Identifying invariant manifold using phase space warping and stochastic interrogation

Advances in the generalization of invariant manifolds to finite time, experimental (or observational) flows have stimulated many recent developments in the approximation of invariant manifolds and Lagrangian coherent structures. This paper explores the identification of invariant manifold like structures in experimental settings, where knowledge of a flow field is absent, but phase space trajectories can be experimentally measured. Several existing methods for the approximation of these structures modified for application when only unstructured trajectory data is available. We find the recently proposed method, based on the concept of phase space warping, to outperform other methods as data becomes limited and show it to extend the finite-time Lyapunov exponent method. This finding is based on a comparison of methods for various data quantities and in the presence of both measurement and dynamic noise.     

Changes in a coflowing jet structure caused by acoustic forcing

In the present work, the changes in the basic turbulence field of an axisymmetric jet in a coflow when acoustic forcing is applied are investigated. The main objective is to discriminate between changes produced by the acoustic forcing and those produced by the presence of solid particles in a two-phase flow. Power spectra of the axial velocities, u(t), are analyzed to reinforce the idea of the existence of a natural frequency. Time-averaged data are used to characterize the basic flow. This basic flow is compared with the flow altered by the acoustic forcing. By smoothing the phase-averaged rms data (mean statistical curve), the general behavior of the instantaneous fluctuations are unveiled and compared with that of the natural (unforced) jet. In this way, it can be seen that rms values in the forced cases are higher than in the unforced ones, due to the contribution of the external forcing. However, once the coherent structures are extinguished, both forced and unforced jets show a similar trend.     

基于非结构/混合网格的高阶精度格式研究进展

尽管以二阶精度格式为基础的计算流体力学(CFD) 方法和软件已经在航空航天飞行器设计中发挥了重要的作用, 但是由于二阶精度格式的耗散和色散较大, 对于湍流、分离等多尺度流动现象的模拟, 现有成熟的CFD 软件仍难以给出满意的结果, 为此CFD 工作者发展了众多的高阶精度计算格式. 如果以适应的计算网格来分类, 一般可以分为基于结构网格的有限差分格式、基于非结构/混合网格的有限体积法和有限元方法,以及各种类型的混合方法. 由于非结构/混合网格具有良好的几何适应性, 基于非结构/混合网格的高阶精度格式近年来备受关注. 本文综述了近年来基于非结构/混合网格的高阶精度格式研究进展, 重点介绍了空间离散方法, 主要包括k-Exact 和ENO/WENO 等有限体积方法, 间断伽辽金(DG) 有限元方法, 有限谱体积(SV) 和有限谱差分(SD) 方法, 以及近来发展的各种DG/FV 混合算法和将各种方法统一在一个框架内的CPR (correctionprocedure via reconstruction) 方法等. 随后简要介绍了高阶精度格式应用于复杂外形流动数值模拟的一些需要关注的问题, 包括曲边界的处理方法、间断侦测和限制器、各种加速收敛技术等. 在综述过程中, 介绍了各种方法的优势与不足, 其间介绍了作者发展的基于"静动态混合重构" 的DG/FV 混合算法. 最后展望了基于非结构/混合网格的高阶精度格式的未来发展趋势及应用前景.     

Uncovering large-scale coherent structures in natural and forced turbulent wakes by combining PIV,POD, and FTLE

Planar velocity data of the unsteady separated flow in the turbulent wake of a circular cylinder obtained by particle image velocimetry (PIV) are analyzed in order to visualize the large-scale coherent structures associated with alternating vortex shedding at a Reynolds number of 2,150. Two different cases are examined: unforced vortex shedding in the natural wake and vortex lock-on incited by forced perturbations superimposed in the inflow velocity. Proper orthogonal decomposition (POD) is employed to reconstruct the low-order wake dynamics from randomly sampled snapshots of the velocity field. The reconstructed flow is subsequently used to determine the evolution of the finite-time Lyapunov exponent (FTLE) fields which identify the Lagrangian coherent structures. The results demonstrate that the combination of methods employed offers a powerful visualization tool to uncover large-scale coherent structures and to exemplify vortex dynamics in natural and forced bluff-body wakes.     

