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Volume 9, Issue 11, November 1997

Linear stability analysis of controlled RayleighBénard convection using shadowgraphic measurement
View Description Hide DescriptionWe conduct a linear stability analysis of RayleighBénard convection in an infinite horizontal layer with active control of the lower boundary heat flux. A simple linear proportional control loop uses a shadowgraph of the convection pattern to actively distribute the constantmean lower boundary heat flux while the upper boundary is kept at a constant temperature. We find it possible to shift the convection threshold by a factor of approximately 3. This is a companion paper to our recent experimental work.

Two and threedimensional instabilities of the cylinder wake in an aligned magnetic field
View Description Hide DescriptionTwo and threedimensional (2D and 3D) instabilities in the wake of a circular cylinder placed in an electrically conducting fluid and subjected to a constant magnetic field aligned with the freestream are investigated numerically. Increasing magnetic fields suppress 2D instability (vortex shedding), whereas 3D instabilities are influenced in a more complex way. In the presence of a magnetic field, 3D instability has been detected below the 2D stability threshold. This can lead to a reversal of the order of instabilities, i.e., 3D instability appears at lower than 2D instability.

Image of absolute instability in a liquid jet
View Description Hide DescriptionThe existence of absolute instability in a liquid jet has been predicted for some time. The disturbance grows in time and propagates both upstream and downstream in an absolutely unstable liquid jet. The image of absolute instability is captured in the NASA 2.2 sec drop tower, and is reported here.

On longitudinal and lateral moment hierarchy in turbulence
View Description Hide DescriptionA quantity of interest, (degree of intermittency between scale hierarchy), which appears in a recent intermittencymodel by She and Lévque [Phys. Rev. Lett. 72, 336 (1994)] is computed from turbulent simulations with TaylorReynolds number in the order of 100. It is found that the lateral and longitudinal values are unequal. For most of the moment hierarchy order , lateral deviates more from K41 (i.e., smaller) than the longitudinal. is also found to be dependent.

Cavity flows of elastic liquids: Twodimensional flows
View Description Hide DescriptionThere is a wealth of experimental and computational results available for the motion of Newtonian fluids in the liddriven cavity geometry, however little is known about the corresponding motion of viscoelasticfluids. We use laser Doppler velocimetry(LDV) and digital particle image velocimetry (DPIV) to probe the dynamics of viscoelasticfluid motion in the classic “liddriven cavity” problem for a range of industrially important aspect ratios using an ideal elastic fluid as the test material. The magnitude of nonNewtonian effects in the cavity are characterized by the dimensionless Deborah number and the experiments span the range . Elastic effects break the symmetry observed in the velocity field of cavity flows of viscous Newtonian fluids at zero Reynolds number. At low De, the flow remains twodimensional but increasing the imposed velocity causes the center of the primary recirculating vortex in the cavity to shift progressively upstream. At larger Deborah numbers, the fluid motion becomes unstable and a threedimensional flow develops. Upon cessation of the forcing boundary motion, a pronounced elastic recoil is observed which leads to a rapid reversal in the direction of the recirculating vortex. This transient motion subsequently decays through viscous dissipative effects on the elastic time scale of the fluid. The kinematics of the localized corner flow near the downstream corner are studied in detail and the distinguishing features of the viscoelastic corner flow with respect to the classic knifeedge problem of Taylor are reported.

Added mass of a disc accelerating within a pipe
View Description Hide DescriptionThe flow of inviscid fluid around a disc in a pipe is computed, and the results are used to determine the added mass of the accelerating disc in the frame in which the mixture velocity is zero. The added mass of an array of discs spaced at regular intervals along the pipe is then computed, and is related to the pressure gradient along the pipe. Some flow profiles are also presented. The results show that the added mass per particle increases as the pipe diameter is reduced relative to the particle size. The added mass per particle decreases as the number density of particles increases, but the added mass per unit length of the pipe nevertheless increases. Thus an increase of either the particle size or number density leads to a tighter coupling between the liquid and the particles; this result should hold for other particle shapes and configurations. Results are also presented for the drift, i.e., the displacement of fluid particles caused by the motion of an isolated disc along the axis of the pipe. If the diameter of the pipe is sufficiently small, the added mass of the disc is modified from that in unbounded fluid, and the background drift at the walls of the pipe can no longer be estimated from the added mass of the disc.

