Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Simulation of three‐dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three‐dimensional (3D) flows over moving objects. By enforcing the non‐slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non‐uniform mesh by the efficient LBM that is based on the second‐order one‐dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Fluid topology optimization: Bio-inspired valves for aircraft engines

This work shows the introduction of three dimensional fluid topology optimization for the design of valves without moving parts that can be manufactured through 3D printing and used in aircraft engines. The design obtained is inherently safer than standard design with moving parts and it resembles closely to biological structures.A new three dimensional fluid topology optimization formulation to achieve such geometries is shown. The novelty of the method is linked to the applicability to realistic high Reynolds numbers, as in aircraft engines. This is the first time that such geometries are obtained for realistic operating conditions that are applicable to gas turbines.The optimal geometries are validated by using Computational Fluid Dynamics simulations.     

Coupled lubrication and Stokes flow finite elements

A method is developed for performing a local reduction of the governing physics for fluid problems with domains that contain a combination of narrow and non‐narrow regions, and the computational accuracy and performance of the method are measured. In the narrow regions of the domain, where the fluid is assumed to have no inertia and the domain height and curvature are assumed small, lubrication, or Reynolds, theory is used locally to reduce the two‐dimensional Navier–Stokes equations to the one‐dimensional Reynolds equation while retaining a high degree of accuracy in the overall solution. The Reynolds equation is coupled to the governing momentum and mass equations of the non‐narrow region with boundary conditions on the mass and momentum flux. The localized reduction technique, termed ‘stitching,’ is demonstrated on Stokes flow for various geometries of the hydrodynamic journal bearing—a non‐trivial test problem for which a known analytical solution is available. The computational advantage of the coupled Stokes–Reynolds method is illustrated on an industrially applicable fully‐flooded deformable‐roll coating example. The examples in this paper are limited to two‐dimensional Stokes flow, but extension to three‐dimensional and Navier–Stokes flow is possible. Copyright © 2003 John Wiley & Sons, Ltd.     

High‐order stable interpolations for immersed boundary methods

The analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Direct forcing is employed. It consists in interpolating boundary conditions from the solid body to the Cartesian mesh on which the computation is performed. Lagrange and least squares high‐order interpolations are considered. The direct forcing IBM is implemented in an incompressible finite volume Navier–Stokes solver for direct numerical simulations (DNS) and large eddy simulations (LES) on staggered grids. An algorithm to identify the body and construct the interpolation schemes for arbitrarily complex geometries consisting of triangular elements is presented. A matrix stability analysis of both interpolation schemes demonstrates the superiority of least squares interpolation over Lagrange interpolation in terms of stability. Preservation of time and space accuracy of the original solver is proven with the laminar two‐dimensional Taylor–Couette flow. Finally, practicability of the method for simulating complex flows is demonstrated with the computation of the fully turbulent three‐dimensional flow in an air‐conditioning exhaust pipe. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order finite difference method for numerical simulations of supersonic turbulent flows

This paper describes a new class of three‐dimensional finite difference schemes for high‐speed turbulent flows in complex geometries based on the high‐order monotonicity‐preserving (MP) method. Simulations conducted for various 1D, 2D, and 3D problems indicate that the new high‐order MP schemes can preserve sharp changes in the flow variables without spurious oscillations and are able to capture the turbulence at the smallest computed scales. Our results also indicate that the MP method has less numerical dissipation and faster grid convergence than the weighted essentially non‐oscillatory method. However, both of these methods are computationally more demanding than the COMP method and are only used for the inviscid fluxes. To reduce the computational cost for reacting flows, the scalar equations are solved by the COMP method, which is shown to yield similar results to those obtained by the MP in supersonic turbulent flows with strong shock waves. Copyright © 2011 John Wiley & Sons, Ltd.     

