Mixed Eulerian-Lagrangian Method Research Articles

Wave Propagation in a Three-Dimensional, Fully-Nonlinear Numerical Wave Tank Based on Multi-transmitting Formula Coupled Damping Zone Method

Abstract The objective of this work is to develop a novel numerical model solving the coupled internal sloshing and external sea-keeping interaction for small/medium liquefied natural gas (LNG) carrier. Currently, the work focuses on simulating wave propagation in a three-dimensional (3D) fully-nonlinear numerical wave tank (NWT) to model the sea. When simulating the nonlinear wave propagation through an unbounded domain in the time domain, it is necessary to truncate the computational domain artificially into a finite domain in order to reduce computational costs. Thus, non-reflecting radiation boundary condition is required for the truncated surface, however, there is no exact non-reflecting condition in existence. In this work, wave propagation in a 3D fully nonlinear NWT is studied based on the fully-nonlinear velocity potential theory. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing algorithm, B-spline, is applied to eliminate the possible saw-tooth instabilities. An effective multi-transmitting formula coupled damping zone (MTF+DZ) radiation condition is employed to minimize wave reflection on the truncated surface. The numerical results are compared with analytical solutions. It is shown that MTF+DZ method can be used for simulating fully-nonlinear, irregular wave propagation.

Modeling of fully nonlinear wave evolution over a submerged bar

A three-dimensional fully nonlinear numerical wave tank (NWT) is developed by using a time-domain higher-order boundary element method (HOBEM). The source generation of waves is adopted to generate input waves. The mixed Eulerian-Lagrangian (MEL) method is utilized to track and refresh the free surface. The model is applied to three examples of wave evolution and harmonic decomposition during propogation over a submerged bar and numerical results are compared with the experimental data and the solutions of Boussinesq model. It shows that the present model agrees better with the measurements than the Boussinesq model and are more adaptable for stronger nonlinear waves.

Nonlinear analysis on wave-plate interaction due to disturbed vertical elastic plate

Hydroelastic behavior of wave-plate interaction due to an initial deflection of a vertical elastic plate is simulated by using a Mixed Eulerian Lagrangian method for the fluid and a Finite Element Method for the plate. An implicit boundary-condition method is developed to solve the coupled motions of fluid and plate. A hybrid wave absorbing beach is installed to prevent the wave reflection from a vertical wall at the end of the wave tank. The 4th order Runge-Kutta scheme with uniform time step is used for time marching. First, numerical results of the elastic vibration of a vertical plate induced by deformation of itself at initial time are validated by comparing with corresponding analytical solution. Then, further comparison between the results from mode-expansion method and FEM is made. Finally, the influences of the edge condition at the top of plate and the plate stiffness on the wave-plate interaction are investigated systematically.

Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave

Micro-cavitation bubbles generated by ultrasound have wide and important applications in medical ultrasonics and sonochemistry. An approximate theory is developed for nonlinear and non-spherical bubbles in a compressible liquid by using the method of matched asymptotic expansions. The perturbation is performed to the second order in terms of a small parameter, the bubble-wall Mach number. The inner flow near the bubble can be approximated as incompressible at the first and second orders, leading to the use of Laplace's equation, whereas the outer flow far away from the bubble can be described by the linear wave equation, also for the first and second orders. Matching between the two expansions provides the model for the non-spherical bubble behaviour in a compressible fluid. A numerical model using the mixed Eulerian–Lagrangian method and a modified boundary integral method is used to obtain the evolving bubble shapes. The primary advantage of this method is its computational efficiency over using the wave equation throughout the fluid domain. The numerical model is validated against the Keller–Herring equation for spherical bubbles in weakly compressible liquids with excellent agreement being obtained for the bubble radius evolution up to the fourth oscillation. Numerical analyses are further performed for non-spherical oscillating acoustic bubbles. Bubble evolution and jet formation are simulated. Outputs also include the bubble volume, bubble displacement, Kelvin impulse and liquid jet tip velocity. Bubble behaviour is studied in terms of the wave frequency and amplitude. Particular attention is paid to the conditions if/when the bubble jet is formed and when the bubble becomes multiply connected, often forming a toroidal bubble. When subjected to a weak acoustic wave, bubble jets may develop at the two poles of the bubble surface after several cycles of oscillations. A resonant phenomenon occurs when the wave frequency is equal to the natural oscillation frequency of the bubble. When subjected to a strong acoustic wave, a vigorous liquid jet develops along the direction of wave propagation in only a few cycles of the acoustic wave.

