Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry

The lattice‐Boltzmann (LB) method, derived from lattice gas automata, is a relatively new technique for studying transport problems. The LB method is investigated for its accuracy to study fluid dynamics and dispersion problems. Two problems of relevance to flow and dispersion in porous media are addressed: (i) Poiseuille flow between parallel plates (which is analogous to flow in pore throats in two‐dimensional porous networks), and (ii) flow through an expansion–contraction geometry (which is analogous to flow in pore bodies in two‐dimensional porous networks). The results obtained from the LB simulations are compared with analytical solutions when available, and with solutions obtained from a finite element code (FIDAP) when analytical results are not available. Excellent agreement is found between the LB results and the analytical/FIDAP solutions in most cases, indicating the utility of the lattice‐Boltzmann method for solving fluid dynamics and dispersion problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Footbridge between finite volumes and finite elements with applications to CFD

The aim of this paper is to introduce a new algorithm for the discretization of second‐order elliptic operators in the context of finite volume schemes on unstructured meshes. We are strongly motivated by partial differential equations (PDEs) arising in computational fluid dynamics (CFD), like the compressible Navier–Stokes equations. Our technique consists of matching up a finite volume discretization based on a given mesh with a finite element representation on the same mesh. An inverse operator is also built, which has the desirable property that in the absence of diffusion, one recovers exactly the finite volume solution. Numerical results are also provided. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical study of an inviscid incompressible flow through a channel of finite length

A two‐dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo‐symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.     

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

A novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.     

Comparison of derivative free Newton‐based and evolutionary methods for shape optimization of flow problems

The object of this study is to investigate two derivative free optimization techniques, i.e. Newton‐based method and an evolutionary method for shape optimization of flow geometry problems. The approaches are compared quantitatively with respect to efficiency and quality by using the minimization of the pressure drop of a pipe conjunction which can be considered as a representative test case for a practical three‐dimensional flow configuration. The comparison is performed by using CONDOR representing derivative free Newton‐based techniques and SIMPLIFIED NSGA‐II as the representative of evolutionary methods (EM). For the shape variation the computational grid employed by the flow solver is deformed. To do this, the displacement fields are scaled by design variables and added to the initial grid configuration. The displacement vectors are calculated once before the optimization procedure by means of a free form deformation (FFD) technique. The simulation tool employed is a parallel multi‐grid flow solver, which uses a fully conservative finite‐volume method for the solution of the incompressible Navier–Stokes equations on a non‐staggered, cell‐centred grid arrangement. For the coupling of pressure and velocity a pressure‐correction approach of SIMPLE type is used. The possibility of parallel computing and a multi‐grid technique allow for a high numerical efficiency. Copyright © 2006 John Wiley & Sons, Ltd.     

An implicit pseudo‐spectral multi‐domain method for the simulation of incompressible flows

A pseudo‐spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method allows the treatment of moderately complex geometries by means of a multi‐domain approach and it is able to cope with non‐constant fluid properties and non‐orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi‐implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure–correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite‐volume discretizations. Copyright © 2003 John Wiley & Sons, Ltd.     

A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamically adaptive tree grid modeling for simulation and visualization of rainwater overland flow

Modeling and visualization of a rainwater overland flow is an important tool for a risk assessment, preparation, evacuation planning, and real‐time forecasting of flood warning. The objective of this research is to develop a numerical software to visualize the rainwater overland flow based on a finite volume method for shallow water equations and in combination with the dynamically adaptive tree grid technique and the dynamic domain‐defining method. The obtained simulations for several experiments were tested and compared with results in literature, both theoretical and experimental results. The comparisons with non‐adaptive grids show that the developed algorithm for simulation is very efficient and has a potential for practical usages, in terms of computational times and accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

A coupled finite volume method for the solution of flow processes on complex geometries

CFD modelling of ‘real‐life’ thermo‐fluid processes often requires solutions in complex three‐dimensional geometries, which can result in meshes containing aspects that are badly distorted. Cell‐centred finite volume methods (CC‐FV), typical of most commercial CFD tools, are computationally efficient, but can lead to convergence problems on meshes that feature cells with highly non‐orthogonal shapes. The control volume‐finite element method (CVFE) uses a vertex‐based approach and handles distorted meshes with relative ease, but is computationally expensive. A combined vertex‐based—cell‐centre technique (CFVM), detailed in this paper, allows solutions on distorted meshes where purely cell‐centred solutions procedures fail. The method utilizes the ability of the vertex‐based approach to resolve the flow field on a distorted mesh, enabling well established cell‐centred physical models to be employed in the solution of other transported quantities. The vertex‐based flow code is verified against a manufactured 3D solution and error norms are compared on meshes with various degrees of distortion. The CFVM method is validated with benchmark solutions for thermally driven flow and turbulent flow. Finally, the method is illustrated on three‐dimensional turbulent flow over an aircraft wing on a distorted mesh where purely cell‐centred techniques fail. The CFVM is relatively straightforward to embed within generic CC based CFD tools allowing it to be employed in a wide variety of processing applications. Copyright © 2006 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.     