扩展有限元法(XFEM)及其应用

扩展有限元法(extended finite element method,XFEM)是1999年提出的一种求解不连续力学问题的数值方法, 它继承了常规有限元法(CFEM)的所有优点, 在模拟界面、裂纹生长、复杂流体等不连续问题时特别有效, 短短几年间得到了快速发展与应用. XFEM与CFEM的最根本区别在于, 它所使用的网格与结构内部的几何或物理界面无关, 从而克服了在诸如裂纹尖端等高应力和变形集中区进行高密度网格剖分所带来的困难, 模拟裂纹生长时也无需对网格进行重新剖分.重点介绍XFEM的基本原理、实施步骤及应用实例等, 并进行必要的评述. 单位分解概念保证了XFEM的收敛, 基于此, XFEM通过改进单元的形状函数使之包含问题不连续性的基本成分, 从而放松对网格密度的过分要求. 水平集法是XFEM中常用的确定内部界面位置和跟踪其生长的数值技术, 任何内部界面可用它的零水平集函数表示. 第2和第3节分别简要介绍单位分解法和水平集法;第4节和第5节介绍XFEM的基本思想、详细实施步骤和若干应用实例, 同时修正了以往文献中的一些不妥之处; 最后, 初步展望了该领域尚需进一步研究的课题.      

Unstructured lattice Boltzmann method in three dimensions

Over the last decade, the lattice Boltzmann method (LBM) has evolved into a valuable alternative to continuum computational fluid dynamics (CFD) methods for the numerical simulation of several complex fluid‐dynamic problems. Recent advances in lattice Boltzmann research have considerably extended the capability of LBM to handle complex geometries. Among these, a particularly remarkable option is represented by cell‐vertex finite‐volume formulations which permit LBM to operate on fully unstructured grids. The two‐dimensional implementation of unstructured LBM, based on the use of triangular elements, has shown capability of tolerating significant grid distortions without suffering any appreciable numerical viscosity effects, to second‐order in the mesh size. In this work, we present the first three‐dimensional generalization of the unstructured lattice Boltzmann technique (ULBE as unstructured lattice Boltzmann equation), in which geometrical flexibility is achieved by coarse‐graining the lattice Boltzmann equation in differential form, using tetrahedrical grids. This 3D extension is demonstrated for the case of 3D pipe flow and moderate Reynolds numbers flow past a sphere. The results provide evidence that the ULBE has significant potential for the accurate calculation of flows in complex 3D geometries. Copyright © 2005 John Wiley & Sons, Ltd.     

Extended finite element modeling of production from a reservoir embedded with an arbitrary fracture network

A numerical scheme based on the eXtended Finite Element Method (XFEM) is proposed to simulate complex fluid flow in a fractured porous reservoir. By enriching the elements fully cut by the fracture and the near‐tip region, the flow mechanism including the tip flux singularity can be exactly represented in the XFEM formulation. Fluid transfer between the matrix and the fractures can be easily coupled, and XFEM also overcomes the sensitivity to the mesh used in the traditional unstructured discretizations, regardless of the complexity of the fracture network. The method is validated for a simple case by the exact analytical solution. Results are compared between XFEM and FEM. Case studies are presented to illustrate the power, efficiency, accuracy, and flexibility of the proposed method for simulating transient productive flow in reservoirs with complex fracture networks.     

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.     

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.     

A two‐phase adaptive finite element method for solid–fluid coupling in complex geometries

In this paper we present a method to solve the Navier–Stokes equations in complex geometries, such as porous sands, using a finite‐element solver but without the complexity of meshing the porous space. The method is based on treating the solid boundaries as a second fluid and solving a set of equations similar to those used for multi‐fluid flow. When combined with anisotropic mesh adaptivity, it is possible to resolve complex geometries starting with an arbitrary coarse mesh. The approach is validated by comparing simulation results with available data in three test cases. In the first we simulate the flow past a cylinder. The second test case compares the pressure drop in flow through random packs of spheres with the Ergun equation. In the last case simulation results are compared with experimental data on the flow past a simplified vehicle model (Ahmed body) at high Reynolds number using large‐eddy simulation (LES). Results are in good agreement with all three reference models. Copyright © 2010 John Wiley & Sons, Ltd.     

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

A boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.     

