Combined Eulerian-Lagrangian Approach Research Articles

Non-linear sloshing characteristics of liquid inside a chamfered bottom tank with submerged internal object under near-and far-fault earthquakes

Cylindrical and rectangular-shaped tanks have a concentration of stagnant impulsive liquid at the bottom corners that do not participate in sloshing. While used as tuned liquid dampers (TLDs), this non-participating fluid adds extra liquid mass to the tank but does not have any role in the vibration control aspect. In the present study, to decrease the stagnant impulsive liquid, a chamfered bottom-shaped tank with a submerged internal object is demonstrated. Due to the high-energy velocity pulse present in near-fault earthquakes, there is a possibility of violent liquid sloshing which may cause unwanted structural damage to the tank. Hence, the present study focuses on the influence of near- and far-fault earthquakes on the non-linear seismic behavior of the tank with submerged internal object. The non-linear sloshing is modeled using the finite element method based on a combined Eulerian-Lagrangian approach. The accuracy of the present non-linear numerical model is checked by verifying the results with the previously published results. In addition to the absolute response, the dynamic convective and impulsive response components are quantified. The proposed non-linear finite element scheme can be utilized in the context of structure-mounted chamfer bottom TLDs in the future.

Investigation of the Seismic Performance of Historical Masonry Aqueducts with Eulerian-Lagrangian Coupled Analysis

ABSTRACT Throughout history, aqueducts have been built to exceed obstacles on the waterway while preserving the energy of water. Our priceless cultural heritage, aqueducts must be protected to be passed on to future generations. Filling the channel with water can increase earthquake resistance due to the damping effect of water. Investigating of fluid and structure interaction of aqueducts through numerical analysis is crucial to understand their behavior under seismic forces. The aqueducts, for which 3D finite element models were prepared, were analyzed using the combined Eulerian-Lagrangian approach, taking into account two main scenarios: with and without fluid-structure interaction. With this approach, it’s aimed to observe the damping effect of the fluid, the sloshing on its surface and possible water overflows. Using the CDP material model for masonry units, the analyzes were diversified with various ground motion records at different Peak Ground Acceleration values. The presence of water in the channel significantly reduced displacements, stresses and damage rates. In addition, it was seen that the spectral acceleration value of the earthquake record was more effective in causing damage rather than the PGA value. Graphics and contour diagrams are given to understand and compare the analysis results of aqueduct models under different conditions.

IGIMP: An implicit generalised interpolation material point method for large deformations

The Material Point Method (MPM) uses a combined Eulerian-Lagrangian approach to solve problems involving large deformations. A problem domain is discretised as material points which are advected on a background grid. Problems are encountered with the original MPM when material points cross between grid cells, and this has been tackled by the development of the Generalised Interpolation MPM, where material points’ domains of influence extend beyond the currently occupied grid cell. In this paper, the Generalised Interpolation Material Point (GIMP) Method has been implemented implicitly in a manner that allows a global stiffness matrix to be constructed similar to that in the Finite Element Method (FEM) by combining contributions from individual elements on the background grid. An updated Lagrangian finite deformation framework has been used to ensure non-linear behaviour within each of the loadsteps. The weighting functions used for this which make the GIMP method different to standard MPM are presented and the implementation is explained. Specific details on computing the deformation gradient to be consistent with the updated Lagrangian framework and the updating of the material point influence domains are outlined, both of which are currently unclear in the published literature. It is then shown through numerical examples that for both small and large deformation elastic and elasto-plastic problems, the implicit GIMP method agrees well with analytical solutions and exhibits convergence properties between that of linear and quadratic FEM.

A large eddy simulation of the breakup and atomization of a liquid jet into a cross turbulent flow at various spray conditions

A three-dimensional large eddy simulation (LES) is conducted to investigate the breakup and atomization of a liquid jet into a cross turbulent flow for several variants of a liquid-gas momentum flux ratio by varying the liquid injection velocity and cross flow temperature. The spray-field dynamics are treated using a combined Eulerian-Lagrangian approach in which the gas phase is discretized using a density-based, finite-volume approach. A Kelvin-Helmholtz and Rayleigh-Taylor (KH-RT) hybrid wave breakup model is implemented to simulate the liquid column and droplet breakup process. While the KH model is applied to the liquid column breakup (primary breakup), the RT model is implemented to the breakup of the small droplets (secondary breakup). The detail flow structures of the counter-rotating vortex pair (CVP) and vortex interaction behind the injector are observed. The spray penetration depth in the crossflow compares well with the experimental data and similar to empirical equations. The Sauter mean diameter (SMD) distribution is analyzed along the flow downstream representing somewhat different, and an analytical correlation model is proposed to pre-evaluate the SMD in the flowfield.

