Coupled Euler-Lagrange simulations of metal fragmentation in pipe bomb configurations.
Abstract not provided.
Gullerud, Arne S.↗
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Lagrangian and Eulerian drag models that are consistent between Euler-Lagrange and Euler-Euler (two-fluid) approaches for homogeneous systems
The undisturbed flow of a particle is of fundamental importance since it controls both the undisturbed flow force and the perturbation force (which includes quasisteady, added-mass, and history forces). Here we use the pairwise interaction extended point particle framework to evaluate the undisturbed flow of each particle through superposition of the perturbation flow induced by all its neighbors. This approach allows calculation of various statistics related to undisturbed fluid velocity under conditions of both stationary and nonstationary particles. In a random distribution of stationary particles, while the macroscale undisturbed flow is slowly varying, the microscale undisturbed flow that arises due to the perturbation flow of neighbors varies substantially from one particle to another and this in turn leads to large variation in the hydrodynamic force exerted on the particles. The effect of particle motion is generally to increase the particle-to-particle variation in the undisturbed fluid velocity of the particles. We observe that this increase is greater for the transverse component than for the streamwise component. As a result, with increasing random particle motion, the distribution of undisturbed fluid-velocity fluctuation becomes isotropic. Three different normalized forces are defined: $Φ^L$ is the Lagrangian normalized force on an individual particle suitable for application in a microscale-informed Euler-Lagrange simulation, $Φ^E$ is the Eulerian normalized average force suitable for application in an Euler-Euler simulation, and $Φ^{LE}$ is the Lagrangian normalized force on an individual particle suitable for application in the standard Euler-Lagrange simulation. In this work, we establish precise relations between these different definitions. The drag laws developed based on particle-resolved direct numerical simulation results and experiments are appropriate for application only as the Eulerian normalized average force. We introduce the force consistency relation and use it to obtain an expression for $Φ^L$, which when applied to each particle and averaged over all the particles equals $Φ^E$. The results are first obtained in the limit of stationary particles and then extended to the general case of nonstationary particles.
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Chaotic dynamics of flexible Euler-Bernoulli beams
Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c{sup 2}) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q{sub 0} and frequency ω{sub p} of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Euler-euler anisotropic gaussian mesoscale simulation of homogeneous cluster-induced gas-particle turbulence
An Euler–Euler anisotropic Gaussian approach (EE-AG) for simulating gas–particle flows, in which particle velocities are assumed to follow a multivariate anisotropic Gaussian distribution, is used to perform mesoscale simulations of homogeneous cluster-induced turbulence (CIT). A three-dimensional Gauss–Hermite quadrature formulation is used to calculate the kinetic flux for 10 velocity moments in a finite-volume framework. The particle-phase volume-fraction and momentum equations are coupled with the Eulerian solver for the gas phase. This approach is implemented in an open-source CFD package, OpenFOAM, and detailed simulation results are compared with previous Euler–Lagrange simulations in a domain size study of CIT. Here, these results demonstrate that the proposed EE-AG methodology is able to produce comparable results to EL simulations, and this moment-based methodology can be used to perform accurate mesoscale simulations of dilute gas–particle flows.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements
Multiphase flow can be difficult to simulate with high accuracy due to the wide range of scales associated with various multiphase phenomena. These scales may range from the size of individual particles to the entire domain of interest. Traditionally, large scale systems can only be simulated using averaging approaches that filter out the locations of individual particles. In this work, the Euler–Lagrange method is used to simulate large-scale dense particle systems in which each individual particle is tracked. In order to accomplish this, the highly scalable spectral element code nek5000 has been extended to handle the multiple levels of multiphase coupling in these systems. These levels include what has been called one-, two-, and four-way coupling. Here, each level has been separated to detail the computational impact of each stage. A binned ghost particle algorithm has also been developed to efficiently handle the challenges of two- and four-way coupling in a parallel processing context. The algorithms and their implementations are then shown to scale to 65,536 Message Passing Interface (MPI) ranks in both the strong and weak limits. After this, validation is performed through simulation of a small-scale fluidized bed. Lastly, a large-scale fluidized bed is simulated with 65,536 MPI ranks and is able to capture the unique physics of the onset of fluidization.