A parallel pressure implicit splitting of operators algorithm applied to flows at all speeds

A parallel implementation of the pressure‐based implicit splitting of operators (PISO) method is described and applied to both compressible and incompressible flows. The treatment of variables at the interfaces between adjacent blocks is highlighted, and, for compressible flow, a straightforward method for the implicit handling of density is described. Steady state and oscillatory flow through a sudden expansion are considered at low speeds for both two‐ and three‐dimensional geometries. Extension of the incompressible method to compressible flow is assessed for subsonic, transonic and supersonic flow through a two‐dimensional bump. Although good accuracy is achieved in these high‐speed flows, including the automatic capturing of shock waves, the method is deemed unsuitable for simulating steady state high‐speed flows on fine grids due to the requirement of very small time steps. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical method for flows in porous and homogenous fluid domains coupled at the interface by stress jump

A numerical method was developed for flows involving an interface between a homogenous fluid and a porous medium. The numerical method is based on the finite volume method with body‐fitted and multi‐block grids. A generalized model, which includes Brinkman term, Forcheimmer term and non‐linear convective term, was used to govern the flow in the porous medium region. At its interface, a shear stress jump that includes the inertial effect was imposed, together with a continuity of normal stress. Furthermore, the effect of the jump condition on the diffusive flux was considered, additional to that on the convective part which has been usually considered. Numerical results of three flow configurations are presented. The method is suitable for coupled problems with regions of homogeneous fluid and porous medium, which have complex geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

An extended finite element method for the simulation of particulate viscoelastic flows

We present an extended finite element method (XFEM) for the direct numerical simulation of the flow of viscoelastic fluids with suspended particles. For moving particle problems, we devise a temporary arbitrary Lagrangian–Eulerian (ALE) scheme which defines the mapping of field variables at previous time levels onto the computational mesh at the current time level. In this method, a regular mesh is used for the whole computational domain including both fluid and particles. A temporary ALE mesh is constructed separately and the computational mesh is kept unchanged throughout the whole computations. Particles are moving on a fixed Eulerian mesh without any need of re-meshing. For mesh refinements around the interface, we combine XFEM with the grid deformation method, in which nodal points are redistributed close to the interface while preserving the mesh topology. Our method is verified by comparing with the results of boundary fitted mesh problems combined with the conventional ALE scheme. The proposed method shows similar accuracy compared with boundary fitted mesh problems and superior accuracy compared with the fictitious domain method. If the grid deformation method is combined with XFEM, the required computational time is reduced significantly compared to uniform mesh refinements, while providing mesh convergent solutions. We apply the proposed method to the particle migration in rotating Couette flow of a Giesekus fluid. We investigate the effect of initial particle positions, the Weissenberg number, the mobility parameter of the Giesekus model and the particle size on the particle migration. We also show two-particle interactions in confined shear flow of a viscoelastic fluid. We find three different regimes of particle motions according to initial separations of particles.     

Mesh adaptation for simulation of unsteady flow with moving immersed boundaries

In this work, an approach for performing mesh adaptation in the numerical simulation of two‐dimensional unsteady flow with moving immersed boundaries is presented. In each adaptation period, the mesh is refined in the regions where the solution evolves or the moving bodies pass and is unrefined in the regions where the phenomena or the bodies deviate. The flow field and the fluid–solid interface are recomputed on the adapted mesh. The adaptation indicator is defined according to the magnitude of the vorticity in the flow field. There is no lag between the adapted mesh and the computed solution, and the adaptation frequency can be controlled to reduce the errors due to the solution transferring between the old mesh and the new one. The preservation of conservation property is mandatory in long‐time scale simulations, so a P1‐conservative interpolation is used in the solution transferring. A nonboundary‐conforming method is employed to solve the flow equations. Therefore, the moving‐boundary flows can be simulated on a fixed mesh, and there is no need to update the mesh at each time step to follow the motion or the deformation of the solid boundary. To validate the present mesh adaptation method, we have simulated several unsteady flows over a circular cylinder stationary or with forced oscillation, a single self‐propelled swimming fish, and two fish swimming in the same or different directions. Copyright © 2012 John Wiley & Sons, Ltd.     

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

This paper presents a local moving least square‐one‐dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function‐vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one‐dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker‐ δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two‐dimensional (2D) Poisson problems, lid‐driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of vortical ideal fluid flow through curved channel

A numerical algorithm to study the boundary‐value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co‐ordinate system. The convergence of the finite‐difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka–Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two‐dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd.