Eulerian Formulation Research Articles

ANALYSIS OF THE WATER IMPACT OF SYMMETRIC WEDGES WITH A MULTI- MATERIAL EULERIAN FORMULATION


 
 
 The two-dimensional hydrodynamic problem of a symmetric wedge vertically impacting in calm water is analysed by using an explicit finite element method based on a multi-material Eulerian formulation. The slam-induced loads on wedges with different deadrise angle at a constant velocity are calculated, including pressure distribution, maximum pressure coefficient, force coefficient and time history of vertical force, which are compared with available theoretical and analytical results. The time evolution of pressure distribution and free surface elevation are presented. Furthermore, the effects of impact velocity are investigated. It shows that this method is capable of predicting the local slamming loads, and as well assessing the effects of the deadrise angle and the impact velocity on the slamming pressure for the wedge-shape section.
 
 

Real-time joint traffic state and model parameter estimation on freeways with fixed sensors and connected vehicles: State-of-the-art overview, methods, and case studies

This paper addresses real-time joint traffic state and model parameter estimation on freeways using data from fixed sensors and connected vehicles. It investigates how the combined usage of both types of sensing data improves the performance of traffic state estimation (TSE) and what role the online model parameter estimation (OMPE) plays therein. The paper first presents a state-of-the-art overview for freeway TSE with mixed sensing, focusing on a few critical issues such as filtering methods, Eulerian and Lagrangian formulation for traffic flow modeling/sensing/estimation, OMPE, and fusion of disparate sensing data, to determine the strengths and weaknesses of various technical paths, and figure out a viable roadmap for future studies. Three representative approaches to the design of freeway traffic state estimators using mixed sensing data are then investigated, which are based on a first-order, a second-order traffic flow model, and a speed-uniformity assumption, respectively. The paper intends to check if the gradual richness of mobile sensing data (in the era of connected vehicles) would compensate the deficiency of first-order models (as compared to second-order models) for TSE; if OMPE would still be essential for TSE in the mixed sensing case compared to the fixed sensing case; if the increasing usage of mobile sensing data would reduce the necessity of OMPE for TSE? The designed traffic state estimators have been evaluated thoroughly using NGSIM data, with the above questions answered.

Elements of future snowpack modeling – Part 2: A modular and extendable Eulerian–Lagrangian numerical scheme for coupled transport, phase changes and settling processes

Abstract. A coupled treatment of transport processes, phase changes and mechanical settling is the core of any detailed snowpack model. A key concept underlying the majority of these models is the notion of layers as deforming material elements that carry the information on their physical state. Thereby an explicit numerical solution of the ice mass continuity equation can be circumvented, although with the downside of virtual no flexibility in implementing different coupling schemes for densification, phase changes and transport. As a remedy we consistently recast the numerical core of a snowpack model into an extendable Eulerian–Lagrangian framework for solving the coupled non-linear processes. In the proposed scheme, we explicitly solve the most general form of the ice mass balance using the method of characteristics, a Lagrangian method. The underlying coordinate transformation is employed to state a finite-difference formulation for the superimposed (vapor and heat) transport equations which are treated in their Eulerian form on a moving, spatially non-uniform grid that includes the snow surface as a free upper boundary. This formulation allows us to unify the different existing viewpoints of densification in snow or firn models in a flexible way and yields a stable coupling of the advection-dominated mechanical settling with the remaining equations. The flexibility of the scheme is demonstrated within several numerical experiments using a modular solver strategy. We focus on emerging heterogeneities in (two-layer) snowpacks, the coupling of (solid–vapor) phase changes with settling at layer interfaces and the impact of switching to a non-linear mechanical constitutive law. Lastly, we discuss the potential of the scheme for extensions like a dynamical equation for the surface mass balance or the coupling to liquid water flow.

Engineering-geological and geoecological aspects of formation of dry dumps on hydrodumps

The paper discusses the issues of increasing the dumping capacity of hydrodumps and their reclamation by filling them with dump embankments from relatively strong overburden. Without the implementation of this measure, the alluvial massifs are inaccessible for a long time for various mining and transport equipment due to the low bearing capacity of the rocks. The specifics of the formation of the composition, state and properties of rocks of alluvial massifs due to the development of fractionation and filtration consolidation processes are noted. The technology of dumping the surface of hydrodump by controlled deformation of the slopes of the first dump layer has been substantiated. A method has been developed for predicting the parameters of intrusion-replacement of alluvial rocks by bulk ones using numerical modeling of the FEM in the Euler formulation. The obtained mechanical properties and regularities of the deformation behavior of technogenic rocks were used to perform geomechanical calculations when justifying the stable parameters of the slopes of the formed natural-technical systems (NTS) “hydrodump+dump” and compaction of embankments to justify the mining-technical stage of reclamation. A geoecological assessment of the impact of the dump NTS on the natural environment has been carried out, which is expressed in the acquisition of land from circulation and the disruption of the natural landscape. The conditions for the formation of technogenic eluvium on the surface of the dump suitable for biological reclamation are considered.

