Volume 6, Issue 11, November 1994
Index of content:

Spatial transport of Reynolds stresses by pressure fluctuations
View Description Hide DescriptionThe spatialtransport of Reynolds stresses by the pressure fluctuations, the so‐called pressure transport, is analyzed by investigating the potential flow equation for the fluctuating pressure. Adopting the convection velocity concept for the unsteady term in the Euler equation, and integrating over space, the pressure–velocity correlation vector is decomposed into a bulk convective transport term (rapid part) and a turbulent diffusion term (slow part). Subsequent modeling of the convection velocity in terms of the intensity of turbulence, skewness, and the velocity scale of large‐scale eddies results in a new computational model for the pressure transport.

Influence of initial conditions on the flow patterns of a shock‐accelerated thin fluid layer
View Description Hide DescriptionPrevious observations of three flow patterns generated by shock acceleration of a thin perturbed, fluid layer are now correlated with asymmetries in the initial conditions. Using a different diagnostic (planar laser Rayleigh scattering) than the previous experiments, upstream mushrooms, downstream mushrooms, and sinuous patterns are still observed. For each experiment the initial perturbation amplitude on one side of the layer can either be larger, smaller, or the same as the amplitude on the other side, as observed with two images per experiment, and these differences lead to the formation of the different patterns.

Ignition of aluminum droplets behind shock waves in water
View Description Hide DescriptionGram quantities of molten aluminumdroplets at temperatures up to 1973 K are forced to interact with water under sustained pressure pulses of up to 40.8 MPa in a hydrodynamicshock tube. Conditions are identified under which the thermal interaction (physical regime) develops into chemical ignition and total combustion events. This, and the absence of catastrophic breakup in the physical regime, raises some very interesting questions with regard to the sequence of events during large‐scale aluminum–water explosions.

Pressure in lattice Boltzmann simulations of flow around a cylinder
View Description Hide DescriptionThe expression for the pressure in lattice Boltzmann realizations of the Navier–Stokes equations involves compressibility effects. The pressure in two‐dimensional steady and unsteady cylinder flow is obtained numerically in a lattice Boltzmann scheme for the first time and compared with experimental and finite‐difference results.

Time‐resolved velocity gradient and optical anisotropy in linear flow by photon correlation spectroscopy
View Description Hide DescriptionA time‐resolved dynamic light scattering technique is presented for measuring the velocity gradient in transient but repeatable flows. The experimental technique is verified via measurements for a Newtonian fluid undergoing a known time‐dependent flow. The method is then applied to the creeping flow of a Newtonian fluid in a corotating two‐roll mill. It is demonstrated that the flow near the stagnation point can be accurately described by an analytical creeping flowsolution for a two‐roll mill in an unbounded fluid. The time dependence of the velocity gradient for a concentrated polymersolution in the startup of the two‐roll mill has also been measured, it is believed, for the first time. The measurement provides direct evidence of the modification of the flow for the viscoelastic polymer liquid, and will ultimately lead to significant insights into the polymer dynamics for concentrated solutions in strong, extension‐like flows. A second significant feature of the dynamic light scattering experiment is that the initial magnitude of the correlation function is related to the degree of optical anisotropy of the polymer molecules, i.e., to the geometric configuration of the polymer chains. Thus, it yields information on the time‐dependent degree of polymer orientation and stretch that is equivalent to birefringence, but is obtained at the ‘‘point’’ occupied by the scattering volume rather than as a two‐dimensional average across the whole fluid as in birefringence. This measurement of polymer configuration is compared with birefringence data for the exact same flow.

Contact‐line motion of shear‐thinning liquids
View Description Hide DescriptionIt is demonstrated that, for the slow advance of a viscous liquid onto a previously dry substrate, the well‐known moving contact line paradox is alleviated for liquids exhibiting power‐law shear‐thinning behavior. In contrast to previous models that allow contact‐line motion, it is no longer necessary to abandon the no‐slip condition at the substrate in the vicinity of the contact point. While the stress is still unbounded at the contact point, the equations of motion are shown to be integrable. A three‐constant Ellis viscosity model is employed that allows a low‐shear Newtonian viscosity, and may thus be used to model essentially Newtonian flows where shear thinning only becomes important in the immediate vicinity of the contact point. Calculations are presented for the model problem of the progression of a uniform coating layer down a vertical substrate using the lubrication approximations. The relationship between viscous heating and shear‐thinning rheology is also explored.

