Volume 12, Issue 2, February 2000
 LETTERS


Selfadaptation and viscous selection in concentrated twodimensional vortex dipoles
View Description Hide DescriptionIn this Letter we deal with 2D direct numerical simulations of concentrated vortex dipoles. We show that various initial dipolar vorticity distributions evolve towards a specific family of dipoles parametrized by the dipole aspect ratio where a is the radius of the vortices based on the vorticity polar moment in half a plane and b is the separation between the vortex centroids. This convergence is achieved through viscous effects. The considered Reynolds numbers are and Moreover, all the dipoles of this family are quasisteady solutions of the Euler equations. Their scatter plots and drift velocities are given for

Internal flow of an electrostatically levitated droplet undergoing resonant shape oscillation
View Description Hide DescriptionExperimental evidence of internal dc flow circulation within an electrostatically levitated, 3.4 mm diameter charged water drop oscillating in shape at a different amplitude is presented. The axisymmetric shape oscillations were excited using a continuously pulsed electrostatic field at the fundamental shape mode frequency of the drop. It was found that the speed of the internal dc flow varied quadratically with respect to the oscillation amplitude without an observable minimum threshold. The implication of the observed internal flow is discussed with respect to the microgravity research facility and environment.

Complex dynamics of combustion flows by direct numerical simulations
View Description Hide DescriptionThe dynamics of a well characterized hydrogen–air combustion flow is studied at high pressures, with detailed chemistry and multicomponent transport, using numerical bifurcation theory, local stability analysis, and timedependent simulations. We report, for the first time, complex dynamics including oscillatory multiplicity, isolas of periodic orbits, and chaotic behavior. The relatively wide range of conditions identified, over which such complex dynamics occurs, could allow for experimental verification of numerical predictions.
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 ARTICLES


Viscous fingering in a shearthinning fluid
View Description Hide DescriptionWe study the Saffman–Taylor instability in a rectangular HeleShaw cell. The driven fluid is a dilute (or semidilute) polymer solution, with a viscosity that exhibits shear thinning. Other nonNewtonian properties such as elastic effects are negligible under the present experimental conditions; the system thus allows for separate investigation of the influence of shear thinning on the instability. The experiments show that, for weak shearthinning, the results for the width of the fingers as a function of the capillary number collapse onto the universal curve for Newtonian fluids, provided the shearthinningviscosity is used to calculate the capillary number. For stronger shear thinning, narrower fingers are found. The experiment allows also for a study of the applicability of Darcy’s law to shear thinning fluids. For Newtonian fluids, this law gives the finger velocity as a function of the pressure gradient. For weakly shearthinning fluids, we find that an effective Darcy’s law, in which the constant viscosity is replaced by the shearthinningviscosity, gives good agreement with the experiments. For stronger shear thinning, the predictions from the effective Darcy’s law deteriorate. Satisfactory agreement with experimental data can be obtained when using a “shearthinning” Darcy’s law, which can be derived using a power law model for the shear rate dependence of the viscosity.

Asymptotic study of the convective parametric instability in HeleShaw cell
View Description Hide DescriptionThis paper concerns a linear study of the convective parametric instability in the case of a Newtonian fluid confined in a HeleShaw cell and submitted to a vertical periodic motion. The gradient of temperature, applied to the fluid layer, is either in the same direction that gravity or in the opposite one. An asymptotic analysis shows that the HeleShaw approximation leads to two linear formulations depending on the order of magnitude of the Prandtl number. For these two asymptotic cases, the convective threshold is determined. It turns out that in the HeleShaw geometrical configuration, parametric oscillations have no influence on the criterion of stability when the Prandtl number is in the order of the unity or very superior to the unity. However, when the Prandtl number is small than unity, the parametric oscillations can affect the convective instability threshold.

Numerical simulation of breakup of a viscous drop in simple shear flow through a volumeoffluid method
View Description Hide DescriptionA spherical drop, placed in a second liquid of the same density, is subjected to shearing between parallel plates. The subsequent flow is investigated numerically with a volumeoffluid (VOF) method. The scheme incorporates a semiimplicit Stokes solver to enable computations at low Reynolds number. Our simulations compare well with previous theoretical, numerical, and experimental results. For capillary numbers greater than the critical value, the drop deforms to a dumbbell shape and daughter drops detach via an endpinching mechanism. The number of daughter drops increases with the capillary number. The breakup can also be initiated by increasing the Reynolds number.