Computer Science↗
Self-induced velocity correction for improved drag estimation in Euler–Lagrange point-particle simulations
Not Available
Balachandar, S. (ORCID:0000000336193695)↗
Generating Euler Diagrams Through Combinatorial Optimization
Abstract Can a given set system be drawn as an Euler diagram? We present the first method that correctly decides this question for arbitrary set systems if the Euler diagram is required to represent each set with a single connected region. If the answer is yes, our method constructs an Euler diagram. If the answer is no, our method yields an Euler diagram for a simplified version of the set system, where a minimum number of set elements have been removed. Further, we integrate known wellformedness criteria for Euler diagrams as additional optimization objectives into our method. Our focus lies on the computation of a planar graph that is embedded in the plane to serve as the dual graph of the Euler diagram. Since even a basic version of this problem is known to be NP‐hard, we choose an approach based on integer linear programming (ILP), which allows us to compute optimal solutions with existing mathematical solvers. For this, we draw upon previous research on computing planar supports of hypergraphs and adapt existing ILP building blocks for contiguity‐constrained spatial unit allocation and the maximum planar subgraph problem. To generate Euler diagrams for large set systems, for which the proposed simplification through element removal becomes indispensable, we also present an efficient heuristic. We report on experiments with data from MovieDB and Twitter. Over all examples, including 850 non‐trivial instances, our exact optimization method failed only for one set system to find a solution without removing a set element. However, with the removal of only a few set elements, the Euler diagrams can be substantially improved with respect to our wellformedness criteria.
Computer Science↗
A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization
We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.
97 MATHEMATICS AND COMPUTING↗
The Euler characteristic as a basis for teaching topology concepts to crystallographers
The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V – E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χ m ) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χ m has an elegant topological interpretation through the concept of orbifolds. Alternatively, χ m can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.
97 MATHEMATICS AND COMPUTING↗
Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem
We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Euler equations and the Sod shock tube problem
The Euler equations are a subset of the magnetohydrodynamic (MHD) equations in the infinitely collisional, unmagnetized limit. MHD modeling is central to many areas of plasma physics, ranging from low-temperature glow discharges to inertial confinement fusion. An important aspect of the Euler equations is their ability to describe states with discontinuities, such as shock waves. A standard benchmark test for numerical implementation of the Euler equations is the Sod shock tube. In this test, the system is initialized at rest with a pressure and density discontinuity, which results in a shock wave traveling into the low-pressure region and a rarefaction wave traveling into the high-pressure region. Starting with the presentation of the Euler equations, a numerical algorithm is presented here to solve these equations in one dimension. This is followed by an overview of the Sod shock tube problem that includes the precise initial setup and the analytic solution. Finally, the analytic solution is compared with results from numerical simulations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Faddeev–Jackiw quantization of topological invariants: Euler and Pontryagin classes
Highlights: •We report the symplectic analysis for Euler and Pontryagin invariants. •We report the Faddeev–Jackiw constraints. •A symplectic tensor is constructed. •The generalized Faddeev–Jackiw brackets are found. •The quantum Faddeev–Jackiw constraints are solved. -- Abstract: The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev–Jackiw context. The Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev–Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Framed 4-graphs: Euler tours, Gauss circuits and rotating circuits
We consider connected finite 4-valent graphs with the structure of opposite edges at each vertex (framed 4-graphs). For any of such graphs there exist Euler tours, in travelling along which at each vertex we turn from an edge to a nonopposite one (rotating circuits); and at the same time, it is not true that for any such graph there exists an Euler tour passing from an edge to the opposite one at each vertex (a Gauss circuit). The main result of the work is an explicit formula connecting the adjacency matrices of the Gauss circuit and an arbitrary Euler tour. This formula immediately gives us a criterion for the existence of a Gauss circuit on a given framed 4-graph. It turns out that the results are also valid for all symmetric matrices (not just for matrices realisable by a chord diagram). Bibliography: 24 titles.
