A mass‐conserving Level‐Set method for modelling of multi‐phase flows

A mass‐conserving Level‐Set method to model bubbly flows is presented. The method can handle high density‐ratio flows with complex interface topologies, such as flows with simultaneous occurrence of bubbles and droplets. Aspects taken into account are: a sharp front (density changes abruptly), arbitrarily shaped interfaces, surface tension, buoyancy and coalescence of droplets/bubbles. Attention is paid to mass‐conservation and integrity of the interface. The proposed computational method is a Level‐Set method, where a Volume‐of‐Fluid function is used to conserve mass when the interface is advected. The aim of the method is to combine the advantages of the Level‐Set and Volume‐of‐Fluid methods without the disadvantages. The flow is computed with a pressure correction method with the Marker‐and‐Cell scheme. Interface conditions are satisfied by means of the continuous surface force methodology and the jump in the density field is maintained similar to the ghost fluid method for incompressible flows. Copyright © 2005 John Wiley & Sons, Ltd.     

GENSMAC3D: a numerical method for solving unsteady three‐dimensional free surface flows

A numerical method for solving three‐dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier–Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user‐friendly input data file is employed to allow almost any arbitrary three‐dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐phase flow interface capturing finite element method

A new interface capturing algorithm is proposed for the finite element simulation of two‐phase flows. It relies on the solution of an advection equation for the interface between the two phases by a streamline upwind Petrov–Galerkin (SUPG) scheme combined with an adaptive mesh refinement procedure and a filtering technique. This method is illustrated in the case of a Rayleigh–Taylor two‐phase flow problem governed by the Stokes equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulation of single‐ and two‐phase flows on sliding unstructured meshes using finite volume method

The paper presents an efficient finite volume method for unstructured grids with rotating sliding parts composed of arbitrary polyhedral elements for both single‐ and two‐phase flows. Mathematical model used in computations is based on the ensemble averaged conservation equations. These equations are solved for each phase and in case of single‐phase flow reduce to the transient Reynolds‐averaged Navier–Stokes (TRANS) equations. Transient flow induced by rotating impellers is thus resolved in time. The use of unstructured grids allows an easy and flexible meshing for the entire flow domain. Polyhedral cell volumes are created on the arbitrary mesh interface placed between rotating and static parts. Cells within the rotating parts move each time step and the new faces are created on the arbitrary interfaces only, while the rest of the domain remain ‘topologically’ unchanged. Implicit discretization scheme allows a wide range of time‐step sizes, which further reduce the computational effort. Special attention is given to the interpolation practices used for the reconstruction of the face quantities. Mass fluxes are recalculated at the beginning of each time step by using an interpolation scheme, which enhances the coupling between the pressure and velocity fields. The model has been implemented into the commercially available CFD code AVL SWIFT (AVL AST, SWIFT Manual 3.1, AVL List GmbH, Graz, Austria, 2002). Single‐phase flow in a mixing vessel stirred by a six‐bladed Rushton‐type turbine and two‐phase flow in aerated stirred vessel with the four‐blade Rushton impeller are simulated. The results are compared with the available experimental data, and good agreement is observed. The proposed algorithm is proved to be both stable and accurate for single‐phase as well as for the two‐phase flows calculations. Copyright 2004 John Wiley & Sons, Ltd.     

A Taylor series‐based finite volume method for the Navier–Stokes equations

A Taylor series‐based finite volume formulation has been developed to solve the Navier–Stokes equations. Within each cell, velocity and pressure are obtained from the Taylor expansion at its centre. The derivatives in the expansion are found by applying the Gauss theorem over the cell. The resultant integration over the faces of the cell is calculated from the value at the middle point of the face and its derivatives, which are further obtained from a higher order interpolation based on the values at the centres of two cells sharing this face. The terms up to second order in the velocity and the terms up to first order in pressure in the Taylor expansion are retained throughout the derivation. The test cases for channel flow, flow past a circular cylinder and flow in a collapsible channel have shown that the method is quite accurate and flexible. Copyright © 2008 John Wiley & Sons, Ltd.     

A semi‐implicit numerical method for the free‐surface Navier–Stokes equations

Semi‐implicit methods are known for being the basis of simple, efficient, accurate, and stable numerical algorithms for simulating a large variety of geophysical free‐surface flows. Geophysical flows are typically characterized by having a small vertical scale as compared with their horizontal extents. Hence, the hydrostatic approximation often applies, and the free surface can be conveniently represented by a single‐valued function of the horizontal coordinates. In the present investigation, semi‐implicit methods are extended to complex free‐surface flows that are governed by the full incompressible Navier–Stokes equations and are delimited by solid boundaries and arbitrarily shaped free‐surfaces. The primary dependent variables are the velocity components and the pressure. Finite difference equations for momentum, and a finite volume discretization for continuity, are derived in such a fashion that, after simple manipulation, the resulting pressure equation yields a well‐posed piecewise linear system from which both the pressure and the fluid volume within each computational cell are naturally derived. This system is efficiently solved by a nested Newton type iterative scheme, and the resulting fluid volumes are assured to be nonnegative and bounded from above by the available cell volumes. The time step size is not restricted by stability conditions dictated by surface wave speed, but can be freely chosen just to achieve the desired accuracy. Several examples illustrate the model applicability to a large range of complex free‐surface flows and demonstrate the effectiveness of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

Parallel solution of three‐dimensional Marangoni flow in liquid bridges

This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three‐dimensional and time‐dependent Navier–Stokes equations on distributed‐memory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non‐linear Navier–Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three‐dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time‐consuming part of the algorithm. A grid‐partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed‐Up is ever smaller than the ideal linear Speed‐Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier–Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic–Newton method for solving incompressible flows

