Arbitrary Lagrangian Eulerian Framework Research Articles

A very accurate Arbitrary Lagrangian–Eulerian meshless method for Computational Aeroacoustics

In this work, we propose a new meshless approach based on a Galerkin discretization of a set of conservation equations on an Arbitrary Lagrangian–Eulerian framework. In particular, we solve the Linearized Euler Equations, using Moving Least Squares as weight functions in the Galerkin discretization. Riemann solvers are introduced in the formulation for the discretization of the convective fluxes. Differently from a purely Lagrangian approach, as it is usual in SPH, the present method is able to work in both Eulerian and Lagrangian configurations, which allows using all the advantages of the Lagrangian approaches in the context of Computational Aeroacoustics.

An adaptive ALE scheme for non-ideal compressible fluid dynamics over dynamic unstructured meshes

This paper investigates the application of mesh adaptation techniques in the Non-Ideal Compressible Fluid Dynamic (NICFD) regime, a region near the vapor-liquid saturation curve where the flow behavior significantly departs from the ideal gas model, as indicated by a value of the fundamental derivative of gasdynamics less than one. A recent interpolation-free finite-volume adaptive scheme is exploited to modify the grid connectivity in a conservative way, and the governing equations for compressible inviscid flows are solved within the Arbitrary Lagrangian Eulerian framework by including special fictitious fluxes representing volume modifications due to mesh adaptation.The absence of interpolation of the solution to the new grid prevents spurious oscillations that may make the solution of the flow field in the NICFD regime more difficult and less robust.Non-ideal gas effects are taken into account by adopting the polytropic Peng-Robinson thermodynamic model. The numerical results focus on the problem of a piston moving in a tube filled with siloxane $\mathrm{MD_4M}$, a simple configuration which can be the core of experimental research activities aiming at investigating the thermodynamic behavior of NICFD flows. Several numerical tests involving different piston movements and initial states in 2D and 3D assess the capability of the proposed adaption technique to correctly capture compression and expansion waves, as well as the generation and propagation of shock waves, in the NICFD and in the non-classical regime.

Numerical approach to study bubbles and drops evolving through complex geometries by using a level set – Moving mesh – Immersed boundary method

The present work proposes a method to study problems of drops and bubbles evolving in complex geometries. First, a conservative level set (CLS) method is enforced to handle the multiphase domain while keeping the mass conservation under control. An Arbitrary Lagrangian-Eulerian (ALE) formulation is proposed to optimize the simulation domain. Thus, a moving mesh (MM) will follow the motion of the bubble, allowing the reduction of the computational domain size and the improvement of the mesh quality. This has a direct impact on the computational resources consumption which is notably reduced. Finally, the use of an Immersed Boundary (IB) method allows to deal with intricate geometries and to reproduce internal boundaries within an ALE framework. The resulting method is capable of dealing with full unstructured meshes. Different problems have been studied to assert the proposed formulation, both involving constricting and non-constricting geometries. In particular, the following problems have been addressed: a 2D gravity-driven bubble interacting with a highly-inclined plane, a 2D gravity-driven Taylor bubble turning into a curved channel, the 3D passage of a drop through a periodically constricted channel, and the impingement of a 3D drop on a flat plate. Good agreement was found for all these cases study, which proves the suitability of the proposed CLS + MM + IB method to study this type of problems.

A staggered procedure for fluid–object interaction with free surfaces, large rotations and driven by adaptive time stepping

The coupling between a rigid body under large rotations and incompressible fluids is investigated within the arbitrary Lagrangian–Eulerian framework. We use here a staggered type of coupling with a predictor/corrector approach for the forces applied to the rigid body. Adaptive time stepping based on feedback control theory imposing a CFL condition on the mesh is investigated. The coupling scheme is first tested on a case illustrating vortex-induced vibrations around a rotating plate. We show the advantages of using the residual-based variational multiscale method for the fluid in the present context. Also, the time-step control and the role of the parameters introduced for the predictor/corrector approach are illustrated using the same test case. A reduced model FPSO ship is then studied, comparing its pitch decay with experimental results. A complex wave–rigid body interaction calculation is finally presented. Results demonstrated the robustness of the predictor/corrector staggered approach with adaptive time-step control for simulating complex interactions of a rigid body under large rotations and free-surface flows.