97 MATHEMATICS AND COMPUTING↗
Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems
The action principle by Low for the classic Vlasov–Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler–Poincaré formulation of Vlasov–Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al, it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler–Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov–Darwin system Sugama et al. The present exposition discusses a generic class of local Vlasov–Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler–Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. Here, we also discuss what happens if no symmetries exist. Finally, two explicit examples will be given—the classic Vlasov–Maxwell and the drift-kinetic Vlasov–Maxwell—and the results expressed in the language of regular vector calculus for familiarity.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Scale Invariance of the Homentropic Inviscid Euler Equations with Application to the Noh Problem
We investigate the inviscid compressible flow (Euler) equations constrained by an ”isentropic” equation of state (EOS), whose functional form in pressure is an arbitrary function of density alone. Under the aforementioned condition, we interrogate using symmetry methods the scale-invariance of the homentropic inviscid Euler equations. We find that under general conditions, we can reduce the inviscid Euler equations into a system of two coupled ordinary differential equations. To exemplify the utility of these results, we formulate two example scale-invariant, self-similar solutions. The first example includes a shock-free expanding bubble scenario, featuring a modified Tait EOS. The second example features the classical Noh problem, coupled to an arbitrary isentropic EOS. In this case, in order to satisfy the conditions set forth in the classical Noh problem, we find that the solution for the flow is given by a transcendental algebraic equation in the shocked density.
97 MATHEMATICS AND COMPUTING↗
Symmetry Determining Equations of the Euler Equations
This purpose of this document is to provide a clear derivation of the symmetry determining equations (SDEs) of the Euler equations to supplement the lecture document. The Euler equations are a coupled system of three first-order quasilinear hyperbolic partial differential equations (PDEs). The Euler equations are a system of partial differential equations (PDEs) for fluid velocity v(t, r), fluid density ρ(t, r), and fluid pressure P(t, r), where t is time and r is the spatial coordinate.
97 MATHEMATICS AND COMPUTING↗
A sweeping positivity-preserving high-order finite difference WENO scheme for Euler equations
We develop a simple, high-order, conservative and robust positivity-preserving sweeping procedure for the density and the nonlinear pressure function in the compressible Euler equations. Using the scaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010), we obtain a non-trivial extension of the scalar sweeping technique in Liu et al. (J Sci Comput 73:1028–1071, 2017) for the positivity of pressure. The sweeping procedure developed in this paper is a post-processing technique, which can be applied to any concave functions of the conserved variables in hyperbolic conservation law systems. Thus, it has applications beyond the Euler equations. This procedure preserves positivity and conservation of physical quantities without destroying the accuracy of the underlying scheme. The algorithm works for general schemes including finite difference, finite volume and discontinuous Galerkin methods; however, in this paper we focus on finite difference weighted essentially non-oscillatory (WENO) methods. As a result, we provide numerical tests of the fifth-order finite difference WENO scheme to demonstrate the accuracy and robustness of the technique.
Compressible Euler equations↗
Self-induced temperature correction for inter-phase heat transfer in Euler-Lagrange point-particle simulation
In a two-way coupled Euler-Lagrange simulation, particles are approximated as point sources and their momentum and energy exchange with the surrounding flow are modeled through point-particle force and heat transfer models. As the particle size increases and approaches the Eulerian grid, the feedback force and heat transfer become large and strongly affect the local flow at the particle location. The fluid velocity and temperature computed in the EL simulation when interpolated to the particle location will be substantially different from the corresponding undisturbed values. In this work we will follow the approach pursued in Balachandar et al. (2018) [1] for self-induced velocity correction to develop an analogous correction procedure for self-induced temperature. We obtain analytical solutions for the self-induced thermal perturbation for a steady uniform cross flow past a Gaussian filtered heat source under both steady and unsteady conditions. The analytical results are extended to finite Peclet number using numerical simulations. The resulting quasi-steady and unsteady models are tested with a simple problem of exponential thermal evolution against the corresponding EL simulation results. We also provide a simple criterion for when the self-induced temperature correction is needed in an Euler-Lagrange simulation.
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