Lagrangian Grid Research Articles

Two-Dimensional Free-Surface Flow Modeling for Wave-Structure Interactions and Induced Motions of Floating Bodies

In this study, the level set (LS) and immersed boundary (IB) methods were integrated into a Navier–Stokes equation two-phase flow solver, to investigate wave-structure interactions and induced motions of floating bodies in two dimensions. The movement of an interfacial boundary of two fluids, even with severe free-surface deformation, is tracked by using the level set method, while an immersed object inside a fluid domain is treated by the IB method. Both approaches can be implemented by solving the Navier–Stokes equations for viscous laminar flows with embedded objects in fluids. For accurate treatment of the solid–liquid phase, appropriate source terms as forcing functions to take into account the hydrodynamic effects on the body boundaries are added into the governing equations. The integrated compact interfacial tracking techniques between the interfaces of gas–liquid phase and the solid–liquid phase allow the use of a combined Eulerian Cartesian and Lagrangian grid system. Problems related to the fluid-structure interactions and induced motions of a floating body, such as (a) a dam-break wave over a dry bed; (b) a dam-break wave over either a submerged semicircular or rectangular cylinder; (c) wave decomposition process over a trapezoid breakwater; (d) a free-falling wedge into a water body; and (e) wave packet interacting with a floating body are selected to test the model performance. For all cases, the computed results are found to agree reasonably well with published experimental data and numerical solutions. For the case of modeling wave decomposition process, improved solutions are obtained. The detailed features of flow phenomena described by the physical variables of velocity, pressure and vorticity are presented and discussed. The present two-phase flow model is proved to have the advantage of simulating the cases with induced severe interfacial oscillations and coupled gas (or air) motions where the single-phase model may miss the contributions of the air motions on the interfaces. Additionally, the proposed method with uses of the LS and IB methods is demonstrated to be able to achieve the reliable predictions of complex flow fields.

МЕТОД РАСПАДА РАЗРЫВОВ В ТРЕХМЕРНОЙ ДИНАМИКЕ УПРУГОПЛАСТИЧЕСКИХ СРЕД

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies with large displacements and deformations based on the method of disintegration of discontinuities according to the Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The Riemann's solver for an elastic medium in the case of an arbitrary stress state are obtained and presented. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact template for moving Eulerian – Lagrangian grids. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can vary. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian – Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing to determine the parameters on the boundary and contact surfaces. The values of the underdetermined parameters near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions and with the experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

Simulating particle sedimentation in a flowing fluid using an immersed boundary–lattice Boltzmann method

ABSTRACTA combination of the lattice-Boltzmann and immersed-boundary methods is used to simulate the behaviour of circular particles in fluids, including their behaviour in stationary and flowing fluids. This method simulates the fluid domain with a regular Eulerian grid, and the non-grid uniform boundaries with a Lagrangian grid. The no-slip boundary condition of the fluid and particle boundaries is enforced by interpolating boundary forces. This simulation uses a direct-force method to achieve a slight deformation of solid particles. The correctness of the method is validated using simulations of the flow past an impulsive cylinder. Finally, the method is used to compare the particles’ behaviour in stationary and flowing fluids. The result shows that a well-known ‘drafting, kissing, and tumbling’ (DKT) case occurs in the particle sedimentation in the stationary viscous fluid. In contrast, kissing occurs in the moving fluid, but no separating.

An Eulerian-Lagrangian-Lagrangian method for solving fluid-structure interaction problems with bulk solids

This paper presents an Eulerian-Lagrangian-Lagrangian (ELL) method to solve the fluid-structure interaction (FSI) problem, in which a bulk solid is immersed in the fluid. The ELL approach uses a small portion of the Lagrangian fluid to wrap the solid body, and they are treated as an enlarged “composite solid”. Then the conventional immersed finite element method (IFEM) is implemented. The velocity condition of the moving interface is explicitly imposed within the unified Lagrangian grids of the composite solid domain, and then smoothly transferred onto the Eulerian fluid nodes nearby the FSI interaction zone. The inaccurate numerical FSI interface in IFEM is hence significantly improved. The idea of warping Lagrangian fluid makes ELL a combination of the body-conforming method and the immersed type method. It is straightforward but offers an excellent physical explanation that makes ELL more realistic. Several numerical examples are computed to demonstrate the superior performance of the proposed approach for solving FSI problems with bulk solids, in comparison with the conventional IFEM.

