Showing papers on "Eulerian path published in 2004"

Embedding Lagrangian Sink Particles in Eulerian Grids

Abstract: We introduce a new computational method for embedding Lagrangian sink particles into a Eulerian calculation. Simulations of gravitational collapse or accretion generally produce regions whose density greatly exceeds the mean density in the simulation. These dense regions require extremely small time steps to maintain numerical stability. Smoothed particle hydrodynamics (SPH) codes approach this problem by introducing nongaseous, accreting sink particles, and Eulerian codes may introduce fixed sink cells. However, until now there has been no approach that allows Eulerian codes to follow accretion onto multiple, moving objects. We have removed that limitation by extending the sink particle capability to Eulerian hydrodynamics codes. We have tested this new method and found that it produces excellent agreement with analytic solutions. In analyzing our sink particle method, we present a method for evaluating the disk viscosity parameter α due to the numerical viscosity of a hydrodynamics code and use it to compute α for our Cartesian adaptive mesh refinement (AMR) code. We also present a simple application of this new method: studying the transition from Bondi to Bondi-Hoyle accretion that occurs when a shock hits a particle undergoing Bondi accretion.

The Eulerian Limit for 2D Statistical Hydrodynamics

Abstract: We consider the 2D Navier–Stokes system, perturbed by a white in time random force, proportional to the square root of the viscosity. We prove that under the limit “time to infinity, viscosity to zero” each of its (random) solution converges in distribution to a non-trivial stationary process, formed by solutions of the (free) Euler equation, while the Reynolds number grows to infinity. We study the convergence and the limiting solutions.

Expected runtimes of evolutionary algorithms for the Eulerian cycle problem

Abstract: Evolutionary algorithms are randomized search heuristics, which are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization. They have successfully been applied to different kinds of arc routing problems. To start the analysis of evolutionary algorithms with respect to the expected optimization time on these problems, we consider the Eulerian cycle problem. We show that a variant of the well-known (1+1) EA working on the important encoding of permutations is able to find an Eulerian tour of an Eulerian graph in expected polynomial time. Altering the operator used for mutation in the considered algorithms, the expected optimization time changes from polynomial to exponential.

A Fast Method for Approximating Invariant Manifolds

Abstract: The task of constructing higher-dimensional invariant manifolds for dynamical systems can be computationally expensive. We demonstrate that this problem can be locally reduced to solving a system of quasi-linear PDEs, which can be efficiently solved in an Eulerian framework. We construct a fast numerical method for solving the resulting system of discretized nonlinear equations. The efficiency stems from decoupling the system and ordering the computations to take advantage of the direction of information flow. We illustrate our approach by constructing two-dimensional invariant manifolds of hyperbolic equilibria in $\R^3$ and $\R^4$.

Predictability of Lagrangian particle trajectories: Effects of smoothing of the underlying Eulerian flow

Abstract: The increasing realism of ocean circulation models is leading to an increasing use of Eulerian models as a basis to compute transport properties and to predict the fate of Lagrangian quantities. There exists, however, a signie cant gap between the spatial scales of model resolution and that of forces acting on Lagrangian particles. These scales may contain high vorticity coherent structures that are not resolved due to computational issues and/or missing dynamics and are typically suppressed by smoothing operators. In this study, the impact of smoothing of the Eulerian e elds on the predictability of Lagrangian particles is e rst investigated by conducting twin experiments that involve release of clusters of synthetic Lagrangian particles into “ true” (unmodie ed) and “ model” (smoothed) Eulerian e elds, which are generated by a QG model with a e ow e eld consisting of many turbulent coherent structures. The Lagrangian errors induced by Eulerian smoothing errors are quantie ed by using two metrics, the difference between the centers of mass (CM) of particle clusters, r, and the difference between scattering of particles around the center of mass, s. The results show that the smoothing has a strong effect on the CM behavior, while the scatter around it is only partially affected. The QG results are then compared to results obtained from a multi-particle Lagrangian Stochastic Model (LSM) which parameterizes turbulent e ow using main e ow characteristics such as mean e ow, velocity variance and Lagrangian time scale. In addition to numerical results, theoretical results based on the LSM are also considered, providing asymptotics of r, s and predictability time. It is shown that both numerical and theoretical LSM results for the center of mass error ( r) provide a good qualitative description, and a quantitatively satisfactory estimate of results from QG experiments. The scatter error ( s) results, on the other hand, are only qualitatively reproduced by the LSM.

Turbulent diffusion in rapidly rotating flows with and without stable stratification

Abstract: Three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity 2Q, chosen vertical here, and the Brunt-Vaisala frequency N, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda. They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos, which also incorporates the wave-vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case B = 2Ω/N = 10, the stratification-dominant case B = 1/10, the non-dispersive case B = 1, and pure stratification B = 0 and pure rotation N = 0

Level-set method for design of multi-phase elastic and thermoelastic materials

Abstract: The problem of designing composite materials with desired mechanical properties is to specify the materials microstructures in terms of the topology and distribution of their constituent material phases within a unit cell of periodic microstructures. In this paper we present an approach based on a multi-phase level-set model for the geometric and material representation and for numerical solution of a least squares optimization problem. The level-set model precisely specifies the material regions and their sharp boundaries in contrast to a raster discretization of the conventional homogenization-based approaches. Combined with the classical shape derivatives, the level-set method yields a computational system of partial differential equations. In using the Eulerian computation scheme with a fixed rectilinear grid and a fixed mesh in the unit cell, the gradient descent solution of the optimization captures the interfacial boundaries naturally and performs topological changes accurately. The proposed method is illustrated with several 2D examples for the synthesis of heterogeneous microstructures of elastic and/or thermoelastic composites composed of two and three material phases.

