A Lagrangian–Eulerian model of particle dispersion in a turbulent plane mixing layer

A Lagrangian–Eulerian model for the dispersion of solid particles in a two‐dimensional, incompressible, turbulent flow is reported and validated. Prediction of the continuous phase is done by solving an Eulerian model using a control‐volume finite element method (CVFEM). A Lagrangian model is also applied, using a Runge–Kutta method to obtain the particle trajectories. The effect of fluid turbulence upon particle dispersion is taken into consideration through a simple stochastic approach. Validation tests are performed by comparing predictions for both phases in a particle‐laden, plane mixing layer airflow with corresponding measurements formerly reported by other authors. Even though some limitations are detected in the calculation of particle dispersion, on the whole the validation results are rather successful. Copyright © 2002 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A particle finite element method applied to long wave run‐up

This paper presents a Lagrangian–Eulerian finite element formulation for solving fluid dynamics problems with moving boundaries and employs the method to long wave run‐up. The method is based on a set of Lagrangian particles which serve as moving nodes for the finite element mesh. Nodes at the moving shoreline are identified by the alpha shape concept which utilizes the distance from neighbouring nodes in different directions. An efficient triangulation technique is then used for the mesh generation at each time step. In order to validate the numerical method the code has been compared with analytical solutions and a preexisting finite difference model. The main focus of our investigation is to assess the numerical method through simulations of three‐dimensional dam break and long wave run‐up on curved beaches. Particularly the method is put to test for cases where different shoreline segments connect and produce a computational domain surrounding dry regions. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

This paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

Fluid–structural interaction with application to rocket engines

The objective of this work is to develop a finite element model for studying fluid–structure interaction. The geometrically non‐linear structural behaviour is considered and based on large rotations and large displacements. An arbitrary Lagrangian–Eulerian (ALE) formulation is used to represent the compressible inviscid flow with moving boundaries. The structural response is obtained using Newmark‐type time integration and fluid response employs the Lax–Wendroff scheme. A number of numerical examples are presented to validate the structural model, moving mesh implantation of the ALE model and complete fluid–structure interaction. Copyright © 1999 John Wiley & Sons, Ltd.     

High‐order time integrators for front‐tracking finite‐element analysis of viscous free‐surface flows

This paper proposes implicit Runge–Kutta (IRK) time integrators to improve the accuracy of a front‐tracking finite‐element method for viscous free‐surface flow predictions. In the front‐tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian–Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second‐order accurate time integrators at best. In this paper, we present a finite‐element ALE formulation in which a consistent evaluation of the mesh velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high‐order fixed mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L‐stable IRK time integrators for the incompressible Navier–Stokes equations to yield high‐order time‐accurate free‐surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth‐order accuracy in time. The proposed free‐surface front‐tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright © 2015 John Wiley & Sons, Ltd.     

A 3D finite element ALE method using an approximate Riemann solution

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann‐like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann‐like problem. Two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well‐known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problem results are presented. Copyright © 2016 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.     

A marker‐and‐cell approach to free surface 2‐D multiphase flows

This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front‐tracking method. The velocity field is computed using a finite‐difference discretization of a modification of the Navier–Stokes equations. These equations together with the continuity equation are solved for the two‐dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

Evaluation of erosion in a two‐way coupled fluid–particle system

In this study, the effects of flow turbulence intensity, temperature, particle sizes and impinging velocity on erosion by particle impact are demonstrated numerically. Underlying turbulent flow on an Eulerian frame is described by the compressible Reynolds averaged Navier–Stokes equations with a RNG k–ε turbulence model. The particle trajectories and particle–wall interactions are evaluated by a Eulerian–Lagrangian approach in a two‐way coupling system. An erosion model considering material weight removal from surfaces is used to predict erosive wear. Computational validation against measured data is demonstrated satisfactorily. The analysis of erosion shows that the prevention of erosion is enhanced by increasing the effects of flow temperature and turbulence intensity and reducing particle inertial momentum. Copyright © 2001 John Wiley & Sons, Ltd.     

