Eulerian Mesh Research Articles

Eulerian Scalar Projection in Lagrangian Point Source Context: An Approximate Inverse Filtering Approach

In many simulations of flows transporting a discrete phase (droplets or particles), point sources are followed numerically in the Lagrangian context over an Eulerian mesh storing the continuous phase variables. The projection of the properties of the moving sources over the Eulerian nodes requires interpolations, which may drastically reduce the overall accuracy of the algorithm and alter the flow physics. From the analytical solution of a scalar field downstream of a source, it is shown that the widely used Lagrange/Euler projection, constructed from the regressive normalized distance between the point source position and the nodes cell, provides results very close to an approximate Gaussian filtering of the exact solution. Then, an approximate deconvolution is discussed to allow for better estimating the Eulerian scalar field.

Particle-laden flows forced by the disperse phase: Comparison between Lagrangian and Eulerian simulations

The goal of the present work is to assess the ability of Eulerian moment methods to reproduce the physics of two-way coupled particle-laden turbulent flow systems. Previous investigations have been focused on effects such as preferential concentration, and turbulence modulation, but in regimes in which turbulence is sustained by an imposed external forcing. We show that in such regimes, Eulerian methods need resolutions finer than nominal Kolmogorov scale in order to capture statistics of particle segregation, but gas and disperse phase velocity variances can be captured with resolutions comparable to the Kolmogorov length. The work is then extended to address the question whether Eulerian methods are suitable in scenarios in which the continuum field of interest (temperature or momentum) is itself primarily driven by particles. To this end we have extended our analysis to the problem of turbulence driven by heated particles (Zamansky et al. PoF 2014) and have assessed capabilities of Eulerian methods in capturing particle segregation, as well as statistics of the temperature and velocity fields. Separate investigations are developed for cases with and without buoyancy driven turbulence. For each case corresponding Lagrangian calculations are developed and convergence of statistics with respect to the number of particles is established. Then the statistically-converged Lagrangian and Eulerian results are compared. Results show that accurate capture of segregation by the Eulerian methods always requires resolutions much higher than the nominal Kolmogorov scale. In scenarios for which a continuum phase is forced by particles, results from Eulerian methods show some sensitivity of predicted continuum statistics to the mesh resolution. This sensitivity was found to be largest for the case of a temperature field forced by hot particles, but without presence of buoyancy. In this case a Eulerian method with nominal Kolmogorov resolution was found to be insufficient for capture of temperature statistics. When additional coupling between particles and continuum phase is introduced by including the buoyancy effects, this sensitivity is suppressed in the temperature field, but some sensitivity to the Eulerian mesh resolution were detected in the momentum fields.

Photoionization feedback in a self-gravitating, magnetized, turbulent cloud

We present a new set of analytic models for the expansion of HII regions powered by UV photoionisation from massive stars and compare them to a new suite of radiative magnetohydrodynamic simulations of turbulent, self-gravitating molecular clouds. To perform these simulations we use the Eulerian adaptive mesh magnetohydrodynamics code RAMSES-RT, including radiative transfer of UV photons. Our analytic models successfully predict the global behaviour of the HII region provided the density and velocity structure of the cloud is known. We give estimates for the HII region behaviour based on a power law fit to the density field assuming that the system is virialised. We give a radius at which the ionisation front should stop expanding ("stall"). If this radius is smaller than the distance to the edge of the cloud, the HII region will be trapped by the cloud. This effect is more severe in collapsing clouds than in virialised clouds, since the density in the former increases dramatically over time, with much larger photon emission rates needed for the HII region to escape a collapsing cloud. We also measure the response of Jeans unstable gas to the HII regions to predict the impact of UV radiation on star formation in the cloud. We find that the mass in unstable gas can be explained by a model in which the clouds are evaporated by UV photons, suggesting that the net feedback on star formation should be negative

A hybrid variational-collocation immersed method for fluid-structure interaction using unstructured T-splines

Summary We present a hybrid variational-collocation, immersed, and fully-implicit formulation for fluid-structure interaction (FSI) using unstructured T-splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational-collocation method. We use T-splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T-splines offer us two main advantages against non-uniform rational B-splines: they can be locally refined and they are unstructured. The generalized-α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T-junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.