Numerical modeling of thermal aging in steady state rolling tires

In this contribution, thermal material degradation in steady state rolling tires is numerically studied. A phenomenological time-dependent thermal damage approach from the field of continuum damage mechanics (CDM) is implemented into a modular simulation environment (Behnke & Kaliske, 2015), which allows the thermo-mechanically coupled analysis of axisymmetric tires in steady state motion via the finite element method within an Arbitrary Lagrangian Eulerian framework. Within the numerical model, rubber compounds of the tire are represented by their temperature-dependent viscoelastic properties at finite deformations. Energy losses as heat source terms, computed on a physical basis from the dissipative material properties (non-equilibrium stresses), are used to predict the heat build-up in the tire under severe service conditions. Model parameters for the time-dependent thermal damage approach are identified from previously published experimental investigations on thermally aged rubber compounds in order to link the temperature increase and the evolution of thermal damage (thermal aging). The thermal damage approach qualitatively captures the irreversible changes of the rubber compounds’ mechanical properties, which, in turn, lead to an alteration of the entire structural response of the thermally damaged tire.

A Multi-Physics Battery Model with Particle Scale Resolution of Porosity Evolution Driven by Intercalation Strain and Electrolyte Flow

We present a coupled continuum formulation for the electrostatic, chemical, thermal, mechanical and fluid physics in battery materials. Our treatment is at the particle scale, at which the active particles held together by carbon-binders, the porous separator, current collectors and the perfusing electrolyte are explicitly modeled. Starting with the description common to the field, in terms of reaction-transport partial differential equations for ions, variants of the classical Poisson equation for electrostatics, and the heat equation, we introduce solid-fluid interaction to the problem. Our main contribution is to model the electrolyte as an incompressible fluid driven by elastic, thermal and lithium intercalation strains in the active material. Our treatment is in the finite strain setting, and uses the Arbitrary Lagrangian-Eulerian (ALE) framework to account for mechanical coupling of the solid and fluid. We present a detailed computational study of the influence of solid-fluid interaction and magnitude of intercalation strain upon porosity evolution, ion distribution and electrostatic potential fields in the cell.

Modelo matemático de la dinámica bidimensional del flujo y concentraciones de gases (O2 y CO2 ) en los sacos alveolares pulmonares

We simulated the dynamics of CO2 transport in the alveolar sacs of the human lung. Using Arbitrary Lagrangian Eulerian (ALE) framework, we control the movement domain for a normal and fast maneuver. The fluid of room air inspired and CO2 concentrations were approximated by Navier-Stokes and convection diffusion equations; the stress-stretch in the wall for different volumes were quantified in equal time of respiration. The expansion for a normal and forced maneuver were represented as 9 and 90% to the initial geometry. The difference of the CO2 was 73x10

A hierarchical sequential ALE poromechanics model for tire‐soil‐water interaction on fluid‐infiltrated roads

SummaryThis paper introduces a hierarchical sequential arbitrary Lagrangian‐Eulerian (ALE) model for predicting the tire‐soil‐water interaction at finite deformations. Using the ALE framework, the interaction between a rolling pneumatic tire and the fluid‐infiltrated soil underneath will be captured numerically. The road is assumed to be a fully saturated two‐phase porous medium. The constitutive response of the tire and the solid skeleton of the porous medium is idealized as hyperelastic. Meanwhile, the interaction between tire, soil, and water will be simulated via a hierarchical operator‐split algorithm. A salient feature of the proposed framework is the steady state rolling framework. While the finite element mesh of the soil is fixed to a reference frame and moves with the tire, the solid and fluid constituents of the soil are flowing through the mesh in the ALE model according to the rolling speed of the tire. This treatment leads to an elegant and computationally efficient formulation to investigate the tire‐soil‐water interaction both close to the contact and in the far field. The presented ALE model for tire‐soil‐water interaction provides the essential basis for future applications, for example, to a path‐dependent frictional‐cohesive response of the consolidating soil and unsaturated soil, respectively. Copyright © 2017 John Wiley & Sons, Ltd.

An interpolation-free ALE scheme for unsteady inviscid flows computations with large boundary displacements over three-dimensional adaptive grids

A novel strategy to solve the finite volume discretization of the unsteady Euler equations within the Arbitrary Lagrangian–Eulerian framework over tetrahedral adaptive grids is proposed. The volume changes due to local mesh adaptation are treated as continuous deformations of the finite volumes and they are taken into account by adding fictitious numerical fluxes to the governing equation. This peculiar interpretation enables to avoid any explicit interpolation of the solution between different grids and to compute grid velocities so that the Geometric Conservation Law is automatically fulfilled also for connectivity changes. The solution on the new grid is obtained through standard ALE techniques, thus preserving the underlying scheme properties, such as conservativeness, stability and monotonicity. The adaptation procedure includes node insertion, node deletion, edge swapping and points relocation and it is exploited both to enhance grid quality after the boundary movement and to modify the grid spacing to increase solution accuracy. The presented approach is assessed by three-dimensional simulations of steady and unsteady flow fields. The capability of dealing with large boundary displacements is demonstrated by computing the flow around the translating infinite- and finite-span NACA 0012 wing moving through the domain at the flight speed. The proposed adaptive scheme is applied also to the simulation of a pitching infinite-span wing, where the bi-dimensional character of the flow is well reproduced despite the three-dimensional unstructured grid. Finally, the scheme is exploited in a piston-induced shock-tube problem to take into account simultaneously the large deformation of the domain and the shock wave. In all tests, mesh adaptation plays a crucial role.

