On the flow of a viscous fluid driven along a channel by suction at porous walls

This paper is a theoretical treatment of the flow of a viscous incompressible fluid driven along a channel by steady uniform suction through porous parallel rigid walls. Many authors have found such flows when they are symmetric, steady and two-dimensional, by assuming a similarity form of solution due to Berman in order to reduce the Navier-Stokes equations to a nonlinear ordinary differential equation. We generalise their work by considering asymmetric flows, unsteady flows and three-dimensional perturbations. By use of numerical calculations, matched asymptotic expansions for large values of the Reynolds number, and the theory of dynamical systems, we find many more exact solutions of the Navier-Stokes equations, examine their stability, and interpret them. In particular, we show that most previously found steady solutions are unstable to antisymmetric two-dimensional disturbances. This leads to a pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, stable time-periodic solutions, stable quasi-periodic solutions, phase locking and chaos in succession as the Reynolds number increases.     

The structure and dynamics of reacting plane mixing layers

The reacting two-dimensional plane mixing layer has been studied in two configurations: a rearward facing step and a two-stream mixing layer. Observations have been made of the steady state behavior, and the unsteady behavior when the flow was forced by a specific acoustic frequency. The steady behavior of the mean properties of the reacting flows is similar to that of non-reacting free shear flows except for the global effects of thermodynamic property changes. The structure of these flows is qualitatively similar to that of non-reacting flows. Vortices form by the two-dimensional Kelvin-Helmholtz instability and grow by subharmonic combination until the mixing layer interacts with the walls. Entrainment is dominated by the two-dimensional vortex motion. Three-dimensional instabilities give rise to secondary vortices which are coherent over several Kelvin-Helmholtz structures and dominate the fine scale mixing process. The mixing transition corresponds to a loss of coherence in the layer. Unsteady behavior occurs when there are resonant interactions with the Kelvin-Helmholtz instability or the instability associated with the recirculation vortex in the rearward facing step flow. Modeling efforts are reported which show promise of simulating the essential features of plane mixing layers.A version of this paper was presented at the ASME Winter Annual Meeting of 1984 and printed in AMD-Vol. 66     

圆柱绕流的非线性动力学

本文利用非线性动力学的概念和方法来研究粘性不可压缩流体的二维圆柱绕流问题．计算了定常流失稳以及出现混沌的临界雷诺数，并估计了混沌解的一些统计特征．     

The route to chaos for the Kuramoto-Sivashinsky equation

We present the results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. Our concern is with the asymptotic nonlinear dynamics as the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic (in time) attractor, to another steady fully modal attractor, to another time-periodic attractor, to a steady tetramodal attractor, to another time-periodic attractor having a full sequence of period-doublings (in the parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the lengths of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period-doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4.669201609.... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.This research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-18605 while the authors were in residence at the Institute of Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. Additional support for the second author was provided by ONR Grant N-00014-86-K-0691 while he was at UCLA.     

Fluid mixing induced by vibrating walls

Recent progress in micro-fluid dynamics has identified an increased demand for efficient mixing of highly viscous fluids in small channels and cavities. One way to do this is through the steady streaming generated by the vibration of solid boundaries. In this paper we investigate the mixing properties of such streaming flows in an infinite channel. A Newtonian fluid is confined within flexible walls with transverse motion in the form of standing waves of small amplitude. The velocity field is determined using a perturbation approach with the slope of the wall as a small parameter [Phys. Fluids 16 (2004) 1822]. Streaming occurs at second order with the formation of cellular flow patterns in the channel. The Lagrangian velocities were found to mimic the Eulerian except for flows at large channel half-widths and low frequencies. Most effective mixing is observed for flows at channel half-widths of similar, or lower, order than the vibratory wavelength and for sufficiently high frequencies.     