WAVETRISK-1.0: an adaptive wavelet hydrostatic dynamical core

Abstract. This paper presents the new adaptive dynamical core wavetrisk. The fundamental features of the wavelet-based adaptivity were developed for the shallow water equation on the β plane and extended to the icosahedral grid on the sphere in previous work by the authors. The three-dimensional dynamical core solves the compressible hydrostatic multilayer rotating shallow water equations on a multiscale dynamically adapted grid. The equations are discretized using a Lagrangian vertical coordinate version of the dynamico model. The horizontal computational grid is adapted at each time step to ensure a user-specified relative error in either the tendencies or the solution. The Lagrangian vertical grid is remapped using an arbitrary Lagrangian–Eulerian (ALE) algorithm onto the initial hybrid σ-pressure-based coordinates as necessary. The resulting grid is adapted horizontally but uniform over all vertical layers. Thus, the three-dimensional grid is a set of columns of varying sizes. The code is parallelized by domain decomposition using mpi, and the variables are stored in a hybrid data structure of dyadic quad trees and patches. A low-storage explicit fourth-order Runge–Kutta scheme is used for time integration. Validation results are presented for three standard dynamical core test cases: mountain-induced Rossby wave train, baroclinic instability of a jet stream and the Held and Suarez simplified general circulation model. The results confirm good strong parallel scaling and demonstrate that wavetrisk can achieve grid compression ratios of several hundred times compared with an equivalent static grid model.

Fragment spatial distribution of prismatic casing under internal explosive loading

Non-cylindrical casings filled with explosives have undergone rapid development in warhead design and explosion control. The fragment spatial distribution of prismatic casings is more complex than that of traditional cylindrical casings. In this study, numerical and experimental investigations into the fragment spatial distribution of a prismatic casing were conducted. A new numerical method, which adds the Lagrangian marker points to the Eulerian grid, was proposed to track the multi-material interfaces and material dynamic fractures. Physical quantity mappings between the Lagrangian marker points and Eulerian grid were achieved by their topological relationship. Thereafter, the fragment spatial distributions of the prismatic casing with different fragment sizes, fragment shapes, and casing geometries were obtained using the numerical method. Moreover, fragment spatial distribution experiments were conducted on the prismatic casing with different fragment sizes and shapes, and the experimental data were compared with the numerical results. The effects of the fragment and casing geometry on the fragment spatial distributions were determined by analyzing the numerical results and experimental data. Finally, a formula including the casing geometry parameters was fitted to predict the fragment spatial distribution of the prismatic casing under internal explosive loading.

Heat transfer between rotating sphere and spherical-surface heat sink

To cool a heated rotating sphere, a heat sink with spherical surface is designed which is situated at a small distance from the sphere; thus, a small curve surface air gap is formed between them which will be used to transfer the heat from the rotating sphere. The numerical simulation method was used to investigate the flow and heat transfer capability of the small air gap. In the entire flow region, a regular Eulerian grid was adopted to solve the modified momentum and energy equations simultaneously. In the region that was occupied by the rotating sphere, a moving Lagrangian grid was used, which tracks the rotational motion of the sphere. A Reynolds function and an energy function were introduced to represent the momentum interaction and thermal interaction between the sphere and the fluid. The heat transfer rate and the flow characteristics of the air gap were presented. The influence of rotating speed and eccentricity as well as the radius of curvature on the flow were quantitatively investigated. It was found that the heat transfer strengths could be adjusted by changing these parameters. It was also observed that the eccentricity and the rotating speed exert significant influences on the heat transfer.

A Transient Coupled Ice Flow‐Damage Model to Simulate Iceberg Calving From Tidewater Outlet Glaciers

AbstractIceberg calving, the detachment of an ice block at the glacier front, is the main process responsible for the dynamic mass loss from the ice sheets to the ocean. Understanding this process is essential to accurately predict ice sheet response to the future climate. We present a transient multiphysics finite‐element model to simulate iceberg break‐off and geometry evolution of a marine‐terminating glacier. The model solves the coupled equations of ice flow, damage mechanics, oceanic melt, and geometry evolution on the same Lagrangian computational grid. A modeling sensitivity analysis shows that the choice of stress measure used for damage evolution strongly influences the resulting calving front geometries. Our analysis suggests that the von Mises stress measures produce the most realistic calving front geometry evolutions for tidewater glaciers. Submarine frontal melt is shown to have a strong impact on the calving front geometry. The presented multiphysics model includes all processes thus far shown to be relevant for the evolution of tidewater glaciers and can be readily adapted for 3‐D and arbitrary bedrock geometries.