Pressure oscillations and instability of working processes in the combustion chambers of solid rocket motors

Metal particles are widely used in space engineering to increase specific impulse and to supress acoustic instability of intra-champber processes. A numerical analysis of the internal injection-driven turbulent gas-particle flows is performed to improve the current understanding and modeling capabilities of the complex flow characteristics in the combustion chambers of solid rocket motors (SRMs) in presence of forced pressure oscillations. The two-phase flow is simulated with a combined Eulerian–Lagrangian approach. The Reynolds-averaged Navier–Stokes equations and transport equations of k–ε model are solved numerically for the gas. The particulate phase is simulated through a Lagrangian deterministic and stochastic tracking models to provide particle trajectories and particle concentration. The results obtained highlight the crucial significance of the particle dispersion in turbulent flowfield and high potential of statistical methods. Strong coupling between acoustic oscillations, vortical motion, turbulent fluctuations and particle dynamics is observed.

Modeling liquid fuel cascades with OpenFOAM

Evaporating liquid cascades resulting from gasoline and liquefied natural gas (LNG) tanks overfilling or rupture of elevated pipes create a source of flammable vapour cloud. Such phenomena were responsible for the formation of the large fuel vapour cloud, the ignition of which resulted in the large scale explosion, in Buncefield [1] on 11 December 2005 at the Hertfordshire Oil Storage Terminal, an oil storage facility located by Hemel Hempstead in Hertfordshire, England. Despite its significance, there lacks adequate models treating the underlying physics of this phenomenon. The present study numerically analyses fuel cascades which are considered as a droplet-laden system. Consideration is given to vapour production inside the cascade due to droplets evaporation and breakup. The solver used here is a modification of the sprayFoam solver which is present in the open source computational fluid dynamics (CFD) toolbox OpenFOAM® [2]. The fuel droplets evaporate during their motion and create a cloud of flammable vapour. In order to capture the characteristics of the hazardous phenomena, the CFD model needs to address the underlying physics with adequate sub-models. In the present study, the multi-phase flow is simulated with a combined Eulerian-Lagrangian approach. The governing equations of the gas phase represent the conservation equations of mass, momentum and energy including the source terms arising from the interaction with the droplets. The Reynolds Averaged Navier-Stokes (RANS) simulation approach was used for its computational efficiency. The Large-Eddy Simulation (LES) would be more robust in handling the interaction of the droplets with the flow but it would be require more computational resource. The particulate phase is simulated through a Lagrangian deterministic or stochastic tracking models to provide particle trajectories and particle concentration. Particular emphasis is given to the effect of impingement of droplets to account for the effect of splashing in the impact region. The study involves developing robust and accurate modelling approaches for the instabilities and aerodynamic breakup in the cascade which contribute to the formation of the cloud, air entrainment and fuel impingement on deflector plates. Suitable sub-models have been implemented in OpenFOAM® to facilitate the study. The predictions are compared with the experimental measurements and CFD predictions previously conducted by Atkinson and Coldrick [3] from the Health and Safety Laboratory (HSL), an agency of the Health and Safety Executive (HSE). The present predictions are found to better capture the interaction between the droplets and the gas phase. Improved agreement with the experimental measurements in the gasoline fuel cascades has also been achieved.

