Eulerian Grid Research Articles

Effects of underwater explosion depth on shock wave overpressure and energy

In this paper, the effects of the water depth on the overpressure (pressure difference between positive shock pressure and hydrostatic pressure) peak and energy of underwater explosion shock waves were analyzed. Two quantitative calculation models were established that accounted for the effect of the water depth, which have theoretical and practical engineering value. A simulated deepwater explosion tank test was first conducted to obtain experimental data of the overpressure peaks and energies of the explosion shock waves generated by 10 and 30 g trinitrotoluene (TNT) explosives in a simulated environment at water depths of 400, 500, and 600 m. A one-dimensional wedge-shaped Euler grid numerical model was established to simulate the underwater explosion using the Autodyn software. The simulation model was validated by the experimental data to prove its accuracy and rationality. Then, numerical simulations were carried out at 13 operating conditions with 30 g of TNT in a full water depth range of 0–5000 m. Based on the simulation data analysis, the calculation models of the overpressure peak and energy flow density of the underwater explosion shock wave were obtained, which contain water depth correction functions. The results show that both the overpressure peak and the shock wave energy decreased with increase in the water depth, but the reduction percentage of the overpressure peak with the water depth was very small. The overpressure peak and energy flow density of the shock wave agreed with the explosion similarity law at all fixed water depths. The proposed calculation models have practical engineering value and generalization ability.

Euler-Lagrange simulation of dense gas-solid flow with local grid refinement

Euler-Lagrange method is powerful for studying dense gas-solid flow, where the Eulerian grid is typically 3–5 times of particle diameter to ensure the accuracy of information mapping between gas and solid phases. This condition limits its applications in simulating reactors with complex geometries which require the use of Eulerian grids that are comparative to or smaller than particle diameter. In this study, eight methods for mapping discrete particle information to continuous fluid field were compared first via the simulation of a packed bed, their pros and cons were analyzed and discussed in detail; A kernel function method (the normalized kernel function method II, NKFMII) was then selected to simulate the hydrodynamics and heat transfer of gas-solid bubbling fluidized beds, the results not only validated NKFMII for interphase information mapping but also indicated that a grid-size-independent solution can be achieved; Finally, in order to balance the computational efficiency and accuracy a hybrid particle centroid method (PCM) and NKFMII method was proposed to achieve the interphase information mapping and then to simulate the hydrodynamics of a fluidized bed with immersed tubes, the ability of local grid refinement and the effectiveness of the coupled PCM and NKFMII method were demonstrated. This study proved that Euler-Lagrange simulation with local grid refinement is able to simulate gas-solid reactors with complex geometries.

Nonlinear simulation of mucociliary clearance: A three- dimensional study

In this study three-dimensional (3D) numerical simulation of mucociliary clearance from the lung has been studied. The major thrust of this investigation is 3D simulation of mucociliary clearance accounting for non-linear viscoelastic characteristics of the mucus layer which is one of the most complex fluids in our body. Using a numerical code, three-dimensional unsteady flow computations have been carried out to study ciliary motion in a two-layer fluid model of the (bronchial) airway surface liquid (ASL) consisting of a Newtonian lower periciliary layer (PCL) and a nonlinear viscoelastic upper mucus layer. The projection finite difference method is used to solve the governing and constitutive equations on a staggered Eulerian grid. The immersed boundary method (IBM) is also employed to study the effect of cilia propulsive force on the fluid flow. The 5-mode nonlinear Giesekus model has been used as the constitutive equation of mucus which is characterized by nonlinear (non-Newtonian viscosity) properties. Numerical results have been devoted to study the effect of various parameters such as cilia beat frequency, cilia length as well as PCL and mucus depth on mucociliary clearance. The present 3D model shows very good agreement with previous experimental studies, while ca. 2.5-fold larger clearance rates were found for 2D models. 3D results show that cilia beating frequency and cilia length are the most critical factors affecting mucociliary clearance, while PCL depth and mucus depth have little and intermediate effect on mucus flow and hence clearance rate respectively.

