Eulerian Reference Research Articles

Numerical Investigation of the Cavitation Effects on the Vortex Shedding from a Hydrofoil with Blunt Trailing Edge

Vortex cavitation can appear in the wake flow of hydrofoils, inducing unwanted consequences such as vibrations or unstable behaviors in hydraulic machinery and systems. To investigate the cavitation effects on hydrofoil vortex shedding, a numerical investigation of the flow around a 2D NACA0009 with a blunt trailing edge at free caviation conditions and at two degrees of cavitation developments has been carried out by means of the Zwart cavitation model and the LES WALE turbulence model which permits predicting the laminar to turbulent transition of the boundary layers. To analyze the dynamic behavior of the vortex shedding process and the coherent structures, two identification methods based on the Eulerian and Lagrangian reference frames have been applied to the simulated unsteady flow field. It is found that the cavitation occurrence in the wake significantly changes the main vortex shedding characteristics including the morphology of the vortices, the vortex formation length, the effective height of the near wake flow and the shedding frequency. The numerical results predict that the circular shape of the vortices changes to an elliptical one and that the vortex shedding frequency is significantly increased under cavitation conditions. The main reason for the frequency increase seems to be the reduction in the transverse separation between the upper and lower rows of vortices induced by the increase in the vortex formation length.

A frame-invariant vector limiter for flux corrected nodal remap in arbitrary Lagrangian–Eulerian flow computations

This article describes a frame-invariant vector limiter for Flux-Corrected Transport (FCT) numerical methods. Our approach relies on an objective vector projection, and, because of its intrinsic structure, the proposed approach can be generalized with ease to higher-order tensor fields.The proposed concept is applied to nodal finite element formulations and the so-called algebraic FCT paradigm, but the ideas pursued here are very general and also apply to more general instantiations of flux-corrected transport.Specifically, we consider the arbitrary Lagrangian–Eulerian (ALE) equations of compressible inviscid flows. In addition to the geometric conservation law (GCL) and the local extreme diminishing (LED) property of the original scalar limiters, the proposed approach ensures frame invariance (objectivity) for vectors. Particularly, we use an ALE strategy based on a two-stage, Lagrangian plus mesh remap (data transfer based on conservative interpolation), in which remap and limiting are performed in a synchronized way. The proposed approach is however of general applicability, is not limited to a specific ALE implementation, and can easily be generalized to computations with standard (monolithic) ALE or Eulerian reference frames.The significance of the frame-invariant limiter for vectors is demonstrated in computations of compressible materials under extreme load conditions. Extensive testing in two and three dimensions demonstrates that the proposed limiter greatly enhances the robustness and reliability of the existing methods under typical computational scenarios.

Carotid Artery Modeling Using the Navier-Stokes Equations for an Incompressible, Newtonian and Axisymmetric Flow

This paper describes two-dimensional (2D) structural and hemodynamics model of the carotid artery and its bifurcation using computational software (CS). The Arbitrary Lagrangian Eulerian (ALE), which was introduced in a finite element system, was utilized as a numerical technique. The structural modeling of the carotid arteries about the bifurcation area was constructed from computed tomography (CT) scans using computer-aided design (CAD) and the Lagrangian formulation was used for the structural domain. The blood was considered as an incompressible Newtonian fluid, and Eulerian reference was applied for its domain. Coupling of the reference systems was carried out on arbitrary computational grid permitting numerical modeling of hemodynamics as governed by 2D axially symmetric incompressible Navier-Stokes equations (NSE). The results for hemodynamic simulations were compared with the physiological blood velocity obtained using the Doppler ultrasound instrument.

A polyhedral/polygonal mesh topology based momentum mapping method for Eulerian/Lagrangian simulations with strong influence/dependence coupling

An improved non-isotropic momentum mapping method for the Eulerian/Lagrangian coupling algorithm is proposed. The concept of strong influence and dependence between the Eulerian and Lagrangian reference frames is the focus of the proposed method. The mapping technique is designed for cell centered control volume based algorithms and applicable to arbitrary polyhedral/polygonal mesh topologies. Eulerian to Lagrangian interpolation methods are reviewed to highlight the importance of consistently treating the influence and dependence between the Eulerian to Lagrangian interpolation technique and the Lagrangian to Eulerian momentum mapping method. A review of non-isotropic momentum source term treatment for the SIMPLE algorithm is discussed along with its importance to the overall algorithm stability. A gas/particulate test case is presented to compare the stability of the proposed momentum mapping method to the commonly used injection technique. The results of the test case demonstrate the importance of consistently treating the interpolation and mapping process and shows that the proposed method leads to a more stable Eulerian/Lagrangian multiphase algorithm.

Coupling of Engineering and Biological Models for Ecosystem Analysis

Robust ecosystem analysis of water resource systems remains elusive. A principle reason is the difficulty in linking engineering models used to simulate physicochemical processes associated with project design or operation with biological models used to simulate biological population attributes. A retrospective shows that each modeling tradition can be generally assigned (with exceptions) into either an Eulerian or Lagrangian reference framework. Eulerian and Lagrangian reference frameworks can be coupled to create a new synthesis, the Coupled Eulerian-Lagrangian Hybrid Ecological Modeling Concept (CEL Hybrid Concept), capable of simulating different ecosystem processes that range widely in spatial and temporal scale. The foundation of the CEL Hybrid Concept is the coupler, a collection of algorithms based on conservation principles that transform and conserve data in a way that allows the two frameworks to share a common information base. The coupling algorithm allows the simulation to aggregate, disaggregate, and translate information, as required by each framework, so that processes that differ substantially in scale can each be adequately simulated. The coupled system is illustrated by linking a fish swim path selection model with a hydrodynamic and water quality model.

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three-dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second-order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge-based local h-refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level-set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure-stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free-surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.

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

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

Analysis of orthogonal metal cutting processes

AbstractThe orthogonal metal cutting process for a controlled contact tool is simulated using a limit analysis theorem. The basic principles are stated in the form of a primal optimization problem with an objective function subjected to constraints of the equilibrium equation, its static boundary conditions and a constitutive inequality. An Eulerian reference co‐ordinate is used to describe the steady state motion of the workpiece relative to the tool. Based on a duality theorem, a dual functional bounds the objective functional of the primal problem from above by a sharp inequality. The dual formulation seeks the least upper bound and thus recovers the maximum of the primal functional theoretically. A finite element approximation of the continuous variables in the dual problem reduces it to a convex programming. Since the original dual problem admits discontinuous solutions in the form of bounded variation functions, care must be taken in the finite element approximation to account for such a possibility. This is accomplished by a combined smoothing and successive approximation algorithm. Convergence is robust from any initial iterate. Results are obtained for a wide range of control parameters including cutting depth, rake angle, rake length and friction. The converged solutions provide information on cutting force, chip thickness, chip stream angle and shear angle which agree well both in values and trend with the published data. But the available data represent only a small subset in the range of parameters exhaustively investigated in this paper.

Lagrangian coordinate transformations in self-similar supernova blast-wave models

The Sedov self-similar solution of the fluid equations provides a description of a spherical supernova blast-wave during its adiabatic phase in a stationary (Eulerian) reference frame. We provide an exact transformation to a Lagrangian coordinate system, a non-inertial frame of reference which follows each individual gas particle. The physical properties of the gas in this comoving reference system are evaluated as a function of time. Numerical methods to facilitate Eulerian-Lagrangian and Lagrangian-Eulerian coordinate transformations are provided.