An immersed boundary method coupled with a dynamic overlapping-grids strategy

An immersed-boundary method is devised within an existing finite-volume dynamic overlapping-grids solver. The immersed-boundary approach allows improved flexibility, removing the constraint of the Eulerian grid to conform to the geometry of solid bodies within the fluid domain. This is beneficial to the regularity of the grid, especially in the near wall regions, improving the stability and accuracy of the overall method. Furthermore, grid generation, which is a very time-consuming, non-automated task in body-fitted techniques, requiring a significant effort in terms of human work-hours, is dramatically simplified. Here the immersed-boundary strategy is utilized together with curvilinear grids capabilities, which are useful to keep cells count under control, a major issue in more conventional immersed-boundary methods using Cartesian grids. The main advantage of the proposed approach is the coupling with a dynamic overlapping-grids methodology. This is especially convenient in presence of moving bodies, since the grid attached to a moving immersed-boundary can follow the body during its motion, with no need to update the position of the Lagrangian grid, discretizing the body surface, relative to the associated Eulerian grid, discretizing the fluid domain in the vicinity of the body. A distance function is defined at each node of the computational grid, based on the position relative to the surface of the immersed-boundary, which is utilized to enforce no-slip boundary conditions via reconstruction of the solution in the vicinity of the body. Thanks to the dynamic overlapping grids, the distance function is computed only once in pre-processing, with no additional cost due to the motion of the immersed-boundary. Here the methodology is discussed in detail and test-cases are presented, featuring both stationary and moving bodies, also in relative motion. Results from present immersed-boundary computations are compared with body-fitted solutions by the same solver and with data from the literature as well.

Analysis of three-dimensional effects in laser driven thin-shell capsule implosions

Three-dimensional (3D) hydrodynamic numerical simulations of laser driven thin-shell gas-filled microballoons have been carried out using the computer code MULTI-3D [Ramis et al., Phys. Plasmas 21, 082710 (2014)]. The studied configuration corresponds to experiments carried at the ORION laser facility [Hopps et al., Plasma Phys. Controlled Fusion 57, 064002 (2015)]. The MULTI-3D code solves single-temperature hydrodynamics, electron heat transport, and 3D ray tracing with inverse bremsstrahlung absorption on unstructured Lagrangian grids. Special emphasis has been placed on the genuine 3D effects that are inaccessible to calculations using simplified 1D or 2D geometries. These include the consequences of (i) a finite number of laser beams (10 in the experimental campaign), (ii) intensity irregularities in the beam cross-sectional profiles, (iii) laser beam misalignments, and (iv) power imbalance between beams. The consequences of these imperfections have been quantified by post-processing the numerical results in terms of capsule nonuniformities (synthetic emission and absorption images) and implosion efficiency (convergence ratio and neutron yield). Statistical analysis of these outcomes allows determination of the laser tolerances that guarantee a given level of target performance.

Evaluation of force term in lattice Boltzmann method with discrete external boundary force for flow over an immersed body

In this paper, lattice Boltzmann method is used for the simulation of two-dimensional flow over an immersed solid body using external boundary force. Fluid simulation is performed using the Eulerian grid for the entire computational domain, while the domain of solid body is discretized using a boundary-fitted Lagrangian grid, imposed on the uniform Eulerian grid. Fluid–solid interaction is modeled using external boundary force (EBF) technique. Two different methods of implementing boundary force are discussed including modification in (a) collision step of LBM and (b) both collision and computation of macroscopic velocity steps. Computations are performed for flow over a stationary circular object as well as particle sedimentation. Results of drag force and sedimentation velocity show good correspondence with previously published ones at Re =1.0 for both methods. No discrepancy in results was observed between two methods for higher Reynolds numbers. Results obtained for sedimentation of an elliptical object show that the type of implementation of boundary force has no significant effect on the predicted location and angular position of the elliptical object. Furthermore, it is observed that more accurate results are obtained by introducing a modification in EBF for which solutions were independent of the number of nodes on the surface boundary.