Pattern study in the 2D solutal convection above a Bridgmantype solidification front
View Description Hide DescriptionWe numerically investigate the twodimensional (2D) convective flow developing in the liquid phase above an alloy growing in the upward Bridgman configuration of directional solidification. Using a timedependent approach, we are able to describe the various cycles of hysteresis that connect the different branches of stable steady solutions. The main trends of the present results show that the bifurcation diagram, composed of the branches, found in previous works for the partition coefficient remains qualitatively valid for for a small frontal width the leading primary bifurcation is subcritical, while a transcritical bifurcation occurs for larger front. We bring the new complementary feature that the subcritical bifurcation becomes supercritical when the front width tends to zero. Furthermore, for an intermediate frontal width, we address the question of the nature of upper stability limits on various stable steady branches. We show that the limit occurs via either a steady secondary bifurcation or a Hopf bifurcation that initiates an unsteady solution branch which is followed up to chaos. The related route is a subharmonic cascade. When following this chaotic branch, a striking relaminarization process towards a steady secondary branch occurs. Finally we shortly investigate the case of a twice larger frontal width, for which several cycles of hysteresis are equally reported.

Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient
View Description Hide DescriptionWe investigate the spreading of passive tracers in closed basins. If the characteristic length scale of the Eulerian velocities is not very small compared with the size of the basin the usual diffusion coefficient does not give any relevant information about the mechanism of spreading. We introduce a finite size characteristic time which describes the diffusive process at scale . When is small compared with the typical length of the velocity field one has , where is the maximum Lyapunov exponent of the Lagrangianmotion. At large the behavior of depends on the details of the system, in particular the presence of boundaries, and in this limit we have found a universal behavior for a large class of system under rather general hypothesis. The method of working at fixed scale makes more physical sense than the traditional way of looking at the relative diffusion at fixed delay times. This technique is displayed in a series of numerical experiments in simple flows.

Molten droplet deposition and solidification at low Weber numbers
View Description Hide DescriptionLow Weber number deposition of small molten droplets on cold targets is of importance in certain dropwise buildup processes, but at this time, critical elements are absent from our theoretical understanding of the deposition process, and prediction from basic principles is not possible. This paper lays down a framework for understanding low Weber number deposition in terms of similarity laws and experimentation. Based on experiments from the highly viscous limit to the inertiadominated limit, correlations are given for the spreading velocity, spreading time scales, postspreading oscillation amplitudes, and oscillation damping time scales. Molten droplets are arrested, and their final solid shape determined, by contact line freezing. In homologous deposition, where the drop and the target are of the same material, the spreading factor is determined principally by the Stefan number, the dimensionless parameter which measures the temperature difference between the fusion point and the target temperature. Some concluding remarks are offered on what needs to be done to accurately compute such deposition processes.

On the pinchoff of a pendant drop of viscous fluid
View Description Hide DescriptionThe pinchoff of a drop of viscous fluid is observed using highspeed digital imaging. The behavior seen by previous authors is observed here; namely, the filament that attaches the drop to the orifice evolves into a primary thread attached to a much thinner, secondary thread by a slight bulge. Here, we observe that the lengths of the primary and secondary threads are reproducible among experiments to within 3% and 10%. The secondary thread becomes unstable as evidenced by wavelike disturbances. The actual pinchoff does not occur at the point of attachment between the secondary thread and the drop. Instead, it occurs between the disturbances on the secondary thread. After the initial pinchoff, additional breaks occur between the disturbances, resulting in several secondary satellitedrops with a broad distribution of sizes. The pinchoff of the thread at the orifice is similar to that at the drop with one main difference: there is no distinct secondary thread. Instead, the primary thread necks down monotonically until wavelike disturbances form, resulting in pinchoff at multiple sites in between. The speed of the tips of the retreating, secondary threads after pinchoff are reported and discussed in the context of various scaling laws.

Visualization of the fluid flow field around a laser generated oscillating bubble
View Description Hide DescriptionWe report on a new method of visualizing qualitatively a fluid flow field around an oscillating cavitation bubble. The near infraredradiation of a Qswitched Nd:YAG laser is focused in deionized water and in a solution of copper sulphate for contrast enhancement to generate a cavitation bubble in free space and in front of a solid boundary. Using highspeed photography and an accurately positioned schlieren knife edge we are able to visualize the heated path of the laser beam and the different stages of the oscillating bubble with high spatial resolution. For the case of a bubble in free space the marked laser path indicates radial fluid flow only. For a bubble in front of a solid boundary the marked laser path clearly shows the motion of the fluid through the bubble during the collapse process. The marked paths are similar to numerically calculated streamline plots.