A time splitting fictitious domain algorithm for fluid–structure interaction problems (A fictitious domain algorithm for FSI)

A Finite Element Method in mixed Eulerian and Lagrangian formulation is developed to allow direct numerical simulations of dynamical interaction between an incompressible fluid and a hyper-elastic incompressible solid. A Fictitious Domain Method is applied so that the fluid is extended inside the deformable solid volume and the velocity field in the entire computational domain is resolved in an Eulerian framework. Solid motion, which is tracked in a Lagrangian framework, is imposed through the body force acting on the fluid within the solid boundaries. Solid stress smoothing on the Lagrangian mesh is performed with the Zienkiewicz–Zhu patch recovery method. High-order Gaussian integration quadratures over cut elements are used in order to avoid sub-meshing within elements in the Eulerian mesh that are intersected by the Lagrangian grid. The algorithm is implemented and verified in two spatial dimensions by comparing with the well validated simulations of solid deformation in a lid driven cavity and periodic elastic wall deformation driven by a time-dependent flow. It shows good agreement with the numerical results reported in the literature. In 3-D the method is validated against previously reported numerical simulations of 3-D rhythmically contracting alveolated ducts.

Implementing Vibration Framework for Simulation of VIV on Rigid Pier by SPH

It is worth mentioning that the effect of waves on fixed and floating platforms is considered as an important element when designing any offshore structures. This paper is developed a numerical model to simulate current and wave interaction with a vertical cylinder as a platform leg using Smooth Particle Hydrodynamics (SPH) method for solving hydrodynamics part and using Finite Element Method (FEM) for structural part. SPH method is Lagrangian meshless based which is accurate enough for free surface modeling in comparison with other Eulerian mesh based methods. Capability of this method to calculate inline and cross flow forces on cylinder taking into consideration different time solution algorithms. The results showed that SPH not only creates much better result for simulating Vortex Induced Vibration (VIV) but also using the predictor-corrector algorithm for time step algorithm can leads to the most accurate results for predicting lift force.

An Euler–Lagrange model for simulating fine particle suspension in liquid flows

This article presents a two-way coupled Euler–Lagrange model to simulate the suspension of fine particles in liquid flows. The goal is to develop a three-dimensional numerical model that is capable of replicating the detailed features of particle-laden turbulent flow. A two-phase fractional-step projection method is developed to ensure mixture incompressibility by solving a modified Poisson equation for pressure, which in turn affects the particle motion through the pressure gradient. An efficient particle-moving algorithm that exchanges particles between Eulerian meshes is developed that automatically retains the necessary particle information at each time step and does not require any Lagrangian particle tracking. A soft-sphere particle collision model is employed to avoid excessive particle overlap and to achieve the random closed packing limit for depositing particles. Since particles are all inherently localized in Eulerian meshes, efficient particle searches can be achieved when calculating particle–particle collision. This model is then used to simulate the gravitational settling of particles, and the results confirm the effect of mixture incompressibility and demonstrate that the model is capable of reaching the random closed packing limit. Numerical examples for the flow problems of particle-induced stratification are conducted, and the model is able to reveal the detailed features of particle-laden flows. Sensitivity on the grid resolution and deviations from the existing single-phase model results are also discussed.

A locally extended finite element method for the simulation of multi-fluid flows using the Particle Level Set method

Abstract The simulation of immiscible two-phase flows on Eulerian meshes requires the use of special techniques to guarantee a sharp definition of the evolving fluid interface. This work describes the combination of two distinct technologies with the goal of improving the accuracy of the target simulations. First of all, a spatial enrichment is employed to improve the approximation properties of the Eulerian mesh. This is done by injecting into the solution space new features to make it able to correctly resolve the solution in the vicinity of the moving interface. Then, the Lagrangian Particle Level Set (PLS) method is employed to keep trace of the evolving solution and to improve the mass conservation properties of the resulting method. While the local enrichment can be understood in the general context of the XFEM, we employ an element-local variant, which allows preserving the matrix graph, and hence highly improving the computational efficiency.

