPG/PSPG schemes gives rise to inconsistency in stabilization terms due to the inability to regenerate the diffusive term from viscous stresses, the numerical accuracy would deteriorate dramatically. To address this issue, projections of convection and pressure gradient terms are constructed and incorporated into the stabilization formulation in our method. This would substantially recover the consistency and free the practitioner from burdensome computations of most items in the residual. Moreover, the XFEM is employed to consider in a convenient way the fluid properties that have interfacial jumps leading to discontinuities in the velocity and pressure fields as well as the projections. A number of numerical examples are analyzed to demonstrate the complete recovery of consistency, the reproduction of interfacial discontinuities and the ability of the proposed projection-based SUPG/PSPG XFEM to model two-phase flows with open and closed interfaces.     

Incompressible Poiseuille flows of Newtonian liquids with a pressure-dependent viscosity

The pressure-dependence of the viscosity becomes important in flows where high pressures are encountered. Applications include many polymer processing applications, microfluidics, fluid film lubrication, as well as simulations of geophysical flows. Under the assumption of unidirectional flow, we derive analytical solutions for plane, round, and annular Poiseuille flow of a Newtonian liquid, the viscosity of which increases linearly with pressure. These flows may serve as prototypes in applications involving tubes with small radius-to-length ratios. It is demonstrated that, the velocity tends from a parabolic to a triangular profile as the viscosity coefficient is increased. The pressure gradient near the exit is the same as that of the classical fully developed flow. This increases exponentially upstream and thus the pressure required to drive the flow increases dramatically.