Accretion and ablation in deformable solids with an Eulerian description: examples using the method of characteristics

Accretion and ablation, i.e., the addition and removal of mass at the surface, are important in a wide range of physical processes, including solidification, growth of biological tissues, environmental processes, and additive manufacturing. The description of accretion requires the addition of new continuum particles to the body, and is therefore challenging for standard continuum formulations for solids that require a reference configuration. Recent work has proposed an Eulerian approach to this problem, enabling side-stepping of the issue of constructing the reference configuration. However, this raises the complementary challenge of determining the stress response of the solid, which typically requires the deformation gradient that is not immediately available in the Eulerian formulation. To resolve this, the approach introduced the elastic deformation as an additional kinematic descriptor of the added material, and its evolution has been shown to be governed by a transport equation. In this work, the method of characteristics is applied to solve concrete simplified problems motivated by biomechanics and manufacturing. Specifically, (1) for a problem with both ablation and accretion in a fixed domain and (2) for a problem with a time-varying domain, the closed-form solution is obtained in the Eulerian framework using the method of characteristics without explicit construction of the reference configuration.

Upwind finite element-PML approximation of a novel linear potential model for free surface flows produced by a floating rigid body

A novel linear potential model is presented to compute free surface flows of incompressible fluids produced by the motion of a floating rigid body in the presence of an underlying non-uniform flow. In particular, the proposed model enables the accurate numerical simulation of the Kelvin wake pattern in a computational domain of reduced size. The governing equations are obtained by using an Arbitrary Lagrangian Eulerian (ALE) formulation which involves the underlying velocity of the fluid around the floating body assuming flat free surface, and a non-dimensional analysis to derive the novel linear system of equations for free surface flows. The discretization of the proposed model is made by a standard Galerkin finite element method, where a SUPG-inspired upwinding strategy has been used in combination with a Perfectly Matched Layer technique which allows truncating the original unbounded fluid domain without introducing spurious reflections in the Kelvin wake pattern. The numerical simulations computed with the proposed approach are compared with the results obtained by the classical linear potential model with uniform underlying flow and also with those from the full incompressible Navier–Stokes equations equipped with the k−ω SST turbulent model. This numerical comparison is discussed in terms of a classical hydrodynamic floating body benchmark involving the Wigley hull.

Performance of an auxetic honeycomb-core sandwich panel under close-in and far-field detonations of high explosive

This paper aims to scrutinise the effectiveness of a re-entrant auxetic honeycomb-core sandwich panel (AHSP) to protect reinforced concrete (RC) slab under close-in and far-field detonations of high explosive. A series of quarter symmetric 3D comprehensive numerical models were developed by using Arbitrary Lagrangian Eulerian (ALE) formulation in LS-DYNA hydrocode. Strain rate effects of the rate sensitive materials were employed in the model to consider dynamic behaviour of the materials during blast loading. The developed numerical model was validated with the documented experimental results. Deformation patterns, energy absorption, overpressure damping, blast pressure deflection, and stress-transmission to the protected structure were investigated with the computational models. Furthermore, the performance of AHSP sacrificial protective structure was compared with an equivalent areal density conventional honeycomb-core sandwich panel (CHSP) structure. The results showed that AHSP not only absorbed higher blast energy but also acted as a pressure deflector by forming a densified concave depression under close-in blast load. AHSP damped higher magnitude of blast overpressure in both close-in and far-field blasts. Moreover, a uniform redistribution of the stress on the protected structure was observed with AHSP protection. The overall response of AHSP was found to be better than CHSP to protect the RC slab.