Parallel embedded boundary methods for fluid and rigid-body interaction

The implementations of an updated body-fitted and a non body-fitted method to deal with the interaction of a fluid and a rigid body are described. The physics of the fluid is modeled by the incompressible Navier–Stokes equations. A parallel fluid solver based on the VMS (Variational Multiscale) Finite Element method serves as the basis for the implementation. For the rigid body movement, the Newton–Euler equations are solved numerically. To account for the interaction, the force that the fluid exerts on the rigid body is determined, on the one hand. On the other hand, the velocity of the rigid body is imposed as a Dirichlet boundary condition on the fluid. A fixed Eulerian mesh discretizes the fluid domain, except for nodes in the vicinity of the rigid body boundary for the case of the updated body-fitted approach. The wet boundary of the rigid body is embedded in the fluid mesh and tracked by a moving surface mesh. It is a distinctive characteristic of the updated body-fitted strategy that, in order to impose velocities on the interface, some of the nodes near the body surface are moved by using a local r-adaptivity algorithm to conform with this surface. By contrast, the non body-fitted approach uses kriging interpolation for velocity prescription over the fluid on the interface. Given that fluid nodes can become solid nodes and vice versa due to the rigid body movement, we have adopted the FMALE approach, a variation of the ALE method to keep the fluid mesh fixed. Algorithms to ensure high performance, like skd-trees to determine if a given spatial point is currently inside the solid, are also used. All these ingredients constitute two approaches that are both computationally efficient and accurate. Numerical experiments are presented to assess their performance comparatively.

The material point method for unsaturated soils

The paper describes a three-phase single-point material point method formulation of coupled flow (water and air) for hydro-mechanical analysis of geotechnical problems involving unsaturated soils. The governing balance and dynamic momentum equations are discretised and adapted to material point method characteristics: an Eulerian computational mesh and a Lagrangian analysis of material points. General mathematical expressions for the terms of the set of governing equations are given. A suction-dependent elastoplastic Mohr–Coulomb model, expressed in terms of net stress and suction variables is implemented. The instability of a slope subjected to rain infiltration, inspired from a real case, is solved and discussed. The model shows the development of the initial failure surface in a region of deviatoric strain localisation, the evolution of stress and suction states in some characteristic locations, the progressive large strain deformation of the slope and the dynamics of the motion characterised by the history of displacement, velocity and acceleration of the unstable mass.

MODELING OF SUPERIOR MESENTERIC ARTERY ANEURYSM USING FLUID–STRUCTURE INTERACTION

The aim of this study was to model the blood flow and predict related hemodynamics characteristics in healthy superior mesenteric artery (SMA) and saccular aneurysm cases. A fluid–structure interaction (FSI) method was performed, using an arbitrary Langrangian–Eulerian mesh. The computational mesh was generated using anatomical data from available human computed tomography (CT)-images. Combining constitution and momentum equations, projection method, the discretized resultant equation were numerically solved for velocity, pressure, shear stress and vortices for healthy/aneurysmal artery. The results including velocity contours, pressure contours, shear rate values, and vortices were obtained and analyzed for three main steps including peak systole, diastole, and end of cardiac cycle. Profiles show the varying velocity and pressure for a pulsatile flow input before and after aneurysms. They also show the formation of single or multiple vortices at aneurysmal area and decrease of wall shear stress with aneurysm enlargement. Furthermore, shear rate values at the neck of aneurysms exceed throughout the entire cardiac cycle. The outcome of the computational analysis is then compared to information available on pressure, vortices and wall shear stress from some clinical findings.

Transient large strain contact modelling: A comparison of contact techniques for simultaneous fluid–structure interaction

Contact between two deformable structures, driven by applied fluid-pressure, is compared for an existing pseudo-transient contact method (the default in the Comsol Multi-physics v3.3 software package) and a new transient method. Application of the new method enables time-dependent and simultaneous Fluid–Structure Interaction (FSI) simulations to be solved. The new method is based on Hertzian contact. It enables truly transient simulations, unlike the default contact method. Both the default and new methods were implemented using a moving Arbitrary-Lagrange–Euler mesh, along with velocity constraints and Lagrange Multipliers to enable simultaneous FSI simulations. The comparison was based on a simple two-dimensional model developed to help understand the opening of a heart valve. The results from the new method were consistent with the steady-state solutions achieved using the default contact method. However, some minor differences in fluid dynamics, structural deformation and contact pressure predicted were obtained. The new contact method developed for FSI simulations enables transient analysis, in contrast to the default contact method that enables steady state solutions only.