SPH numerical simulation of non-steady sand ripple wind-sand flow structure.

The Smoothed Particle Hydrodynamics (SPH) method is used to numerically simulate the unsteady sand ripple wind-sand flow movement in a small scale. Compared with the traditional grid-based numerical method, the SPH method uses discrete independent particles to represent the solution domain, get rid of the constraints of the computational grid, and completely solve various problems caused by large deformation or distortion of the grid based on the Euler grid method. Because it also includes the Lagrangian dynamics method, it can easily track the trajectory of any single particle, so the SPH method has unique advantages when dealing with the two-phase flow coupling of wind and sand. This paper expounds the theoretical basis and calculation method of SPH method and focuses on the design idea of applying the core factors in the SPH formula to the numerical simulation program. By discretizing the SPH numerical model of the sand ripple wind-sand two-phase flow, the temporal and spatial changes of the sand particle group on the slope surface during the wind-sand movement are simulated. The formation and development process of macro-wind-sand flow and the saltation law of micro-typical particles are analyzed, which are consistent with the existing wind tunnel experiment results, and the accuracy of SPH numerical simulation method is verified. The results show that the calculation accuracy of this method is high, and it provides a scientific theoretical basis for our further research and prevention of sand damage.

A statistical analysis of velocity and acceleration fluctuations of inertial particles in particle-laden turbulent Couette flow

Dynamics of the particle phase in a particle-laden turbulent flow is strongly influenced by the fluctuating velocity and vorticity field of the fluid phase. The present work mainly focuses on exploring the statistics of velocity and acceleration of the particle phase in a particle-laden turbulent Couette flow. Direct numerical simulations have been performed for particle-laden turbulent Couette flow with two different Reynolds numbers, Reδ=750 and 1300, in the presence of sub-Kolmogorov sized inertial particles for multiple Stokes numbers (Stokes number ≫1). The inter-particle and wall-particle interactions have also been considered to be elastic. We report the distribution functions for the linear and rotational velocities and accelerations in the presence of particle roughness. From the particle equation of rotational motion, we arrive at the expression where the fluctuating angular acceleration αi′ of the particle is expressed as the ratio of a linear combination of fluctuating rotational velocities of particle (ωi′) and fluid angular velocity (Ωi′) to the particle rotational relaxation time τr. The analysis is done using probability density function plots and Jensen–Shannon divergence-based method to assess the similarity between the particle net rotational acceleration distributions f(αi′), with (i) the distributions of particle acceleration component arises from fluctuating fluid angular velocity computed in the particle-Lagrangian frame f((Ω′i/τr)pl), (ii) fluctuating particle angular velocity f(ω′i/τr), and (iii) the fluid angular velocity f((Ω′i/τr)e) computed in the fluid Eulerian grids. The analysis leads to the conclusion that for a wide range of Reynolds and Stokes numbers, f(αi′) can be represented with a Gaussian white noise with a pre-estimated strength that can be calculated from the temporal decorrelation correlation of fluid-phase angular velocity fluctuations at Eulerian grid (Ω′i/τr)e.

Evaluating the impacts of temperature on a bubbling fluidized bed biomass gasification using CPFD simulation model

Biomass gasification and subsequently fuel and chemicals synthesis allow for a seamless transition from fossil raw materials to a renewable economy. A computational particle fluid dynamics (CPFD) simulation model is developed for biomass gasification in a bubbling fluidized bed reactor with steam as the fluidizing gas. CPFD methodology incorporates the multiphase particle-in-cell (MP-PIC) method and the particle parceling algorithm to solve the gas phase as a continuous fluid in a Eulerian grid of cells where particles are modeled as discrete Lagrangian points. The model has been validated by experimental results and shows good agreement. Reactor temperature effects as a key process operating parameters have been investigated and the results of the product gas composition on the temperature of 950, 1000, 1050, 1100, 1150, and 1200 K are shown. The results illustrate that at a higher temperature, the product of CO and H2 is promoted whereas the generation of CO2 and CH4 is diminished.