A 3D parallel particle-in-cell solver for extreme wave interaction with floating bodies

Floating structures are widely used for vessels, offshore platforms, and recently considered for deep water floating offshore wind system and wave energy devices. However, modelling complex wave interactions with floating structures, particularly under extreme conditions, remains an important challenge. Following the three-dimensional (3D) parallel particle-in-cell (PIC) model developed for simulating wave interaction with fixed bodies, this paper further extends the methodology and develops a new 3D parallel PIC model for applications to floating bodies. The PIC model uses both Lagrangian particles and Eulerian grid to solve the incompressible Navier-Stokes equations, attempting to combine both the Lagrangian flexibility for handling large free-surface deformations and Eulerian efficiency in terms of CPU cost. The wave-structure interaction is resolved via inclusion of a Cartesian cut cell method based two-way strong fluid-solid coupling algorithm that is both stable and efficient. The numerical model is validated against 3D experiments of focused wave interaction with a floating moored buoy. Good agreement between the numerical and experimental results has been achieved for the motion of the buoy and the mooring force. Additionally, the PIC model achieves a CPU efficiency of the same magnitude as that of the state-of-the-art OpenFOAM® model for an extreme wave-structure interaction scenario.

Eulerian–Lagrangian flow-vegetation interaction model using immersed boundary method and OpenFOAM

This paper presents a novel coupled flow-vegetation interaction model capable to resolve the flow and motion of flexible vegetation with large deflections simultaneously on a hybrid Eulerian–Lagrangian grid. The hydrodynamics model is based on a Navier–Stokes flow solver with the Volume of Fluid surface capturing method in OpenFOAM. The deforming and moving vegetation are tracked through a series of Lagrangian grids attached to the vegetation and embedded in the computational domain of OpenFOAM. The governing equations for the flexible vegetation motion are based on a slender rod theory and are solved by a Finite Element Method on a vegetation-following Lagrangian grid. The hydrodynamics and vegetation models are coupled through the vegetation-induced hydrodynamic forces using a novel diffused immersed boundary method to avoid excessive fluid grid refinements around the vegetation. The standard two-equation k − ε turbulence model is extended for vegetated flow by incorporating additional closure terms of vegetation effects. The newly developed coupled flow-vegetation model is validated against experiments for both individual single-stem vegetation and a vegetation patch in a large-scale wave flume. Model results of asymmetric displacement of the vegetation motion, wave height decay, and wave kinematics within and outside the vegetation patch are in good agreement with measurements. The drastical change in wave radiation stress by the presence of vegetation patch leads to a strong current jet near the top of vegetation patch, which in turn drives a local circulation pattern around the vegetation patch. The effect of vegetation bending stiffness on the model results and empirical drag and inertia coefficients for calculating the hydrodynamic forces are examined in detail. Maximum turbulence energy is observed close to the bed and the top of vegetation patch just before and after the wave crest/trough, where the relative flow velocity to vegetation is large.

Simulation of 2D MHD Processes Using Lagrangian Unstructured Grids by the Example of Simulating Operation of Explosive Opening Switches

Simulation of 2D MHD Processes Using Lagrangian Unstructured Grids by the Example of Simulating Operation of Explosive Opening Switches

An Eulerian–Lagrangian–Lagrangian method for solving thin moving rigid body immersed in the fluid

It is convention in computational mechanics to use Eulerian grids for fluids and Lagrangian grids for solids. In this paper, a novel method called ELL using both Lagrangian and Eulerian grids for fluids is presented in a carefully designed manner to handle complex fluid–structure interaction (FSI) problems with moving thin rigid body. The key idea is to use a small portion of the fluid to wrap the thin body. This wrapping fluid is described in Lagrangian coordinate, the same way the solid is described. They are treated as one “composite solid” thus the immersed domain method (IDM) can be implemented. In this way, the moving interface boundary conditions can be explicitly imposed with the help of the unified Lagrangian grids in the interaction zone. The initial fluid domain can still be treated as usual Eulerian grids based on IDM theory by introducing a fictitious fluid. The FSI interface can be accurately modeled since the wrapping fluid and the solid are always attached together. This effectively solves the major “blur interface” problem in the conventional immersed boundary method (IBM). To demonstrate this ELL idea, a number of benchmark examples have been studied in comparison with the results in the open literature. Finally, a 2D flapping wing is simulated to show the potential of ELL in engineering application. It is observed that ELL method proves to be good in performance.