Spin coating of viscoelastic and nonvolatile fluids over a planar disk
View Description Hide DescriptionSpin coating of two commercially used polymer solutions is studied both theoretically and experimentally. Physical and rheological characterization of these solutions indicates that under the spinning conditions currently used they behave as nonvolatile, viscoelastic fluids with constant viscosity and elasticity. The corresponding Reynolds (Re) and Deborah (De) numbers are up to order unity. The theoretical analysis demonstrates and explains why, at very short times after the inception of impulsive spinning, the velocity and stress fields in such fluids develop in an oscillatory manner. The amplitude of these oscillations increases with the ratio of the retardation parameter to the Deborah number, whereas their damping rate gets smaller as De increases. Since these oscillations dissipate very rapidly, and before substantial thinning of the film takes place, the thinning rate, velocity, and shear stress components do not deviate eventually from those of a Newtonian fluid. Such a complete explanation of similar experimental findings has not been offered before. The radial normal stress component does increase considerably over its Newtonian value, and this explains certain ‘‘experimental practices.’’ Similar oscillatory development early on occurs even at higher Re, as long as Re∼De, but it is dissipated again, this time because of the abrupt thinning of the film. The theoretical results are in good agreement with experimental measurements of ‘‘dry film’’ thickness and with dynamical measurements of ‘‘wet film’’ thickness during spinning, which are reported herein for the first time. Care must be taken in reporting ‘‘dry film’’ thickness because the commercial solutions under study retain part of the solvent after ‘‘soft baking’’ over a hotplate. Complete solvent removal produces dry films, but requires treatment in a vacuum oven, higher temperatures, and longer heating times.

Static shape of an acoustically levitated drop with wave–drop interaction
View Description Hide DescriptionThe static shape of a drop levitated and flattened by an acoustic standing wave field in air is calculated, requiring self‐consistency between the drop shape and the wave. The wave is calculated for a given shape using the boundary integral method. From the resulting radiation stress on the drop surface, the shape is determined by solving the Young–Laplace equation, completing an iteration cycle. The iteration is continued until both the shape and the wave converge. Of particular interest are the shapes of large drops that sustain equilibrium, beyond a certain degree of flattening, by becoming more flattened at a decreasing sound pressure level. The predictions for flattening versus acoustic radiation stress, for drops of different sizes, compare favorably with experimental data.

Experimental study of streaming flows associated with ultrasonic levitators
View Description Hide DescriptionSteady‐state acoustic streaming flow patterns have been observed during the operation of a variety of resonant single‐axis ultrasonic levitators in a gaseous environment and in the 20–37 kHz frequency range. Light sheetillumination and scattering from smoke particles have revealed primary streaming flows which display different characteristics at low and high sound pressure levels. Secondary macroscopic streaming cells around levitated samples are superimposed on the primary streaming flow pattern generated by the standing wave. These recorded flows are quite reproducible, and are qualitatively the same for a variety of levitator physical geometries. An onset of flow instability can also be recorded in nonisothermal systems, such as levitated spot‐heated samples when the resonance conditions are not exactly satisfied. A preliminary qualitative interpretation of these experimental results is presented in terms of the superposition of three discrete sets of circulation cells operating on different spatial scales. These relevant length scales are the acoustic wavelength, the levitated sample size, and finally the acoustic boundary layer thickness. This approach fails, however, to explain the streaming flow‐field morphology around liquid drops levitated on Earth. Observation of the interaction between the flows cells and the levitated samples also suggests the existence of a steady‐state torque induced by the streaming flows.

Bifurcation diagrams of axisymmetric liquid bridges subjected to axial electric fields
View Description Hide DescriptionThe stability of dielectric liquid bridges between plane parallel electrodes when an electric potential difference is applied between them is studied for an axisymmetric configuration regarding arbitrary volume, axial gravity, and unequal coaxial anchoring disks attached to the electrodes. The stability is determined from the bifurcation diagrams related to the static problem. Two mathematical approaches are presented which are different in scope. First, the Lyapunov–Schmidt projection technique is applied to give the liquid bridge bifurcation diagrams for the bridge considered as an imperfect cylindrical one. The imperfection parameters, i.e., the relative difference of radii to the mean diameter, the deviation from the cylindrical volume, and the gravitational Bond number, are assumed to be small. Second, a Galerkin/finite element technique is used to obtain numerically bifurcation diagrams for arbitrary values of all the parameters. Agreement between both methods is good for small enough values of the imperfection parameters. The effect of the polarization charges existing at the free surface is highlighted. As in the absence of applied electric field, the gravitational Bond number and the relative difference of radii separately decrease the stability of the liquid column, but both effects conveniently combined may cancel out.