In this paper we present a comparative study of three non-linear schemes for solving finite element systems of Navier–Stokes incompressible flows. The first scheme is the classical Newton–Raphson linearization, the second one is the modified Newton–Raphson linearization and the last one is a new scheme called the asymptotic–Newton method. The relative efficiency of these approaches is evaluated over a large number of examples. © 1997 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.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical study of fully developed laminar flows in pipes with two planar curvatures

The spectral element method is applied on unstructured tetrahedral elements to solve the Navier–Stokes equations for fully developed laminar flow in pipes with two planar curvatures. Specific implementations of the spectral element method to double curved pipes and parallelization are described. Previous studies on flows in pipes focused on constant curvature or torsion geometries, as well as pipes with varying curvature. This study focuses on the periodic variation of both the curvature as well as torsion by analysing a pipe having two planar curvatures. The effects of the three parameters defining the pipe are studied to isolate the curvature and torsion effect on the magnitude and angle of the secondary flow. Furthermore, the geometric effects on the wall shear stress are studied, as it is an important fluid flow property, especially in blood flows. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐resolution p‐adaptive DG simulations of flows with moving shocks

High‐order accurate DG discretization is employed for Reynolds‐averaged Navier–Stokes equations modeling of complex shock‐dominated, unsteady flow generated by gas issuing from a shock tube nozzle. The DG finite element discretization framework is used for both the flow field and turbulence transport. Turbulent flow in the near wall regions and the flow field is modeled by the Spalart–Allmaras one‐equation model. The effect of rotation on turbulence modeling for shock‐dominated supersonic flows is considered for accurate resolution of the large coherent and vortical structures that are of interest in high‐speed combustion and supersonic flows. Implicit time marching methodologies are used to enable large time steps by avoiding the severe time step limitations imposed by the higher order DG discretizations and the source terms. Sufficiently high mesh density is used to enable crisp capturing of discontinuities. A p ? type refinement procedure is employed to accurately represent the vortical structures generated during the development of the flow. The computed solutions showed qualitative agreement with experiments. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical simulation of flows around two circular cylinders by mesh‐free least square‐based finite difference methods

In this paper, the mesh‐free least square‐based finite difference (MLSFD) method is applied to numerically study the flow field around two circular cylinders arranged in side‐by‐side and tandem configurations. For each configuration, various geometrical arrangements are considered, in order to reveal the different flow regimes characterized by the gap between the two cylinders. In this work, the flow simulations are carried out in the low Reynolds number range, that is, Re=100 and 200. Instantaneous vorticity contours and streamlines around the two cylinders are used as the visualization aids. Some flow parameters such as Strouhal number, drag and lift coefficients calculated from the solution are provided and quantitatively compared with those provided by other researchers. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulating hydrostatic and non‐hydrostatic oceanic flows

The thin aspect ratio of oceanic basins is simultaneously a complication to contend with when developing ocean models and an opportunity to simplify the equations of motion. Here we discuss these two aspects of this geometric feature in the context of hydrostatic and non‐hydrostatic ocean models. A simple analysis shows that the horizontal viscous operator in the hydrostatic primitive equations plays a central role in the specification of boundary conditions on the lateral vertical surfaces bounding the domain. The asymptotic analysis shows that for very thin aspect ratios the leading‐order flow cannot be closed unless additional terms in the equations are considered, namely either the horizontal viscous forces or the non‐hydrostatic pressure forces. In either case, narrow boundary layers must be resolved in order to close the circulation properly. The computational cost increases substantially when non‐hydrostatic effects are taken into account. Copyright © 2008 John Wiley & Sons, Ltd.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

Numeric simulation of pollutant dispersion by a control‐volume based on finite element method

The dispersion of pollutants in the environment is an issue of great interest as it directly affects air quality, mainly in large cities. Experimental and numerical tools have been used to predict the behavior of pollutant species dispersion in the atmosphere. A software has been developed based on the control‐volume based on the finite element method in order to obtain two‐dimensional simulations of Navier–Stokes equations and heat or mass transportation in regions with obstacles, varying position of the pollutant source. Numeric results of some applications were obtained and, whenever possible, compared with literature results showing satisfactory accordance. Copyright © 2010 John Wiley & Sons, Ltd.     

Comment on ‘generalized potential flow theory and direct calculation of velocities applied to the numerical solution of the Navier–Stokes and the Boussinesq equations’

In a recent paper a generalized potential flow theory and its application to the solution of the Navier–Stokes equation are developed.¹ The purpose of this comment is to show that the analysis presented in that paper is in general not correct. We note that the theoretical development of Reference 1 is in fact an extension—although not cited—of some work first done by Hawthorne for steady inviscid flow.² Hawthorne's solution is correct, and his analysis, which we briefly describe, provides a useful introduction to this note.     

A numerical model for aerated‐water wave breaking

This work presents a numerical model designed for the simulation of water‐wave impacts on a structure when aeration of the liquid phase is considered. The model is based on a multifluid Navier–Stokes approach in which all fluids are assumed compressible. The numerical method is based on a finite volume algorithm in space and a second order Runge–Kutta method in time. A validation of this model is performed. It shows a good accuracy for acoustic and shock wave propagation in a bubbly liquid and for wave breaking. Copyright © 2011 John Wiley & Sons, Ltd.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

Colocated schemes for the incompressible Navier–Stokes equations on non‐smooth grids for two‐dimensional problems

The accuracy of colocated finite volume schemes for the incompressible Navier–Stokes equations on non‐smooth curvilinear grids is investigated. A frequently used scheme is found to be quite inaccurate on non‐smooth grids. In an attempt to improve the accuracy on such grids, three other schemes are described and tested. Two of these are found to give satisfactory results. Copyright © 2000 John Wiley & Sons, Ltd.