Resonant coupling of oscillating gas or vapor bubbles in water: An experimental study
View Description Hide DescriptionResonant coupling of oscillating gas or vapor bubbles in water is experimentally studied using a twofrequency acoustic apparatus. First, the threshold amplitude of resonant coupling between the volume and axisymmetric shape oscillations is measured for gas bubbles in the shape modes, and for vapor bubbles in the modes, and 5, where n is the mode number. The result qualitatively agrees with a theoretical prediction based on a phasespace analysis. Secondly, the efficiency of two coupling conditions, two–one (2:1) and one–one (1:1) resonant conditions is compared for the and 5 modes. The result of the comparison can be generalized as follows: When n is an even number, the (1:1) coupling condition has better energy transfer efficiency than (2:1). The relationship reverses when n is an odd number, i.e., the (2:1) coupling condition has better energy transfer efficiency than (1:1).

Experiments on twodimensional vortex patterns
View Description Hide DescriptionThe evolution of a strongly magnetized electron system is identical to that of an ideal twodimensional (2D) fluid; an electron column is equivalent to a fluidvortex. We have studied the stability of 2D vortex patterns with electron columns confined in a Malmberg–Penning trap. The following cases are presented: the stability of Nvortices arranged in a ring; the stability of Nvortices arranged in a ring with a central vortex; the stability of more complicated vortex patterns.

Twofluid TaylorCouette flow with countercurrent axial flow: Linear theory for immiscible liquids between corotating cylinders
View Description Hide DescriptionWe computationally investigate the stability of a pair of radially stratified immiscible liquids undergoing countercurrent axial flow in the annular gap between rapidly corotating coaxial cylinders: twofluid TaylorCouette flow with counterflow. A simple analysis determines conditions under which a nearly cylindrical interface is maintained in the presence of counterflow (i.e., axial pressure gradients). Stability analysis reveals that for small axial Reynolds numbers, the flow is slightly stabilized against TaylorCouetteinstability, consistent with results for a single phase. At axial Reynolds numbers greater than about ten, however, the flow is susceptible to a (generally nonaxisymmetric) KelvinHelmholtz instability, which precedes the TaylorCouette mode. Furthermore, new results are presented for the case without axial flow. A bifurcation to vortices that corotate with their counterparts in the other phase is found. Finally, limitations of the generalized Rayleigh criterion developed in our earlier work are elucidated. In particular, we show how it fails if one of the fluid layers is very thin.

Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories
View Description Hide DescriptionThe turbulent Rayleigh–Taylor instability is investigated over a comprehensive range of fluid density ratio and different acceleration histories using the Linear Electric Motor. The mixing layer is diagnosed with backlit photography and laserinduced fluorescence. For a constant acceleration, the bubble (2) and spike (1) amplitudes are found to increase as with and with For temporally varying accelerations this can be generalized to using rather than the displacement For impulsive accelerations, S remains constant during the coast phase and the amplitudes obey a power law with and with These values of and compare favorably with numerical simulations and mix models. The average diameter at the mixing front for bubbles is found to increase as in qualitative agreement with “merger” models, but the associated is two times larger than the terminal velocity of an isolated bubble. The spikes become relatively narrow at large R, yet they still grow as

Measurements of surfacewave damping in a container
View Description Hide DescriptionFor surface waves in brimful right circular cylinders with clean interfaces and pinned contact lines, damping rates and frequencies are measured for the six lowest frequency surfacewave modes over a twodecade range of the inverse Reynolds numberC. Asymptotic calculations that include viscous dissipation in both Stokes boundary layers and the bulk show good agreement with measurements for small C; the theory typically overpredicts the measured damping rates for C large. Our measurements suggest interfacial contamination may account for a previously reported discrepancy between theory and experiment.

Relations between intermittency and structure function exponents in turbulence
View Description Hide DescriptionWe derive from the Navier–Stokes equation an exact equation satisfied by the dissipation rate correlation function, which we study in the equal time limit, for homogeneous, isotropic turbulence. We exploit its mathematical similarity to the corresponding equation derived from the onedimensional stochastic Burgers equation to show that the main intermittency exponents are and where the ζ’s are exponents of velocity structure functions and is a dynamical exponent characterizing the fourth order structure function. We discuss the role of sweeping and Galilean invariance in determining the intermittency exponents.