The effect of a uniform magnetic field on the onset of steady Marangoni convection in a layer of conducting fluid with a prescribed heat flux at its lower boundary
View Description Hide DescriptionIn this paper a combination of analytical and numerical techniques are used to analyze the effect of a uniform vertical magnetic field on the onset of steady Marangoni convection in a horizontal layer of quiescent, electrically conductingfluid with a uniform vertical temperature gradient subject to a prescribed heat flux at its rigid lower boundary. Critical Marangoni numbers for the onset of instability are calculated which are significantly different from those calculated previously in the case of an isothermal lower boundary. Analytical results for the behavior of the critical Marangoni number in the asymptotic limit of large magnetic field strength are also obtained. It is concluded that the magnetic field always has a stabilizing effect on the onset of steady Marangoni convection, but that when the free surface is deformable situations with a sufficiently large Marangoni number will always have unstable modes no matter how strong the applied magnetic field is.

Analysis of velocity data taken in Surface Tension Driven Convection Experiment in microgravity
View Description Hide DescriptionSome velocity field results from the Surface Tension Driven Convection Experiment (STDCE) that was conducted aboard the USML‐1 Spacelab in 1992 are reported. 10 cSt silicone oil was placed in an open circular container (10 cm wide×5 cm deep) and heated either by a cylindrical heater (1.11 cm diam) placed along the centerline or by a CO_{2} laser to induce thermocapillary flow. Tests were conducted under varieties of powers, laser beam diameters, and free‐surface shapes. The flow field was studied by flow visualization and the data were analyzed by a PIV technique. The results from the velocity measurement are presented and the effects of heating mode and free‐surface shape on the flow are discussed. The results are also compared with a numerical analysis conducted in conjunction with the experiment. Good agreement is shown.

An investigation of recirculating flow in a driven cavity
View Description Hide DescriptionThe problem of steady incompressible laminar shear‐driven flow in a two‐dimensional quarter‐circular cavity is discussed. An analytical solution is presented to the continuity and momentum equations at leading order for the case when the Reynolds number (Re) is much smaller than one. In addition, the O(Re) correction is computed numerically, and comparison of the two‐term expansion for the streamfunction with a full numerical computation of the Navier–Stokes equations yields good agreement for a nominal value of Re as large as 100. Solutions for Re≫1 are also presented; these indicate several extra features that are not present in the classical driven‐cavity problem, pertaining in particular to the flow direction and to the manner in which the governing equations should be correctly scaled.

Stochastic characteristics of a point‐source plume diffusion field upstream of a sphere
View Description Hide DescriptionThis research aimed to clarify the disturbance generated by a sphere in a point‐source diffusing plume developing in grid‐generated turbulence. Of special concern is the effect of the distortion of the mean stream near the forward stagnation point on the diffusion. Some results from statistical processing of experimental data are given. It was found that, because of the strong distortion of the mean stream, molecular diffusion in the vicinity of the stagnation point is greatly accelerated. As a result, as the stagnation point is approached, the conditioned probability density function for near‐zero concentration shows a remarkable slump. This result is explained qualitatively with the help of the probability density function (PDF) budget equation. A new concept, smeariness, is defined and used to effectively explain the experimental results.

Nonlinear analysis of instability modes in the Taylor–Dean system
View Description Hide DescriptionThe linear and weakly nonlinear stability of flow in the Taylor–Dean system is investigated. The base flow far from the boundaries, is a superposition of circular Couette and curved channel Poiseuille flows. The computations provide for a finite gap system, critical values of Taylor numbers, wave numbers and wave speeds for the primary transitions. Moreover, comparisons are made with results obtained in the small gap approximation. It is shown that the occurrence of oscillatory nonaxisymmetric modes depends on the ‘‘anisotropy’’ coefficient in the dispersion relation, and that the critical Taylor number changes slightly with the azimuthal wave number for large absolute values of rotation ratio. The weakly nonlinear analysis is made in the framework of the Ginzburg–Landau equations for anisotropic systems. The primary bifurcation towards stationary or traveling rolls is supercritical when Poiseuille component of the base flow is produced by a partial filling. An external pumping can induce a subcritical bifurcation for a finite range of rotation ratio. Special attention is also given to the influence of anisotropyproperties on the phase dynamics of bifurcated solution (Eckhaus and Benjamin–Feir conditions).