Numerical analysis of fluid flow through radial diffusers in the presence of a chamfer in the feeding orifice with a mixed Eulerian–Lagrangian method

This paper reports an experimentally validated numerical analysis of fluid flow through a radial diffuser comprising two concentric and parallel disks. The flow is supplied axially by a feeding orifice placed in the back disk and becomes radial after being deflected by the front disk. The main purpose of the study is to assess the effect of a chamfer at the exit of the feeding orifice. Due to the irregular flow geometry, a mixed Eulerian–Lagrangian method is employed to numerically solve the fluid flow. The model is validated by comparing the numerical results with experimental data for pressure distribution on the front disk, as a function of the gap between the disks and the Reynolds number. Results for pressure on the front disk and effective flow and force areas show that the flow is significantly affected by chamfer angles as small as 5°.

A mixed Eulerian–Lagrangian method for modelling metal extrusion processes

A mixed Eulerian–Lagrangian approach for the computational modelling of metal extrusion processes in complex three dimensional geometries is presented. The approach involves the representation of the workpiece as a pseudo-fluid, and requires the solution of non-Newtonian fluid flow equations in an Eulerian context, using a free-surface algorithm to track its extreme deformation during its extrusion. The solid mechanics equations associated with the tools are solved in a conventional Lagrangian context. Thermal interactions between the workpiece and tools are modelled and a fluid–structure interaction technique is employed to capture the effect of the fluid traction load imposed by the workpiece on the tools, and especially the subsequent adaption of the Eulerian mesh to account for the impact of die deformation. Two extrusion test cases are investigated and the results obtained show the potential of the model with regard to representing the physics of the process, the advantages of the model over a more loosely coupled approach, and the parallel scalability of the resulting software.

A numerical and experimental study of a collapsing bubble-induced droplet ejector

This paper aims to study a novel drop-on-demand droplet generation mechanism in which the oscillation and deformation of a non-equilibrium bubble in close proximity to a free surface induce an axisymmetric liquid spike on the free surface. The evolution of the liquid spike and its deformation due to the effect of surface tension force lead to the formation of a droplet. The free surface can be accorded by either a circular hole on a horizontal flat plate or by the top opening/nozzle of a vertical cylinder. A high-speed camera capable of obtaining images at a frame rate of 15,000 fps is utilized to observe the droplet formation process. Numerical simulations corresponding to the experiments are performed using the boundary integral spatial solution coupled with the time integration, i.e., a mixed Eulerian–Lagrangian method. In the experiments the bubble is generated using a very low voltage (only 55 V) in contrast to the relatively much higher voltages usually employed in reported works. This is very attractive from a safety viewpoint and accords great simplification of the setup. A comparison is made between the numerical and experimental results. A reasonable agreement has been found. The influences of the main design parameters, namely, the bubble-free surface distance and the dimension of the hole/nozzle on the bubble dynamics and on the droplet formation process are discussed and the conditions of the bubble dynamics under which a satellite-free droplet can be generated are sought. Furthermore, the effects of different geometries, namely, the horizontal flat plate and the vertical cylinder on the bubble dynamics and on the droplet features are examined. One important feature of the proposed actuation mechanism is the capability of producing droplets much smaller than the nozzle size. The possible applications of this mechanism are those where the accurate direction of the ejected droplet is of great importance such as inkjet printing.

A Eulerian-Lagrangian Method Applied to Fluid Flow in Lid-Driven Cavities With Irregular Bottom Walls

In this work, a mixed Eulerian-Lagrangian method is employed to explore fluid flows in two-dimensional, lid-driven, irregular-bottom cavities. The methodology can handle fluid flows in the presence of irregularly shaped solid boundaries. A fixed Cartesian grid is used in the discretization of the momentum and mass equations, precluding the need to generate a grid to accommodate the non-Cartesian walls. Special treatment is required to deal with the control volumes located at the fluid–solid interface, an approach that is described herein. The discretization of the governing equations uses the finite-volume method with a collocated, i.e., cell-centered arrangement of velocity and pressure. The procedure is verified by solving two-dimensional lid-driven cavity flows for different Reynolds numbers in standard cavity shapes such as semicircular, rectangular, and square, with or without an irregular bottom wall. Results for the velocity components along the geometric centerline, stream function patterns, and pressure contours are presented and discussed. Excellent congruence with benchmark solutions is obtained.

