Volume 15, Issue 2, February 2003
 LETTERS


A simple model for selfsustained oscillations in homogeneous shear flow
View Description Hide DescriptionGeneration of the largescale coherent vortical structures in homogeneous shear flow couples dynamical processes of energy and enstrophy production. In the large rate of strain limit, the simple estimates of the contributions to the energy and enstrophy equations result in a dynamical system, describing experimentally and numerically observed selfsustained nonlinear oscillations of energy and enstrophy. It is shown that the period of these oscillations is independent of the box size and the energy and enstrophy fluctuations are strongly correlated.

Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box
View Description Hide DescriptionHighresolution direct numerical simulations (DNSs) of incompressible homogeneous turbulence in a periodic box with up to grid points were performed on the Earth Simulator computing system. DNS databases, including the present results, suggest that the normalized mean energy dissipation rate per unit mass tends to a constant, independent of the fluid kinematic viscosity ν as The DNS results also suggest that the energy spectrum in the inertial subrange almost follows the Kolmogorov scaling law, where is the wavenumber, but the exponent is steeper than −5/3 by about 0.1.
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 ARTICLES


Control of vortex breakdown by addition of nearaxis swirl
View Description Hide DescriptionWe present a new method for controllingvortex breakdown (VB) via addition of co or counterrotation near the axis. Corotation is adequate to totally suppress VB, whereas counterrotation increases the number and size of VB “bubbles” and makes the flow unsteady. We study these effects in a closed cylindrical container, in which a rotating end disc drives the base flow; an independently rotating central rod (with rod radius≪disk radius) is employed to control VB. This flow, being free of ambient disturbances, is well suited for understanding both the VB mechanism and its control; the present work appears to be the first to study VB control. We develop and explain our control strategy using flow visualization and simple analytical reasoning. Our results suggest that an additional co or counterrotation, applied near the vortex axis, can be effective in suppressing or enhancing VB in practical flows.

The velocityscalar cross spectrum of stretched spiral vortices
View Description Hide DescriptionThe stretchedspiral vortexmodel is used to calculate the velocityscalar cross spectrum for homogeneous, isotropic turbulence in the presence of a mean scalar gradient. The only nonzero component of the cospectrum is that contributed by the velocity component in the direction of the imposed scalar gradient while the quadrature spectrum is identically zero, in agreement with experiment. For the velocity field provided by the stretchedspiral vortex, the velocityscalar spectrum can be divided into two additive components contributed by the velocity components along the vortex axis, and in the plane normal to this axis, respectively. For the axial velocity field, a new exact solution of the scalar convectiondiffusion equation is found exhibiting scalar variation in the direction of the vortex tube axis. An asymptotic expression was found for the cospectrum contributed by this solution and the axial velocity, with the leading order term showing a range. This term is produced by the winding of the initial axial velocity field by the axisymmetric vortex core. The next order term gives a range, and arises from the lowest order effect of the nonaxisymmetric vorticity on the evolution of the axial velocity. Its coefficient can be of either sign or zero depending on the initial conditions. The contribution to the cospectrum from the velocity in the plane of the vortex is also calculated, but no universal high wave number asymptotic form is found. The integrals are evaluated numerically and it is found that the resulting cospectrum does not remain of one sign. Its form depends on the choice of the vortex core velocity profile and time cutoff in the spectral integrals. The onedimensional cospectrum contributed by the axial velocity is compared with the experimental data of Mydlarski and Warhaft [J. Fluid Mech. 358, 135–175 (1998)].

Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields
View Description Hide DescriptionThe Hamiltonian structure of the inhomogeneous layer models for geophysicalfluid dynamics devised by Ripa [Geophys. Astrophys. Fluid Dyn. 70, 85 (1993)] involves the same Poisson bracket as a Hamiltonian formulation of shallow water magnetohydrodynamics in velocity, height, and magnetic flux function variables. This Poisson bracket becomes the Lie–Poisson bracket for a semidirect product Lie algebra under a change of variables, giving a simple and direct proof of the Jacobi identity in place of Ripa’s long outline proof. The same bracket has appeared before in compressible and relativistic magnetohydrodynamics. The Hamiltonian is the integral of the three dimensional energy density for both the inhomogeneous layer and magnetohydrodynamic systems, which provides a compact derivation of Ripa’s models.