A numerical study of particle-laden flow around an obstacle: flow evolution and Stokes number effects

Three-way coupled numerical simulations of particle-laden fluid flow around a square obstacle are performed by a coupled finite-volume and discrete-element method, taking account of the collisions between particles. The fluid flow is described by the Eulerian formalism, while the Lagrangian formalism is used for the solid particles. The dynamics of each phase and the modulation mechanism of the flow field by the inertial particles are explored. The Reynolds number based on the side length of the obstacle is 100, which leads to a typical periodic flow pattern of vortex shedding. Under the action of particles, the flow tends to be chaotic even in the upstream region, and the vortex shedding occurs earlier. The intensity of the vortex is attenuated due to particle dissipation. At sufficiently small Stokes numbers, the particles exhibit flow tracer behavior with relatively high fidelity. However, as the Stokes number and mass loading increase, the interaction between the particles and fluid becomes important, especially in the core region of the channel. With the increase of the particle response time, the spanwise velocity fluctuation is attenuated, while the influence on the streamwise velocity fluctuation seems complicated. No particles are observed to hit the rear surface of the obstacle or deposit in the domain during the flow. When the Stokes number and mass loading are elevated from 0.0036 and 0.065 to 0.5 and 9.024, respectively, the distribution of the particles near the channel centerline varies from being filled in the shedding vortices to surrounding them. Meanwhile, the particle-free zones centered on these vortices become larger. For the Stokes number and mass loading up to 1.5 and 27.073, respectively, the particles are scattered in the downstream area, and the flow pattern of the Kármán vortex street is modified. Within the range of the Stokes number considered, the drag and lift coefficients, and the Strouhal number also show a certain trend of variation.

Assessment of two wind gust injection methods: Field velocity vs. split velocity method

The objective of the present paper is to revisit two well-known wind gust injection methods in a consistent manner and to assess their performance based on different application cases. These are the field velocity method (FVM) and the split velocity method (SVM). For this purpose, both methods are consistently derived pointing out the link to the Arbitrary Lagrangian Eulerian formulation and the geometric conservation law. Furthermore, the differences between FVM and SVM are worked out and the advantages and disadvantages are compared. Based on a well-known test case considering a vertical gust hitting a plate and a newly developed case taking additionally a horizontal gust into account, the methods are evaluated and the deviations resulting from the disregard of the feedback effect in FVM are assessed. The results show that the deviations between the predictions by FVM and SVM are more pronounced for the horizontal gust justifying the introduction of this new test case. The main reason is that the additional source term in SVM responsible for the feedback effect of the surrounding flow on the gust itself nearly vanishes for the vertical gust, whereas it has a significant impact on the flow field and the resulting drag and lift coefficients for the horizontal gust. Furthermore, the correct formulation of the viscous stress tensor relying on the total velocity as done in case of SVM plays an important role, but is found to be negligible for the chosen Reynolds number of the present test cases. The study reveals that SVM is capable of delivering physical results in contradiction to FVM. It paves the way for investigating further complex gust configurations (e.g., inclined gusts) and practical applications towards coupled fluid–structure interaction simulations of engineering structures impacted by wind gusts.

Lattice Boltzmann method for fluid–structure interaction in compressible flow

We present a two-way coupled fluid–structure interaction scheme for rigid bodies using a two-population lattice Boltzmann formulation for compressible flows. An arbitrary Lagrangian–Eulerian formulation of the discrete Boltzmann equation on body-fitted meshes is used in combination with polynomial blending functions. The blending function approach localizes mesh deformation and allows treating multiple moving bodies with a minimal computational overhead. We validate the model with several test cases of vortex induced vibrations of single and tandem cylinders and show that it can accurately describe dynamic behavior of these systems. Finally, in the compressible regime, we demonstrate that the proposed model accurately captures complex phenomena such as transonic flutter over an airfoil.

High order weight‐adjusted discontinuous Galerkin methods for wave propagation on moving curved meshes

AbstractThis article presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian–Eulerian formulation to map the wave equation from a time‐dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled and inverted at each time step when using explicit time‐stepping methods. We avoid this step by utilizing an easily invertible weight‐adjusted approximation. The resulting semi‐discrete weight‐adjusted DG scheme is provably energy stable up to a term that (for a fixed time interval) converges to zero with the same rate as the optimal error estimate. Numerical experiments using both polynomial and B‐spline bases verify the high order accuracy and energy stability of proposed methods.

New insights into soil arching behaviour in column-supported embankments

The observation of failure surfaces within column-supported embankments is critical to understanding how the embankment stresses are transferred towards the column heads. In this study, finite element simulations utilising a strain-softening constitutive model, nonlocal regularisation, and the Arbitrary Lagrangian–Eulerian formulation are used to examine these failure surfaces over various embankment geometries. This methodology offers insights into the nature of the failure mechanism, the development of a plane of equal settlement, and the influence of the subsoil settlement profile. Depending on the embankment geometry, the results indicate a punching failure, inverted general bearing failure, or a localised failure develops. The transition between punching and inverted general bearing failure is found to be closely related to the establishment of a plane of equal settlement within the embankment. The height of the plane of equal settlement and the range of failure mechanisms that develop were largely insensitive to the nature of the subsoil settlement profiles simulated. These findings have implications for the practical design of efficient embankments and the effective design of future experimental studies.

