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Volume 13, Issue 7, July 2001
 LETTERS


Lowgravity sideways doublediffusive instabilities
View Description Hide DescriptionInstabilities of a doublediffusive system in a slot are considered and relevance to the microgravity environment is identified. Formulation of the problem including the effects of gjitter is given, and a twotimescale analysis, valid in the limit of highfrequency modulation, is employed to derive the equations governing the mean field. Twodimensional linear stability results show that the effective rapid gravity modulation is destabilizing, resulting in a dramatically wider range of parameters where this doublediffusive instability is expected to prevail in suitable space experiments.

Viscous nonlinear theory of Richtmyer–Meshkov instability
View Description Hide DescriptionWe propose a quantitative prediction of the effect of viscosity on the weakly nonlinear impulsive Richtmyer–Meshkov instability between two fluids of arbitrary density and viscosity. This theory is based on an asymptotic analysis of the Navier–Stokes equations using singular perturbation techniques. The law obtained for interface deformation does not agree with former theoretical predictions of the effect of viscosity [K. O. Mikaelian, Phys. Rev. E 47, 375 (1993)], but compares very well with direct numerical simulations we performed using a fronttracking code developed in our laboratory. Application of this law to typical experimental parameters gives a formal demonstration of the relevance of inviscid models for the description of typical shocktube experiments; at the same time, however, it shows that care should be taken with regard to viscosity in the case of impulsive experiments performed with liquids.

 ARTICLES


Peristaltically driven channel flows with applications toward micromixing
View Description Hide DescriptionFlows driven by a transverse, small amplitude traveling wave propagating along the boundary of a closed rectangular container are examined. Highfrequency motions are the primary focus of interest, although lowfrequency results are discussed also. Using asymptotic analysis appropriate for high frequencies, the steady, timeindependent (streaming) flow is computed analytically and compared with results of the exact calculation. The boundarylayer structure is delineated and average Eulerian and Lagrangianflow characteristics are compared. Experiments confirming the major qualitative highfrequency findings are reported in an Appendix. The results could be useful for modeling peristaltically operated microelectromechanical systemsdevices where fluid motion needs to be produced without internal moving mechanical components.

Unsteady flow of thin liquid film on a disk under nonuniform rotation
View Description Hide DescriptionDevelopment of thin liquid film under nonuniform rotation has been studied numerically by using finite difference technique under the assumption of planar interface. For impulsive rotation of the disk an anomalous behavior of the rate of film thinning with the variation of the Reynolds number is observed in a different time zone. It is also shown that the azimuthal velocity field develops into the entire depth faster with the smaller impulsive rotation. A physical explanation for the above observation is provided. Further it is found that faster rate of thinning can be obtained if the spinner starts impulsively and then increases its spinning rate continuously.

Models for Marangoni drying
View Description Hide DescriptionMarangoni drying is a new ultraclean drying process, which relies on surfacetension gradient forces, socalled Marangoni stresses. This method is of particular use in the semiconductor industry wherein obtaining ultraclean surfaces is of paramount importance. The present work provides a mathematical description of this novel process involving four coupled partial differential equations, derived in the thinlayer approximation, for the film thickness, the concentration of chemicals in the air, at the air–liquid interface and in the bulk of the liquid film.Numerical solution of these equations yields prediction of typical profiles that accompany the spreading and drying processes. Particular attention is aimed at the prediction of the minimum film thickness as a function of system parameters with a view to optimizing the drying process.

Slip boundary condition on an idealized porous wall
View Description Hide DescriptionSlip boundary condition on a porous wall is investigated by considering a viscousflow near an idealized porous wall based on the Stokes’ approximation. The idealized porous wall is composed of a large number of parallel and equidistant thin semiinfinite plates. The flow is assumed to be a simple shear flow far from the plates and stagnant deep inside the channels of the plates. The slip velocity on the idealized porous wall is calculated by solving a Wiener–Hopf equation. From the streamline patterns shown, it is found that Moffatt’s infinite sequence of viscouseddies developed between the two adjacent plates. Pressure and shear stress distributions on the plates are shown and local flow near the edge of the plate is discussed. Laminar shear flow at the farfield unidirectional along the edges of the plates is also considered.

