Volume 11, Issue 12, December 1999
 LETTERS


A note on subharmonic instabilities
View Description Hide DescriptionWhen a fluid system is subject to timeperiodic forcing, it is well known that it may exhibit both harmonic and subharmonic instabilities, the classic example being Faraday oscillations. When the forcing is confined to a periodic shearing motion, however, it has been observed that the subharmonic response is absent. The underlying mathematical feature that unifies these systems is a conjugatetranslation symmetry [A. C. Or, J. Fluid Mech. 335, 213 (1997)]. We show that any subharmonic solutions of periodically driven systems with conjugatetranslation symmetry must have Floquet multipliers with multiplicity greater than one. The effect of this constraint is that subharmonic solutions are very difficult to locate within the system’s parameter space and, more importantly, that phase locking cannot occur for such systems.
 Top

 ARTICLES


Computational and experimental analysis of dynamics of drop formation
View Description Hide DescriptionDynamics of formation of a drop of a Newtonian liquid from a capillary tube into an ambient gas phase is studied computationally and experimentally. While this problem has previously been studied computationally either (a) using a set of onedimensional equations or (b) treating the dynamics as that of irrotational flow of an inviscid fluid or creeping flow, here the full nonlinear, transient Navier–Stokes system subject to appropriate initial and boundary conditions is solved in two dimensions to analyze the dynamics at finite Reynolds numbers. The success of the computations rests on a finite element algorithm incorporating a multiregion mesh which conforms to and evolves with the changing shape of the drop. The new algorithm is able to capture both the gross features of the phenomenon, such as the limiting length of a drop at breakup and the volume of the primary drop, and its fine features, such as a microthread that develops from a main thread or a neck in a viscousdrop approaching breakup. The accuracy of the new calculations is verified by comparison of computed predictions to old and new experiments. With the new algorithm, it is shown for the first time that the interface of a viscousdrop can overturn before the drop breaks. Calculations have also been carried out to determine the range of parameters over which algorithms that treat the drop liquid as inviscid and the flow inside it as irrotational can accurately predict the dynamics of formation of drops of low viscosity liquids. Limiting lengths of drops and primary drop volumes are computed over a wide range of the parameter space spanned by the relevant dimensionless groups.

Wake measurements for flow around a sphere in a viscoelastic fluid
View Description Hide DescriptionThe flow field around a sphere falling at its terminal velocity in a column of viscoelastic nonshearthinning fluid is experimentally measured with digital particle imagevelocimetry. The working fluid is an extensively characterized, monodisperse, polystyrene based Boger fluid. The sphere radius relative to the radius of the column of fluid is small The Weissenberg number ranges from 0.5 to 14 over which the sphere experiences a drag increase up to 8 times that of the Newtonian flow. The flow field is investigated in detail for 0.5 to 2.5. A length and width scale is defined for the wake. Over this range of the wake is found to grow linearly with and become selfsimilar in a transverse crosssection of the axial component of the velocity. Streamlines along with extension and rotation rates along those streamlines are also determined.

Electrically driven convection in a thin annular film undergoing circular Couette flow
View Description Hide DescriptionWe investigate the linear stability of a thin, suspended, annular film of conducting fluid with a voltage difference applied between its inner and outer edges. For a sufficiently large voltage, such a film is unstable to radially driven electroconvection due to charges which develop on its free surfaces. The film can also be subjected to a Couette shear by rotating its inner edge. This combination is experimentally realized using films of smectic A liquid crystals. In the absence of shear, the convective flow consists of a stationary, azimuthally onedimensional (1D) pattern of symmetric, counterrotating vortex pairs. When Couette flow is applied, an azimuthally traveling pattern results. When viewed in a corotating frame, the traveling pattern consists of pairs of asymmetric vortices. We calculate the neutral stability boundary for arbitrary radius ratio α and Reynolds number of the shear flow, and obtain the critical control parameter and the critical azimuthal mode number The Couette flow suppresses the onset of electroconvection, so that The calculated suppression is compared with experiments performed at α=0.56 and 0⩽Re⩽0.22.