Numerical Simulation of Particle Dispersion in a Spatially Developing Mixing Layer

Although there have been several numerical studies on particle dispersion in mixing layers, most of them have been conducted for temporally evolving mixing layers. In this study, numerical simulations of a spatially developing mixing layer are performed to investigate particle dispersion under various conditions. The full compressible Navier--Stokes equations are solved with a high-order compact finite difference scheme, along with high-order time-integration. Accurate non-reflecting boundary conditions for the fluid flow are used, and several methods for introducing particles into the computational domain are tested. The particles are traced using a Lagrangian approach assuming one-way coupling between the continuous and the dispersed phases. The study focuses on the roles of the large-scale vortex structures in particle dispersion at low, medium and high Stokes numbers, which highlights the important effects of interacting vortex structures in nearby regions in the spatially developing mixing layer. The effects of particles with randomly distributed sizes (or Stokes numbers) are also investigated. Both instantaneous flow fields and statistical quantities are analyzed, which reveals essential features of particle dispersion in spatially developing free shear flows, which are different from those observed in temporally developing flows. The inclusion of the gravity not only modifies the overall dispersion patterns, but also enhances stream-crossing by particles. Received 7 June 2001 and accepted 19 February 2002     

A potential-flow,deformable-body model for fluid–structure interactions with compact vorticity: application to animal swimming measurements

This paper presents an approach to quantify the unsteady fluid forces, moments and mass transport generated by swimming animals, based on measurements of the surrounding flow field. These goals are accomplished within a framework that is independent of the vorticity field, making it unnecessary to directly resolve boundary layers on the animal, body–vortex interactions, or interactions among vortex lines in the wake. Instead, the method identifies Lagrangian coherent structures in the flow, whose dynamics in flows with compact vorticity are shown to be well approximated by potential flow concepts, especially the Kirchhoff and deformation potentials from deformable body theory. Examples of the application of these methods are given for pectoral fin locomotion of the bluegill sunfish and undulatory swimming of jellyfish, and the methods are validated by analysis of a canonical starting vortex ring flow. The transition to a Lagrangian approach toward animal swimming measurements suggests the possibility of implementing recently developed particle tracking (vis-à-vis DPIV) techniques for fully three-dimensional measurements of animal swimming.     

Eulerian spectrum of finite-time Lyapunov exponents in compound channels

Fluid flows reveal a wealth of structures, such as vortices and barriers to transport. Usually, either an Eulerian or a Lagrangian frame of reference is employed in order to detect such features of the flow. However, the two frameworks detect structures that have different properties. Indeed, common Eulerian diagnostics (Hua-Klein and Okubo-Weiss criterion) employed in order to detect vortices do not always agree with Lagrangian diagnostics such as finite-time Lyapunov exponents. Besides, the former are Galilean-invariant whereas the latter is objective. However, both the Lagrangian and the Eulerian approaches to coherent structure detection must show some links under any inertial-frame. Compound channels flows have been accurately studied in the past, both from a Lagrangian and an Eulerian point of view. The features detected do not superimpose: Eulerian vortices do not coincide with barriers to transport. The missing link between the two approaches is here recovered thanks to a spectral analysis.

     

Experimental determination of three-dimensional finite-time Lyapunov exponents in multi-component flows

We present an experimental approach for estimating finite-time Lyapunov exponent fields (FTLEs) in three-dimensional multi-component or multi-phase flows. From time-resolved sequences of particle images, we directly compute the flow map and coherent structures, while avoiding and outperforming the computationally costly numerical integration. Performing this operation independently on each flow component enables the determination of three-dimensional Lagrangian coherent structures (LCSs) without any bias from the other components. The locations of concurrent LCSs for different flow elements (e.g., passive tracers, inertial particles, bubbles, or active particles) can provide new insight into the interpenetrating FTLE structure in complex flows.     

Two- and Four-Way Coupled Euler–Lagrangian Large-Eddy Simulation of Turbulent Particle-Laden Channel Flow

Large-eddy simulations (LES) of a vertical turbulent channel flow laden with a very large number of solid particles are performed. The motivation for this research is to get insight into fundamental aspects of co-current turbulent gas-particle flows, as encountered in riser reactors. The particle volume fraction equals about 1.3%, which is relatively high in the context of modern LES of two-phase flows. The channel flow simulations are based on large-eddy approximations of the compressible Navier–Stokes equations in a porous medium. The Euler–Lagrangian method is adopted, which means that for each individual particle an equation of motion is solved. The method incorporates four-way coupling, i.e., both the particle-fluid and particle–particle interactions are taken into account. The results are compared to single-phase channel flow in order to investigate the effect of the particles on turbulent statistics. The present results show that due to particle–fluid interactions the mean fluid profile is flattened and the boundary layer is thinner. Compared to single-phase turbulent flow, the streamwise turbulence intensity of the gas phase is increased, while the normal and spanwise turbulence intensities are reduced. This finding is generally consistent with existing experimental data. The four-way coupled simulations are also compared with two-way coupled simulations, in which the inelastic collisions between particles are neglected. The latter comparison clearly demonstrates that the collisions have a large influence on the main statistics of both phases. In addition, the four-way coupled simulations contain stronger coherent particle structures. It is thus essential to include the particle–particle interactions in numerical simulations of two-phase flow with volume fractions around one percent.     