Drop breakup in the flow through fixed beds via stochastic simulation in model Gaussian fields
View Description Hide DescriptionShaqfeh and Koch have shown that the flow through a dilute disordered fixed bed of fibers produces large polymer conformation change beyond a certain critical flow rate [J. Fluid Mech. 244, 17 (1992)]. We now examine the effect of this flow on the shape and breakup of viscous drops. Because the flow through a dilute fixed bed is equivalent to a certain anisotropic Gaussian flow field, we follow our previous paper and reproduce a model of the flow through a spectral expansion where the wave number vectors are chosen from statistical distributions which ensure that the desired velocity field will be realized [Phys. Fluids 9, 1222 (1997)]. We examine the dynamics of model drop shapes, averaged over the Gaussian statistics of the flow field, by synthesizing a large number of flow realizations. The drop surface is modeled using the first, second, and third order small deformation theories which can accurately predict critical conditions in classical strong flows. While the first order model yields a bounded average drop shape for all flow conditions, the second and third order models demonstrate that the flow through fixed beds is indeed “strong” since beyond a certain value of the poresize capillary number, large average drop deformation occurs and the average drop shape becomes unbounded (“drop breakup”). This critical condition is determined for various viscosity ratios and fixed bed particle volume fractions. Similar to a simple shear flow, we find that there is a critical viscosity ratio, beyond which breakup is not observed in the fixed bed for any In addition, the critical condition is shown to depend heavily on the transient nature of the flow in the bed since approximately half of the flow fields in which drop breakup occurs would not break an initially spherical drop at any if they were steady. For supercritical capillary numbers, we define conditions under which the unbounded drop shapes fragment into smaller droplets and we examine the drop breakup rates as a percentage of the drop population.

Coalescence limited by hydrodynamics
View Description Hide DescriptionWe consider an assembly of liquid drops imbedded in another immiscible liquid of similar viscosity. It is shown that a coalescence between two drops induces another coalescence when the average distance between the drops is less than a threshold value, resulting in a “chain reaction” of coalescences. The threshold value is calculated using a “shell” model that is based on the boundary integral approach. Another “manydrop” model is developed to test the shell approximation. We show that, although the shell model is adequate, its results can be improved by lowering the shell surface tension.

The effect of surface roughness on flow structures in a neutrally stratified planetary boundary layer flow
View Description Hide DescriptionThe effects of surface roughness on the structures of a neutrally stratified planetary boundary layerflow are investigated by the largeeddy simulation technique. Our numerical model, which assumes horizontal periodicity, shows that the growth of an internal boundary layer (IBL) in response to an abrupt change of surface roughness (either smoothtorough transition or roughtosmooth transition) obeys the power of the time, similar to that along the downwind fetch. A sudden increase or decrease in the surface shear stress during the transition is also observed. A quadrant analysis shows that during the transition, ejections and sweeps are altered significantly. Flow visualization further illustrates that the distribution density and the strength of coherent vortical structures and ejection eddies increase substantially during the smoothtorough transition. Conversely, these parameters decrease in the roughtosmooth transition. The mean velocity profile has an inflection point at the IBL top, but the coherent vortical motions and ejection eddies affected by the change of the roughness are inside the IBL, suggesting that this inflection point is more static than dynamic. We also compare the quasisteady coherent flowstructures of different surface roughness values after the transition period. Streak spacing appears to increase with increasing surface roughness. Ejection eddies and vortical structures increase in scale as well as in strength as the surface roughness increases. The correlation between drag coefficient and flowstructures in boundary layerflows is discussed.

Threedimensional instability of the shear layer over a circular cylinder
View Description Hide DescriptionThe instability of the shear layer separated from a circular cylinder was investigated for Reynolds numbers, based on cylinder diameter, from 3500 to 53 000. For this, the most amplified instability frequencies were found and compared against available results for this range. Based on this initial set of measurements, was chosen for a detailed study of the instability of the separated shear layer to 3D disturbances. This involved the excitation of plane Kelvin–Helmholtz waves using an acoustic source, and pairs of oblique waves of equal but opposite angles produced by roughness which was distributed evenly along the cylinder span at the separation line. The sensitivity of the shear layer instability to the initial 2D disturbance amplitude and 3D disturbance wave angle was documented. The results showed that there exists a most amplified 3D wave angle (spanwise wave length) for the separated shear layer. The sensitivity to the wave angle agreed well with the linear 3D stabilityanalysis of a Stuart vortex by Pierrehumbert and Widnall and the numerical simulations of a free shear flow by Rogers and Moser.