Thermal chaotic mixing of power-law fluids in a mixer with alternately rotating walls

We investigate the enhancement of both mixing and heat transfer in a two-rod mixer for highly viscous non-Newtonian fluids. The mixer is composed of two vertical circular rods in a cylindrical tank. Chaotic flows are obtained by imposing the temporal modulations of the rotational velocities of the walls. We study the effects of different stirring protocols, which lead to non-chaotic and chaotic flows, on the efficiency of both mixing and heat transfer for three different rheological fluid behaviors: shear-thinning, Newtonian and shear-thickening. For this purpose, we use statistical indicators that characterize the mean value of the fluid temperature and its homogenization. We find that chaotic mixing is suitable for shear-thickening fluids for which we observe a clear enhancement of the thermal mixing (heat extraction and homogenization). This is due to the increase in the apparent fluid viscosity in the vicinity of the rotating walls. This aspect confirms the relevance of chaotic mixing for highly viscous fluids.     

Predicting transport by Lagrangian coherent structures with a high-order method

Recent developments in identifying Lagrangian coherent structures from finite-time velocity data have provided a theoretical basis for understanding chaotic transport in general flows with aperiodic dependence on time. As these theoretical developments are extended and applied to more complex flows, an accurate and general numerical method for computing these structures is needed to exploit these ideas for engineering applications. We present an unstructured high-order hp/spectral-element method for solving the two-dimensional compressible form of the Navier–Stokes equations. A corresponding high-order particle tracking method is also developed for extracting the Lagrangian coherent structures from the numerically computed velocity fields. Two different techniques are used; the first computes the direct Lyapunov exponent from an unstructured initial particle distribution, providing easier resolution of structures located close to physical boundaries, whereas the second advects a small material line initialized close to a Lagrangian saddle point to delineate these structures. We demonstrate our algorithm on simulations of a bluff-body flow at a Reynolds number of Re = 150 and a Mach number of M = 0.2 with and without flow forcing. We show that, in the unforced flow, periodic vortex shedding is predicted by our numerical simulations that is in stark contrast to the aperiodic flow field in the case with forcing. An analysis of the Lagrangian structures reveals a transport barrier that inhibits cross-wake transport in the unforced flow. The transport barrier is broken with forcing, producing enhanced transport properties by chaotic advection and consequently improved mixing of advected scalars within the wake.     

多相流局部混合型质点网格法

提出模拟多相流的局部混合型质点网格法,该方法能稳定地模拟高浓度流体颗粒两相 流. 在每一个颗粒团尺度的欧拉网格下(本文称之为欧拉微元),基于Lagrangian 追踪原理,可直接估计体积内颗粒的总量,从而准确求出欧拉微元和控制容积内颗粒的 浓度(即颗粒在容积内的体积含量). 同时,假设在新的时间步下,颗粒在欧拉微元里充分 混合,形成新颗粒团. 作者对竖直和倾斜容器中单粒径颗粒沉降和竖直容器中双粒径颗粒双 峰悬浮液沉降过程进行了计算,结果与实测数据相符.     

Convection in a low Prandtl number fluid layer rotating about a vertical axis

Two- and three-dimensional convection flows in a horizontal layer of a low Prandtl number fluid heated from below and rotating about a vertical axis are studied numerically with a Galerkin method. Solutions for subcritical steady finite amplitude convection and convection in the form of standing oscillations are obtained. Parameter regimes that appear to be attainable in laboratory experiments have been emphasized. The stability of subcritical two-dimensional steady convection has been investigated and three-dimensional chaotic states of convection have been found.     

Lagrangian Transport and Chaotic Advection in Two-Dimensional Anisotropic Systems

Enhanced geothermal systems (EGSs) are a promising concept to make geothermal power generation available in many regions worldwide. However, in current EGSs generally only a fraction of the geothermal reservoir is effectively accessed by the production fluid, resulting in suboptimal performance. Recent studies in the literature on a two-dimensional Darcy flow in a circular reservoir driven by reoriented injector–producer wells demonstrated that well configurations and pumping schemes designed on the basis of chaos theory enable efficient distribution of the production fluids throughout the entire reservoir. Key to this is accomplishment of chaotic advection, i.e. the rapid dispersion and stretching of material fluid regions, by a “proper” flow forcing. However, these studies concern isotropic reservoirs, while geothermal reservoirs typically are highly anisotropic. Our theoretical/computational study expands on said studies by investigating the impact of anisotropy on fundamental aspects of the Lagrangian transport of production fluids. This reveals that anisotropy generically eliminates key organizing mechanisms in the Lagrangian transport, viz. symmetries, and thus tends to promote disorder and, inherently, chaotic advection. However, symmetries are partially preserved—and thus order and coherence partially restored—in non-generic cases such as pumping schemes employing an even number of injector–producer wells and well configurations aligned with the anisotropy. Symmetry associated with well alignment in fact appears crucial to an intriguing “order within chaos” observed only in such cases: prolonged confinement of fluid to subregions of chaotic areas.     