Time domain three-dimensional fully nonlinear computations of steady body–wave interaction problem

Three-dimensional fully nonlinear waves generated by moving disturbances with steady forward speed without motions are solved using a mixed Eulerian–Lagrangian method in terms of an indirect boundary integral method and a Runge–Kutta time marching approach which integrates the fully nonlinear free surface boundary conditions with respect to time. A moving computational window is used in the computations by truncating the fluid domain (the free surface) into a computational domain. The computational window maintains the computational domain and tracks the free surface profile by a node-shifting scheme applied within it. An implicit implement of far field condition is enforced automatically at the truncation boundary of the computational window. Numerical computations are applied to free surface waves generated by Wigley and Series 60 hulls for the steady problem. The present numerical results are presented and compared with existing linear theory, experimental measurements, and other numerical nonlinear computations. The comparisons show satisfactory agreements for these hydrodynamic problems.

FNPF Analysis of Stochastic Experimental Fluid-Structure Interaction Systems

A two-dimensional fully nonlinear potential flow model is employed to investigate nonlinear stochastic responses of an experimental fluid-structure interaction system that includes both single-degree-of-freedom surge-only and two-degree-of-freedom surge-heave coupled motions. Sources of nonlinearity include free surface boundary, fluid-structure interaction, and large geometry in the structural restoring force. Random waves performed in the tests include nearly periodic, periodic with band-limited noise, and narrow band. The structural responses observed can be categorized as nearly deterministic (harmonic, sub- and super-harmonic), noisy periodic, and random. Transition phenomena between coexisting response attractors are also identified. An implicit boundary condition upholding the instantaneous equilibrium between the fluid and structure using a mixed Eulerian-Lagrangian method is employed. Numerical model predictions are calibrated and validated via the experimental results under the three types of wave conditions. Extensive simulations are conducted to identify the response characteristics and the effects of random perturbations on nonlinear responses near primary and secondary resonances.

Numerical simulation of nonlinear wave and force generated by a wedge-shape wave maker

Nonlinear waves and forces induced by a wedge-shape wave maker were simulated in a potential-theory-based fully nonlinear 2D Numerical Wave Tank (NWT). The NWT is developed in a time domain by using Boundary Element Method (BEM) including Mixed Eulerian–Lagrangian method (MEL) and Runge–Kutta 4th-order (RK4) integration as a time marching process. For ensuring accurate nonlinear free surface both material-node and semi-Lagrangian approach are independently developed for crosschecking. The acceleration-potential scheme is used for obtaining accurate time derivative of velocity potential. The developed NWT is utilized to calculate water particle velocity and a series of higher-harmonic force components on the wave maker. The added-mass and radiation-damping coefficients of the wave maker are also obtained from the least-square method. The simulation results are compared with the experimental and numerical results of other researchers. To compare the relative importance of free-surface and body-surface nonlinearities, a body nonlinear formulation is independently developed. Force by body nonlinear method is in good agreement with fully nonlinear result in case of low body-stroke frequency.

A comparison of modelling methods for polydispersed wet‐steam flow

AbstractSpontaneous nucleation of water droplets in moist air or steam may result in droplet spectra which are complex in shape and which span a broad range of sizes. This is particularly true if the flow is transonic or supersonic with shock waves present, or if an already droplet‐laden flow re‐expands to give secondary or tertiary nucleations. Computation of such flows requires careful modelling of the size distributions if two‐phase behaviour is to be accurately predicted. In this paper, three methods are presented for treating size distributions and growth of the liquid phase in condensing steam: a mixed Eulerian–Lagrangian method, a fully Eulerian method, and a method based on moments of the droplet spectra. These are compared by computing condensing flow within a one‐dimensional supersonic nozzle under conditions that yield very different types of size spectra. Copyright © 2003 John Wiley & Sons, Ltd.

