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Volume 12, Issue 12, December 2000
 LETTERS


Selfamplification of the field of velocity derivatives in quasiisotropic turbulence
View Description Hide DescriptionWe report results of direct numerical simulations of the Navier–Stokes equations regarding the role of selfamplification of the field of velocity derivatives, i.e., involving both vorticity and strain. The main result is that even at rather moderate values of the Reynolds number the selfamplification totally dominates the process of production of the field of velocity derivatives. The role of external forcing in this process is negligible. This dominance occurs not only in the mean, but practically pointwise throughout the whole flow field. The property of selfamplification possesses a number of quantitative universal properties which are independent of the details of forcing.

 ARTICLES


Inhomogeneous viscosity fluid flow in a widegap Couette apparatus: Shearinduced migration in suspensions
View Description Hide DescriptionThe first portion of this twopart paper investigates the short time evolution of a low Reynolds number flow characterized by a spatially inhomogeneous viscosity within the annular domain between two widely separated concentric circular cylinders undergoing relative rotation. The viscosity is regarded as a material property and as such is convected with the fluid. Any possible “diffusion” of this viscosity is supposed negligible, at least in the short times of interest in our calculation. The initial viscosity field, assumed to be only slightly inhomogeneous, is expanded into a Fourier series with respect to the polar angle, and the contributions of the zeroth and first harmonics are subsequently addressed. Approximate shorttime analytic solutions for the velocity and viscosity fields are obtained. In the second part of this paper the results of the preceding analysis are employed in an attempt to gain insight into the experimentally observed shearinduced migration of particles in suspensions being sheared in a widegap Couette apparatus. The connection of the inhomogeneous viscosity problem studied in the first part to such shearinduced migration phenomena lies in the assumption that the local viscosity of a suspension of (nonBrownian) particles is functionally dependent only upon the local suspended particle volumetric fraction. In such circumstances, the local transport of suspended particles corresponds to a concomitant transport of the local suspensionviscosity and viceversa. Subject to the foregoing interpretation and limited by algebraic tractability to short times, a global radial migration is predicted. It increases with an increase in the annular gap size between the cylinders and depends upon the phase angle between the rotating outer and inner cylinders, but not upon their relative circumferential velocity—a conclusion consistent with experimental observations. Further, to leading order, particle migration is found to be independent of purely radialviscosity disturbances (the zeroth harmonic) and to arise entirely from coupling between circumferential disturbances in the velocity and viscosity (i.e., particle concentration) fields. The solution also indicates that the high shearrate region, proximate to the inner wall, may either become less viscous on average (thereby predicting net radial migration away from the high shear rate region) or, conversely, more viscous (corresponding to migration toward the high shear rate region migration). The latter case arises in circumstances that involve a large positive radial gradient in the viscosity’s first harmonic.

Regular and singular latetime asymptotes of potential motion of fluid with a free boundary
View Description Hide DescriptionWe study theoretically a potential motion of fluid with a free boundary and with no external forces. We derive and integrate the equations describing the flow evolution from the initial to a highly nonlinear stage in 2D and 3D cases and study the influence of the initial conditions in a wide range of parameters. It is shown that at late time a nonlinear structure of bubbles and spikes is generated and the structure is determined by the initial velocity field. The local dynamics of highly symmetric 3D flows has a universal form when expressed in dimensionless units. Bubbles and spikes conserve a nearcircular contour, and at a fixed length scale, the 3D solutions depend significantly on the flow symmetry. We show that the influence of the initial conditions could result in nonuniqueness of solutions describing the regular bubble.

Modeling the splash of a droplet impacting a solid surface
View Description Hide DescriptionA numerical model is used to simulate the fingering and splashing of a droplet impacting a solid surface. A methodology is presented for perturbing the velocity of fluid near the solid surface at a time shortly after impact. Simulation results are presented of the impact of molten tin, water, and heptane droplets, and compared with photographs of corresponding impacts. Agreement between simulation and experiment is good for a wide range of behaviors. An expression for a splashing threshold predicts the behavior of the molten tin. The results of water and especially heptane, however, suggest that the contact angle plays an important role, and that the expression may be applicable only to impacts characterized by a relatively low value of the Ohnesorge number. Various experimental data of the number of fingers about an impacting droplet agree well with predictions of a previously published correlation derived from application of Rayleigh–Taylor instability theory.