Vorticity-based polynomial adaptation for moving and deforming domains

This paper introduces a novel non-dimensional vorticity-based polynomial adaptation indicator for moving and deforming domains using a high-order unstructured spatial discretization. We verify the utility of this approach when applied to the Arbitrary Lagrangian–Eulerian (ALE) form of the compressible Navier–Stokes equations for a range of applications on moving and deforming domains. Specifically, we verify the ALE implementation by performing simulations of an Euler Vortex (EV), and then, illustrate the accuracy and efficiency of the adaptation routine by performing simulations of flow over an oscillating circular cylinder with two different flow settings, dynamic stall of a 2D NACA 0012 airfoil undergoing heaving and pitching motions, shallow dynamic stall of a 3D SD 7003 airfoil undergoing heaving and pitching motions, and flow over a Vertical Axis Wind Turbine (VAWT) composed of two NACA 0012 airfoils. Results demonstrate that the non-dimensional vorticity indicator can track regions of interest, such as vortices and boundary layers, and yields a significant reduction in degrees of freedom when paired with polynomial adaptation.

Heave Compensation Dynamics for Offshore Drilling Operation

In this study, dynamic response analysis of a heave compensation system for offshore drilling operations was conducted based on multibody dynamics. The efficiency of the heave compensation system was computed using simulation techniques and virtually confirmed before being applied to drilling operations. The heave compensation system was installed on a semi-submersible and comprises several interconnected bodies with various joints. Therefore, a dynamics kernel based on multibody dynamics was developed to perform dynamic response analysis. The recursive Newton–Euler formulation was adopted to construct the equations of motion for the multibody system. Functions of the developed dynamics kernel were verified by comparing them with those from other studies. Hydrostatic force, linearized hydrodynamic force, and pneumatic and hydraulic control forces were considered the external forces acting on the platform of the semi-submersible rig and the heave compensation system. The dynamic simulation was performed for the heave compensation system of the semi-submersible rig for drilling operations up to 3600 m water depth. From the results of the simulation, the efficiency of the heave compensation system was evaluated to be approximately 96.7%.

Effect of NACA0012 Airfoil Pitching Oscillation on Flow Past a Cylinder

The flow past a cylinder is a classical problem in flow physics. In a certain range of Reynolds number, there will be Karman vortex street phenomenon in the wake of a cylinder, which will greatly increase the pressure drag of the cylinder. By controlling the vortex shedding phenomenon, drag reduction of the cylinder could be effectively realized. In this paper, a NACA0012 airfoil with pitching oscillation is placed downstream of the cylinder. Based on the tight coupling method, kinematics equations and Navier–Stokes equations in the arbitrary Lagrangian–Eulerian form are solved. Firstly, the effect of airfoil oscillation period and the distance between airfoil leading edge and cylinder center (x/D) are studied respectively, especially considering the aspects of vortex shedding and drag reduction effect. Besides, the vortex interaction in the flow field around the airfoil and cylinder is analyzed in detail. It is found that the NACA0012 airfoil with pitching oscillation can change the period of vortex shedding. Moreover, it can also increase the drag reduction rate to as high as 50.5%, which presents a certain application prospect in the engineering drag reduction field, e.g., for launch vehicles, ship masts, submarine pipelines, etc.

Fluid–structure modelling for material deformation during cavitation bubble collapse

The dynamics of cavitation bubbles and their interaction with solids has been studied for long, mostly on the macro-scale of sheet and vortex cavitation. Erosion takes place on small, concentrated areas of solid walls exposed to cavitation. One-way fluid structure interaction (FSI), in which the feedback of material deformation on the fluid is not considered, provides an adequate understanding of the underlying mechanism. However, in practical applications the solid material motion and deformation can significantly affect the bubble dynamics. This will, in principle, alter the structural loading, stresses and erosion in the solid. In this paper, we will report comparisons of one-way and two-way coupled FSI modelling using our finite volume based fluid solver YALES211http://yales2.coria-cfd.fr. and finite element based solid solver Cast3M.22http://www-cast3m.cea.fr. In two-way FSI coupling, the feedback of the solid wall deformation is considered in the fluid domain. To do so, we develop a compressible fluid solver with Arbitrary Lagrangian–Eulerian formulation to equip it with moving mesh capabilities in order to maintain a continuous fluid–solid domain. We present results using a step-wise coupling approach for two-way FSI and analyse the effects due to structural response on bubble dynamics and damping of impact pressures by different materials. Finally, we discuss our understanding on the need for one-way and two-way FSI modelling based on structural characteristics which can significantly affect the material erosion by nearby cavitation bubbles.

