Volume 8, Issue 12, December 1996
Index of content:

Thermocapillary instabilities with system rotation
View Description Hide DescriptionInfluence of rotation on thermocapillary instabilities is demonstrated from linear stability analysis of a parallel flow model relevant to crystal growth. It is shown that system rotation has a dramatic influence on transitions at parameter values typical of the microgravity environment of an orbiting space laboratory. Thus the Coriolis force must be included in the design of future space experiments.

Attractors of finite‐sized particles: An application to enhanced separation
View Description Hide DescriptionIt is shown in the context of the periodically driven eccentric annular system that small rigid spherical particles advected in a time‐periodic bounded flow can be represented by a dissipative dynamical system with simple or strange attractors. The behavior of such particles differs significantly from the patterns in the fluid flow driving the dynamical system. This may have implications for transport, mixing, and separation in multiphase flows as well as in predicting the dispersion of trace substances in natural fluid media. Numerical results suggest that the sensitivity of the limit sets to parameter values may be used to separate species of particles differing slightly in the value of some physical property.

A blinking rotlet model for chaotic advection
View Description Hide DescriptionThe Stokes flow due to a rotlet in a circle is determined. The solution shows that for a certain position of the rotlet, the flow has a second stagnation point symmetrically placed inside the circle. Thus, a ‘‘blinking rotlet’’ model can be constructed in which the rotlet that is ‘‘off’’ does not disturb the flow. This model seems preferable to the ‘‘blinking vortex’’ flow when discussing chaotic advection by a Stokes flow, and is useful for comparisons with recent experimental and computational investigations of this phenomenon in a cylindrical tank with two rotating cylinders.

Density variations in a one‐dimensional granular system
View Description Hide DescriptionIn this work we examine a system of inelastic particles confined to move on a line between an elastic wall and a heat source. Solving a Boltzmann equation for this system leads to an analytic expression for steady state behavior. Numerical simulations show that the system is in fact capable of simultaneously displaying both the uniform density of the analytic solution, and a state in which the particles are collected into a cluster adjacent to the elastic wall. The boundary conditions for the Boltzmann treatment are then reworked to provide a theoretical description of how smooth particle distributions and clumping phenomena can coexist. From this, we gain a prediction for the time scale of clump formation in this system.

Retarded motion of bubbles in Hele–Shaw cells
View Description Hide DescriptionThe motion of bubbles in a Hele–Shaw cell driven by a surrounding fluid or by gravity has been studied. Assuming that the surrounding fluid wets the solid wall and that the bubble surface is rigid due to the surfactant influence, the translational velocity of an elliptic bubble is estimated. The result indicates that the bubble velocity can decrease by an order of magnitude compared to the prediction of Taylor and Saffman [Q. J. Mech. Appl. Math. 12, 265 (1959)] due to the surfactant influence. The retarded bubble velocity is apparently in reasonable agreement with the experimental observations of Kopf‐Sill and Homsy [Phys. Fluids 31, 18 (1988)], suggesting that the puzzling observations by them are likely to be due to the surface active contaminants.

Streaming generated in a liquid bridge due to nonlinear oscillations driven by the vibration of an endwall
View Description Hide DescriptionIt has been observed that streaming can be generated in a liquid bridge supported by two endwalls, with one wall vibrating to excite its capillary oscillations [Anilkumar et al., J. Appl. Phys. 73, 4165 (1993); Mollot et al., J. Fluid Mech. 255, 411 (1993)]. The finding has been applied to suppress thermocapillary convection in crystal growth using the float zone technique [Grugel et al., J. Cryst. Growth 142, 209 (1994)]. In this work we shall explain the mechanism that drives the streaming, for low streaming velocities, in terms of an ‘‘acceleration shear’’ of the free surface of the liquid column. The results compare favorably with experiments with long columns. We have found that for a low viscosity liquid, the streaming pattern changes sign when the oscillation is adjusted from one side of a resonant peak to the other. This allows for more flexibility in the application of the streaming to counteract thermocapillary convection, during crystal growth in a float zone. We have also found that for low viscosities, the streaming velocity is inversely proportional to viscosity.