Steady and oscillatory thermocapillary convection generated by a bubble
View Description Hide DescriptionIn this article, we study steady and oscillatory thermocapillary and natural convective flowsgenerated by a bubble on a heated solid surface. The dynamic characteristics of the timedependent convection are captured using a combined numericalexperimental approach. The index of refraction fringe distribution patterns constructed numerically by taking an inverse Abel transform of the computed temperature fields are compared directly to the experimental Wollaston prism (WP) interferograms for both steadystate and oscillatory convection. The agreement between numerical predictions and experimental measurements is excellent in all cases. It is shown that below the critical Marangoni number, steadystate conditions are attainable. With increasing Ma, there is a complete transition from steady state up to a final nonperiodic fluctuating flow regime through several complicated symmetric and asymmetric oscillatory states. The most prevalent oscillatory mode corresponds to a symmetric up and down fluctuation of the temperature and flow fields associated with an axially traveling wave. Careful examination of the numerical results reveals that the origin of this class of convective instability is closely related to an intricate temporal coupling between largescale thermal structures which develop in the fluid in the form of the cold return flow and the temperature sensitive surface of the bubble. Gravity and natural convection play an important role in the formation of these thermal structures and the initiation of the oscillatory convection. Consequently, at lowg, the time evolution of the temperature and flow fields around the bubble are very different from their 1g counterparts for all Marangoni numbers.

Transitions from Taylor vortex flow in a corotating Taylor–Couette system
View Description Hide DescriptionThe stability of the Taylor vortex flow in the periodic Taylor–Couette problem with corotating cylinders is examined. Transitions to twisted and wavy twisted vortices and wavy inflow and outflow boundary flows are considered. Marginal stability curves for the transition from Taylor to twisted and wavy twisted vortices have been calculated. The azimuthal wave number and the phase velocity at their onset have also been obtained. To compare with experiments and previous numerical works for the narrow gap approximation, the case of radius ratio 0.883 is analyzed in detail. An explanation for the increase in the azimuthal wave number of the twisted vortices as the Reynolds number of the inner cylinder is increased is provided. The velocity fields of twisted vortices, wavy twisted vortices, wavy inflow, and wavy outflow boundary flows at their onset are also shown.

Chaotic advection and relative dispersion in an experimental convective flow
View Description Hide DescriptionLagrangian motion in a quasitwodimensional, timedependent, convective flow is studied at different Rayleigh numbers. The particle tracking velocimetry technique is used to reconstruct Lagrangian trajectories of passive tracers. Dispersion properties are investigated by means of the recently introduced finite size Lyapunov exponent analysis. Lagrangian motion is found to be chaotic with a Lyapunov exponent which depends on the Rayleigh number as The power law scaling is explained in terms of a dimensional analysis on the equation of motion. A comparative study shows that the fixed scale method makes more physical sense than the traditional way of looking at the relative dispersion at fixed times.

Transient recirculation in a slowly varying tube impulsively rotated about its axis with constant angular velocity
View Description Hide DescriptionFor time viscous fluid is in slow flow through a long straight axially symmetric tube whose radius, varies slowly with axial distance, When the tube is impulsively rotated about its axis with angular velocity, at which angular speed it is thereafter maintained. During the transition from zero angular velocity, when to solid body rotation, when the flow in the tube can briefly exhibit striking physical behavior, markedly different from the flow in the stationary tube. We present a linearization of the Navier–Stokes equations, valid when the Blasius parameter ε, which governs the magnitude of the inertial forces, tends to zero and the swirl parameter, λ, which is the ratio of a representative tube wall velocity, to a representative axial velocity, tends to infinity, with the product held fixed. An analytic solution suitable for computation and valid for suitably large is presented and streamlines are plotted for a typical diverging and a typical converging tube at time when The relevance of the results to the phenomenon of vortex breakdown in tubes is discussed.

Numerical study of singularity formation in a class of Euler and Navier–Stokes flows
View Description Hide DescriptionWe study numerically a class of stretched solutions of the threedimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudospectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier–Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the twodimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.