On multiple states in singlelayer flows
View Description Hide DescriptionFor free surface flows over obstacles in a channel of constant width, there is a range of parameter values where two steady flow states are possible, with the state that is actually obtained being determined by the past history. Specifically, one of these flow states is wholly supercritical (i.e., no waves can propagate upstream against the flow) over the obstacle. The other has a hydraulic jump that travels to upstream infinity, and the flow undergoes a subcritical (i.e., waves can propagate in both directions) to supercritical transition at the obstacle crest. A new, third steady solution is described here, in which a hydraulic jump is stationary over the upstream face of a long obstacle. This new solution is contiguous with the other two, and in a sense, lies between them. It is shown that this new solution is unstable, in that if the stationary jump is displaced to a location with a slightly different bottom height, it will move further in the same direction. By this criterion, jumps are unstable on upslope flow, and stable on downslope flow. These properties, and the general character of hysteresis implied by these multiple hydraulic equilibria, have been tested with two series of experiments. The new solution is ordinarily not found because of its instability, but it can be viewed by manually balancing the unstable jump. Comparisons were also made between the observed abrupt transitions between flow states, and the predictions of hydraulic theory. Qualitatively the agreement is quite good, with differences attributable to experimental factors that are not contained in twodimensional long wave hydraulics.

Thermocapillary migration of long bubbles in polygonal tubes. II. Experiments
View Description Hide DescriptionWe study experimentally the thermocapillary migration of a long gas bubble in a horizontal pipe of rectangular cross section. An imposed axial temperature gradient produces a gradient of surface tension leading to a steady migration of the bubble towards the hotter region. The bubble velocity is found to be independent of bubble length for sufficiently long bubbles, and to vary linearly with the temperature gradient. For pipes of small vertical dimension, gravity is negligible and the measurements of the bubble velocity are in good agreement with the zero gravity theory of Mazouchi and Homsy [Phys. Fluids 13, 1594 (2001)]. In larger pipes, gravity is no longer negligible and the bubble is observed to move faster than that theory predicts. A simple calculation taking into account the combined effect of buoyant rise and thermocapillary stress accounts for these results.

On the physical mechanisms of twoway coupling in particleladen isotropic turbulence
View Description Hide DescriptionThe objective of the present study is to analyze our recent direct numerical simulation (DNS) results to explain in some detail the main physical mechanisms responsible for the modification of isotropic turbulence by dispersed solid particles. The details of these twoway coupling mechanisms have not been explained in earlier publications. The present study, in comparison to the previous DNS studies, has been performed with higher resolution and considerably larger number (80 million) of particles, in addition to accounting for the effects of gravity. We study the modulation of turbulence by the dispersed particles while fixing both their volume fraction, and mass fraction, for three different particles classified by the ratio of their response time to the Kolmogorov time scale: microparticles, critical particles, large particles, Furthermore, we show that in zero gravity, dispersed particles with (denoted here as “ghost particles”) modify the spectra of the turbulence kinetic energy and its dissipation rate in such a way that keeps the decay rate of the turbulence energy nearly identical to that of particlefree turbulence, and thus the twoway coupling effects of these ghost particles would not be detected by examining only the temporal behavior of the turbulence energy of the carrier flow either numerically or experimentally. In finite gravity, these ghost particles accumulate, via the mechanism of preferential sweeping resulting in the stretching of the vortical structures in the gravitational direction, and the creation of local gradients of the drag force which increase the magnitudes of the horizontal components of vorticity. Consequently, the turbulence becomes anisotropic with a reduced decay rate of turbulence kinetic energy as compared to the particlefree case.

Dean vorticesinduced enhancement of mass transfer through an interface separating two immiscible liquids
View Description Hide DescriptionTwofluid Dean vortex flow in a coiled pipe with vanishing torsion, and its effect on the mass transfer through the liquid–liquid interface of two immiscible fluids are studied numerically. The liquids are stratified by gravity, with the denser one occupying the lower part of the pipe. The Navier–Stokes equations in both fluid layers are solved numerically by the finite volume method. The results reveal a detailed structure of the transverse flow (the Dean vortices) in coiled pipes with the dimensionless curvature 0.1. Both cocurrent and countercurrent axial flows in the fluid layers are considered. Using the flow fields predicted, the mass transferequation is solved. It is shown that the mass transfer of a passive scalar (say, a protein with the Schmidt number of the order of through the interface can be significantly enhanced by the Dean vortices, so that the mass transfer rate can be increased by three to four times. This makes the Dean vortex flow an effective tool for mass transfer enhancement at the liquid–liquid interface. It is shown that the Dean flow provides a stronger mixing than the Taylor–Couette flow. It is also shown that there exists an optimal axial flow rate in terms of this enhancement. The optimal flow corresponds to the value of the Dean number of about 180. In the countercurrent flow case the Dean vortices can split, which has a negative effect on the mass transfer enhancement. Both the cocurrent and countercurrent axial flows yield a similar enhancement effect on the interfacial mass transfer rate. The problem is related to the search for novel bioseparator devices.