Full‐Scale Large Eddy Simulation of a 185 t Ladle Optimizing a Hybrid Eulerian Grid and a Discrete Bubble Mapping Approach

Numerical simulation is a vital tool for the continuous improvement of all steelmaking processes. In previous works, the authors have dedicated different approaches to optimize modeling of fluid flow in the ladle. This includes a sensitivity analysis of validation techniques, assembling of a validation database, and a model optimization on the water model scale. Herein, the final part of this project is presented at which the optimized model is scaled to the real process. Associated problems are identified and solutions are proposed. To achieve a tradeoff between large eddy simulation (LES) grid requirements and current computational capacities, a hybrid meshing strategy is optimized. To overcome the intrinsic drawbacks of the Euler–Lagrange model, a mapping approach for phase decoupling is implemented and made publicly available on GitHub. The upscaling of semi‐empiric submodels is discussed. Good agreement with experimental mixing time measurements is found, indicating the success of upscaling. A first time‐averaged flow field of a full‐scale 185 t ladle with the LES model is presented. Due to the extreme computing requirements, these results may currently be used as a benchmark, but the trend toward increasing computing capacities may make LES models state of the art in the future.

Beam elements with frictional contact in the material point method

AbstractThis article presents a novel approach to modeling beam elements that accommodate large displacements/rotations in the context of the material point method (MPM). The MPM is a hybrid Lagrangian/Eulerian approach for solving solid mechanics problems involving extremely large deformations and rotations. A solid body is discretized into a set of Lagrangian material points, called particles in the MPM. The equations of motion are solved on a fixed Eulerian background grid. The beam particle developed herein consists of two end nodes, each possessing three translational and three rotational degrees of freedom (DOF) in 3D. The end nodes are tracked and define the evolving beam particle domain. As in the conventional MPM, the linear momentum is conserved on the background grid, but here we also use this grid to enforce C1 continuity between beam particles by mapping angular velocities and accelerations for the beam rotational DOF to/from the background grid. The contact interactions between the beam particles are treated using multiple velocity fields on the background grid. Spatial nodes representing the spatial extent of the beam particle are introduced, which allows for collision detection and associated frictional contact. The effectiveness of the proposed approach is demonstrated through a series of numerical examples. Beam structures subjected to large displacements and rotations are compared with analytical/numerical solutions, in which good agreement is obtained.

Development of Smoothed Particle Hydrodynamics for Simulation of Flow and Contaminant Transport on Natural Urban Terrain and Streams

Smoothed Particle Hydrodynamics (SPH) method is proposed, as an alternative mesh-free approach to model all components of rainfall, surface runoff, fluid flow and contaminant transport with the representation of contaminant and fluid, as particles. By doing so, contaminant particles can be traced for the motion within runoff or fluid flow, even in the form of minute concentration which is difficult to render in conventional Eulerian grid methods. Weakly compressible SPH (WCSPH) is selected with cubic spline kernel, and the incorporation of Large Eddy Simulation (LES) representing turbulence effect. Various SPH diffusion formulations have been reviewed and selected. The selected SPH formulation for contaminant concentration is validated against analytical diffusion equation with boundary conditions of solid wall or free surface. The validated method is applied to calculate the overland flow and the contaminant transport on a model terrain and a real terrain geometry. The real terrain is a part of the city of Teluk Intan in Perak, Malaysia and is simulated using digital elevation model (DEM) data retrieved from Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) Global Water Body Dataset (ASTWBD) for ground elevation and channel surface.

