Mixed Eulerian-Lagrangian Scheme Research Articles

Non-linear finite element analysis of seismically excited sloped wall tank

The non-linear seismic behavior of a ground-supported sloped wall tank is analyzed in the present study using a finite element model. A combination of three-node triangular and four-node quadrilateral finite elements is used to discretize the liquid domain. The velocity potential formulation is adopted for the liquid domain with a mixed Eulerian-Lagrangian scheme. By comparing the current numerical results with the existing experimental results, the competency of the present non-linear model is verified. The fundamental sloshing frequencies obtained from the present non-linear model are found closer to the experimental results compared to the linear counterpart. Non-linear seismic characteristics of liquid sloshing inside a ground-supported sloped wall tank are investigated under six different earthquakes categorized as low-, intermediate-, and high-frequency contents. Also, the dynamic impulsive and convective response components of hydrodynamic base shear force and pressure are quantified successfully. The comparison of the non-linear hydrodynamic response with the linear counterpart shows a significant difference in results which reasonably clarifies the limited capabilities of the linear model for response prediction as well as designing sloped wall tanks subjected to seismic excitation. The developed non-linear numerical model in the present study can be utilized for tuning and designing structure-coupled sloped wall TLDs.

Time domain simulations of piston-like resonant flow in the gap of an oscillating twin-hull using inviscid and viscous flow solvers

A numerical study of a semi-circular twin-hull section under heave oscillation is presented. Two different time domain simulation methods were used: Firstly, a boundary element method based on potential theory which incorporates fully nonlinear free surface boundary conditions by using a mixed Eulerian Lagrangian scheme. Secondly, a finite volume method in combination with a volume of fluid scheme for capturing the free surface. In order to evaluate the effects of viscosity, simulations with the finite volume method were carried out using inviscid as well as viscous flow assumptions. Hydrodynamic mass and damping coefficients were derived from first order Fourier coefficients and validated against results from linear theory and experiments. A detailed comparison of nonlinear results from both simulation methods was carried out in vicinity of the piston-mode resonance frequency and a very good agreement was found. The phase angles corresponding to the Fourier coefficients varied strongly with frequency, which went along with a phase shift of the fluid motion in the gap. Influence of viscosity on the flow was found to be present but had relatively low impact on the hydrodynamic forces.

A Mixed Eulerian–Lagrangian scheme for scalar transport

The tracking of pollutants in gas and liquid is a major problem to address in atmospheric environment, air quality characterization, and industrial material processes. A Lagrangian scheme devoted to the approximation of the advection term in an advection–diffusion equation is proposed to deal with small diffusivity values or large Peclet numbers. The Lagrangian scheme is used in practice as an Eulerian method discretized on Lagrangian marker points. Advection and diffusion of a circular concentration spot in a vortex flow are considered for validation purpose. The resulting mixed Eulerian–Lagrangian scheme reduces the numerical diffusion to almost computer error and provides better results than other Eulerian classical schemes of the literature. The scheme is finally illustrated in a natural convection situation.

A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global $$L^2$$ projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.

Application of ALE to nonlinear wave radiation by a non-wall-sided structure

Water waves generated by the forced oscillation of a 3D non-wall-sided structure are analyzed based on the fully nonlinear potential-flow theory. In order to track an exact position of the free surface, Arbitrary Lagrangian-Eulerian (ALE) scheme is used in the computation. The feature of ALE scheme is that by introducing a prescribed path (curve) for each fluid marker on the free surface, the movement of fluid marker representing the deformation of free surface is confined along a path. With properly-designed configuration of those paths, this scheme brings about several advantages compared with the Mixed Eulerian-Lagrangian scheme and the Semi-Lagrangian scheme which are extensively used. In the computation, a higher-order boundary element method (HOBEM) and the 4th-order Runge-Kutta method are adopted as the solver of an initial boundary value problem (BVP). In the validation, the radiated waves generated by a truncated circular cylinder with flare oscillating in heave or surge are studied. Regarding the calculation of hydrodynamic forces, the temporal derivative of the velocity potential (ϕt) is evaluated in an exact but simple manner, i.e. ϕt is obtained by solving a reconstructed BVP without evaluating second-order derivatives of the velocity potential. Besides hydrodynamic forces, wave profiles at specific points and wave run-ups are also computed. Comparisons of the present results against published results of corresponding computations are made. To evaluate capability of the proposed ALE scheme, bodies with curved large flare are used in the computation.