Accelerations in isotropic and homogeneous turbulence and Taylor’s hypothesis
View Description Hide DescriptionThe validity of Taylor’s hypothesis is analyzed by comparing the root mean square (rms) values of full (Lagrangian) and inertial accelerations in an isotropic and homogeneous turbulent flow. Full, local, and inertial accelerations in turbulence were decomposed into solenoidal and potential components, which made it possible to avoid dealing, at least directly, with the pressuregradient term in the Navier–Stokes equation. The evaluations of the correlation functions and spectra of the accelerations are presented. These evaluations have been obtained using the Batchelor [Proc. Cambridge Philos. Soc. 47, 359 (1951)] longitudinal structure function that describes statistical properties of the turbulent velocity field. This function is equally valid for both inertial and dissipative subranges. It was shown that the ratio of the rms values of the full and inertial accelerations depends on the Reynolds number only and decreases at large as At of about 20 this ratio is close to 0.72. At of 1000 the ratio is less than 0.1. The validity of Taylor’s hypothesis depends on the ratio of the rms values of the accelerations. The results indicate that Taylor’s hypothesis is valid for large (exceeding about 1000) and becomes questionable at below 100. At large the full acceleration in homogeneous and isotropic turbulence turned out to be independent of the Reynolds number.

Estimation of the Kolmogorov constant by direct numerical simulation of a continuous scalar
View Description Hide DescriptionThe coefficient which determines the effective turbulent diffusion in velocity space, is fundamental in Lagrangianmodeling. Others, for example, Yeung and Pope [J. Fluid Mech. 207, 531 (1989)] have investigated this coefficient numerically with direct numerical simulation (DNS) by analyzing the dispersion of discrete particles. In our paper we estimate the coefficient by using DNS to study the initial evolution of continuous passive scalar fields. Using an equivalent probability density function (pdf) and conditional moment analyses we examine the initial transient behavior for passive scalar mixing, with both linear and Gaussiantype initial distributions in the velocity space. Our estimates of are found to be consistent with the data found in the literature.

Linear predictive filtering in a numerically simulated turbulent flow
View Description Hide DescriptionA numerical investigation is made into the use of linear predictive (Wiener) filters to predict flow quantities in the nearwall region of a turbulent boundary layer for use in active control algorithms. Optimal filters for the prediction of Reynolds stress and fluctuating streamwise velocity components using wallshear stress information are developed and their dependence on the number and location of shear sensors is explored. It is found that a densely populated, wallbased, sensor system can predict the nearwall Reynolds stress with good accuracy, and that 76% of the optimal performance can be achieved with as few as four sensors whose locations coincide with the strongest weights of the Wiener filter derived using a very dense network of input sensors.

Successive generation of sounds by shock–strong vortex interaction
View Description Hide DescriptionThe development of a flow field and the generation of sound due to the interaction between a vortex ring and a shock wave are studied numerically. The axisymmetric, unsteady, compressible Navier–Stokes equations are solved by a finite difference method. The results show that, when the vortex ring moves in the opposite direction to the shock wave, the interaction produces reflected shock waves, first. Then, the reflected shock wavesinteract with the vortex ring, and new rarefaction and compression waves are produced. The new compression waves interact with the vortex ring again, resulting in the further generation of rarefaction and compression waves. As the strength of the vortex ring is increased, this process is repeated and the pressure waves are generated successively.

On a small structure in velocity field within a contact region
View Description Hide DescriptionIn a contact region (contact surface in Euler terms) connecting the two uniform regions in a flow field such as caused by the sudden break of a membrane in a shocktube, there exists a small structure in velocity field, which has been reported earlier by the calculation based on the Boltzmann equation of the BGK type. Probably, the accepted view would be that the velocity within this region is uniform. Here this velocity structure is investigated thoroughly based on not only the Boltzmann equation but also the Navier–Stokes equations. Actually the velocity field has a hump or a pit, the width of which is found to be the same as the thickness of the region. The magnitude of the hump or the pit decreases with time, inversely proportional to the square root of time. At the socalled tailoring condition at which the contact region has so far been thought not to manifest itself, actually it does exist; the velocity structure does not appear of course but the temperature and, hence, the density still have structures, although they are small in magnitude.