Simultaneous particle image velocimetry and infrared imagery of microscale breaking waves
View Description Hide DescriptionWe report the results from a laboratory investigation in which microscale breaking waves were detected using an infrared (IR) imager and twodimensional (2D) velocity fields were simultaneously measured using particle image velocimetry (PIV). In addition, the local heat transfer velocity was measured using the controlled flux technique. To the best of our knowledge these are the first measurements of the instantaneous 2D velocity fields generated beneath microscale breaking waves. Careful measurements of the water surface profile enabled us to make accurate estimates of the nearsurface velocities using PIV. Previous experiments have shown that behind the leading edge of a microscale breaker the cool skin layer is disrupted creating a thermal signature in the IR image [Jessup et al., J. Geophys. Res. 102, 23145 (1997)]. The simultaneously sampled IR images and PIV data enabled us to show that these disruptions or wakes are typically produced by a series of vortices that form behind the leading edge of the breaker. When the vortices are first formed they are very strong and coherent but as time passes, and they move from the crest region to the back face of the wave, they become weaker and less coherent. The nearsurface vorticity was correlated with both the fractional area coverage of microscale breaking waves and the local heat transfer velocity. The strong correlations provide convincing evidence that the wakes produced by microscale breaking waves are regions of high nearsurface vorticity that are in turn responsible for enhancing air–water heat transfer rates.

The spatial correlations in the velocities arising from a random distribution of point vortices
View Description Hide DescriptionThis paper is devoted to a statistical analysis of the velocity fluctuations arising from a random distribution of point vortices in twodimensional turbulence. Exact results are derived for the correlations in the velocities occurring at two points separated by an arbitrary distance. We find that the spatialcorrelation function decays extremely slowly with the distance. We discuss formal analogies with the statistics of the gravitational field in stellar systems.

Statistical equilibrium theory for axisymmetric flows: Kelvin’s variational principle and an explanation for the vortex ring pinchoff process
View Description Hide DescriptionThermodynamics of vorticity density fields (ω/r) in axisymmetric flows are considered, and the statistical equilibrium theories of Miller, Weichman, and Cross [Phys. Rev. A 45, 2328 (1992)], Robert and Sommeria [J. Fluid Mech. 229, 291 (1991)], and Turkington [Comm. Pure Appl. Math. 52, 781 (1999)] for the twodimensional flows in Cartesian coordinates are extended to axisymmetric flows. It is shown that the statistical equilibrium of an axisymmetric flow is the state that maximizes an entropy functional with some constraints on the invariants of motion. A consequence of this argument is that only the linear functionals of vorticity density, e.g., energy and total circulation, are conserved during the evolution of an axisymmetric inviscid flow to the statistical equilibrium. Furthermore, it is shown that the final equilibrium state satisfies Kelvin’s variational principle; the mean field profiles maximize the energy compatible with the resulting dressed vorticity density. Finally, the vortex ring pinchoff process is explained through statistical equilibrium theories. It appears that only a few invariants of motion (the kinetic energy, total circulation, and impulse) are important in the pinchoff process, and the higher enstrophy densities do not play a significant role in this process.

Threedimensional structure and decay properties of vortices in shallow fluid layers
View Description Hide DescriptionRecently, several laboratory experiments on vortex dynamics and quasitwodimensional turbulence have been performed in thin (stratified) fluid layers. Commonly, it is tacitly assumed that vertical motions, giving rise to a threedimensional character of the flow, are inhibited by the limited vertical dimension. However, shallow water flows, which are vertically bounded by a noslip bottom and a free surface, necessarily possess a threedimensional structure due to the shear in the vertical direction. This shear may lead to significant secondary circulations. In this paper, the threedimensional (3D) structure and the decay properties of vortices in shallow layers of fluid, both homogeneous and stratified, have been studied in detail by 3D direct numerical simulations. The quasitwodimensionality of these flows is an important issue if one is interested in a comparison of experiments of this type with purely twodimensional theoretical models. The influence of several flow parameters, like the depth of the fluid and the Reynolds number, has been investigated. In general, it can be concluded that the flow loses its twodimensional character for larger fluid depth and larger Reynolds number. Furthermore, it is possible to construct a regime diagram that allows the assessment of the parameter regime, where the flow can be considered as quasitwodimensional. It is found that the presence of secondary circulations within a planar vortex flow results in a deformation of the radial profile of axial vorticity. In the limiting case of quasitwodimensional flow, the vorticity profiles can be scaled according to a simple diffusion model. In a twolayer stratified system, the decay is reduced and threedimensional motions are significantly inhibited compared to the corresponding flows in a homogeneous layer.

Rotating magnetohydrodynamic freeshear flows. I. Linear stability analysis
View Description Hide DescriptionThe results of a study are presented of the linear stability of a magnetohydrodynamic planar freeshear flow subject to system rotation and a uniform magnetic field which are both parallel or antiparallel to the shear flowvorticity vector. The results show that, with anticyclonic rotation, the effect of the uniform magnetic field is to enlarge the unstable region of the rotating shear flow, while the growth rate of the most unstable disturbance is left unchanged. With cyclonic rotation, on the other hand, the effect of the magnetic field is to strongly stabilize the flow. The spatial structure of the anticyclonic magnetic shear/Coriolis instability mode is investigated. Considering the Rossby number at which the spatial extent of the most unstable disturbance becomes minimum, it is found that this Rossby number decreases as the uniform magnetic field is intensified.