Quantifying tracer dynamics in moving fluids: a combined Eulerian-Lagrangian approach

Eulerian models coupling physics and biology provide a powerful tool for the study of marine systems, complementing and synthesizing in situ observations and in vitro experiments. With the monotonic improvements in computing resources, models can now resolve increasingly complex biophysical interactions. Quantifying complex mechanisms of interaction produces massive amounts of numerical data that often require specialized tools for analysis. Here we present an Eulerian-Lagrangian approach to analyzing tracer dynamics in moving fluids. As an example of its utility, we apply this tool to quantifying plankton dynamics in oceanic mesoscale coherent structures. In contrast to Eulerian frameworks, Lagrangian approaches are particularly useful for revealing physical pathways, and the dynamics and distributions of tracers along these trajectories. Using a well-referenced Lagrangian tool, we develop a method to assess the variability of biogeochemical properties (computed using an Eulerian model) along particle trajectories. We discuss the limitations of this new method, given the different biogeochemical and physical timescales at work in the Eulerian framework. We also use Lagrangian trajectories to track coherent structures such as eddies, and we analyze the dynamics of the local ecosystem using two techniques: (i) estimating biogeochemical properties along trajectories, and (ii) averaging biogeochemical properties over dynamic regions (e.g., the eddy core) defined by ensembles of similar trajectories. This hybrid approach, combining Eulerian and Lagrangian model analyses, enhances the quantification and understanding of the complex planktonic ecosystem responses to environmental forcings; it can be easily applied to the dynamics of any tracer in a moving fluid.

Mechanical Erosion of Graphite Nozzle in Solid-Propellant Rocket Motor

A detailed theoretical/numerical framework is established to study the mechanical erosion of graphite-nozzle materials in solid rocket motors with aluminized ammonium perchlorate/hydroxyl-terminated polybutadiene composite propellants. The analysis is based on a combined Eulerian–Lagrangian approach for treating multiphase motor flowfields. The multicomponent reacting gas-phase dynamics is formulated using the conservation equations of mass, momentum, and energy in the Eulerian framework. Turbulence closure is achieved using the standard two-equation model. The dispersed phase, consisting of aluminum and alumina droplets, is treated in the Lagrangian framework. Combustion of aluminum droplets to aluminum-oxide smoke is considered. Two empirical correlations are first calibrated and then employed to predict the mechanical-erosion rate of the nozzle surface. The estimated erosion rates fall within the range of the available experimental data. Mechanical erosion is prevalent in the convergent section of the rocket nozzle due to the particle impingement on the nozzle surface. No such erosion, however, is observed at the nozzle throat or downstream because the droplet trajectories move away from the nozzle surface in those regions.

Modeling of Water-Spray Application in the Forced Dispersion of LNG Vapor Cloud Using a Combined Eulerian–Lagrangian Approach

The safety and security of liquefied natural gas (LNG) facilities has prompted the need for continued study of LNG mitigation systems. Water-spray curtains are widely recognized as an effective mea...

Evaluation of shear stress accumulation on blood components in normal and dysfunctional bileaflet mechanical heart valves using smoothed particle hydrodynamics

Evaluating shear induced hemodynamic complications is one of the major concerns in design of the mechanical heart valves (MHVs). The monitoring of these events relies on both numerical simulations and experimental measurements. Currently, numerical approaches are mainly based on a combined Eulerian–Lagrangian approach. A more straightforward evaluation can be based on the Lagrangian analysis of the whole blood. As a consequence, Lagrangian meshfree methods are more adapted to such evaluation. In this study, smoothed particle hydrodynamics (SPH), a fully meshfree particle method originated to simulate compressible astrophysical flows, is applied to study the flow through a normal and a dysfunctional bileaflet mechanical heart valves (BMHVs). The SPH results are compared with the reference data. The accumulation of shear stress patterns on blood components illustrates the important role played by non-physiological flow patterns and mainly vortical structures in this issue. The statistical distribution of particles with respect to shear stress loading history provides important information regarding the relative number of blood components that can be damaged. This can be used as a measure of the response of blood components to the presence of the valve implant or any implantable medical device. This work presents the first attempt to simulate pulsatile flow through BMHVs using SPH method.