Vortex morphologies on reaccelerated interfaces: Visualization, quantification and modeling of one and twomode compressible and incompressible environments
View Description Hide DescriptionWe examine the vortex dynamics and interfacial evolution of “reacceleration” and “reshock” for single interface densitystratified fluid (Richtmyer–Meshkov) environments. In the former case, we simulate, visualize, and quantify the parameter range of the free falling tank laboratory experiment of J. Jacobs and C. F. Niederhaus, where the impulsive subsequent acceleration of the interface occurs after several rolls from the initial impulsive acceleration. We interpret the rapid onset of chaotic motion in the rollup region as due to the formation of intertwined closelaying layers of oppositesigned vorticity. In the latter case, we make compressible simulations at low Mach number and compare with incompressible simulations. We juxtapose the results and find an excellent agreement in large and intermediate size features and their magnitudes before the “reshock.” At the “reshock,” discrepancies arise due to highly compressible regime. The simulations were made with incompressible vorticitybased methods, vortexincell (VIC) and vortex blob (VB), and compressible secondorder Godunov codes.

Stability of shock wave reflections in nonequilibrium steady flows and hysteresis
View Description Hide DescriptionIn the present work we have addressed the issue of the stability of shock wave reflection in the presence of vibrational and chemical relaxation phenomena and its relation with the occurrence of the hysteresis. In order to better understand the physics of the shock wave reflections we have first formulated an evolution equation for the entropy of a mixture of gases in thermal and chemical nonequilibrium by invoking the shifting equilibrium assumption and the concepts of irreversible thermodynamics, and assuming (i) that all diatomic molecules behave as harmonic oscillators; and (ii) finite rate chemistry. A perturbation analysis of the total entropy evolution equation has then been carried out to analyze the stability of shock wave configurations (either regular or Mach) both for ideal and real gases. The analysis shows that a Mach reflection is more stable than a regular one; furthermore, its stability is enhanced by nonequilibrium effects. In order to clarify the occurrence of the hysteresis phenomenon in light of the conclusions reached through the stability analysis, we have also carried out multidimensional simulations (both at flight and wind tunnel conditions) by developing a pseudotransient procedure to span a (hysteresis) loop dual solution domain Mach reflection domain dual solution domain. The simulations show that the total entropy of the system exhibits an abrupt change along the path dual solution domain Mach reflection domain, while it is continuous along the reverse path. An argument is then developed to prove that hysteresis is the natural consequence of the different stability properties of regular and Mach reflections and the Prigogine minimum total entropy production principle.

 BRIEF COMMUNICATIONS


On the deviatoric normal stress on a slip surface
View Description Hide DescriptionA general formula for the deviatoric normal stress on a slip surface for incompressible flows is derived and its application is discussed.

Doubleperiodic arrays of vortices
View Description Hide DescriptionAnalytical solutions to the sinhPoisson equation are discussed. This equation plays a role in the theory of vortex dynamics [Mallier and Maslowe, Phys. Fluids A 5, 1074 (1993)] and in the discussion of the most probable states of inviscid twodimensional flows in fluids and plasmas [Montgomery and Joyce, Phys. Fluids17, 1139 (1974)]. We present a family of doubleperiodic solutions on a rectangular grid. In limiting cases these solutions reproduce Mallier–Maslowe vortex streets and arrays of Greenhill’s point vortices.

Mixing in twodimensional vortex interactions
View Description Hide DescriptionWe examine the mixing of a passive tracer initially contained within several widely separated vortex patches in a twodimensional, nearly inviscid, incompressible flow. The initial vortex positions and sizes are chosen so that they collapse toward a common center, resulting in a strong interaction. The area of tracer ejected from the vortices is found to be well correlated with the departure of each tracer contour from an ideal elliptical shape. This result appears to be more widely applicable, and, in particular, it may be useful for quantifying smallscale mixing in realistic atmospheric and oceanic flows.

The criteria for the onset of doublediffusive instabilities at a vertical boundary
View Description Hide DescriptionWhen a body of fluid with a vertical salinity gradient is heated at a vertical boundary instabilities can sometimes appear. Several criteria for when these instabilities should be observed have been proposed, based on experiments and theory. Four different criteria will be examined which would appear to be mutually incompatible. Each of the experimentally derived criteria provides a good measure for the onset of instability in the corresponding experiments. When compared to the theoretical predictions the experimentally derived criteria are all shown to be compatible with the theory if the constraints of the experiments are included, but all would be expected to fail if the parameters of the respective experiments were extended.