A monolithic arbitrary Lagrangian–Eulerian‐based finite element strategy for fluid–structure interaction problems involving a compressible fluid

AbstractIn this work, we present a new monolithic finite element strategy for solving fluid–structure interaction problems involving a compressible fluid and a hyperelastic structure. In the Lagrangian limit, the time‐stepping strategy that we propose conserves the total energy, and linear and angular momenta. Detailed proofs with numerical validations are provided. We use a displacement‐based Lagrangian formulation for the structure, and a velocity‐based arbitrary Lagrangian–Eulerian mixed formulation with appropriately chosen interpolations for the various field variables to ensure stability of the resulting numerical procedure. A hybrid formulation is used to prevent locking of thin structures. Apart from physical variables such as displacement, velocity, and so forth, no new variables are introduced in the formulation. The use of the exact tangent stiffness matrix ensures that the algorithm converges quadratically within each time step. A number of benchmark examples have been solved to illustrate the good performance of the proposed method.

Numerical modeling of liquid sloshing in flexible tank with FSI approach

PurposeThe paper aims to present a numerical modeling procedure for the analysis of liquid sloshing in a flexible tank subjected to an external excitation, with taking into account the effects of fluid–structure interaction (FSI).Design/methodology/approachA numerical model based on coupling a two-phase flow solver and an elastic solid solver is developed in OpenFOAM code. The Arbitrary Lagrangian–Eulerian formulation is adopted for the two-phase Navier–Stokes equations in a moving domain. The volume of fluid (VOF) method is applied for the air–liquid interface tracking. The finite volume method is used for the spatial discretization of both the fluid and the structure dynamics equations. The FSI coupling problem is solved by an explicit coupling scheme. The model is validated for linear and nonlinear sloshing cases. Then, it is used to analyze the effects of the liquid sloshing on the dynamic response of the tank and the effects of the tank flexibility on the liquid sloshing.FindingsThe obtained results show that the flexibility of the tank walls amplifies the amplitude of the sloshing and increases the fluctuation period of the air–liquid interface. Furthermore, it is found that the bending moment acting on the tank walls may be underestimated when rigid walls assumption is adopted as usually done in sloshing tank modeling. Also, tank walls flexibility causes a phase shift in the free surface dynamic response.Originality/valueA review of previous studies on liquid sloshing in flexible tanks revealed that FSI effects have not been clearly and comprehensively analyzed for large-amplitude liquid sloshing. Many physical and numerical aspects of this problem still require clarifications and enhancements. The added value of the present work and its originality lie in the investigation of large-amplitude liquid sloshing in flexible tanks by using a staggered coupling approach. This approach is carried out by an original combination of a linear solid solver with a two phase fluid solver in OpenFOAM code. In addition, FSI effects on some response quantities, identified and analyzed herein, have not been found in the previous works.

A computationally efficient formulation of the governing equations for unit operation design

A computationally-efficient numerical method that uses a Pseudo-Eulerian formulation (PEF) for the design calculation of unit operations is presented and validated. This method is applicable to any unit operation that can be modelled using a system of ODEs. Performing the design of a unit operation in the PEF is tenfold faster than with the conventional Eulerian formulation (EF). The mathematical equivalence between the PEF and the EF is demonstrated by proving that the solution of different unit operation design problems provides the same numerical result independently of the formulation. It is shown that reducing the computation of the unit operation design problems also speeds the computation time of a process design or an optimization flowsheet. Additionally, as opposed to other computationally efficient methods for unit operation design, the PEF allows the accurate estimation of the concentration or temperature profiles of complex unit operations such as a multiphase multicomponent reactor system.

Role of fluid-structure interaction in free convection in square open cavity with double flexible oscillating fins

The problem of transient free convection in an open cavity containing two elastic thin fins is studied in this paper. The left surface is heated, and the right surface is in opening, which is in contact with the environment air. The dimensionless forms of the governing equations for continuity, momentum and heat transfer in the fluid and structures domains are written in Arbitrary Lagrangian–Eulerian (ALE) formulation. The governing equations are solved by the finite element method (FEM). The governing parameters considered are the oscillating fin direction, fin amplitude and length with several flexibility values and aperture sizes. The results show that a maximum heat transfer enhancement occurs at the suitable parameters between the aperture size and the fins oscillating direction.