Unsteady development of a deformable bubble rising in a quiescent liquid

An accurate finite-volume based numerical method for the simulation of an isothermal two-phase flow, consisting of a rising deformable bubble translating in a quiescent, unbounded liquid, is presented. This direct simulation method is built on a sharp interface concept and developed on an Eulerian, Cartesian fixed-grid with a cut-cell scheme and marker points to track the moving interface. The unsteady Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the evolving interface shape and movement of the bubble. The highlights of this method are that it utilizes a combined Eulerian–Lagrangian approach, and is capable of treating the interface as a sharp discontinuity. A fixed underlying grid is used to represent the control volume. The interface, however, is denoted by a separate set of marker particles which move along with the interface. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. This numerical scheme can handle a wide range of density and viscosity ratios. The bubble is assumed to be spherical and at rest initially, but deforms as it rises through the liquid pool due to buoyancy. Additionally, the flow is assumed to be axisymmetric and incompressible. The bubble deformation and dynamic motion are characterized by the Reynolds number, the Weber number, the density ratio and the viscosity ratio. The effects of these parameters on the translational bubble dynamics and shape are given and the physical mechanisms are explained and discussed. Results for the shape, velocity profile and various forces acting on the bubble are presented here as a function of time until the bubble reaches terminal velocity. The range of Reynolds numbers investigated is 1 < Re < 100, and that of Weber number is 1 < We < 10.

Modeling mass loading effects in industrial cyclones by a combined Eulerian–Lagrangian approach

Cyclone separation is studied by means of numerical simulations. While the gas flow is modeled by a modified Reynolds stress (RS) model, the behavior of the particles is pictured by a combined Eulerian–Lagrangian approach. A mono-disperse Eulerian particle phase is utilized to account for inter-particle collisions, while the effects of fractional separation and particle-wall collisions are considered by poly-disperse Lagrangian particles. The above particle models interact in two ways. On the one hand, the Lagrangian particles determine the local mean diameter of the substitute Eulerian particle class. On the other hand, especially in regions of high particle concentration, the Eulerian particle phase exerts an additional collisional force onto the Lagrangian particle trajectories. An industrial cyclone is chosen as a test case and the numerical results are evaluated with respect to pressure drop as well as to global and fractional separation efficiency. In this context the influence of the cyclone’s mass loading and wall roughness is highlighted. Simulations indicate that the separation efficiency improves with increasing mass loading until an excess loading is reached while at the same time the pressure drop is reduced. Furthermore, it can be shown that rough walls lead to a reduction of separation efficiency while simultaneously the pressure drop decreases. The simulations results are compared with both an analytic theory of Muschelknautz [Die Berechnung von Zykonabscheidern fur Gase. Chem Ing Techn 44, (1+2):63–71, 1972] as well as with real plant measurements.

Droplet–Droplet Interactions in Spray Drying by Using 2D Computational Fluid Dynamics

A 2D axisymmetric model of the spray drying process is presented. The two-phase flow theoretical model is based on a combined Eulerian-Lagrangian approach and takes binary interactions (coalescence or bouncing) between spray droplets into consideration. Validation of the model (incorporated in FLUENT 6.3.26) demonstrated good agreement and consistency with the literature data. The results of transient simulations showed that droplet–droplet interactions displace the region of heat and mass transfer from the central core toward the periphery of the drying chamber. It was also found that insulation of the spray dryer can substantially affect temperature and humidity patterns, whereas its influence on the velocity flow field is less marked.

Stability analysis of Eulerian-Lagrangian methods for the one-dimensional shallow-water equations

In this paper stability and error analyses are discussed for some finite difference methods when applied to the one-dimensional shallow-water equations. Two finite difference formulations, which are based on a combined Eulerian-Lagrangian approach, are discussed. In the first part of this paper the results of numerical analyses for an explicit Eulerian-Lagrangian method (ELM) have shown that the method is unconditionally stable. This method, which is a generalized fixed grid method of characteristics, covers the Courant-Isaacson-Rees method as a special case. Some artificial viscosity is introduced by this scheme. However, because the method is unconditionally stable, the artificial viscosity can be brought under control either by reducing the spatial increment or by increasing the size of time step. The second part of the paper discusses a class of semi-implicit finite difference methods for the one-dimensional shallow-water equations. This method, when the Eulerian-Lagrangian approach is used for the convective terms, is also unconditionally stable and highly accurate for small space increments or large time steps. The semi-implicit methods seem to be more computationally efficient than the explicit ELM; at each time step a single tridiagonal system of linear equations is solved. The combined explicit and implicit ELM is best used in formulating a solution strategy for solving a network of interconnected channels. The explicit ELM is used at channel junctions for each time step. The semi-implicit method is then applied to the interior points in each channel segment. Following this solution strategy, the channel network problem can be reduced to a set of independent one-dimensional open-channel flow problems. Numerical results support properties given by the stability and error analyses.