Spatial simulation of the instability of channel flow with local suction/blowing
View Description Hide DescriptionA direct numerical simulation was made of instability in a spatially evolving channel flow. A local surface suction/blowing was imposed at the upper wall A Tollmien–Schlichting (TS) wave was superimposed on the laminar channel flow at the inflow. At the outflow, the buffer domain technique was applied to suppress the reflection of outgoing waves. The influence of the local suction/blowing on the linear and nonlinear instabilities of the flow was examined. It was found that the local suction/blowing increases the disturbance energy significantly in the interaction zone for subcritical (Re=5000) and supercritical (Re=10 000) cases. The effects of the blowing strength and the initial TS wave amplitude on the subcritical channel flow were scrutinized. Two regimes of the wave/flow interaction were found by varing i.e., “monotonic” and “vortex splitting” regimes.

Shear instability of twofluid parallel flow in a Hele–Shaw cell
View Description Hide DescriptionWe study experimentally the parallel flow in a Hele–Shaw cell of two immiscible fluids, a gas and a viscousliquid, driven by a given pressure gradient. We observe that the interface is destabilized above a critical value of the gas flow and that waves grow and propagate along the cell. The experimental threshold corresponds to a velocity difference of the two fluids in good agreement with the inviscid Kelvin–Helmholtz instability, while the wave velocity corresponds to a pure viscous theory deriving from Darcy’s law. We report our experimental results and analyze this instability by the study of a new equation where the viscous effects are added to the Euler equation through a unique drag term. The predictions made from the linear stability analysis of this equation agree with the experimental measurements.

Spatiotemporal dynamics of forced periodic flows in a confined domain
View Description Hide DescriptionNonlinear dynamics of a localized linear array of vortices is investigated by numerical simulations. The setup consists of a thin fluid layer (electrolyte) enclosed in a rectangular box and driven by the injection of homogeneous electric currents in an alternating magnetic field. The spectral model simulates the Navier–Stokes equations in two dimensions with steady forcing and linear bottom friction. The model provides an accurate representation of the evolution of flow pattern. Fourier decomposition of the streamfunction shows that a subharmonic instability occurs in the same symmetry subspace as that of a basic flow in the primary instability regime and a shear flow mode appears in a different symmetry subspace in the secondary instability regime. The exploration of temporal behavior shows that the system produces Hopftype bifurcations. Chaos and frequency locking due to the mutual interaction between the unstable modes are observed. A general scenario of the dynamics of forced periodic flows is discussed by the amplitude equations, which are modeled using formal group theoretical techniques.

Lattice gas automaton simulation of acoustic streaming in a twodimensional pipe
View Description Hide DescriptionThe lattice gas automaton fluidmodelling technique is used to study acoustic streaming phenomena arising from the interaction of sound waves with noslip boundaries in a twodimensional pipe. It is demonstrated that this fluid simulation tool is able to reproduce the general form of the acoustic streaming flow field observed by Andrade [Proc. R. Soc. London 175, 1 (1883)] and predicted by Lord Rayleigh [Philos. Trans. R. Soc. London 175, 1 (1884)]. The differences that are evident may be due to the restrictions on the computing resource which prohibits the use of a lattice large enough to fulfill the conditions used by Rayleigh in his derivation of the acoustic streaming flow field.

Flowfield characteristics of an aerodynamic acoustic levitator
View Description Hide DescriptionA droplet held in a singleaxis ultrasonic levitator will principally sustain a certain external blowing along the levitation axis, which introduces the possibility of investigating heat and/or mass transfer from the droplet under conditions which are not too remote from those in spray systems. The focus of the present work is on the influence of the acoustic field on the external flow. More specifically, an axisymmetric submerged gas jet in an axial standing acoustic wave is examined, both in the absence and presence of a liquid droplet.Flow visualization is first presented to illustrate the global flow effects and the operating windows of jet velocities and acoustic powers which are suitable for further study. An analytic and numeric solution, based on the parabolic boundary layer equations are then given for the case of no levitated droplet, providing quantitative estimates of the acoustic field/flow interaction. Detailed velocity measurements using a laser Doppler anemometer verify the analytic results and extend these to the case of a levitated droplet. Some unresolved discrepancy remains in predicting the maximum velocity attainable before the droplet is blown out of the levitator. Two methods are developed to estimate the sound pressure level in the levitator by comparing flowfield patterns with analytic results. These results and observations are used to estimate to what extent acoustic aerodynamic levitators can be used in the future for investigating transport properties of individual droplets.