A weakly nonlinear analysis of the dynamics of a viscous flow in a symmetric channel with a sudden expansion
View Description Hide DescriptionA weakly nonlinear analysis of the dynamics of a twodimensional, laminar flow in a long channel with a sudden expansion and the transition from symmetric to asymmetric states is presented. The asymptotic analysis is based on a study of the unsteady Navier–Stokes equations around the critical Reynolds number, where a bifurcation occurs. It explores the special nonlinear interactions between the unsteady, convective, and viscous effects. The analysis results in an ordinary, nonlinear, firstorder differential equation (similar to the Landau equation) which describes the evolution of the perturbation’s amplitude as function of Re near The analytical solution shows that when the symmetric state is stable. However, when the symmetric state in the channel loses its stability and evolves into an asymmetric state. The flow evolution, as described by the nonlinear model, shows agreement with time history plots from simulations using the unsteady Navier–Stokes equations. The linear stability characteristics of both the symmetric and asymmetric states are also found from the nonlinear approach and match with the previous results. The analysis provides new insight into the previous experimental and numerical results and sheds light on the nonlinear transition of a viscousflow in an expanding channel.

Velocity field for Taylor–Couette flow with an axial flow
View Description Hide DescriptionThe flow in the gap between an inner rotating cylinder concentric with an outer stationary cylinder with an imposed pressuredriven axial flow was studied experimentally using particle imagevelocimetry(PIV) in a meridional plane of the annulus. The radius ratio was and the aspect ratio was Velocity vector fields for nonwavy toroidal and helical vortices show the axial flow winding around vortices. When the axially averaged axial velocity profile is removed from the velocity field in a meridional plane, the velocity field looks much like it would with no imposed axial flow except that the vortices translate axially and the distortion of the azimuthal velocity contours in meridional plane related to the vortices is shifted axially by the axial flow. The velocity vector fields for wavy vortices also show axial flow winding around the vortices. Again, removing the axial velocity profile results in a flow that appears similar to that with no axial flow. The path of the vortices is generally axial, but the vortices periodically move retrograde to the imposed axial flow due to the waviness of the vortices. The axial velocity of helical vortices, both nonwavy and wavy, is twice the rotational frequency of the inner cylinder indicating a coupling between the axial translation of the vortices and the cylinder rotation. Little fluid transport between vortices occurs for nonwavy vortices, but there is substantial transport between vortices for wavy vortexflow, much like supercritical cylindrical Couette flow with no axial flow.

Secondary breakup of axisymmetric liquid drops. I. Acceleration by a constant body force
View Description Hide DescriptionThe secondary breakup of liquid drops,accelerated by a constant body force, is examined for small density differences between the drops and the surrounding fluid. Two cases are examined in detail: a density ratio close to unity where the Boussinesq approximation is valid) and a density ratio of ten. A finite difference/front tracking numerical technique is used to solve the unsteady Navier–Stokes equations for both the drops and the surrounding fluid. The breakup is controlled by the Eötvös number (Eo), the Ohnesorge number (Oh), and the viscosity and density ratios. If viscous effects are small (small Oh), the Eötvös number is the main controlling parameter. In the Boussinesq limit, as Eo increases the drops break up in a backward facing bag, transient breakup, and a forward facing bag mode. At a density ratio of ten, similar breakup modes are observed, with the exception that the forward facing bag mode is replaced by a shear breakup mode. Similar breakup modes have been seen experimentally for much larger density ratios. Although a backward facing bag is seen at low Oh, where viscous effects are small, comparisons with simulations of inviscid flows show that the bag breakup is a viscous phenomenon, due to boundary layer separation and the formation of a wake. At higher Oh, where viscous effects modify the evolution, the simulations show that the main effect of increasing Oh is to move the boundary between the different breakup modes to higher Eo. The results are summarized by “breakup maps” where the different breakup modes are shown in the Eo–Oh plane for different values of the viscosity and the density ratios.

Shape and stability of doubly connected axisymmetric free surfaces in a cylindrical container
View Description Hide DescriptionThe equilibrium and stability of a liquid that partially fills a cylindrical container with planar ends are examined. It is assumed that the free surface is axisymmetric and does not cross the symmetry axis of the container. Particular attention is given to the case where gravity is parallel to the cylinder’s axis, and where the free surface has one contact line on the lateral cylindrical wall and the other on one of the planar ends. The equilibrium configuration of such a surface is determined by the wetting angle, α, the Bond number, B, and the relative volume, of the annular region bounded by the free surface and the solid container. Shapes of stable and critical surfaces have been analyzed, and the stability regions for arbitrary Bond numbers have been obtained in the plane. The shape and stability problems for a zero gravity configuration with both contact lines on the lateral wall of the cylinder are also studied. In addition, the stability of a free surface with at least one contact line coinciding with the edge formed by the lateral wall and a planar end is discussed.