Computational Modeling and Analysis of Fluid Structure Interaction in Micromixers with Deformable Baffle

Abstract A passive micromixer with obstacles in the form of deformable baffles is examined numerically. The model deploys an Arbitrary Lagrangian-Eulerian framework with Fluid-structure interaction coupled with a diffusion–advection model. Numerical analysis is carried out in the Reynolds number [Re] range of 0.01≤Re≤300. The objective of the present study is to enhance mixing between two component flow streams in a microchannel encompassing a deformable baffle. In the present work, the baffle deforms only due to the dynamic force of fluids. No external forces are applied. To exemplify the effectiveness of the present design, water and a suspension of curcumin drug loaded nanoparticles are taken as two fluids. Mixing index based on the variance of the local concentration of the suspension is employed to appraise the mixing performance of the micromixer. The introduction of the deformable baffle in a micromixer proliferates the mixing performance with minimal pressure drop over the tested Reynolds number range.

Effective and energy-preserving time discretization for a general nonlinear poromechanical formulation

We consider a general nonlinear poromechanical model, formulated based on fundamental thermodynamics principle, suitable for representing the coupling of rapid internal fluid flows with large deformations of the solid, and compatible with a wide class of constitutive behavior. The objective of the present work is to propose for this model a time discretization scheme of the partitioned type, to allow the use of existing time schemes – and possibly separate solvers – for each component of the model, i.e. for the fluid and the solid. To that purpose, we adapt and extend an earlier proposed approach devised for fluid-structure interaction in an Arbitrary Lagrangian-Eulerian framework. We then establish an energy estimate for the resulting time scheme, in a form that is consistent with the underlying energy principle in the poromechanical formulation, up to some numerical dissipation effects and some perturbations that we have carefully identified and assessed. In addition, we provide some numerical illustrations of our numerical strategy with test problems that present typical features of large strains and rapid fluid flows, and also a case of singular transition related to total drainage. An example of challenging application envisioned for this model and associated numerical coupling scheme concerns the perfusion of the heart.

Unsteady aerodynamics of flapping flight - A fluid-structure interaction study of fore-hind wing phase difference

Flights of dragonflies, various insects and birds have been a subject of active research that may offer insight towards enhanced aerodynamic performance at low Reynolds numbers. To that end, we mimick the flapping biomechanics of a dragonfly by two thin flat airfoils plunging in tandem with each other. In the present study, we aim to investigate the effect of difference in flapping phase between fore and hind wings towards their aerodynamic performances. We computationally simulate incompressible, viscous, laminar flow around two thin flat airfoils that are purely plunging, at a Strouhal number of 0.25 and Reynolds number of 6500, using a flow solver in an Arbitrary Lagrangian-Eulerian framework. Kinematics of both fore and hind wing flapping followed a similar sinusoidal function but with relative phase angle difference to each other, that were varied between -50° to +50° including two cases were phase difference is 0° (i.e. in-phase fore-hind wing flapping) and +90° (i.e. fore wing lags hind wing by 90°). Numerical results indicate that maximum lift and drag forces for each fore and hind wings occur at phase angle of -40° and that power efficiency of tandem wings are better at phase angles when hind wing leads the fore wing, with maximum power efficiency occurring at a fore-hind wing phase difference of +30°. The complex fore-hind wing vortex interaction indicate likely benefit on the hind wing as it interacts with the fore wing at different phase angles.

A computational study on effect of pitch difference in pure plunging tandem wings

Flapping wing in tandem configuration may offer enhanced aerodynamic performance at low Reynolds number, in which micro air vehicles operate. The present study aims to investigate the effect of fore-hind wing pitch difference on the aerodynamic performance of tandem wings. To that end, two-dimensional, laminar flow around two thin flat airfoils that are sinusoidally plunging in phase with each other, were computationally simulated at a Reynolds number of 10000, using a flow solver in an Arbitrary Lagrangian-Eulerian framework. The fore wing pitch angle was fixed to 10°, while the hind wing pitch angle was varied between -10°, 0°, 10° and 20°. Numerical results shows that aerodynamic performance of the fore wing may be affected by the hind wing pitch angle and that tandem wings may offer improved lift to drag efficiency at some optimal fore-hind wing pitch difference compared to twice the results of a similar single wing case. In addition, the complex fore-hind wing vortex interaction is also affected by the hind wing pitch angle.