Approximations for turbulent energy and temperature variance dissipation rates in grid turbulence
View Description Hide DescriptionUsing a combination of a transverse vorticity probe and a pair of parallel cold wires, simultaneous approximations, denoted as and to the energy dissipation rate ε and the temperature variance dissipation rate are obtained in decaying grid turbulence at a Taylor microscale Reynolds number of about 52. While there are important differences between the spectra of either or and those of their isotropic counterparts and the correlation between and is as small as that between and The large discrepancies, which exist in the literature, for the correlation coefficient between the locally averaged values of ε and reflect a dependence on the flow type as well as on the Reynolds number. Whereas is strongly correlated with the correlation between and is weak. The correlation between and is comparable to that between and The effect different choices of ε and have on the refined similarity hypothesis (RSH) (Kolmogorov, 1962) and its extension (RSHP) to a passive scalar is also examined. By reference to a nearly complete ε obtained with a threecomponent vorticity probe, RSH is more closely satisfied by than In contrast, RSHP appears to be approximately satisfied, regardless of which approximations are used for ε and

A lowdimensional approach to nonlinear planeCouette flow of viscoelastic fluids
View Description Hide DescriptionThe nonlinear stability of the onedimensional plane Couette flow is examined for a Johnson–Segalman fluid. The velocity and stress are represented by symmetric and antisymmetric Chandrasekhar functions in space. The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Both inertia and normal stress effects are included. For given Reynolds number and viscosity ratio, two critical Weissenberg numbers are found at which an exchange of stability occurs between the Couette and other steady flows. The critical points coincide with the two extrema of the stress/rateofstrain curve. At low (high) Reynolds number, the flow decays monotonically (oscillatorily) toward the steadystate solution. The number and stability of the nontrivial branches around the critical points are examined using the method of multiple scales. Comparison between the approximate and the numerical branches leads to excellent agreement in the vicinity of the critical points. The influence of the higherorder modes is assessed, showing loworder convergence and good accuracy when the flow profiles are compared against existing finiteelement results.

The initialvalue problem for a modeled boundary layer
View Description Hide DescriptionThe stability of a basic parallel flow of a viscous incompressible fluid has been traditionally investigated by linearizing the Navier–Stokes equations and using the method of normal modes in order to determine the temporal behavior. In this paper we adopt a method introduced by Lord Kelvin and used much in recent years, solving the equations instead as an initialvalue problem and thereby the full dynamics, both the early transient and the long time asymptotic, can be determined directly. The basic flow investigated is a boundarylayer modeled by a piecewise linear profile. Various initial distributions of velocity and vorticity are used as a basis of fundamental solutions to represent more general initial disturbances of the basic flow. The simplicity of the model permits the problem to be solved by use of a movingcoordinate system and of matched asymptotics in the limit of small viscosity. The solutions are remarkably explicit, although complicated and substantiate the increasingly common view that linear disturbances, although decaying ultimately according to the linear theory, may grow so much transiently as to excite nonlinear growth and effective instability; in particular, streamwise vortices are found to be strongly amplified. Although not done here, the method can be applied to other parallel shear flows.

Absolute and convective instabilities of a swirling jet/wake shear layer
View Description Hide DescriptionThe absolute (AI)/convective (CI) nature of the instability is determined in the family of swirling jet/wake shear layers considered by Martin and Meiburg [Phys. Fluids 6, 424 (1994)] and Lim and Redekopp [Eur. J. Mech. B/Fluids 17, 165 (1998)]. This idealized model includes as essential ingredients both the centrifugal instability associated with the swirl difference and the Kelvin–Helmholtz instability associated with the swirl and axial velocity differences between the core and the outer flow. Centrifugally stabilizing or destabilizing swirl differences are found to promote AI, but a centrifugally destabilizing configuration is more effective in triggering such a transition. For sufficiently large swirl differences, both coflowing jets and wakes may become AI. In the case of jets, a centrifugally destabilizing swirl difference first brings about AI via the axisymmetric mode in a large range of mean swirl values. By contrast, a centrifugally stabilizing swirl difference triggers AI via the helical mode In the case of wakes, a centrifugally destabilizing swirl difference leads to AI via the bending mode whereas a centrifugally stabilizing swirl difference triggers AI via various negative helical modes etc.