Thermal separation in nearaxis boundary layers with intense swirl
View Description Hide DescriptionSwirling flows have a wide range of applications and exhibit a variety of interesting features. Gas cooling near the axis in these flows, the socalled Ranque–Hilsch effect, is one of them. To gain insight into this phenomenon, we have analyzed the thermal, nearaxis boundary layer of a gas jet driven by a class of conical inviscid quasiincompressible flows whose axial and azimuthal velocity components, and and stagnation temperature, behave near the axis as and where and are the axial and radial coordinates, is the Squire number directly related to the swirl strength, is any real number such as is a reference temperature, and and are arbitrary dimensional constants; is assumed to be positive while may be either positive or negative. To simplify the boundary layer analysis, low Mach numberflows with small relative variations in the gas density have been considered. Radial profiles of axial and azimuthal velocity components, and static and stagnation temperatures are found to depend on the Squire parameter the Prandtl number, Pr, and the rest of the parameters of the problem. Even for the case of inviscid vortices with positive values of for which the stagnation temperature increases towards the axis, is found that the stagnation temperature decreases substantially in the vortex core for some range of values of both and Pr (Ranque–Hilsch effect) when the effect of both heat conduction and the work done by viscous forces are taken into account. It is also found that there exists an optimum value for which the cooling effect reaches a sharp maximum and that small deviations of from reduce drastically the cooling effect. The appropriate tuning of can be dramatically important for the efficient operation of Ranque–Hilsch tubes. The influence of the Prandtl number and the rest of the parameters of the problem has been also considered.

Fully nonlinear global modes in slowly varying flows
View Description Hide DescriptionWe study the existence of nonlinear solutions of the real Ginzburg–Landau amplitude equation, with varying coefficients when the solution is subject to a boundary condition at These solutions, called nonlinear global modes, are explicitly obtained from a matched asymptotic expansion when nonlinear effect dominates over the inhomogeneity. The dynamics of this model is believed to mimic the behavior of strongly nonlinear but weakly nonparallel basic flow (basic flow varying in the streamwise direction). For the model, we derive scaling laws for the amplitude of nonlinear global modes and for the position of the maximum that explain for the first time the experimental observations of GoujonDurand et al. [Phys. Rev. E 50, 308 (1994)] and the numerical simulations of Zielinska and Wesfreid [Phys. Fluids 7, 1418 (1995)] of the wake behind bluff bodies.

Fourvortex motion with zero total circulation and impulse
View Description Hide DescriptionThe problem of four interacting point vortices on the unbounded plane with vanishing total circulation and vanishing impulse is reduced to a threebody problem analogous to the threevortex problem on the unbounded plane. A “phase plane analysis” using trilinear coordinates, similar to that used for the threevortex problem, is presented and used to discuss details of the motion. The methodology and results complement earlier analyses of the same problem by Eckhardt and Rott.

Vortices in rotating systems: Centrifugal, elliptic and hyperbolic type instabilities
View Description Hide DescriptionThis paper is devoted to the effects of rotation on the linear dynamics of twodimensional vortices. The asymmetric behavior of cyclones and anticyclones, a basic problem with respect to the dynamics of rotating flows, is particularly addressed. This problem is investigated by means of linear stability analyses of flattened Taylor–Green vortices in a rotating system. This flow constitutes an infinite array of contrarotating onesigned nonaxisymmetric vorticity structures. We address the stability of this flow with respect to threedimensional shortwave perturbations via both the geometrical optics method and via a classical normal mode analysis, based on a matrix eigenvalue method. From a physical point of view, we show that vortices are affected by elliptic, hyperbolic and centrifugal instabilities. A complete picture of the shortwave stability properties of the flow is given for various levels of the background rotation. For Taylor–Green cells with aspect ratio we show that anticyclones undergo centrifugal instability if the Rossby number verifies Ro>1, elliptic instability for all values of Ro except 0.75<Ro<1.25 and hyperbolic instability. The Rossby number is here defined as the ratio of the maximum amplitude of vorticity to twice the background rotation. On the other hand, cyclones bear elliptic and hyperbolic instabilities whatever the Rossby number. Besides, depending on the Rossby number, rotation can either strengthen (anticyclonic vortices) or weaken elliptic instability. From a technical point of view, in this article we bring an assessment of the links between the shortwave asymptotics and the normal mode analysis. Normal modes are exhibited which are in complete agreement with the shortwave asymptotics both with respect to the amplification rate and with respect to the structure of the eigenmode. For example, we show centrifugal eigenmodes which are localized in the vicinity of closed streamlines in the anticyclones; elliptical eigenmodes which are concentrated in the center of the cyclones or anticyclones; hyperbolic eigenmodes which are localized in the neighborhood of closed streamlines in cyclones.

Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence?
View Description Hide DescriptionThis paper investigates the dynamics of tracer gradient for a twodimensional flow. More precisely, the alignment of the tracer gradient vector with the eigenvectors of the strainrate tensor is studied theoretically and numerically. We show that the basic mechanism of the gradient dynamics is the competition between the effects due to strain and an effective rotation due to both the vorticity and to the rotation of the principal axes of the strainrate tensor. A nondimensional criterion is derived to partition the flow into different regimes: In the strain dominated regions, the tracer gradient vector aligns with a direction different from the strain axes and the gradient magnitude grows exponentially in time. In the straineffective rotation compensated regions, the tracer gradient vector aligns with the bisector of the strain axes and its growth is only algebraic in time. In the effective rotation dominated regions, the tracer gradient vector is rotating but is often close to the bisector of the strain axes. A numerical simulation of 2D (twodimensional) turbulence clearly confirms the theoretical preferential directions in strain and effective rotation dominated regions. Effective rotation can be dominated by the rotation rate of the strain axes, and moreover, proves to be larger than strain rate on the periphery of vortices. Taking into account this term allows us to improve significantly the Okubo–Weiss criterion. Our criterion gives the correct behavior of the growth of the tracer gradient norm for the case of axisymmetric vortices for which the Okubo–Weiss criterion fails.

Note on forced Burgers turbulence
View Description Hide DescriptionA putative powerlaw range of the probability density of velocity gradient in highReynoldsnumber forced Burgers turbulence is studied. In the absence of information about shock locations, elementary conservation and stationarity relations imply that the exponent in this range satisfies if dissipation within the powerlaw range is due to isolated shocks. A generalized model of shock birth and growth implies if initial data and forcing are spatially homogeneous and obey Gaussian statistics. Arbitrary values can be realized by suitably constructed homogeneous, nonGaussian initial data and forcing.

Calculations of longitudinal and transverse velocity structure functions using a vortex model of isotropic turbulence
View Description Hide DescriptionThe longitudinal structure function (LSF) and the transverse structure function (TSF) in isotropic turbulence are calculated using a vortexmodel. The vortexmodel is composed of the Rankine and Burgers vortices which have the exponential distributions in the vortexReynolds number and vortex radii. This model exhibits a power law in the inertial range and satisfies the minimal condition of isotropy that the secondorder exponent of the LSF in the inertial range is equal to that of the TSF. Also observed are differences between longitudinal and transverse structure functions caused by intermittency. These differences are related to their scaling differences which have been previously observed in experiments and numerical simulations.

A nonlinear mechanism for receptivity of freestream disturbances
View Description Hide DescriptionNumerical experiments on the interaction of simple vortical freestream disturbances with a laminar boundary layer are presented. Both spatial and temporal direct numerical simulations (DNS) have been performed for two types of freestream disturbances. A linear and a nonlinear receptivity mechanism were identified. The nonlinear mechanism was found to force streaks inside the boundary layer similar to those found in experiments on freestream turbulence and it performed equally well for disturbances elongated in the streamwise direction as for and oblique freestream disturbances. The boundary layer response caused by the nonlinear mechanism was, depending on the initial disturbance energy, comparable to that of the linear mechanism, which was only efficient for freestream streamwise vortices. A parameter study revealed that the wall normal velocity component of the freestream disturbances is more important for the investigated receptivity mechanisms than the streamwise component. The identified boundary layer receptivity mechanism, in which threedimensional disturbances in the freestream continuously force streaks inside the boundary layer, may explain differences between experimental results and previously suggested theories for the origin of streaks in boundary layers subjected to freestream turbulence.

