Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Depth‐averaged two‐dimensional curvilinear explicit finite analytic model for open‐channel flows

A depth‐averaged two‐dimensional model has been developed in the curvilinear co‐ordinate system for free‐surface flow problems. The non‐linear convective terms of the momentum equations are discretized based on the explicit–finite–analytic method with second‐order accuracy in space and first‐order accuracy in time. The other terms of the momentum equations, as well as the mass conservation equation, are discretized by the finite difference method. The discretized governing equations are solved in turn, and iteration in each time step is adopted to guarantee the numerical convergence. The new model has been applied to various flow situations, even for the cases with the presence of sub‐critical and supercritical flows simultaneously or sequentially. Comparisons between the numerical results and the experimental data show that the proposed model is robust with satisfactory accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

Development of a generalized multi‐layer model for 3‐D simulation of free surface flows

A mathematical model was developed for three‐dimensional (3‐D) simulation of free surface flows. In this model, the flow depth is divided into a number of layers and shallow water equations are integrated in each layer to derive the hydrodynamic equations. To give a general form to this model, each layer is assumed to be non‐horizontal with varying thickness in the flow domain. A non‐orthogonal curvilinear coordinate system is employed in the model, to allow for flexibility in dealing with the irregular geometry of natural watercourses. Due to the similarity in governing equations, two‐dimensional (2‐D) depth averaged programs can be developed into a multi‐layer model. The development for a depth averaged program and its numerical scheme is described in this paper. Experimental data and semi‐analytical solutions are used to evaluate the performance of the model. Three different cases of open channel flow are tested: 1‐flow in a straight open channel, 2‐the flow development region in a channel, and 3‐flow in a meandering channel. It is shown that the model has the capability to predict velocity distribution and secondary flows in complex 3‐D flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

An implicit two‐dimensional non‐hydrostatic model for free‐surface flows

An implicit finite volume model in sigma coordinate system is developed to simulate two‐dimensional (2D) vertical free surface flows, deploying a non‐hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier–Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection–diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri‐diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non‐hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free‐surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

In the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.     

A fixed‐grid b‐spline finite element technique for fluid–structure interaction

We present a fixed‐grid finite element technique for fluid–structure interaction problems involving incompressible viscous flows and thin structures. The flow equations are discretised with isoparametric b‐spline basis functions defined on a logically Cartesian grid. In addition, the previously proposed subdivision‐stabilisation technique is used to ensure inf–sup stability. The beam equations are discretised with b‐splines and the shell equations with subdivision basis functions, both leading to a rotation‐free formulation. The interface conditions between the fluid and the structure are enforced with the Nitsche technique. The resulting coupled system of equations is solved with a Dirichlet–Robin partitioning scheme, and the fluid equations are solved with a pressure–correction method. Auxiliary techniques employed for improving numerical robustness include the level‐set based implicit representation of the structure interface on the fluid grid, a cut‐cell integration algorithm based on marching tetrahedra and the conservative data transfer between the fluid and structure discretisations. A number of verification and validation examples, primarily motivated by animal locomotion in air or water, demonstrate the robustness and efficiency of our approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of unsteady multidimensional free surface motions by level set method

This paper presents a numerical method that couples the incompressible Navier–Stokes equations with the level set method in a curvilinear co‐ordinate system for study of free surface flows. The finite volume method is used to discretize the governing equations on a non‐staggered grid with a four‐step fractional step method. The free surface flow problem is converted into a two‐phase flow system on a fixed grid in which the free surface is implicitly captured by the zero level set. We compare different numerical schemes for advection of the level set function in a generalized curvilinear format, including the third order quadratic upwind interpolation for convective kinematics (QUICK) scheme, and the second and third order essentially non‐oscillatory (ENO) schemes. The level set equations of evolution and reinitialization are validated with benchmark cases, e.g. a stationary circle, a rotating slotted disk and stretching of a circular fluid element. The coupled system is then applied to a travelling solitary wave, and two‐ and three‐dimensional dam breaking problems. Some interesting free surface phenomena are revealed by the computational results, such as, the large free surface vortices, air entrapment and splashing of the water surge front. The computational results are in excellent agreement with theoretical predictions and experimental data, where they are available. Copyright © 2003 John Wiley & Sons, Ltd.     

Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

A three‐dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds‐averaged Navier–Stokes equations with a non‐hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co‐ordinate system, with a semi‐implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five‐diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non‐hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind‐induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

In this paper, the domain‐free discretization method (DFD) is extended to simulate the three‐dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior‐dependent points are updated at each time step by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of ‘osculating plane’ is adopted, with which the local DFD can be easily implemented for the three‐dimensional case. Geometry‐adaptive tetrahedral mesh is employed for three‐dimensional calculations. Finally, we validate the DFD method for three‐dimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary‐conforming grid was displayed and the results show that the present DFD results compare very well with the reference data. Copyright © 2009 John Wiley & Sons, Ltd.     

An implicit numerical algorithm for solving non‐hydrostatic free‐surface flow problems

Details are given of the development of a two‐dimensional vertical numerical model for simulating unsteady free‐surface flows, using a non‐hydrostatic pressure distribution. In this model, the Reynolds equations and the kinematic free‐surface boundary condition are solved simultaneously, so that the water surface elevation can be integrated into the solution and solved for, together with the velocity and pressure fields. An efficient numerical algorithm has been developed, deploying implicit parameters similar to those used in the Crank–Nicholson method, and generating a block tri‐diagonal algebraic system of equations. The model has been applied to simulate a range of unsteady flow problems involving relatively strong vertical accelerations. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.