A multi-dimensional finite volume cell-centered direct ALE solver for hydrodynamics

In this paper we describe a second order multi-dimensional scheme, belonging to the class of direct Arbitrary Lagrangian–Eulerian (ALE) methods, for the solution of non-linear hyperbolic systems of conservation law. The scheme is constructed upon a cell-centered explicit Lagrangian solver completed with an edge-based upwinded formulation of the numerical fluxes, computed from the MUSCL-Hancock method, to obtain a full ALE formulation. Numerical fluxes depend on nodal grid velocities which are either set or computed to avoid most of the mesh problems typically encountered in purely Lagrangian simulations. In order to assess the robustness of the scheme, most results proposed in this paper have been obtained by computing the grid velocities as a fraction of the Lagrangian nodal velocities, the ratio being set before running the test case. The last part of the paper describes preliminary results about the triple point test case run in the ALE framework by computing the grid velocities with the fully adaptive Large Eddy Limitation (L.E.L.) method proposed in [1]. Such a method automatically computes the grid velocities at each node defining the mesh from the local characteristics of the flow. We eventually discuss the advantages and the drawback of the coupling.

VISCOELASTIC LINEAR AND NONLINEAR ANALYSIS OF STEADY STATE ROLLING RUBBER WHEELS: A COMPARISON

ABSTRACT The influence of a linear viscoelastic model (A) and a nonlinear viscoelastic model (B) for the representation of the temperature-, time- and amplitude-dependent behavior of steady state rolling rubber wheels (e.g., tires) is discussed and highlighted through the example of a so-called Grosch wheel. A viscoelastic benchmark material at large strains is proposed and represented with the help of model A and model B, where equality between the two models is obtained for the small strain regime. The model parameters are given in detail. To represent the steady state motion of axisymmetric rubber wheels, the Arbitrary Lagrangian Eulerian (ALE) framework is used within the finite element method (FEM). First, the simulation approach and the results for steady state rolling viscoelastic axisymmetric rubber wheels are verified by a comparison with the solution results of a commercial simulation approach for linear viscoelasticity at isothermal conditions. Second, the simulation results using model A and model B are compared to highlight the influence of nonlinear relaxation characteristics. Third, temperature effects are assessed based on a thermo-mechanically coupled simulation approach. Finally, the results for models A and B are compared to assess advantages and disadvantages of the model approaches and their significance for the representation of the temperature-, time- and amplitude-dependent material behavior. As global objective quantities, the temperature rise due to rolling and the rolling loss power, as well as the vertical stiffness and the stress state in the rubber component under investigation, are computed and compared.

Modeling and Finite Element Formulation for Acoustic Problems Including Rotating Domains

A finite element formulation for the efficient numerical simulation of sound in computational domains, including rotating regions, is presented. The mathematical description is based on an arbitrary Lagrangian–Eulerian framework and results in a convective wave equation for the scalar acoustic potential. Numerically, the capability of nonconforming grids is explored by applying a Nitsche-type mortaring between stationary and rotating regions. The formulation can be applied to classical acoustics (stagnant fluid) as well as moving fluids in the case of aeroacoustics. The validation with the analytical solution of a rotating point source in three dimensions demonstrates the accuracy and robustness of the developed numerical scheme. Additionally, the convergence study shows that the intersection mesh operations needed in the nonconforming setting do not deteriorate the accuracy of the numerical solution.

A phase-field/ALE method for simulating fluid–structure interactions in two-phase flow