The Influence of Periodic Excitation on a Turbulent Separation Bubble

Turbulent separation limits the performance in many engineering applications, for example creating pressure losses in diffuser like flows or stall on aircraft wings. In the present study the turbulent boundary layer flow over a flat plate separating due to an adverse pressure gradient is studied as a model problem and the effect of periodic excitation in both time and space is investigated through direct numerical simulations. Linear stability analysis is used to analyse the sensitivity of the flow with respect to time-periodic excitations. The dependence on position, amplitude and frequency of the forcing is investigated. For a certain frequency range at sufficiently high amplitudes, it is possible to eliminate the separated region. Furthermore, three-dimensional effects are studied by applying a steady spanwise forcing as well as a both time-dependent and spanwise varying forcing. A forcing varying in spanwise direction is shown to be the most effective in eliminating the separated region, whereas two-dimensional time-periodic excitation was not as efficient as it was expected.     

Instantaneous three-dimensional flow visualization of a supersonic mixing layer

An imaging technique capable of time-resolved, three-dimensional visualizations of compressible flows is described and applied to a supersonic mixing layer. The three-dimensional planar imaging system uses a custom high-speed camera to acquire 10 successive planar images through the mixing layer at a rate of 10⁷ frames per second. Mixed fluid in the layer is visualized by Mie scattering of a laser light sheet from condensed alcohol droplets. After collection, the planar images are corrected for distortions and stacked to form data volumes. Comparative visualizations at low and moderate convective Mach numbers (M _c = 0.43 and 0.62) are used to examine the effects of compressibility on large-scale structure in mixing layers. The visualizations graphically reveal the shift from two-dimensional spanwise rollers to three-dimensional structure, such as oblique and V-shaped bands, with increasing compressibility. Additionally, direct comparison between the high- and low-speed edges of the mixing layer shows the high-speed interface to be smoother than its lowspeed counterpart.The authors would like to acknowledge Dr. Michael Miller and Dr. Jerry Seitzman for many helpful discussions, and William Urban for assistance in performing the experiments. This work is supported by the Air Force Office of Scientific Research, Aerospace Sciences Directorate, J. Tishkoff technical monitor.     

Development of a microfluidic device for simultaneous mixing and pumping

We conducted experimental and numerical studies aimed at developing a microfluidic device capable of simultaneous mixing while pumping. The proposed multifunctional device makes use of alternating current electroosmotic flow and adopts an array of planar asymmetric microelectrodes with a diagonal or herringbone shape. The pumping performance was assessed in terms of the fluid velocity at the center of the microchannel, obtained by micro PIV. To assess the mixing, flow visualizations were carried out over the electrodes to verify the lateral flows. The mixing degree was quantified in terms of a mixing efficiency obtained by three-dimensional numerical simulations. The results showed that simultaneous mixing and pumping was achieved in the channels with diagonal or herringbone electrode configurations. A herringbone electrode configuration showed better pumping compared with a reference, as well as enhanced mixing.     

On identification of flow separation and vortices in internal periodic flows

To permit simplified analysis of complex time-dependent flows, possible relationship between the near-wall flow, flow separation and vortices are studied numerically for a flow in a constricted two-dimensional channel. The pulsating incoming wave-form consists of a steady flow, followed by a half-sinus flow superimposed on the steady component. One pair of vortices is created in each cycle, one vortex near each wall. The vortices propagate downstream in the next cycles, promoting flow separation as they move. Existing flow separation criteria were not found to be uniformly valid. A relation between the near-wall flow and the vortical system exists only during the steady incoming flow phase of the cycle. It seems that local criteria of flow separation cannot be found for complex internal pulsating flow fields. However, the vorticity field can be utilized, even in complex time-periodic flows, for identifying vortices that have been formed by the roll-up of shear layers.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments

A direct method for analyzing diapycnal mixing in a stably stratified fluid (Winters et al., 1995) has been applied to the stably stratified shear layer. The diapycnal flux and mixing efficiency are computed as functions of time, whatever the turbulent activity in the fluid. The mixing properties of two- and three-dimensional numerical simulations of the Boussinesq equations are analyzed and compared. The interest of the former simulations is to emphasize the fundamental role of three-dimensional effects in fluid mixing and to quantify it. We focus on the influence of stratification (measured by the minimum Richardson number J) and changes in Prandtl number on the overall mixing that occurs as the computed flows evolve from unstable initial conditions.