On the numerical design of a new type of 4 K GM/PT hybrid refrigerators

In this paper we developed and designed, based on theoretical considerations, a new type of 4 K GM/PT hybrid refrigerators. The upper warm stage of the hybrid refrigerator is a typical GM refrigeration cycle, and the cold stage is a pulse tube refrigerator (PTR), on which is thermodynamically coupled the upper warm stage. Four different types of phase shifting assembly: (1) a cold auxiliary piston that is connected to the displacer of the upper GM refrigerator stage, (2) an orifice with reservoir encircled the cold head of the upper stage, (3) an orifice with double-inlet, and (4) in combination with the cold auxiliary piston, orifice and double-inlet, has been proposed and analyzed for the adjustment of the phase shift between the gas mass flow and pressure in the pulse tube. Numerical simulation is performed to understand the unique thermo-physical features, to reveal the time-dependent dynamic parameters and to quantify the overall cooling performance of the hybrid refrigerator. We rely on a one-dimensional, unsteady compressible flow numerical model that is based on a mixed Eulerian–Lagrangian method developed by the present author. The model will be first applied to analyze the cooling performance of the hybrid refrigerator with different types of phase shifting assembly. In what following, it is used to simulate the dynamic parameters in the cold stage of the pulse tube cycle. Next, optimization of the structure parameters and geometrical configuration of the new refrigerator will be presented. Finally, the influences of different hybrid regenerative materials on the cooling capacity of the new hybrid GM/PTR will be also discussed.

Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. Part 2. Nonlinear waves and forces on a body

The mixed-Eulerian–Lagrangian method using high-order boundary elements, described in Xue et al. (2001) for the simulation of fully nonlinear three-dimensional wave–wave and wave–body interactions, is here extended and applied to the study of two nonlinear three-dimensional wave–body problems: (a) the development of bow waves on an advancing ship; and (b) the steep wave diffraction and nonlinear high-harmonic loads on a surface-piercing vertical cylinder. For (a), we obtain convergent steady-state bow wave profiles for a flared wedge, and the Wigley and Series 60 hulls. We compare our predictions with experimental measurements and find good agreement. It is shown that upstream influence, typically not accounted for in quasi-two-dimensional theory, plays an important role in bow wave prediction even for fine bows. For (b), the primary interest is in the higher-harmonic ‘ringing’ excitations observed and quantified in experiments. From simulations, we obtain fully nonlinear steady-state force histories on the cylinder in incident Stokes waves. Fourier analysis of such histories provides accurate predictions of harmonic loads for which excellent comparisons to experiments are obtained even at third order. This confirms that ‘ringing’ excitations are directly a result of nonlinear wave diffraction.

Computational study of a 4 K two-stage pulse tube cooler with mixed Eulerian–Lagrangian method

A new mixed Eulerian–Lagrangian numerical model for simulating and visualizing the internal processes and the variations of dynamic parameters of a two-stage pulse tube cooler (PTC) operating at 4 K temperature region has been developed. We use the Lagrangian method, a moving grid, to follow the exact tracks of gas particles as they move with pressure oscillation in the pulse tube to avoid any numerical false diffusion. The Eulerian approach, a fixed computational grid, is used to simulate the variations of dynamic parameters in the regenerator. A variety of physical factors, such as real thermal properties of helium, multi-layered magnetic regenerative materials, pressure drop and heat transfer in the regenerator, and heat exchangers, are taken into account in this model. The time-variations of gas temperature, pressure, mass flow rate, enthalpy flow in a cycle, in first- and second-stage regenerators are presented in the paper. More attention is paid to the effects of different regenerative materials on the performance of the 4 K two-stage PTC.

Non-linear simulations of wave-induced motions of a floating body by means of the mixed Eulerian-Lagrangian method

A non-linear calculation method based on the mixed Eulerian-Lagrangian (MEL) method is presented for wave-induced motions of a two-dimensional floating body. Attention is focused on an effective calculation of the hydrodynamic force associated with the temporal derivative of the velocity potential in the Bernoulli pressure equation. Unlike other existing methods, the acceleration field can be computed simultaneously with the velocity field, which contributes greatly to a reduction in the computation time. Computations are performed for a wall-sided model and a flared model, and numerical results of the waves at upwave and downwave positions and the body motions (sway, heave and roll) are compared with corresponding experiments. The overall agreement is very good, confirming the validity of the present method. The parametric oscillation in roll, observed for the flared model, is also discussed.