Steady streaming in an oscillatory inviscid flow
View Description Hide DescriptionSteady streaming, arising within a fluctuating flow field, is commonly associated with attenuation due to viscosity. In this paper we show that such streaming may be induced within an inviscid fluid when the underlying fluctuating flow is the result of a nonconservative body force. In such a situation the induced streaming is significantly greater than that in a viscous fluid brought about by a fluctuating conservative body force. The vehicle we choose to illustrate this is a unidirectional buoyancy force, otherwise known as gjitter. In particular the flow about a heated sphere in an otherwise gravityfree environment is analyzed. The relationship with the analogous viscousflow problem is investigated.

A long range spherical model and exact solutions of an energy enstrophy theory for twodimensional turbulence
View Description Hide DescriptionThe equilibrium statistical mechanics of the energy–enstrophy theory for the twodimensional (2D) Euler equations is solved exactly. A family of lattice vortex gas models for the Euler equations is derived and shown to have a welldefined nonextensive continuum limit. This family of continuousspin lattice Hamiltonians is shown to be nondegenerate under different point vortex discretizations of the Euler equations. Under the assumptions that the energy, total circulation and the enstrophy (mean squared vorticity) are conserved, this lattice vortex gas model is equivalent to a long range version of Kac’s exactly solvable spherical model with logarithmic interaction. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematically equivalent to Kac’s spherical constraint. This spherical model is shown to have a free energy that is analytic in the properly scaled inverse temperatures in the range Phase transitions occur at the positive value and Spin–spin correlations are calculated giving twopoint vorticity correlations that are important to the study of turbulence. There are exactly three distinct phases in the energyenstrophy theory for 2D flows, namely (a) an uncorrelated high positive temperature phase, (b) an antiferromagnetic checkerboard vorticity pattern at low positive temperature, and (c) a highly correlated physical domain scale vorticity pattern (for instance, a large positive vorticity region surrounded by a sea of negative vorticity) at negative temperatures. The boundary agrees with the known numerical and analytical results on the occurrence of coherent or ordered structures at negative temperatures. The critical temperature is new, as is the corresponding checkerboard low positive temperature phase. Physical interpretations of the results in this paper are obtained.

Random Taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence
View Description Hide DescriptionThe properties of acceleration fluctuations in isotropic turbulence are studied in direct numerical simulations (DNS) by decomposing the acceleration as the sum of local and convective contributions and or alternatively as the sum of irrotational and solenoidal contributions and The main emphasis is on the nature of the mutual cancellation between and which must occur in order for the acceleration (a) to be small as predicted by the “random Taylor hypothesis” [Tennekes, J. Fluid Mech. 67, 561 (1975)] of small eddies in turbulent flow being passively “swept” past a stationary Eulerian observer. Results at Taylorscale Reynolds number up to 240 show that the randomTaylor scenario accompanied by strong antialignment between the vectors and is indeed increasingly valid at higher Reynolds number. Mutual cancellation between and also leads to the solenoidal part of a being small compared to its irrotational part. Results for spectra in wave number space indicate that, at a given Reynolds number, the random Taylor hypothesis has greater validity at decreasing scale sizes. Finally, comparisons with DNS data in Gaussian random fields show that the mutual cancellation between and is essentially a kinematic effect, although the Reynolds number trends are made stronger by the dynamics implied in the Navier–Stokes equations.

An inertial range crossover in structure functions
View Description Hide DescriptionExperimental and numerical data within the traditional inertial subrange defined by the thirdorder structure function is used to study higherorder scaling exponents for the longitudinal and transverse structure functions. For these exponents converge only over larger scales, where is between η and λ and has an dependence. Below these scales, scaling exponents cannot be determined for any of the structure functions without resorting to procedures such as extended selfsimilarity (ESS). With ESS, different longitudinal and transverse higherorder exponents are obtained that are consistent with earlier results. The relationship of these statistics to derivative and pressure statistics, to turbulent structures and to length scales is discussed.