An added-mass partition algorithm for fluid–structure interactions of compressible fluids and nonlinear solids

We describe an added-mass partitioned (AMP) algorithm for solving fluid–structure interaction (FSI) problems involving inviscid compressible fluids interacting with nonlinear solids that undergo large rotations and displacements. The computational approach is a mixed Eulerian–Lagrangian scheme that makes use of deforming composite grids (DCG) to treat large changes in the geometry in an accurate, flexible, and robust manner. The current work extends the AMP algorithm developed in Banks et al. [1] for linearly elasticity to the case of nonlinear solids. To ensure stability for the case of light solids, the new AMP algorithm embeds an approximate solution of a nonlinear fluid–solid Riemann (FSR) problem into the interface treatment. The solution to the FSR problem is derived and shown to be of a similar form to that derived for linear solids: the state on the interface being fundamentally an impedance-weighted average of the fluid and solid states. Numerical simulations demonstrate that the AMP algorithm is stable even for light solids when added-mass effects are large. The accuracy and stability of the AMP scheme is verified by comparison to an exact solution using the method of analytical solutions and to a semi-analytical solution that is obtained for a rotating solid disk immersed in a fluid. The scheme is applied to the simulation of a planar shock impacting a light elliptical-shaped solid, and comparisons are made between solutions of the FSI problem for a neo-Hookean solid, a linearly elastic solid, and a rigid solid. The ability of the approach to handle large deformations is demonstrated for a problem of a high-speed flow past a light, thin, and flexible solid beam.

Time-domain simulation of large-amplitude wave–structure interactions by a 3D numerical tank approach

A time-domain 3D Rankine panel method based on a simplified variant of the mixed Eulerian–Lagrangian scheme under certain approximations is developed for studying steep nonlinear waves interacting with practical ship and offshore configurations at zero speed. Appropriate techniques have been developed that enable the method to produce very long-duration simulation results. Two levels of time-domain computations are performed: (1) a fully linear formulation where all external forces are computed on the mean wetted surface, and (2) an approximate nonlinear computation where the hydrodynamics interaction forces (diffraction and radiation forces) are determined on the mean surface and the forces arising from the incident steep waves and hydrostatic restoring forces are determined based upon the exact wetted surface under the nonlinear incident wave. Numerical computations for three practical marine structures, the barge, the S175 hull, and the semisubmersible are presented. The linear computations for which very long-duration simulations are achievable from the present method are validated against results from other available methods. As the method is developed for stationary floating bodies undergoing oscillation about their mean location, it cannot be applied for a fully unrestrained body which can freely drift. In absence of physical restraints, the approximate nonlinear calculation requires imposition of artificial constraints partially or fully restraining the horizontal motions. Very long-duration simulations under the influence of steep nonlinear large-amplitude waves for all the three structures considered could be achieved. Comparative studies between different force and motion components in large-amplitude waves from linear and the approximate nonlinear computations are made to bring out the influence of the incident wave nonlinearities on these structures. It is found that the nonlinearities of the forces and motions are strongly dependent on the above water hull geometry. Compared to a small water-plane area hull (the semisubmersible), or a wall-sided hull (the barge), a flared hull (S175) results in pronounced nonlinear features in the forces and motion time-histories.

Wave resonances in a narrow gap between two barges using fully nonlinear numerical simulation

The traditional potential flow theory to describe fully nonlinear waves is reformulated by separating the contributions from incident and scattered waves, in order to improve the computational efficiency. The nonlinear incoming wave is specified explicitly and the modified nonlinear free surface boundary conditions for the scattered wave are expressed in the full Lagrangian description. At each time step only the scattered wave is solved using a mixed Eulerian–Lagrangian scheme by a higher-order boundary element method. The accuracy of the newly developed model is illustrated by comparisons with existing experimental and numerical data in the case of wave diffraction around an array of circular cylinders. Wave resonances in the gap between two side-by-side barges in beam seas, as in Molin et al. [1], are simulated with the barges subjected to regular waves. To clearly understand the gap resonant responses, long time simulations are performed to achieve final steady states, and the resonant mode shapes of the gap surface are presented. The gap free surface RAOs (Response Amplitude Operators) in the case of mild waves are found to agree well with linear calculations. The nonlinear effects on the resonant response due to the free surface conditions are then investigated. The first resonant frequency is found to shift but the peak value is not changed much with increasing incoming wave steepness, which is known as stiff/soft spring behavior of a nonlinear system. Through the investigation of barges with different drafts, the stiff and soft spring behaviors are identified.

NUMERICAL AND EXPERIMENTAL STUDY ON WAVE-CURRENT INTERACTIONS OVER A SUBMERGED BAR

Wave-current interactions are simulated by a fully-nonlinear numerical wave tank (NWT). The model is developed based on the time-domain higher-order boundary element method (HOBEM). The mixed Eulerian-Lagrangian scheme is adopted to track the transient free surface with the fourth-order Runga-Kutta method for refreshing wave profile and velocity potential at the next time step. The meshes are regridded at each time step to avoid numerical instabilities caused by mesh motions. An image Green function is utilized to increase the speed of computation. Using the developed NWT, nonlinear wave-current interactions (1) without body; (2) with a submerged vertical bar for various wave and current conditions have been simulated. The physical experiment on evolution of waves in uniform current is carried out in a wave flume at Dalian University of Technology. The flume is 69.0m long, 3.0m wide and 1.8m deep. The experimental setup is shown in Fig.1. The flume is equipped with a hydraulically driven, irregular wave generator at one end, and a wave absorber at the other. The still-water depth (h) used in this study was 0.6m, and then a vertical submerged bar was installed along the bottom of flume. It was 0.4m height and 0.5m width. In order to record the wave transformation and propagation clearly, five wave gauges were installed. The incident wave amplitude is varied from 0.025m to 0.054m, and incident wave periods are 1s and 2s, respectively. Currents are introduced by a bidirectional centrifugal pump. Following currents flow in the same direction as wave propagation, and its velocity was set for positive, and the opposite situation represented opposing currents. An acoustic Doppler velocimeter (ADV) was used to measure the current velocity and the mean velocity U0=0.2m/s was chosen. The numerical results and experimental data are compared well with each other. The effects of currents and submerged bar on wave profile, higher-harmonic waves and reflection coefficient are further studied, respectively.