Channel flow induced by wall injection of fluid and particles
View Description Hide DescriptionThe Taylor flow is the laminar singlephase flow induced by gas injection through porous walls, and is assumed to represent the flow inside solid propellant motors. Such a flow is intrinsically unstable, and the generated instabilities are probably responsible for the thrust oscillations observed in the aforesaid motors. However particles are embedded in the propellants usually used, and are released in the fluid by the lateral walls during the combustion, so that there are two heterogeneous phases in the flow. The purpose of this paper is to study the influence of these particles on stability by comparison with stability results from the singlephase studies, in a plane twodimensional configuration. The particles are supposed to be chemically inert and of a uniform size. In order to carry out a linear stability study for this flow modified by the presence of particles, the mean particle velocity field is first determined, assuming that only the gas exerts forces on the particles. This field is sought in a selfsimilar form, which imposes a limit on the size of the particles. However, the particle mass concentration cannot be obtained in a selfsimilar form, but can only be described by a partial differential equation. The mean flowcharacteristics being determined, the spectrum of the discretized linear stability operator shows first that particle addition does not trigger any new “dangerous” modes compared with the singlephase flow case. It also shows that the most amplified mode in the case of the singlephase flow remains the most amplified mode in the case of the twophase flow. Moreover, the addition of particles acts continuously upon stability results, behaving linearly with respect to the particle mass concentration when the latter is small. The linear correction to the monophasic mode, as well as the evolution of the modes with weak values of the particle mass concentration at the wall, are shown to be proportional to the ejection velocity of the particles. Then, the evolution of the eigenmodes from a given injection speed of the particles to another one is deduced by affinity, all other parameters being fixed. With a fixed Stokes number, stability results for a finite Reynolds number and results for the inviscid flow bring together when augmenting the particle mass concentration at the wall. Therefore, by knowing singlephase flow results and the evolution of stability characteristics of the twophase flow in the inviscid case, it is easy to determine whether particleladen Taylor flow is more or less stable than the monophasic Taylor flow for large particle mass concentration.

The joint cascade of energy and helicity in threedimensional turbulence
View Description Hide DescriptionThreedimensional (3D) turbulence has both energy and helicity as inviscid constants of motion. In contrast to twodimensional (2D) turbulence, where a second inviscid invariant—the enstrophy—blocks the energy cascade to small scales, in 3D there is a joint cascade of both energy and helicity simultaneously to small scales. It has long been recognized that the crucial difference between 2D and 3D is that enstrophy is a nonnegative quantity whereas the helicity can have either sign. The basic cancellation mechanism which permits a joint cascade of energy and helicity is illuminated by means of the helical decomposition of the velocity into positively and negatively polarized waves. This decomposition is employed in the present study both theoretically and also in a numerical simulation of homogeneous and isotropic 3D turbulence. It is shown that the transfer of energy to small scales produces a tremendous growth of helicity separately in the + and − helical modes at high wave numbers, diverging in the limit of infinite Reynolds number. However, because of a tendency to restore reflection invariance at small scales, the net helicity from both modes remains finite in that limit. Since energy and helicity are not separately conserved in the + and − modes, there are four “fluxlike” quantities for both invariants, which correspond to transfer either out of large scales or into small scales and either to + helical or to − helical modes. The helicity fluxes out of large scales in the separate + and − channels are not constant in wave number up to the Kolmogorov dissipation wave number but only up to a smaller wave number recently identified by Ditlevsen and Giuliani [Phys. Fluids 13, 3508 (2001); Phys. Res. E 63, 036304 (2001)]. However, contrary to their argument, the net helicity flux is shown to be constant all the way up to the Kolmogorov wave number: there is no shorter inertial range for helicity cascade than for energy cascade. The transfer of energy and helicity between + and − modes, which permits the joint cascade, is shown to be due to two distinct physical processes, advection and vortex stretching.