Nonlocality and intermittency in threedimensional turbulence
View Description Hide DescriptionNumerical simulations are used to determine the influence of the nonlocal and local interactions on the intermittency corrections in the scaling properties of threedimensional turbulence. We show that neglect of local interactions leads to an enhanced smallscale energy spectrum and to a significantly larger number of very intense vortices (“tornadoes”) and stronger intermittency (e.g., wider tails in the probability distribution functions of velocity increments and greater anomalous corrections). On the other hand, neglect of the nonlocal interactions results in even stronger smallscale spectrum but significantly weaker intermittency. Thus, the amount of intermittency is not determined just by the mean intensity of the small scales, but it is nontrivially shaped by the nature of the scale interactions. Namely, the role of the nonlocal interactions is to generate intense vortices responsible for intermittency and the role of the local interactions is to dissipate them. Based on these observations, a new model of turbulence is proposed, in which nonlocal (rapid distortion theorylike) interactions couple large and small scale via a multiplicative process with additive noise and a turbulentviscositymodels the local interactions. This model is used to derive a simple version of the Langevin equations for smallscale velocity increments. A Gaussian approximation for the large scale fields yields the Fokker–Planck equation for the probability distribution function of the velocity increments. Steady state solutions of this equation allows one to qualitatively explain the anomalous corrections and the skewness generation along scale. A crucial role is played by the correlation between the additive and the multiplicative (largescale) process, featuring the correlation between the stretching and the vorticity.

Decaying and kicked turbulence in a shell model
View Description Hide DescriptionDecaying and periodically kicked turbulence are analyzed within the Gledzer–Ohkitani–Yamada shell model, to allow for sufficiently large scaling regimes. Energy is transferred towards the small scales in intermittent bursts. Nevertheless, mean field arguments are sufficient to account for the ensemble averaged energy decay or the parameter dependencies for the ensemble averaged total energy in the kicked case. Within numerical precision, the inertial subrange intermittency remains the same, whether the system is forced or decaying.

On velocity structure functions and the spherical vortex model for isotropic turbulence
View Description Hide DescriptionWe investigate a stochastic model for homogeneous, isotropic turbulence based on Hill’s spherical vortex. This is an extension of the method of Synge and Lin [Trans. R. Soc. Can. 37, 45 (1943)], to the calculation of higher evenorder velocity structure functions. Isotropic turbulence is represented by a homogeneous distribution of eddies, each modeled by a spherical vortex. The cascade process of eddybreakdown is incorporated into the statistical model through an average over an assumed lognormal distribution of vortex radii. We calculate the statistical properties of the model, in particular ordern velocity structure functions defined by rankn tensors for the ensemble average of a set of incremental differences in velocity components. We define where denotes the ensemble average. Specifically and the longitudinal component of are calculated directly from the spherical vortex ensemble. Matching the longitudinal components of and with experimental results fixes two independent model parameters. The lateral and mixed components of and the longitudinal component of are then model predictions.

Conditional modeelimination and the subgridmodeling problem for isotropic turbulence
View Description Hide DescriptionIt is shown analytically that different subgrid eddyviscosities are required to characterize the evolution of the velocity field, the energy spectrum and the dissipation rate, respectively. Also, a subgrid viscosity calculated recursively from the conditionally averaged “subgrid–subgrid” stress can only be used to renormalize the dissipation relation. It is also shown, by direct numerical simulation, that the spectral correlations of subgrid and resolved stresses are dominated by “resolved–subgrid” terms, in agreement with existing conclusions from energy transfer or other more global methods. We also note the low level of correlation between the exact subgrid stress and the eddy–viscosity model for all explicit–scales wave numbers. Lastly, we report results from a largeeddy simulation(LES) based on a recursively calculated effective viscosity and note that the results are about as good as for other subgrid models but underline the need to take account of phase effects as well as the dissipation rate.

A dynamic subfilterscale model for plane parallel flows
View Description Hide DescriptionWe present a dynamic model of the subfiltered scales in plane parallel geometry using a generalized, stochastic rapid distortion theory (RDT). This new model provides expressions for the turbulent Reynolds subfilterscale stresses via estimates of the subfilter velocities rather than velocity correlations. Subfilterscale velocities are computed using an auxiliary equation which is derived from the Navier–Stokes equations using a simple model of the subfilter energy transfers. It takes the shape of a RDT equation for the subfilter velocities, with a stochastic forcing. An analytical test of our model is provided by assuming deltacorrelation in time for the supergrid energy transfers. It leads to expressions for the Reynolds stresses as a function of the mean flow gradient in the plane parallel geometry and can be used to derive mean equilibrium profiles both in the nearwall and core regions. In the nearwall region we derive a general expression for the velocity profile which is linear in the viscous layer and logarithmic outside. This expression involves two physical parameters: the von Karman constant and the size of the viscous layer (which can be computed via a numerical implementation of our model). Fits of experimental profiles using our general formula provides reasonable values of these parameters to the size of the viscous layer is about 15 wall units). In the core region, we find that the shape of the profile depends on the geometry of the flow; it ranges from algebraic in channel flow, to exponential in the bulk of boundary layers, and linear in plane Couette flow. This classification is consistent with Oberlack’s system, which is based on symmetry arguments. Fits of boundary layer flows or channel flows at different Reynolds number over the whole flow region are performed using our results, and are found to be in very good agreement with available data.