FINITE ELEMENT ANALYSIS OF NON-LINEAR FLUID STRUCTURE INTERACTION IN HYDRODYNAMICS USING MIXED LAGRANGIAN-EULERIAN METHOD

Fluid structure Interaction (FSI) is important in analysis of: (1) Cardio-vascular dynamics; (2) Underwater/ Offshore structures; (3) Aircraft wings and turbine blade designs; (4) Design of tall structures/buildings; (5) high speed hard disk drives etc. At present, such capabilities are being incorporated, if any, only to a limited extent in commercial CFD codes leaving the engineers to develop their own codes. In the current study, a hydro-elastic problem of an underwater structure has been considered. Traditionally, the hydrodynamics and the structural dynamics are solved using finite difference/boundary element methods and finite element method (FEM) respectively. Moreover, the governing equation for fluid and structure are usually written in Eulerian and Lagrangian reference frames - posing further difficulties for coupling the two systems. In this research, both the fluid and structural systems are solved by the finite element method using a mixed Eulerian-Lagrangian scheme, where, fluid mesh moves and adapts to new free surface and structural positions. A full non-linear free surface implementation is considered. A mesh adaptation using Laplacian smoothing is performed to reduce the need for re-meshing the domain frequently. The scheme is validated with solutions available in the literature and extended to the present FSI problem with non-linear free surface boundary condition.

Numerical simulation of the starting flow down a step

The starting flow down a step in an open channel is simulated using a vortex method. A mixed Eulerian-Lagrangian scheme based on the vortex-in-cell algorithm and random walk diffusion is used with a non-uniform mesh. The comparison of numerical and experimental instantaneaous streamline plots show good agreement, especially for early stages of the flow development. It is also shown that, during intermediate stages of the flow development and for the range of Reynolds numbers investigated, the distance from the step to the reattachment point of the main vorticity domain increases linearly with the dimensionless time. Again, good agreement is obtained between experimental data and computed solutions.

Numerical Studies of Separated Flow from Bodies with Sharp Corners by the Vortex in Cell Method

The Navier-Stokes equations are solved around arbitrary cross-sections in the laminar flow regime. A mixed Eulerian-Lagrangian scheme (Vortex-In-Cell Method) is used in combination with the Operator Splitting Technique. The main feature of this numerical model is the generation of discrete vortices all around the body surface; this is achieved in the frame of potential theory in combination with a conformal transformation of the body contour. Problems occur at sharp corners, since infinite velocity is predicted from the ideal fluid calculations. The treatment of the singular behavior will consist of a truncation of the mapping derivative at the corners. It is shown by applications for a square and a diamond that this artificial cutting-off does not strongly affect the mass coefficient and the drag force due to pressure. The drag force due to skin friction is more affected. An additional treatment is necessary in the immediate vicinity of the corners of a flat plate in a cross flow. This will consist of "slowing down" the vortices to avoid unrealistic motions due to convection.

Numerical simulations of nonlinear axisymmetric flows with a free surface

A numerical method is developed for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-Lagrangian scheme under the assumption of potential flow. Taking advantage of axisymmetry, Rankine ring sources are used in a Green's theorem boundary-integral formulation to solve the field equation; and the free surface is then updated in time following Lagrangian points. A special treatment of the free surface and body intersection points is generalized to this case which avoids the difficulties associated with the singularity there. To allow for long-time simulations, the nonlinear computational domain is matched to a transient linear wavefield outside. When the matching boundary is placed at a suitable distance (depending on wave amplitude), numerical simulations can, in principle, be continued indefinitely in time. Based on a simple stability argument, a regriding algorithm similar to that of Fink & Soh (1974) for vortex sheets is generalized to free-surface flows, which removes the instabilities experienced by earlier investigators and eliminates the need for artificial smoothing. The resulting scheme is very robust and stable.For illustration, three computational examples are presented: (i) the growth and collapse of a vapour cavity near the free surface; (ii) the heaving of a floating vertical cylinder starting from rest; and (iii) the heaving of an inverted vertical cone. For the cavity problem, there is excellent agreement with available experiments. For the wave-body interaction calculations, we are able to obtain and analyse steady-state (limit-cycle) results for the force and flow field in the vicinity of the body.