Lagrangian versus Eulerian correlations and transport scaling

Abstract: The effect of the shape of the Eulerian correlation of an electrostatic turbulence on the scaling of the diffusion coefficient is studied using the decorrelation trajectory method. We show that a strong influence appears due to trajectory trapping when the Kubo number is larger than 1.

Challenges of Simulating Droplet Coalescence within a Spray

Abstract: This article reports various challenges that have been encountered in the process of developing validated Lagrangian and Eulerian models for simulating particle agglomeration within a spray dryer. These have included the challenges of accurately measuring droplet coalescence rates within a spray, and modeling properly the gas–droplet and droplet-droplet turbulence interactions. We have demonstrated the relative versatility and ease of implementation of the Lagrangian model compared with the Eulerian model, and the accuracy of both models for predicting turbulent dispersion of droplets and the turbulent flow-field within a simple jet system. The Lagrangian and Eulerian droplet coalescence predictions are consistent with each other, which implies that the numerical aspects of each simulation are handled properly, suggesting that either approach can be used with confidence for future spray modeling. However, it is clear that considerable research must be done in the area of particle turbulence model...

Eulerian network model of air traffic flow in congested areas

Abstract: We derive an Eulerian network model applicable to air traffic flow in the National Airspace System. The model relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE), which contains a velocity control term inside the divergence operator. We relate the PDE to aircraft count, which is a key metric in air traffic control. Using the method of characteristics, we construct an analytical solution to the LWR PDE for the case in which the control depends only on space (and not time). We validate our model against real air traffic data (ETMS data), by first showing that the Eulerian description enables good aircraft count predictions, provided a good choice of numerical parameters is made. Finally, we show some predictive capabilities of the model.

Adjoint-based constrained control of Eulerian transportation networks: application to air traffic control

Abstract: We use an Eulerian network model of the airspace to simulate air traffic in congested areas of airspace. The model relies on a set of coupled first order hyperbolic partial differential equations (PDEs), obtained from the original Lighthill-Whitham-Richards (LWR) traffic model. The Jameson-Schmidt-Turkel (JST) is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against known analytical solutions. Linear numerical schemes are discarded because of their poor performance, thus prohibiting the use of linear optimization for controlling the network. Instead, the adjoint problem of the linearized network control problem is computed. The constraints of the problem are enforced using a logarithmic barrier method. Simulations are run with real air traffic data to demonstrate the applicability of the method for traffic management. Scenarios involving several airports between Chicago and the East Coast are investigated.

Eulerian method for computing multivalued solutions of the Euler-Poisson equations and applications to wave breaking in klystrons.

Abstract: We provide methods of computing multivalued solutions to the Euler-Poisson system and test them in the context of a klystron amplifier. An Eulerian formulation capable of computing multivalued solutions is derived from a kinetic description of the Euler-Poisson system and a moment closure. The system of the moment equations may be closed due to the special structure of the solution in phase space. The Eulerian moment equations are computed for a velocity modulated electron beam, which has been shown by prior Lagrangian theories to break in a finite time and form multivalued solutions. The results of the Eulerian moment equations are compared to direct computation of the kinetic equations and a Lagrangian method also developed in the paper. We use the Lagrangian formulation for the explicit computation of wave breaking time and location for typical velocity modulation boundary conditions.

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

Abstract: This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second-order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least-squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure-stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure-computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free-surface problems. Copyright © 2004 John Wiley & Sons, Ltd.

Note on counting Eulerian circuits

Abstract: We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P.

Generalized eulerian coordinates for relativistic fluids: hamiltonian rest-frame instant form, relative variables, rotational kinematics

Abstract: We study the rest-frame instant form of a new formulation of relativistic perfect fluids in terms of new generalized Eulerian configuration coordinates. After the separation of the relativistic center of mass from the relative variables on the Wigner hyper-planes, we define orientational and shape variables for the fluid, viewed as a relativistic extended deformable body, by introducing dynamical body frames. Finally we define Dixon's multipoles for the fluid.

Eulerian analysis of the dispersion of evaporating polydispersed sprays in a statistically stationary turbulent flow

Abstract: Direct numerical simulations of a statistically stationary spatially decaying turbulence were performed to study the dispersion of evaporating droplets in a non-homogeneous flow High-order finite-difference methods have been used to solve the compressible gas equations along with a Lagrangian solver for the polydispersed spray to produce a reliable reference solution The key and novel point in this paper is to provide a comprehensive analysis from an Eulerian point of view of the turbulent dispersion of evaporating polydispersed sprays with various mean Stokes numbers For this purpose, our work focuses on simplified evaporation laws and one-way coupling to isolate some of the key physical phenomena to be captured by an Eulerian description, which would be hidden by the intricacy of the numerous couplings occurring with more complex models A special emphasis is laid on the polydispersed character of the spray in the particular chosen configuration It is shown that the dynamics of the droplets at the g

The E t -Construction for Lattices, Spheres and Polytopes

Abstract: We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein et al. (6). We also construct for each d ≥ 3 an infinite family of (d − 2)-simplicial 2-simple d-polytopes, thus solving a problem of Grunbaum (9).