Computational modeling of multiphase viscoelastic and elastoviscoplastic flows

SummaryIn this paper, a three‐dimensional numerical solver is developed for suspensions of rigid and soft particles and droplets in viscoelastic and elastoviscoplastic (EVP) fluids. The presented algorithm is designed to allow for the first time three‐dimensional simulations of inertial and turbulent EVP fluids with a large number particles and droplets. This is achieved by combining fast and highly scalable methods such as an FFT‐based pressure solver, with the evolution equation for non‐Newtonian (including EVP) stresses. In this flexible computational framework, the fluid can be modeled by either Oldroyd‐B, neo‐Hookean, FENE‐P, or Saramito EVP models, and the additional equations for the non‐Newtonian stresses are fully coupled with the flow. The rigid particles are discretized on a moving Lagrangian grid, whereas the flow equations are solved on a fixed Eulerian grid. The solid particles are represented by an immersed boundary method with a computationally efficient direct forcing method, allowing simulations of a large numbers of particles. The immersed boundary force is computed at the particle surface and then included in the momentum equations as a body force. The droplets and soft particles on the other hand are simulated in a fully Eulerian framework, the former with a level‐set method to capture the moving interface and the latter with an indicator function. The solver is first validated for various benchmark single‐phase and two‐phase EVP flow problems through comparison with data from the literature. Finally, we present new results on the dynamics of a buoyancy‐driven drop in an EVP fluid.

Enhancement of a 2D front-tracking algorithm with a non-uniform distribution of Lagrangian markers

The 2D front tracking method is enhanced to control the development of spurious velocities for non-uniform distributions of markers. The hybrid formulation of Shin et al. (2005) [7] is considered. A new tangent calculation is proposed for the calculation of the tension force at markers. A new reconstruction method is also proposed to manage non-uniform distributions of markers. We show that for both the static and the translating spherical drop test case the spurious currents are reduced to the machine precision. We also show that the ratio of the Lagrangian grid size Δs over the Eulerian grid size Δx has to satisfy Δs/Δx>0.2 for ensuring such low level of spurious velocity. The method is found to provide very good agreement with benchmark test cases from the literature.

Load Balancing for Parallel Multiphase Flow Simulation

This paper presents a scalable dynamic load balancing scheme for a parallel front-tracking method based multiphase flow simulation. In this simulation employing both Lagrangian and Eulerian grids, processes operating on Lagrangian grid are susceptible to load imbalance due to moving Lagrangian grid points (bubbles) and load distribution based on spatial location of bubbles. To load balance these processes, we distribute load keeping in view both current processor load distribution and bubble spatial locality and remap interprocess communication. The result is a uniform processor load distribution and predictable and less expensive communication scheme. Scalability studies on the Hazel Hen supercomputer demonstrate excellent scaling with exponential savings in execution time as the problem size becomes increasingly large. While moderate speedup is observed for strong scaling, speedup of up to 30% is achieved over nonload-balanced version when simulating 13824 bubbles on 4096 cores for weak scaling studies.

A conservative MHD scheme on unstructured Lagrangian grids for Z-pinch hydrodynamic simulations

A new algorithm to model resistive magnetohydrodynamics (MHD) in Z-pinches has been developed. Two-dimensional axisymmetric geometry with azimuthal magnetic field Bθ is considered. Discretization is carried out using unstructured meshes made up of arbitrarily connected polygons. The algorithm is fully conservative for mass, momentum, and energy. Matter energy and magnetic energy are managed separately. The diffusion of magnetic field is solved using a derivative of the Symmetric-Semi-Implicit scheme, Livne et al. (1985) [23], where unconditional stability is obtained without needing to solve large sparse systems of equations. This MHD package has been integrated into the radiation-hydrodynamics code MULTI-2D, Ramis et al. (2009) [20], that includes hydrodynamics, laser energy deposition, heat conduction, and radiation transport. This setup allows to simulate Z-pinch configurations relevant for Inertial Confinement Fusion.

An adaptive reconstruction for Lagrangian, direct-forcing, immersed-boundary methods

Lagrangian, direct-forcing, immersed boundary (IB) methods have been receiving increased attention due to their robustness in complex fluid-structure interaction problems. They are very sensitive, however, on the selection of the Lagrangian grid, which is typically used to define a solid or flexible body immersed in a fluid flow. In the present work we propose a cost-efficient solution to this problem without compromising accuracy. Central to our approach is the use of isoparametric mapping to bridge the relative resolution requirements of Lagrangian IB, and Eulerian grids. With this approach, the density of surface Lagrangian markers, which is essential to properly enforce boundary conditions, is adapted dynamically based on the characteristics of the underlying Eulerian grid. The markers are not stored and the Lagrangian data-structure is not modified. The proposed scheme is implemented in the framework of a moving least squares reconstruction formulation, but it can be adapted to any Lagrangian, direct-forcing formulation. The accuracy and robustness of the approach is demonstrated in a variety of test cases of increasing complexity.