A numerical study of the solidification process of a retracting fluid filament

In this study, the retraction and solidification of a fluid filament are studied by a front-tracking method/finite difference scheme. The interface between two phases is handled by connected points (Lagrangian grid), which move on a fixed grid domain (Eulerian grid). The Navier-Stokes and energy equations are solved to simulate the problem. Initially, the fluid filament has a shape as half of a cylindrical capsule contact with a cold flat surface. We consider the effect of the aspect ratio (Ar) on the solidification of the fluid filament. It is found that an increase in the aspect ratio (Ar) in the range of 2 – 14 causes the retraction length to increase. The rate of the solidification of a fluid filament decreases when the Ar ratio increases. The solidification time, the solidification height and the tip angle of the fluid filament under the influence of the aspect ratio are also considered. After complete solidification, a small protrusion on the top of the solidified fluid filament is found.

Mixed Mesh Finite Volume Method for 1D Hyperbolic Systems with Application to Plug-Flow Heat Exchangers

We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed.

The Cauchy-Lagrange method for 3D-axisymmetric wall-bounded and potentially singular incompressible Euler flows

We treat the incompressible, and axisymmetric Euler equations for a three-dimensional cylindrical domain with boundaries. The equations are solved by the novel Cauchy-Lagrange algorithm (CLA), which uses the time-analyticity of the Lagrangian trajectories of an incompressible Euler flow and computes the time-Taylor coefficients of the Lagrangian map via recursion relations. This semi-Lagrangian algorithm uses a pseudo-spectral type approach in space by approximating the flow fields by Chebyshev-Fourier polynomials. New methods are presented to solve the resulting Poisson problems directly for their second-order space derivatives. The flow fields, known on the Lagrangian trajectories after one time-step, are interpolated back onto the Eulerian grid to start a new recursion cycle. The time-step is only limited by the radius of convergence and, thus, independent of any Courant-Friedrichs-Lewy (CFL) condition. This allows to advance the flow with larger time-steps independently of the mesh. Stationary and swirl-free flows are used to thoroughly test our implementation for the given geometry. In this work, our ultimate goal is to apply the CLA to a flow, which might develop finite time singularities, resulting in a loss of smoothness. This demands either 3D, or 2D with swirl, and may well require the presence of solid boundaries, as indicated in recent numerical work (Luo and Hou (2014) [45]).

SPH–FV coupling algorithm for solving multi-scale three-dimensional free-surface flows

In two previous papers we presented an algorithm for coupling the Finite Volume (FV) method for the solution of 2D Navier–Stokes equations discretized on block structured Eulerian grids with the weakly-compressible Lagrangian Smoothed Particle Hydrodynamics (SPH) method. This coupling procedure exploits the SPH method to discretize flow regions close to free-surfaces and the Finite Volume approach to resolve both the bulk flow and the wall regions, where grid stretching can be favourably used. The information exchange between the two numerical schemes is established through overlapping zones. In the present paper this coupling paradigm is extended to a 3D framework. To this purpose, the extension of the algorithms for particle creation/deletion on the interfaces are described and issues related to free-surface intersection with each sub-domain boundary are addressed. Moreover, a new coupling procedure that simplifies the algorithm is proposed and tested. Effectiveness and accuracy achieved by the coupled solver are tested on challenging problems involving large free surface deformations and vorticity generation like the 3D flow past a cylinder below a free surface, the breaking wave generated by a ship bow in forward motion and the impact of a flat plate on the water surface.

LaPIV using multigrid warping and proxy regularization

The Lagrangian Particle Image Velocimetry (LAPIV) method was firstly proposed in Yang et al. (2019) as a prototype approach to achieve the goal of accurate and efficient reconstruction of 3D Eulerian velocity field of fluid flow from multi-view particle images. After validating against synthetic datasets, the prototype has already shown significant advantages in revealing more small scale flow structures than other stateof-the-art Eulerian velocity estimation methods, such as TomoPIV (Scarano, 2013) and VIC# (Jeon et al., 2019). However, at this early stage, LAPIV can not be easily applied to other datasets. In the current work, we focus on extending LAPIV to operational search by incorporating several essential and wellestablished paradigms: multi-resolution, warping, and proxy regularization. Recent approaches, Lasinger et al. (2019) and Cornic et al. (2020), function in the same vein as LAPIV, aiming at reconstructs the dense Eulerian volumetric flow directly from multi-view particle-seeded images Another pipeline consists of firstly reconstructing the Lagrangian flow using the Lagrangian Particle Tracking (LPT), then optimally interpolating the Lagrangian flow to Eulerian grids, taking into account the Eulerian dynamics constraints as in Flowfit (Gesemann, 2020) and VIC# (Jeon et al., 2019). If Eulerian flow is required, LAPIV is the preferred approach due to its simplicity and ability to utilize the original rich image features.