We present a phase-field/ALE method for simulating fluid–structure interactions (FSI) in two-phase flow. We solve the Navier–Stokes equation coupled with the Cahn–Hilliard equation and the structure equation in an arbitrary Lagrangian Eulerian (ALE) framework. For the fluid solver, a spectral/hp element method is employed for spatial discretization and backward differentiation for time discretization. For the structure solver, a Galerkin method is used in Lagrangian coordinates for spatial discretization and the Newmark-β scheme for time discretization. The mesh is updated from the initial configuration by a harmonic mapping constructed from the velocity of the interface between the fluid and the structure subdomains. To test the accuracy of the phase-field approach of this multi-physics method, we first simulate two-phase co-annular laminar flow in a stationary pipe and compare the results with the analytical solution. To test the accuracy of the FSI solver, we simulate a pipe conveying single-phase flow and compare the results with an existing validated code (Newman and Karniadakis, 1997). Finally, we present two numerical simulations of FSI in two-phase flow, specifically, in a flexible pipe conveying two fluids that induce self-sustained oscillations, and in external cross flow past a circular cylinder that modifies the classical vortex street due to a Kelvin–Helmholtz instability. These three-dimensional simulations demonstrate the capability of the method in dealing with FSI problems in two-phase flow with moving grids as well as its robustness and efficiency in handling different fluids with large contrast in physical properties.

A variational multiscale finite element method for monolithic ALE computations of shock hydrodynamics using nodal elements

We present a monolithic arbitrary Lagrangian–Eulerian (ALE) finite element method for computing highly transient flows with strong shocks. We use a variational multiscale (VMS) approach to stabilize a piecewise-linear Galerkin formulation of the equations of compressible flows, and an entropy artificial viscosity to capture strong solution discontinuities. Our work demonstrates the feasibility of VMS methods for highly transient shock flows, an area of research for which the VMS literature is extremely scarce. In addition, the proposed monolithic ALE method is an alternative to the more commonly used Lagrangian+remap methods, in which, at each time step, a Lagrangian computation is followed by mesh smoothing and remap (conservative solution interpolation). Lagrangian+remap methods are the methods of choice in shock hydrodynamics computations because they provide nearly optimal mesh resolution in proximity of shock fronts. However, Lagrangian+remap methods are not well suited for imposing inflow and outflow boundary conditions. These issues offer an additional motivation for the proposed approach, in which we first perform the mesh motion, and then the flow computations using the monolithic ALE framework. The proposed method is second-order accurate and stable, as demonstrated by extensive numerical examples in two and three space dimensions.

Residual-based turbulence models and arbitrary Lagrangian–Eulerian framework for free surface flows

We present a residual-based turbulence model for problems with free surfaces. The method is derived based on variational multiscale ideas that assume a decomposition of the solution fields into overlapping scales that are termed as coarse and fine scales. The fine scales are further split hierarchically into fine-scales level-I and fine-scales level-II. The hierarchical variational problems that govern the two fine-scale components are modeled employing bubble functions approach. The model for level-II scales is variationally embedded in the mixed field level-I problem to yield a stable level-I formulation. Subsequently, the model for level-I scales that in fact constitutes the fine-scale turbulence model is then variationally injected in the coarse-scale variational form. A significant feature of the method is that it does not contain any embedded tunable parameters. To accommodate the moving boundaries we cast the formulation in an arbitrary Lagrangian–Eulerian frame of reference. The free surface boundary condition is imposed weakly which results in a formulation that conserves the volume of the fluid. A variety of benchmark problems show the accuracy and range of applicability of the proposed formulation and results are compared with published data. A wavy bed problem is investigated to show the interaction of turbulence generated at the bottom surface with the free surface thereby leading to irregular free surface elevations.

Numerical investigation of the forward and backward travelling waves through an undulating propulsor: performance and wake pattern

Recently, the mechanisms of natural undulatory locomotion of aquatic animal swimming have become one of the most significant issues for the researchers, swimmers and engineers. This study aims to elucidate and compare the propulsive vortical signature and performance of backward (negative undulation) and forward (positive undulation) travelling waves through a typical fishlike propulsor by a systematic numerical study. The numerical approach uses a pressure-based finite volume method solver to solve Navier–Stokes equations in an arbitrary Lagrangian–Eulerian framework domain containing a two-dimensional NACA 0012 foil moving with prescribed kinematics. Some of the important findings are: (1) time-dependent forces and pitching moments show that both mechanisms suffer from the energy losses; however, the negative undulations show less energy losses, (2) Strouhal number of undulations shows a transition point from drag-dominated regime to thrust-dominated regime for both mechanisms and a transition point from thrust-dominated regime to drag-dominated regime for positive undulation, (3) the consumed power plot shows that at lower and medium St, both mechanisms consume closely same powers, while, at higher St, the negative undulation consumes more power, (4) the efficiency plots show that the efficiency of the positive undulation is close to zero, while, the negative undulation reaches considerably higher performances, (5) low St vortical patterns of the mechanisms show some similarity, while, the patterns at medium and high St reveal appreciable discrepancies, and (6) the most significant difference between the vortical patterns of the mechanisms is appearing more regular and stronger vortices in the wake of the negative undulation.