In three dimensions, the flow dynamics exhibit three successive stages, each with different mixing properties. During the first stage, a primarily two-dimensional Kelvin–Helmholtz instability develops and the mixing efficiency is high (the flux Richardson number Rf_b ranges between 0.37 and 0.68, decreasing as J increases). The second stage is characterized by the development of small-scale three-dimensional instabilities. These motions result in significantly higher diapycnal flux than during the first stage but in only moderate mixing efficiency (Rf_b0.32), as the rate of kinetic energy dissipation is also high during this stage. Finally, the turbulent activity is progressively expulsed toward the outer regions of the shear layer and decays in time while the central region relaminarizes. During this final stage, Rf_b approaches an asymptotic value close to 0.25 and the diapycnal diffusivity displays a clear functional dependence on a gradient Richardson number Ri_b of the form Ri_b⁻².

As expected, the two-dimensional flows are unable to reproduce the mixing properties of the flow, except during the first stage. During the subsequent turbulent regime, both the diapycnal flux and the dissipation rate of kinetic energy are too small (because, for the latter quantity, of the nonlinear enstrophy conservation constraint). The final stage consists in a quasi-stationary weakly turbulent regime, for which the diapycnal diffusivity behaves as Ri_b⁻¹. It should be noted that, despite these differences, Rf_b relaxes toward the 0.25 value found in three dimensions.     

Experimental and numerical investigation of thermal chaotic mixing in a T-shaped microchannel

In this paper, thermal chaotic mixing characteristics of two miscible fluids in a T-shaped microchannel are investigated experimentally and numerically. In the experiments, fluorescent dye Acid Yellow and Rhodamine B was employed to show the mass mixing behavior and temperature field, respectively. Power input and flow rate were studied to investigate the thermal mixing characteristics in the microchannel. The mixing efficiency increases with increasing power input, while decreases with increasing flow rate. A numerical simulation of conjugate forced convection-conduction heat and mass transfer was employed to investigate the thermal chaotic mixing processes in the T-shaped microchannel. The measured mixing efficiency versus applied voltage and flow rate were compared with numerical simulation results, which showed reasonably agreement.     

Flow and mixing characteristics of pulsed confined opposed jets in turbulent flow regime

A numerical study is performed on a two-dimensional confined opposed-jet configuration to gain basic understanding of the flow and mixing characteristics of pulsed turbulent opposed-jet streams. The sinusoidal pulsating flows with different temperature are imposed at opposed-jet inlets, which are mixed with each other in a confined flow channel. The current mathematical model taking the effect of temperature-dependent thermo-physical properties of fluid into account can present a good prediction for opposed-jet streams compared with experimental data. The numerical results indicate that introduction of temperature difference between opposed jet flows can lead to an asymmetric flow field immediately after jet impact, and the sinusoidal flow pulsations can effectively enhance mixing rate of opposed jets. Parameter studies are conducted for optimization of pulsed opposed jets. The effect of Reynolds number and flow pulsation as well as the configuration geometry on the mixing performance are discussed in detail. Examination of the flow and thermal field shows that the mixing rate is highly dependent on the vortex-induced mixing and residence time of jet fluid in the exit channel.     

Performance of a multigrid calculation procedure in three-dimensional sudden expansion flows

The performance of a recently developed calculation procedure for steady incompressible flows is assessed in a variety of three-dimensional sudden expansion type flows representative of those encountered in several types of industrial equipment. The calculation procedure, called here BLIMM (for block-implicit multigrid method), is based on a coupled solution of the three-dimensional momentum and continuity equations in primitive variables, using the multigrid technique. Different Reynolds numbers and finite difference grids are considered for each flow situation. The rates of convergence and the computational times are reported for each case.