An FEM-Level-set Numerical Model for Potential Flow with Free Surface

A Level-set and Finite Element Method (FEM) based two-dimensional numerical model is proposed for the free surface flows in the frame of potential theory. The FEM is used to solve the governing equation of potential flow while the Level-set method is used to capture the free surface. A new numerical technique is proposed to treat the partial elements that are cut by the free surface. Fully nonlinear boundary condition is used on the free surface. During the simulation, Eulerian meshes are adopted and only the fluid phase is calculated. A linear sloshing problem is firstly used as a benchmark to check the accuracy of the FEM solver. The numerical results are compared with the theoretical solution, and good agreements are obtained. Then the dam breaking flow with violent free surface is used to validate the proposed FEM-Level-set model. The comparisons show that present model has good numerical stability, and is able to produce reasonable prediction on the free surface evolution. Finally a fully nonlinear sloshing problem is simulated. The numerical results of the present model agree well with those obtained by the model based on boundary element method.

An Improved Momentum-exchanged Immersed Boundary-based Lattice Boltzmann Method for Incompressible Viscous Thermal Flows

An improved momentum-exchanged immersed boundary-based lattice Boltzmann method (MEIB-LBM) is proposed for incompressible viscous flows in this paper. In present work, we come back to the intrinsic feature of LBM, which uses the density distribution function as a dependent variable to evolve the flow field, and use the non-equilibrium density distribution function correction at the neighbour Euler mesh points to satisfy the non-slip boundary condition on the immersed boundary. The improvements for the original MEIB-LBM are that the intrinsic feature of LBM is kept and the flow penetration is removed. Examples, including force convection over a stationary heated circular cylinder for heat flux condition, and natural convection with a buoyant circle particle in viscous fluid, are provided to validate the present method.

Euler–Lagrange simulation of high pressure shock waves

This paper presents the numerical simulation of overdriven detonation (or O.D.D.) that occurs when a high velocity object impacts an explosive. The pressure and the velocity at this state are higher than those of the Chapman–Jouguet (C–J) state. First, before the simulation of this event, a study of PBX air blast by using multi-material Eulerian method is presented. Pressure peaks are computed for several distances from the explosive. Second, the O.D.D. phenomenon is modeled by the Euler–Lagrange penalty coupling, which permits to couple a Lagrangian mesh of the flyer plate to multi-material Eulerian mesh of explosives and air. This coupling gives us the high detonation velocities in the acceptor explosive and demonstrates that it is able to handle shock–structure interaction problems.

A NURBS-based immersed methodology for fluid–structure interaction

We introduce an isogeometric, immersed, and fully-implicit formulation for fluid–structure interaction (FSI). The method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids, which are a common case in various fields, such as, for example, biomechanics. In our FSI method, we utilize an Eulerian mesh on the whole domain and a Lagrangian mesh on the solid domain. The Lagrangian mesh is arbitrarily located on top of the Eulerian mesh in a non-conforming fashion. Due to the formulation of our problem, based on the Immersed Finite Element Method (IFEM), we do not need mesh update or remeshing algorithms. The fluid–structure interface is the boundary of the Lagrangian mesh, but cuts arbitrarily the Eulerian mesh. The generalized- α method is used for time discretization and NURBS-based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains. The information transfer between the two meshes is carried out using the NURBS functions, which avoids the use of the so-called discretized delta functions. The higher order and especially the higher continuity of NURBS functions allow us to deal with severe mesh distortion in the Lagrangian mesh in comparison with classical C0 linear piecewise functions as we prove numerically. Our numerical solutions attain good agreement with theoretical data for free-falling objects in two and three dimensions, which confirms the feasibility of our methodology.

A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant

Surface active agents play a significant role in interfacial dynamics of multiphase systems.While the understanding of their behavior is crucial to many important practical applications, realistic mathematical modeling and computer simulation represent an extraordinary task. By employing a front-tracking method with Eulerian adaptive mesh refinement capabilities in concert with a finite volume scheme for solving an advection–diffusion equation constrained to a moving and deforming interface, the numerical challenges posed by the full three-dimensional computer simulation of transient, incompressible two-phase flows with an insoluble surfactant are efficiently and accurately tackled in the present work. The individual numerical components forming the resulting methodology are here combined and applied for the first time. Verification tests to check the accuracy and the simulation of the deformation of a droplet in simple shear flow in the presence of an insoluble surfactant are performed, the results being compared to laboratory experiments as well as to other numerical data. In all the cases considered, the methodology presents excellent conservation properties for the total surfactant mass (even to machine precision under certain circumstances).