High-order overset grid method for detecting particle impaction on a cylinder in a cross flow

An overset grid method was developed to investigate the interaction between a particle-laden flow and a circular cylinder. The method is implemented in the Pencil Code, a high-order finite-difference code for compressible flow simulation. High-order summation-by-parts operators were used at the cylinder boundary, and both bi-linear Lagrangian and bi-quadratic spline interpolation were used to communicate between the Cartesian background grid and the body-conformal cylindrical grid. The performance of the overset grid method was assessed to benchmark cases of steady and unsteady flows past a cylinder. Results show high-order accuracy and good agreement to the literature. Particle-laden flow simulations were performed, with inertial point particles impacting on a cylinder. The simulations reproduced results from the literature at a significantly reduced cost. Further, an investigation into blockage effects on particle impaction revealing that the previously published DNS data is less accurate than assumed for particles with very small Stokes numbers.     

The onset of dynamic stall revisited

Dynamic stall on a helicopter rotor blade comprises a series of complex aerodynamic phenomena in response to the unsteady change of the blade’s angle of attack. It is accompanied by a lift overshoot and delayed massive flow separation with respect to static stall. The classical hallmark of the dynamic stall phenomenon is the dynamic stall vortex. The flow over an oscillating OA209 airfoil under dynamic stall conditions was investigated by means of unsteady surface pressure measurements and time-resolved particle image velocimetry. The characteristic features of the unsteady flow field were identified and analysed utilising different coherent structure identification methods. An Eulerian and a Lagrangian procedure were adopted to locate the axes of vortices and the edges of Lagrangian coherent structures, respectively; a proper orthogonal decomposition of the velocity field revealed the energetically dominant coherent flow patterns and their temporal evolution. Based on the complementary information obtained by these methods the dynamics and interaction of vortical structures were analysed within a single dynamic stall life cycle leading to a classification of the unsteady flow development into five successive stages: the attached flow stage; the stall development stage; stall onset; the stalled stage; and flow reattachment. The onset of dynamic stall was specified here based on a characteristic mode of the proper orthogonal decomposition of the velocity field. Variations in the flow field topology that accompany the stall onset were verified by the Lagrangian coherent structure analysis. The instantaneous effective unsteadiness was defined as a single representative parameter to describe the influence of the motion parameters. Dynamic stall onset was found to be promoted by increasing unsteadiness. The mechanism that results in the detachment of the dynamic stall vortex from the airfoil was identified as vortex-induced separation caused by strong viscous interactions. Finally, a revised criterion to discern between light and deep dynamic stall was formulated.     

Direct numerical simulation of particle transport by hairpin vortices in a laminar boundary layer

The transport of solid particles by coherent wall structures is studied here. This phenomenon is present in numerous environmental and engineering flows. The flow above a wall-mounted hemisphere is used for generating hairpin vortices in a laminar boundary layer in a controlled way. By means of direct numerical simulation (DNS) of the fluid flow and simultaneous Lagrangian tracking of particles, the influence of hairpin vortices on solid particles released in the wake of the obstacle is analyzed.     

High-order resolution Eulerian–Lagrangian simulations of particle dispersion in the accelerated flow behind a moving shock

This paper presents a computational study of the two-dimensional particle-laden flow developments of bronze particle clouds in the accelerated flow behind a moving normal shock. Particle clouds with a particle volume concentration of 4% are arranged initially in a rectangular, triangular and circular shape. Simulations are performed with a recently developed high-order resolution Eulerian–Lagrangian method that approximates the Euler equations governing the gas dynamics with the improved high order weighted essentially non-oscillatory (WENO-Z) scheme, while individual particles are traced in the Lagrangian frame using high-order time integration schemes. Reflected shocks form ahead of all the cloud shapes. The detached shock in front of the triangular cloud is weakest. At later times, the wake behind the cloud becomes unstable, and a two-dimensional vortex-dominated wake forms. Separated shear layers at the edges of the clouds pull particles initially out of the clouds that are consequently transported along the shear layers. Since flows separated trivially at sharp corners, particles are mostly transported out of the cloud into the flow at the sharp front corner of the rectangular cloud and at the trailing corner of the triangular cloud. Particles are transported smoothly out of the circular cloud, since it lacks sharp corners. At late times, the accelerated flow behind the running shock disperses the particles in cross-stream direction the most for the circular cloud, followed by the rectangular cloud and the triangular cloud.     

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.     