An accurate finite element scheme with moving meshes for computing 3D‐axisymmetric interface flows

An accurate finite element scheme for computing 3D‐axisymmetric incompressible free surface and interface flows is proposed. It is based on the arbitrary Lagrangian Eulerian (ALE) approach using free surface/interface‐resolved moving meshes. Key features like the surface force, consisting of surface tension and the local curvature, and jumps in the density and viscosity over different fluid phases are precisely incorporated in the finite element formulation. The local curvature is approximated by using the Laplace–Beltrami operator technique combined with a boundary approximation by isoparametric finite elements. A new approach is used to derive the 3D‐axisymmetric form from the variational form in 3D‐Cartesian coordinates. Several test examples show the high accuracy and the robustness of the proposed scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Fluid–solid interaction problems with thermal convection using the immersed element‐free Galerkin method

In this work, the immersed element‐free Galerkin method (IEFGM) is proposed for the solution of fluid–structure interaction (FSI) problems. In this technique, the FSI is represented as a volumetric force in the momentum equations. In IEFGM, a Lagrangian solid domain moves on top of an Eulerian fluid domain that spans over the entire computational region. The fluid domain is modeled using the finite element method and the solid domain is modeled using the element‐free Galerkin method. The continuity between the solid and fluid domains is satisfied by means of a local approximation, in the vicinity of the solid domain, of the velocity field and the FSI force. Such an approximation is achieved using the moving least‐squares technique. The method was applied to simulate the motion of a deformable disk moving in a viscous fluid due to the action of the gravitational force and the thermal convection of the fluid. An analysis of the main factors affecting the shape and trajectory of the solid body is presented. The method shows a distinct advantage for simulating FSI problems with highly deformable solids. Copyright © 2009 John Wiley & Sons, Ltd.     

Multi‐material closure model for high‐order finite element Lagrangian hydrodynamics

We present a new closure model for single fluid, multi‐material Lagrangian hydrodynamics and its application to high‐order finite element discretizations of these equations 1 . The model is general with respect to the number of materials, dimension and space and time discretizations. Knowledge about exact material interfaces is not required. Material indicator functions are evolved by a closure computation at each quadrature point of mixed cells, which can be viewed as a high‐order variational generalization of the method of Tipton 2 . This computation is defined by the notion of partial non‐instantaneous pressure equilibration, while the full pressure equilibration is achieved by both the closure model and the hydrodynamic motion. Exchange of internal energy between materials is derived through entropy considerations, that is, every material produces positive entropy, and the total entropy production is maximized in compression and minimized in expansion. Results are presented for standard one‐dimensional two‐material problems, followed by two‐dimensional and three‐dimensional multi‐material high‐velocity impact arbitrary Lagrangian–Eulerian calculations. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.     

An assessment of a parallel,finite element method for three‐dimensional,moving‐boundary flows driven by capillarity for simulation of viscous sintering

A parallel, finite element method is presented for the computation of three‐dimensional, free‐surface flows where surface tension effects are significant. The method employs an unstructured tetrahedral mesh, a front‐tracking arbitrary Lagrangian–Eulerian formulation, and fully implicit time integration. Interior mesh motion is accomplished via pseudo‐solid mesh deformation. Surface tension effects are incorporated directly into the momentum equation boundary conditions using surface identities that circumvent the need to compute second derivatives of the surface shape, resulting in a robust representation of capillary phenomena. Sample results are shown for the viscous sintering of glassy ceramic particles. The most serious performance issue is error arising from mesh distortion when boundary motion is significant. This effect can be severe enough to stop the calculations; some simple strategies for improving performance are tested. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element solution of three‐dimensional turbulent flows applied to mold‐filling problems

This paper presents a finite element solution algorithm for three‐dimensional isothermal turbulent flows for mold‐filling applications. The problems of interest present unusual challenges for both the physical modelling and the solution algorithm. High‐Reynolds number transient turbulent flows with free surfaces have to be computed on complex three‐dimensional geometries. In this work, a segregated algorithm is used to solve the Navier–Stokes, turbulence and front‐tracking equations. The streamline–upwind/Petrov–Galerkin method is used to obtain stable solutions to convection‐dominated problems. Turbulence is modelled using either a one‐equation turbulence model or the κ–ε two‐equation model with wall functions. Turbulence equations are solved for the natural logarithm of the turbulence variables. The change of dependent variables allows for a robust solution algorithm and good predictions even on coarse meshes. This is very important in the case of large three‐dimensional applications for which highly refined meshes result in untreatable large numbers of elements. The position of the flow front in the mold cavity is computed using a level set approach. Finally, equations are integrated in time using an implicit Euler scheme. The methodology presents the robustness and cost effectiveness needed to tackle complex industrial applications. Copyright © 2000 John Wiley & Sons, Ltd.     

A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.