Transient growth in circular pipe flow. I. Linear disturbances
View Description Hide DescriptionThe behavior of very low‐amplitude disturbances in a circular pipe is considered. Direct simulation of the Navier–Stokes equations is used to compute the evolution of two‐ and three‐dimensional waves and the results are found to be in good agreement with solutions to the Orr–Sommerfeld equation for Hagen–Poiseuille flow. Transient growth mechanisms are also investigated computationally, in which case it is found that the growth of disturbances with large but finite streamwise wavelength exhibits a very rich structure of temporal evolution depending on the particular initial condition chosen. Comparison with recent results reported by Bergström on optimal disturbances is also given. In Part II of this study these findings will be extended to the nonlinear development of like disturbances.

Transient growth in circular pipe flow. II. Nonlinear development
View Description Hide DescriptionThe Navier–Stokes equations for circular pipe flow are integrated using direct numerical simulation for the case of transitional Reynolds number. Previous work on linear disturbances (reported in Part I) is exploited for the simulation of low to moderate amplitude disturbances where it is found that the transient growth mechanism persists in the nonlinear development with the evolution attributable to the linear mechanism remaining of considerable significance. A hypothesis of Trefethen et al. [Science 261, 578 (1993)] concerning the role of nonlinearity in the transition process and ultimately in turbulence is elucidated and given support. It is suggested that nonlinearity is essential in continually perturbing the eigenmodes of the flow in such a way that each mode is never permitted to relax to its least stable eigenstate (damped in the subcritical case). In this way, the linear growth mechanism can be regarded as an underpinning component of the general nonlinear feedback insofar as it is the only part which can extract energy from the mean flow and thus yield a net increase in disturbance energy. The physical aspects of the flow simulations are consistent with puff formation where, using a pair of helical waves as initial data, a sharp trailing front is formed naturally; axisymmetric ring vortices are generated and the general flow characteristics are in broad agreement with experiment.

Centrifugal instability of pulsed flow
View Description Hide DescriptionThe stability of a pulsed flow in a Taylor–Couette geometry with both cylinders rotating at the same angular velocity Ω(t)=Ω_{0} cos (ωt) is investigated. The first experimental evidence showing that the flow is less unstable in the limit of low and high frequency while destabilization is maximum for an intermediate frequency ω_{0} is reported. A detailed analysis of the restabilization at frequencies just above ω_{0} reveals a behavior not accounted for by previous theoretical analysis. Thus, the linear stability analysis is reconsidered by using a different implementation of the Floquet theory and a satisfactory agreement with the present experimental results is found.

Timing of vortex formation from an oscillating cylinder
View Description Hide DescriptionThe instantaneous structure of the near‐wake of a cylinder subjected to forced oscillations is examined using particle imaging, which leads to representations of the streamline patterns and distributions of vorticity. As the frequency of excitation of the cylinder is increased relative to the inherent vortex formation frequency, the initially formed concentration of vorticity moves closer to the cylinder until a limiting position is reached; at this position, the vorticity concentration abruptly switches to the opposite side of the cylinder. This process induces abrupt changes of the topology of the corresponding streamline patterns; such topological patterns alone, however, do not properly suggest the existence and rearrangement of the vorticity concentrations. Moreover, this vorticity‐switching concept persists to high values of Reynolds number, where the values of the mean base pressure coefficient and vortex formation length differ substantially from those at low Reynolds number. The switching mechanism is not significantly altered, either in an instantaneous or ensemble‐averaged sense, by the presence of small‐scale Kelvin–Helmholtz vortices that coexist with the large‐scale (Kármán) vortices.

On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe
View Description Hide DescriptionThe bifurcation structure is presented for an axisymmetric swirling flow in a constricted pipe, using the pipe geometry of Beran and Culick [J. Fluid Mech. 242, 491 (1992)]. The flow considered has been restricted to a two‐dimensional parameter space comprising the Reynolds number Re and the relative swirl V _{0} of the incoming swirling flow. The bifurcation diagram is constructed by solving the time‐dependent axisymmetric Navier–Stokes equations. The stability of the steady results presented by Beran and Culick, obtained from a steady axisymmetric Navier–Stokes code, has been confirmed. Further, the steady solution branch has also been extended to much larger V _{0} values. At larger V _{0}, a stable unsteady solution branch has been identified. This unsteady branch coexists with the previously found stable steady solution branch and originates via a turning point bifurcation. The bifurcation diagram is of the type described by Benjamin [Proc. R. Soc. London Ser. A 359, 1 (1978)] as the canonical unfolding of a pitchfork bifurcation. This type of bifurcation structure in the two‐dimensional parameter space (Re,V _{0}), suggests the possibility of hysteresis behavior over some part of parameter space, and this is observed in the present study. The implications of this on the theoretical description of vortex breakdown and the search for a criterion for its onset are discussed.