3D-PTV flow measurements of Newtonian and non-Newtonian fluid blending in a batch reactor in the transitional regime

Lagrangian trajectories obtained through 3D Particle Tracking Velocimetry (3D-PTV) measurements have been used to visualize the flow field of Newtonian and non-Newtonian fluids in a flat-bottomed vessel. The vessel, of diameter T=180 mm, was equipped with a 6-blade Rushton turbine of diameter D=T/3 and four baffles of width b=T/10. The experiments were carried out in the transitional flow regime (73≤Re≤1,257). The velocities and Lagrangian accelerations in the flows have been calculated from the time-resolved tracer coordinates. Non-Newtonian fluids exhibited a smaller impeller flow number compared to Newtonian fluids. The distributions of shear rate have been obtained via interpolation of the Lagrangian velocity data in a 3D Eulerian grid. In the impeller region, the mean shear rate was, to a first approximation, proportional to the impeller rotational speed, although a more detailed analysis revealed influences of both rheology and Reynolds number. The mean Lagrangian acceleration scaled with the mean shear rate raised to the power of 1.8.

A remeshed vortex method for mixed rigid/soft body fluid–structure interaction

We outline a 2D algorithm for solving incompressible flow–structure interaction problems for mixed rigid/soft body representations, within a consistent framework based on the remeshed vortex method. We adopt the one-continuum formulation to represent both solid and fluid phases on an Eulerian grid, separated by a diffuse interface. Rigid solids are treated using Brinkman penalization while an inverse map technique is used to obtain elastic stresses in the hyperelastic solid phase. We test our solver against a number of benchmark problems, which demonstrate physical accuracy and first to second order convergence in space and time. Benchmarks are complemented by additional investigations that illustrate the ability of our numerical scheme to capture essential fluid–structure interaction phenomena across a variety of scenarios involving internal muscular actuation, self propulsion, multi-body contact, heat transfer and rectified viscous streaming effects. Through these illustrations, we showcase the ability of our solver to robustly deal with different constitutive models and boundary conditions, solve disparate multi-physics problems and achieve faster time-to-solutions by sidestepping CFL time step restrictions.

An experimental investigation of sediment kinematics and multi‐scale propagation for laboratory bed‐load dunes

AbstractThe dynamics of river bedforms is still not well understood and detailed experimental investigation may provide significant insight into the process mechanisms. In this study, results are presented for a laboratory experiment that returned long time series of sediment kinematics for bed‐load dunes. The run was performed in a closed conduit but, in the light of prior literature, the results are also representative of dunes under open‐channel flows. A flow rate of 1.4 times the threshold for sediment transport was used, and the experiment lasted long enough to measure the sediment kinematics for four full dunes. This enabled a stability of mean values to be obtained for the key properties; concentration of moving sediment, sediment velocity and sediment transport rate, that were measured over a Eulerian grid. At a measuring location, all the properties present an oscillation pattern that resembles the passage of the dunes, also including higher‐frequency oscillations (due to the flow turbulence) superimposed onto the low‐frequency ones related to the dunes. The time evolution of the sediment transport rate is more similar to that of the sediment concentration than to that of the sediment velocity, which varies comparatively less. A multi‐scale propagation is demonstrated considering the propagation of the dunes, of the sediment grains, and of quick ‘sediment gusts’ triggered by the flow field. Taking advantage of a Taylor‐like hypothesis, that is assumed to be valid as a first‐order approximation, a mean temporal evolution of the sediment–kinematics properties is determined. Although limited to a single hydro‐dynamic condition, and thus in need of further testing, the present results are of interest for a number of applications, including bedform analysis, live‐bed scour processes, and quantification of sediment transport rates by measuring surrogate quantities.