Measurement of shock structure and shock–vortex interaction in underexpanded jets using Rayleigh scattering
View Description Hide DescriptionThe density field of underexpanded supersonic free jets issuing from a choked circular nozzle was measured using a Rayleigh scatteringbased technique. This reliable and nonintrusive technique is particularly suitable for highspeed flows and is fundamentally superior to the intrusive probes and particlebased techniques such as laser Doppler velocimetry. A continuous wave laser and photon counting electronics were employed for time and phaseaveraged density measurements. The use of dustfree air for the entrained flow allowed measurements in the shear layer region. The free jets were produced in the plenum to ambient pressure ratio range of 1.88–5.75, which corresponded to a fully expanded Mach number range of A comparative study of schlieren photographs and timeaveraged density data provided insight into the shockcell structures. The radial profiles obtained at various axial stations covering a downstream distance of 10 jet diameters show the development of the jet shear layer and the decay of the shock–cells. The supersonic free jets produced screech sound. A phaseaveraged photon counting technique, using the screech tone as the trigger source, was used to measure the unsteady density variation. The phaseaveraged density data show the evolution of the largescale turbulent vortices that are found to be modulated periodically along the flow direction. A comparison with previously obtained data showing nearfield pressure fluctuation and convective speed of the organized vortices reveals many interesting dynamics. All quantities show regular spatial modulation. The locations of local maxima in density fluctuations are found to coincide with the high convective speed and the antinode points in the nearfield pressure fluctuation. Interestingly, the periodicity of modulation is found to be somewhat different from the shock spacing. Instead it shows that the standing wave system, known to exist in the nearfield pressure fluctuation, extends into the jet shear layer. The standing wave is formed between the downstream moving Kelvin–Helmholtz instability waves and the upstream propagating part of sound waves. A detailed field measurement of the unsteady density fluctuation was conducted for the and 1.42 jets for which the nearfield pressure fluctuation data were obtained previously. The phasematched, combined plots of the density fluctuation present inside the jet flow, and the pressure fluctuation present just outside the jet boundary provide a charming insight into the shock–vortex interaction leading to the sound wave generation.

Dynamic inverse modeling and its testing in largeeddy simulations of the mixing layer
View Description Hide DescriptionWe propose new identities for dynamic subgrid modeling in largeeddy simulation involving an explicit filter and its inverse. Exact defiltering of a class of numerical realizations of the tophat filter is developed. The approach is applied to largeeddy simulation of the temporal mixing layer. Smagorinsky’s model is adopted as base model and the results are compared to the standard dynamic eddyviscosity model as well as to filtered DNS (direct numerical simulation) results. The difference between the results of the two models for the present application is found to be quite small. This is explained by performing a sensitivity analysis with respect to the dynamic coefficient, which hints towards a “selfrestoring” response underlying the observed robustness of the physical predictions. Using DNS data the validity of the assumption that the model coefficients are independent of filter width is tested and found to favor the inverse modeling procedure. The computational effort of the dynamic inverse model is 15% smaller than of the standard dynamic eddyviscosity model.

The subgridscale estimation model on nonuniform grids
View Description Hide DescriptionThe subgridscale estimation procedure developed previously using onedimensional top hat filters on uniform grids is generalized to nonuniform grids. The method is evaluated in large eddy simulations of turbulent channel flow performed on a grid which is nonuniform in the wallnormal direction.

A twotimescale turbulence model for compressible flows: Turbulence dominated by mean deformation interaction
View Description Hide DescriptionThe multipletimescale concept is applied to develop a turbulencemodel for compressible flows.Transport equations for the turbulent kinetic energies and the energy transfer rates are linked to each domain of the turbulent spectrum. The model coefficients are calibrated, with respect to simple flows, by using a new method which takes advantage of the spectral character of the model. One innovation of this method is to use, as a component, the CG model [V. M. Canuto and I. Goldman, Phys. Rev. Lett. 54, 430 (1985)] which gives the large scale spectrum as a function of the instabilitygeneratingturbulence. Then, the twotimescale model, with its complete set of coefficients, has been successfully applied to the simulation of plane mixing layers and homogeneous shear flows. A significant issue of this work is the study of the behavior of the twotimescale model when a shock waveinteracts with a homogeneous turbulence. We first compare model results with experimental data for a 2.8 Mach number interaction [D. Alem, Ph.D. thesis, Université de Poitiers, 1995]. The decrease of the integral length scale, predicted by the linear analysis, is reproduced with the twotimescale model, which, moreover, recovered the rate of reduction measured by Alem. The amplification of the turbulence level through the shock wave is also consistent with the measurements. Then, we confront our results with a direct numerical simulation of the shock–turbulence interaction at [S. Lee et al., J. Fluid Mech. 251, 533 (1993)]. The spectrum of the turbulence injected in the inflow region of the direct numerical simulation appeared to be far from the freely decaying state. The twotimescale model, which accounts for the spectral nonequilibrium effects, is able to recover the spatial decrease of turbulence in the inflow region whereas a singletimescale model fails. Moreover, the profiles for the turbulent kinetic energy and its dissipation rate over all the calculation domain are much better reproduced with the twotimescale model than with the primary model.