Computation of solitary waves during propagation and runup on a slope

A numerical time-simulation algorithm for analysing highly nonlinear solitary waves interacting with plane gentle and steep slopes is described by employing a mixed Eulerian–Lagrangian method. The full nonlinear free surface conditions are considered here in a Lagrangian frame of reference without any analytical approximations, and thus the method is valid for very steep waves including overturning. It is found that the runup height is crucially dependent on the wave steepness and the slope of the plane. Pressures and forces exerted on impermeable walls of different inclinations (slopes) by progressive shallow water solitary waves are studied. Strong nonlinear features in the form of pronounced double peaks are visible in the time history of pressure and force signals with increasing heights of the oncoming solitary waves. The effect of nonlinearity is less pronounced as the inclination of the wall decreases with respect to the bottom surface.

A Time-Domain Nonlinear Simulation Method for Wave-Induced Motions of a Floating Body

A nonlinear calculation method based on the Mixed Eulerian Lagrangian method is presented for wave-induced motions of a 2-D floating body. Attention is placed on an effective calculation of the hydrodynamic force associated with the temporal derivative of the velocity potential in Bernoulli's pressure equation. Unlike other existing methods, the acceleration field can be computed simultaneously with the velocity field, which contributes greatly to reduction of the computation time. By use of Green's second identity, the new method is explained as an extension from the mode decomposition method, and close relations between the two methods are emphasized. Computations are performed for a wall-sided model and a flared model, and numerical results of the waves at upwave and downwave positions and the body motions (sway, heave, and roll) are compared with corresponding experiments. The overall agreement is very good, confirming validity of the present method. Discussion is also made on the parametric oscillation in roll, observed for the flared model.

Full-Nonlinear Simulations of Hydrodynamic Forces on a Heaving Two-Dimensional Body

A full-nonlinear computation technique based on the Mixed Eulerian-Lagrangian method is presented and validated quantitatively for a fundamental 2-D radiation problem. The time marching is performed with the 4-th order Runge-Kutta-Gill algorithm, and at each time step the integral equations for the velocity and acceleration fields are solved by applying a higher-order boundary-element method with quadratic isoparametric elements. Stable simulations over a large number of periods are possible, because an efficient absorbing beach is used to avoid wave reflections from an outer boundary and regridding is performed to avoid the cluster of nodal points on the free surface.Computed results are compared with linear-theory results of the Green function method in the time and frequency domains, measurements of the time history of waves generated by a wavemaker, and Fourier-analyzed results of hydrodynamic forces acting on an ellipse and a wedge. All of these results are in excellent agreement, confirming the validity of the calculation method.

Dispersion of a continuously injected, nonlinearly adsorbing solute in chemically or physically heterogeneous porous formations

In this paper we consider transport of a nonlinearly adsorbing solute in a two-dimensional porous medium. The solute is continuously injected along a line source with a size equal to the transverse dimension of the domain. Adsorption is described by the Freundlich isotherm. The domain is assumed to be spatially variable and variation of hydraulic conductivity and adsorption coefficient is considered separately. Solute transport is solved by a mixed Eulerian-Lagrangian method. It is shown that nonlinear adsorption adds an extra requirement of taking into account pore scale dispersion while using Eulerian-Lagrangian solution methods. Solute dispersion is characterized in terms of spatial moments, where the pdf is the derivative of the transversely averaged concentration field. Results of numerical calculations are obtained for nonlinear and linear adsorption. It is shown that for large displacements an increase of pore scale dispersion leads to a decrease of average front spreading in case of nonlinear adsorption. The nonlinearity of adsorption opposes the local spreading in the longitudinal direction, whereas transverse spreading decreases the average front spreading. For larger displacements the front variance increases linearly with time. If transverse mixing is of the order of magnitude of the scale of heterogeneity, the front variance approaches a constant value, i.e a traveling wave becomes apparent.