Planktonic Communities and Chaotic Advection in Dynamical Models of Langmuir Circulation

A deterministic mechanism for the production of plankton patches within a typical medium scale oceanic structure is proposed and investigated. By direct numerical simulation of a simple model of Langmuir circulation we quantify the effects of unsteady flows on planktonic communities and demonstrate their importance. Two qualitatively different zones within the flow are identified: chaotic regions that help to spread plankton and locally coherent regions, that do not mix with the chaotic regions and which persist for long periods of time. The relative importance of these regions to both phytoplankton and zooplankton is investigated, taking into account variations in plankton buoyancy. In particular, species-specific retention zone structure is discussed in relation to variations in environmental forcing.     

Towards high-fidelity erosion prediction: On time-accurate particle tracking in turbomachinery

Erosion and fouling caused by ingested particles causes performance degradation and safety issues in turbomachinery components. Simulating these processes is a complex multiphysics and multiscale problem which has not reached a satisfactory level of maturity yet. The current state of the art approach is based on RANS solutions, which provide an averaged carrier phase on which the particles are advanced in an a posteriori manner. Upon wall impact, the particle quantities are then fed to the erosion and rebound models. In this work, we present as an alternative to this approach an Eulerian/Lagrangian simulation framework of high-order accuracy in space and time for the time-resolved prediction of particle motion in complex flows. We apply it to the LES of a particle-laden flow over a T106C low pressure turbine linear cascade. We then contrast time-averaged and time-accurate flow fields as carrier phases for the particles and highlight how the associated modeling assumptions influence the solution. Based on the particle Stokes number, we identify characteristic regimes and their interaction with the flow phase. By a detailed comparison of the particle statistics, we highlight the effects of turbulent small scale behavior and define the modeling challenges associated with finding accurate particle closure models for time-averaged simulations. This framework constitutes a first step towards high-fidelity erosion prediction for turbomachinery applications.     

Flow visualization of interactions among large coherent structures in an axisymmetric jet

Interactions between large coherent structures are visualized with both schlieren photography in two air jets and dye photography in a water jet. The density difference needed for the schlieren technique is provided by an electrically heated wire ring surrounding the jet. External forcing with either single axisymmetric, single non-symmetric, combined axisymmetric or combined non-symmetric modes was applied. It was found that forcing the jet with a pair of different spinning modes leads to azimuthal distortions of the mean flow. This observation confirms and explains existing hotwire data. Simultaneous excitation with two axisymmetric modes may produce structures of higher modes or even cause structurally undistinguishable development. Streamwise structures are observed both in the unforced jet and in the axisymmetrically forced jet. They do not seem to be caused by a Görtier instability from the concave curvature of the conventional nozzle, since they were also found in a jet flow from a specially designed nozzle with only convex contraction surface.Supported in part by the National Science Foundation under Grant No. MSM 8900086 and by the Deutsche Forschungsgemeinschaft DFG, Fi178/34-1     

Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents

Multiphase flows are ubiquitous in our daily lifeand engineering applications.It is important to investigatethe flow structures to predict their dynamical behaviors effectively.Lagrangian coherent structures(LCS) defined bythe ridges of the finite-time Lyapunov exponent(FTLE) isutilized in this study to elucidate the multiphase interactionsin gaseous jets injected into water and time-dependent turbulent cavitation under the framework of Navier-Stokes flowcomputations.For the gaseous jets injected into water,the highlightedphenomena of the jet transportation can be observed by theLCS method,including expansion,bulge,necking/breaking,and back-attack.Besides,the observation of the LCS revealsthat the back-attack phenomenon arises from the fact that theinjected gas has difficulties to move toward downstream region after the necking/breaking.For the turbulent cavitatingflow,the ridge of the FTLE field can form a LCS to capturethe front and boundary of the re-entraint jet when the adverse pressure gradient is strong enough.It represents a barrier between particles trapped inside the circulation regionand those moving downstream.The results indicate that theFTLE field has the potential to identify the structures of multiphase flows,and the LCS can capture the interface/barrieror the vortex/circulation region.     

Chaotic mixing and mixing efficiency in a short time

Several studies of the chaotic motion of fluid particles by two-dimensional time-periodic flows or three-dimensional steady flows, called Lagrangian chaos, are first introduced. Secondly, some of the studies on efficient mixing caused by Lagrangian chaos, called chaotic mixing, are reviewed with discussion of several indices for the estimation of mixing efficiency. Finally, several indices to estimate the efficiency of mixing in a short time, such as those related to transport matrices, stable and unstable manifolds of hyperbolic periodic points of Poincaré maps, and lines of separation, are explained by showing examples of mixing by two-dimensional time-periodic flows between eccentric rotating cylinders and mixing by three-dimensional steady flows in a model of static mixers.