Statistical analysis of coherent vortices near a free surface in a fully developed turbulence
View Description Hide DescriptionThe dynamics of coherent vortices, their interactions with an unsheared gas–liquid interface, i.e., free surface, and their contribution to turbulentheat transfer has been investigated in a fully developed turbulence using the results from a direct numerical simulation. Fully resolved free surfaceturbulence simulations were performed at Reynolds numbers of and based on the wall shear velocity and water depth. Passive heat transfer at a Prandtl number of is enforced by imposing a constant temperature difference between the bottom noslip boundary and free surface. Instantaneous turbulent flow realizations are stored and used to establish a database from which the statistical properties of the flow can be established. The threedimensional twopoint correlations between the total heat flux at the free surface and the subsurface hydrodynamics are evaluated to determine the spatial extent of the coherent vortices which contribute to the enhancement of heat transport at the free surface. A conditional averaging technique is also used to explore the structure of the typical coherent vortices which promote heat transfer at the interface. The twopoint correlation technique reveals ringlike coherent vortices in the subsurface region, which are comprised of a vortex pair and spanwise vortices. The conditional averaging strategy is also applied to an intense ejection (second quadrant) event to examine dynamics of coherent vortices and their development. The results of the statistical analysis near the second quadrant event reveals a hairpinlike vortex, known as a bursting eddy, which is generated in the nearwall region and advects toward the free surface. The eddy then changes its shape to a ringlike structure as it approaches very near to the free surface. Backward tracking of the ringlike vortex shows that its origin is clearly a nearwall hairpinlike bursting vortex associated with the ejection of lowspeed fluid in the direction normal to the wall. The interaction of the ringlike vortex with the free surface produces a splat event, which involves fluid impingement onto the free surface from the turbulent bulk, an event which intensifies the local temperature gradient at the free surface. It is also found that turbulent shear layers, which can be identified as regions of intense spanwise vorticity, impact onto the free surface, accompanying the interaction of the ringlike vortex with the free surface.

Stability analysis of conducting jets under ac radial electric fields for arbitrary viscosity
View Description Hide DescriptionA temporal linear modal stability analysis is presented for conducting viscousliquid jets flowing with nonzero velocity relative to an ambient gas and subjected to an ac radial electric field. Parametric resonance between natural dc frequencies and the frequency (or multiple) of the imposed ac field eventually leads to destabilization of the jet for perturbations with wave numbers in the stable domain. In this way, it is possible to obtain drops of smaller size. The main result is the extension of the stability analysis to liquids of arbitrary viscosity using a dynamical approach, instead of previous variational models valid for slightly viscous liquids. The effect of the outer gas in relative motion is taken into account in the framework of currently available semiempirical theories. A brief discussion of the dispersion relation for dc fields is included as the natural starting point for the discussion of the ac case. Use of the 1D averaged model for axisymmetric perturbations, an alternative to the 3D approach, allows a complete determination, in this particular case, of the distribution and nature of roots of the dispersion relation in the complex plane. The theoretical study presented here is ready to be compared to future experiments in the Rayleigh and first windinduced regime, as no relevant instability mechanisms have been excluded; namely, capillary instability, viscous damping, quasielectrostatic pressure effects, Kelvin–Helmholtz instability corrected to account for the gas viscosity, and finally, parametric resonance.

Oscillatory and chaotic thermocapillary convection in a halfzone liquid bridge
View Description Hide DescriptionThermocapillarydriven convection in a halfzone liquid bridge was investigated experimentally. The induced flows were categorized into several regimes mainly through the pattern of the suspended particle motion in the bridge and the surface temperature variation. Special attention was paid to the flow structures far beyond the critical condition. Chaotic and turbulent flows were realized in this configuration. They were distinguished from the periodic oscillatory flow by applying the pseudophasespace reconstruction from the time series of the surface temperature variation.

The effect of outer cylinder rotation on Taylor–Couette flow at small aspect ratio
View Description Hide DescriptionWe present the results of a combined experimental and numerical study of Taylor–Couette flow where both inner and outer cylinders rotate. Excellent quantitative agreement has been obtained between finiteelement calculations and experimental measurements for a range of aspect ratios and rotation rates. Counterrotation was found to enhance the breaking of the reflectional symmetry about the midplane of the apparatus, corotation to suppress symmetry breaking. For the region of parameter space explored, increasing the corotation of the outer cylinder had a qualitatively similar effect to increasing the aspect ratio, and vice versa.

Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers
View Description Hide DescriptionWe present an experimental study of gravity driven films flowing down sinusoidal bottom profiles of high waviness. We find vortices in the valleys of the undulated bottom profile. They are observed at low Reynolds numbers down to the order of The vortices are visualized employing a particle image velocimeter with fluorescent tracers. It turns out that the vortices are generated beyond a critical film thickness. Their size tends to a finite value for thick films. The critical film thickness depends on the waviness of the bottom undulation, the inclination angle, and on the surface tension but not on the Reynolds number. Increasing the waviness, a second vortex can be generated.

On the hydrodynamic stability of channel flow with cross flow
View Description Hide DescriptionWe study plane channel flow, with a homogeneous cross flow through porous walls, mainly with respect to the stability to twodimensional wave disturbances. Since the stability of a shear flow depends both on the velocity distribution and the Reynolds number we partly investigated this flow under the conditions that the flowReynolds number was constant. The flow exhibits some interesting and unexpected stability characteristics. The effect of the cross flow was for certain parameter regions stabilizing and for others destabilizing. The latter result is in contrast to previous studies.

Thermocapillary convection with moving contact points
View Description Hide DescriptionMarangoni convection in a cavity subject to two types of heating, i.e., heating through the sidewalls and heating by a point source from above, has been investigated. The assumed wetting conditions permit motion of the interface along the sidewalls subject to a constant contact angle constraint. The analysis considers complete interface deformation effects. The results determined for large Biot and zero Marangoni numbers show the existence of limit points beyond which steady, continuous interface cannot exist. The limit points define the maximum capillary number Ca and the maximum cavity length L permitted. The permitted values of the Reynolds number may be bounded from below and from above depending on the values of other transport parameters and on the form of the external heating. Interface approaching bottom of the cavity leading, most likely, to the formation of a dry spot, represents the main factor limiting the existence of steady convection. The topology of the flow field is similar to the case when the wetting conditions result in the fixed location of the contact points while the topology of the interface is qualitatively different. The change of the contact conditions from the fixed contact points to the fixed contact angles results in a significant reduction of the range of parameters that guarantee the existence of a continuous interface.

Thermal convection below a conducting lid of variable extent: Heat flow scalings and twodimensional, infinite Prandtl number numerical simulations
View Description Hide DescriptionTheoretical heat flow scalings are presented for free thermal convection occurring below a conducting lid. Cases in which the lid covers the full extent of the convecting fluid and in which the lid has a variable lateral extent are treated. The scaling predictions are tested against a suite of twodimensional numerical simulations that solve the full mathematical equations describing infinite Prandtl number thermal convection occurring below a conducting lid. The scaling predictions and simulation results show reasonably good agreement for Rayleigh numbers greater than The scaling predictions are also tested against previous numerical simulations of finite Prandtl number thermal convection below a laterally uniform lid. Scaling predictions and simulations results again show reasonable agreement.

Interaction between Ekman pumping and the centrifugal instability in Taylor–Couette flow
View Description Hide DescriptionThe endwalls in a Taylor–Couette cell introduce adjacent boundary layers that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman layers and the Taylor vortices near transition from nonvortical to vortical flow via direct numerical simulation using a spectral method. We consider a radius ratio of in a short annulus having a lengthtogap ratio of To analyze the nature of the interaction between the vortices and the endwall layers, three endwall boundary conditions were considered: fixed endwalls, endwalls rotating with the inner cylinder, and stressfree endwalls. Below the critical Taylor number, endwall vortices for rotating endwalls are more than twice the strength of the vortices for fixed endwalls. This trend continues well above the transition to vortical flow, consistent with a simple force balance analysis near the endwalls. Stressfree endwalls result in endwall vortices that are similar in strength to those for rotating endwalls above the critical Taylor number. The endwall conditions significantly change the bifurcation diagram based on the radial velocity near the center of the annulus. For stressfree endwall conditions, the bifurcation is quite sharp, although only one fork of the bifurcation results unless the initial conditions are specifically set to favor the other fork. For rotating and fixed endwalls, there is a continuous transition from a featureless flow to a vortical flow due to the endwall vortices.