Free surface tracking with polynomial reconstruction and error correction

A non‐linear method, PREC, for computation of the movement of a free surface is proposed here. The method is composed of three steps: identifying the free surface by using a non‐linear function from the volume fraction matrix, updating the volume fraction matrix using a volume projection method with error correction, and treatment of the results using overshooting or undershooting. Identification of the free surface includes using a polynomial function with 2, 4, or 8 coefficients for one‐, two‐, or three‐dimensional problems, respectively. The polynomial reconstruction involves non‐negligible numerical error. The second advection step includes a linear projection method in space and time. Advection of the volume fraction matrix is computed from the occupying volume of the mesh at the previous time step. At the new time step, the error at each grid point is assumed to be similar to the error at the previous time step and is used for correction. Overshooting or undershooting develops around the free surface mesh points due to the solution's finite time increment. The third step includes truncating the numerical overshooting or undershooting volumes, i.e. isotropic spreading of the excess fluid volumes. The PREC method is evaluated for a one‐dimensional flow case and several two‐dimensional simple flow cases with circular sections (cases include transition parallel to a coordinate, transition with an intersection angle to a coordinate, and rotation). The results from the present method are compared with analytical solutions and results from a donor‐cell VOF method. As a result of these comparisons, the PREC method is validated. Copyright © 2008 John Wiley & Sons, Ltd.     

CFD‐based optimization of aerofoils using radial basis functions for domain element parameterization and mesh deformation

A novel domain element shape parameterization method is presented for computational fluid dynamics‐based shape optimization. The method is to achieve two aims: (1) provide a generic ‘wrap‐around’ optimization tool that is independent of both flow solver and grid generation package and (2) provide a method that allows high‐fidelity aerodynamic optimization of two‐ and three‐dimensional bodies with a low number of design variables. The parameterization technique uses radial basis functions to transfer domain element movements into deformations of the design surface and corresponding aerodynamic mesh, thus allowing total independence from the grid generation package (structured or unstructured). Independence from the flow solver (either inviscid, viscous, aeroelastic) is achieved by obtaining sensitivity information for an advanced gradient‐based optimizer (feasible sequential quadratic programming) by finite‐differences. Results are presented for two‐dimensional aerofoil inverse design and drag optimization problems. Inverse design results demonstrate that a large proportion of the design space is feasible with a relatively low number of design variables using the domain element parameterization. Heavily constrained (in lift, volume, and moment) two‐dimensional aerofoil drag optimization has shown that significant improvements over existing designs can be achieved using this method, through the use of various objective functions. Copyright © 2008 John Wiley & Sons, Ltd.     

Simulation of oscillatory flow in an aortic bifurcation using FVM and FEM: A comparative study of implementation strategies

Cardiovascular diseases are one of the major causes of long‐term morbidity and mortality in human beings. The nearly epidemic increase in prevalence of such diseases poses a serious threat to public health and calls for efficient methods of diagnosis and treatment. Non‐invasive diagnostic procedures such as MRI are often used in this context; however, these are limited in terms of spatial and temporal resolution and do not provide information on time‐dependent pressures and wall shear stresses—key quantities considered to be partially responsible for the formation and development of related pathologies. The present study is concerned with the numerical simulation of oscillatory flow through the abdominal aortic bifurcation. Computational fluid dynamics simulation of oscillatory flow in a branched geometry at high Reynolds numbers poses considerable challenges. The present study reports a detailed comparison of simulations performed with a finite volume and a finite element method, two approaches with significant differences in their discretization strategy, treatment of boundary conditions and other numerical aspects. Both solvers were parallelized, using loop parallelization of the BiCGStab linear solver for the finite volume and domain decomposition based on the Schur complement method for the finite element technique. The experience gained with these two approaches for the solution of flow in a bifurcation forms the focus of this study. Although similar results were obtained for both methods, the computation time required for convergence was found to be significantly smaller for the finite element approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

A new numerical procedure for solving the two‐dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well‐established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity–pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first‐ and second‐order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well‐known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward‐facing step, and lid‐driven cavity flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.