Validation of acousticanalogy predictions for sound radiated by turbulence
View Description Hide DescriptionPredicting sound radiated by turbulence is of interest in aeroacoustics, hydroacoustics, and combustionnoise. Significant improvements in computertechnology have renewed interest in applying numerical techniques to predict sound from turbulent flows. One such technique is a hybrid approach in which the turbulence is computed using a method such as direct numerical simulation (DNS) or large eddy simulation(LES), and the sound is calculated using an acoustic analogy. In this study, sound from a turbulent flow is computed using DNS, and the DNS results are compared with acousticanalogy predictions for mutual validation. The source considered is a threedimensional region of forced turbulence which has limited extent in one coordinate direction and is periodic in the other two directions. Sound propagates statistically as a plane wave from the turbulence to the far field. The cases considered here have a small turbulentMach number so that the source is spatially compact; that is, the turbulence integral scale is much smaller than the acoustic wavelength. The scaling of the amplitude and frequency of the farfield sound for the problem considered are derived in an analysis based on Lighthill’s acoustic analogy. The analytical results predict that the farfield sound should exhibit “dipoletype” behavior; the rootmeansquare pressure in the acoustic far field should increase as the cube of the turbulentMach number. The acoustic power normalized by the turbulent dissipation rate is also predicted to scale as turbulentMach number cubed. Agreement between the DNS results and the acousticanalogy predictions is good. This study verifies the ability of the Lighthill acoustic analogy to predict sound generated by a threedimensional, turbulent source containing many length and time scales.

Effects of shallowness on the development of freesurface mixing layers
View Description Hide DescriptionThe development of two shallow mixing layers with different water depths is analyzed experimentally by means of laser Doppler anemometry. The experiments show that bottom friction plays an important role in the growth of the mixing layer width and in the strength and dimensions of the large quasi twodimensional turbulence structures therein. It is found in this study that the initial growth rate of both mixing layers is similar to what has been found for deep water plane mixing layers. Further downstream the reduction of the growth rate can be ascribed to the decrease of the velocity difference between the two ambient streams in combination with the suppression of the growth of the large turbulence structures. In the most shallow mixing layer considered, the influence of the bottom friction is dominant, impeding the further growth of the mixing layer width. It is demonstrated that the reduced mixing layer growth is related to a loss of coherence in the large turbulence structures. This loss of coherence also reduces the characteristic lengthscale that establishes the lateral mixing of matter and momentum in the mixing layer. Eventually the water depth becomes the dominant length scale that determines the characteristic motion of the turbulence in that case. From the energy density spectra of the turbulencefluctuations and from the phase relation between the two velocity components in the horizontal plane it is concluded that large structures contribute most to the exchange of momentum in the mixing layer and thus to the Reynoldsstresses.

Numerical studies of flow over a circular cylinder at
View Description Hide DescriptionFlow over a circular cylinder at Reynolds number 3900 is studied numerically using the technique of large eddy simulation. The computations are carried out with a highorder accurate numerical method based on Bsplines and compared with previous upwindbiased and central finitedifference simulations and with the existing experimental data. In the very near wake, all three simulations are in agreement with each other. Farther downstream, the results of the Bspline computations are in better agreement with the hotwire experiment of Ong and Wallace [Exp. Fluids 20, 441–453 (1996)] than those obtained in the finitedifference simulations. In particular, the power spectra of velocity fluctuations are in excellent agreement with the experimental data. The impact of numerical resolution on the shear layer transition is investigated.

Prediction of turbulent free shear flow statistics using a simple stochastic model
View Description Hide DescriptionOnedimensional turbulence, a stochastic simulation of turbulent flow evolution based on application of a mixinglengthtype hypothesis to individual turbulenteddies, is used to predict transverse profiles of singlepoint statistics up to third order for two timedeveloping planar free shear flows, a mixing layer and a wake. Comparison of computed results to statistics obtained from direct numerical simulations of these flows indicates that the model, despite its simplicity, captures important features of turbulent free shear flow structure. Implications concerning the possible universality of some aspects of turbulentshear flow are discussed.