Asynchronous distributed-memory task-parallel algorithm for compressible flows on unstructured 3D Eulerian grids

We discuss the implementation of a finite element method, used to numerically solve the Euler equations of compressible flows, using an asynchronous runtime system (RTS). The algorithm is implemented for distributed-memory machines, using stationary unstructured 3D meshes, combining data-, and task-parallelism on top of the Charm++ RTS. Charm++’s execution model is asynchronous by default, allowing arbitrary overlap of computation and communication. Task-parallelism allows scheduling parts of an algorithm independently of, or dependent on, each other. Built-in automatic load balancing enables continuous redistribution of computational load by migration of work units based on real-time CPU load measurement. The RTS also features automatic checkpointing, fault tolerance, resilience against hardware failure, and supports power-, and energy-aware computation. We demonstrate scalability up to 25×109 cells at O(104) compute cores and the benefits of automatic load balancing for irregular workloads. The full source code with documentation is available at https://quinoacomputing.org.

A numerical method for the solid mechanics with Euler variables

Based on the requirement of field engineering, we proposed a new numerical method for analyzing the solid mechanics with Euler variables. By replacing the Lagrange variables with the Euler variables, there will be three new items appearing in the equation. We use the fixed Euler grid to calculate solid mechanics, and use the boundary of material regiment to describe the movement of the boundary of material, after the boundary of material regiment moves beyond the Euler element grid size, new Euler element grids will appear and the corresponding old Euler element grids which have been out of the region of material will disappear. By this way, the movement of solid could be calculated in the Euler fixed grids. Then we did some test cases, we get the result that the new method is right and could be used to calculate the solid mechanic problems.

Anisotropic fluid dynamical simulations of heavy-ion collisions

We present VAH, a (3+1)–dimensional simulation that evolves the far-from-equilibrium quark-gluon plasma produced in ultrarelativistic heavy-ion collisions with anisotropic fluid dynamics. We solve the hydrodynamic equations on an Eulerian grid using the Kurganov–Tadmor algorithm in combination with a new adaptive Runge–Kutta method. Our numerical scheme allows us to start the simulation soon after the nuclear collision, largely avoiding the need to integrate it with a separate pre-equilibrium dynamics module. We test the code's performance by simulating on the Eulerian grid conformal and non-conformal Bjorken flow as well as conformal Gubser flow, whose (0+1)–dimensional solutions are precisely known. Finally, we compare non-conformal anisotropic hydrodynamics to second-order viscous hydrodynamics in central Pb+Pb collisions and find that the former's longitudinal flow profile responds more consistently to the fluid's gradients along the spacetime rapidity direction. Program summaryManuscript Title: Anisotropic fluid dynamical simulations of heavy-ion collisionsAuthors: Mike McNelis, Dennis Bazow, Ulrich HeinzProgram Title:VAHLicensing provisions: GPLv3Programming Language: C++Computer: Laptop, desktop, clusterOperating System: GNU/Linux distributions, Mac OS XGlobal memory usage: 1.2 GB (for a 129×129×63 grid)Keywords: Ultrarelativistic heavy-ion collisions, quark-gluon plasma, relativistic hydrodynamics, computational fluid dynamicsClassification: 12 Gases and Fluids, 17 Nuclear physicsExternal routines/libraries: GNU Scientific Library (GSL)Nature of problem: Modeling the far-from-equilibrium dynamics of quark-gluon plasma produced in ultrarelativistic heavy-ion collisions.Solution method: Kurganov–Tadmor algorithm, adaptive stepsize methodRunning time: A (3+1)–d non-conformal anisotropic fluid dynamical simulation of a central Pb+Pb collision on a 129×129×63 grid takes about 530s for an Intel Xeon E5-2680 v4 multi-core processor with OpenMP acceleration (see Sec. 5.3).