A continuum‐discrete model using Darcy's law: formulation and verification

SummaryThis paper presents a numerical scheme for fluid‐particle coupled discrete element method (DEM), which is based on poro‐elasticity. The motion of the particles is resolved by means of DEM. While within the proposition of Darcian regime, the fluid is assumed as a continuum phase on a Eulerian mesh, and the continuity equation on the fluid mesh for a compressible fluid is solved using the FEM. Analytical solutions of traditional soil mechanics examples, such as the isotropic compression and one‐dimensional upward seepage flow, were used to validate the proposed algorithm quantitatively. The numerical results showed very good agreement with the analytical solutions, which show the correctness of this algorithm. Sensitivity studies on the effect of some influential factors of the coupling scheme such as pore fluid bulk modulus, volumetric strain calculation, and fluid mesh size were performed to display the accuracy, efficiency, and robustness of the numerical algorithm. It is revealed that the pore fluid bulk modulus is a critical parameter that can affect the accuracy of the results. Because of the iterative coupling scheme of these algorithms, high value of fluid bulk modulus can result in instability and consequently reduction in the maximum possible time‐step. Furthermore, the increase of the fluid mesh size reduces the accuracy of the calculated pore pressure. This study enhances our current understanding of the capacity of fluid‐particle coupled DEM to simulate the mechanical behavior of saturated granular materials. Copyright © 2014 John Wiley & Sons, Ltd.

Blade vibration triggered by low load and high back pressure

Load and condenser pressure play an important role in overall life of turbine blades especially in the last stages of low pressure (LP) turbine. LP last-stage blades have greatest potential to stall flutter during low steam flow and high backpressure. Exhaust loss refers to the energy lost to the condenser as a result of the steam velocity going into the condenser. The faster the steam leaves the last stage blades, the greater the energy loss. Since the backpressure is a function of the plant cooling system, and the flow is a function of the initial plant design, the only cost-effective way to reduce the velocity is by increasing the area. This is achieved by increasing the length of last stage bucket.The two primary forces acting on the blades are the steady centrifugal force due to rotation and the fluctuating steam bending force. The best estimation of dynamic stresses in the blades due to the two known sources of vibration works out to be a small portion of the allowable value. But still, incidences of blade failure are regularly reported. Published reports on compilation of blade failures in power plants say that cause for large number of failure is not fully understood. One such cause on which very little is published is self excitation in long blades. Self excitation sustain until the condition remain conducive to vibration inside the turbine. In plants operating at low load and high back pressure the blades are most susceptible to self excitation wherein the blade vibrates in its lower mode. The paper deals with monitoring self excitation in an operating power plant. A Single long blade has been modeled to steady fluid structure interaction by Lagrangian and Eulerian meshing approach during low load and high back pressure. The effect of the steam inlet angle on the blade has been studied. The model simulates low steam flow condition to show increased blade response thus validating observation made in operating power plant.

Stress constrained shape and topology optimization with fixed mesh: A B-spline finite cell method combined with level set function

In this paper, we develop an efficient and flexible design method that integrates the B-spline finite cell method (B-spline FCM) and the level set function (LSF) for stress constrained shape and topology optimization. Any structure of complex geometry is embedded within an extended, regular and fixed Eulerian mesh no matter how the structure is optimized. High-order B-spline shape functions are further implemented to ensure precisions of stress analysis and sensitivity analysis. Meanwhile, level set functions, i.e., implicit functions are used to enable topological changes of the considered structure through smooth boundary variations. Involved parameters rather than the conventional discrete form of LSF are directly taken as design variables to facilitate the numerical computing process. To be specific, the LSF is constructed by means of R-functions that incorporate cubic splines as implicit functions to offer flexibilities for shape optimization within the framework of fixed mesh, while the compactly supported radial basis functions (CS-RBFs) are employed as implicit functions for stress constrained topology optimization. It is shown the proposed FCM/LSF method is a convenient approach that makes it possible to calculate stress and stress sensitivities with high precision. Representative examples of shape and topology optimization with and without stress constraints are solved with success demonstrating the advantages of the FCM/LSF method.