Suppression of instability in a liquid film flow
View Description Hide DescriptionThe stability of a viscousliquid filmflow down an inclined plane that oscillates in the direction parallel to the flow is analyzed by use of a Chebyshev series solution with the Floquet theory. When the inclined plane is stationary, it is known that the onset of the film instability manifests itself as long surface waves [J. Fluid Mech. 554, 505 (1957); Phys. Fluids 6, 321 (1963)] or relatively short shear waves [‘‘Critical angle of shear wave instability in a film,’’ to appear in J. Appl. Mech.; J. Eng. Math. 8, 259 (1974); Phys. Fluids 30, 983 (1987)], depending on the angle of inclination. It is demonstrated that the unstable film can be stabilized by use of appropriate amplitudes and frequencies of the plate oscillation to suppress the shear waves as well as the long waves. The ranges of amplitude and frequency in which the film can be stabilized depend on the flow parameter.

A stability study of the developing mixing layer formed by two supersonic laminar streams
View Description Hide DescriptionAn inviscid, parallel, spatial linear stabilityanalysis is performed on both bounded and semibounded developing mixing layers formed between a laminar Mach 8 stream and another at Mach 3. Three unstable modes have been found in the initial mixing zone of the unbounded flow. In the downstream region of this flow the slowest of these modes becomes stable, and the remaining two correspond to the fast and slow modes of the self‐similar mixing layer. For the semibounded flow, a wall in the slow stream introduces a series of acoustic modes which replace the fast mode of the unbounded flow. For both the unbounded and the semibounded flows the largest growth rates belong to the slowest mode which resides in the developing region and is insensitive to the presence of the wall. The existence of this mode has been detected in a wind‐tunnel experiment, at frequency ranges suggested by, and with maximum growth rates in agreement with, the theory.

Steady spatial oscillations in a curved duct of square cross‐section
View Description Hide DescriptionPressure driven flow of an incompressible Newtonian fluid in a spiral duct of square cross‐section was studied both experimentally and numerically. The duct has a curvature ratio (R _{ c }=R/a, where R is the radius of curvature and a is the duct dimension) of 15.1 at the inlet and spirals inwards for nine turns at a uniform rate. A one‐component laser‐Doppler anemometer was used to measure streamwise velocities. The flow development was determined for Dean number, Dn, of 100, 125, 150, 180 and 250, based on the radius at the flow inlet [Dn=Re/(R _{ c })^{1/2}, where Re is the Reynolds number,v _{θ} ^{′} a/ν]. Steady oscillations in the streamwise direction between 2‐cell and 4‐cell states, first predicted by Sankar et al. [Phys. Fluids 31, 1348 (1988)], were observed for Dean numbers between 139 and 240. No time dependent flow phenomena were observed. The experimental data are in very good agreement with the numerical simulations, which were based on the parabolized steady three‐dimensional Navier–Stokes equations. The results are consistent with calculations by Winters [J. Fluid Mech. 180, 343 (1987)] that predict the existence of a region where no stable two‐dimensional solutions exist.

Instability and breakdown of internal gravity waves. I. Linear stability analysis
View Description Hide DescriptionWe have performed three‐dimensional linear stability analysis, based on Floquet theory, to study the stability of finite amplitude internal gravity waves. This analysis has been used to compute instability growth rates over a range of wave amplitudes and propagation angles, especially waves above and below overturning amplitude, and identifies several new characteristics of wave instability. Computation of instability eigenfunctions has allowed us to analyze the energetics of the instability and to clarify the paths of energy transfer from the base wave to the instability. We find that the presence of wave overturning has no qualitative effect on the wave instability, except for the limiting case when the wavenumber vector is vertical. Instabilities which are nearly two‐dimensional are closely related to second‐order wave–wave interactions. But the three‐dimensional instabilities, more prominent at higher wave amplitudes, may be caused by higher order resonance interactions. The energetics of the instabilities range from being shear driven to being driven by ‘‘density gradient’’ production (the potential energy analog of ‘‘shear’’ production); this characteristic is strongly dependent on wave propagation angle and the three‐dimensionality of the instability.

Wave flow of rivulets on the outer surface of an inclined cylinder
View Description Hide DescriptionIn this paper we present the results of an experimental study of wavy rivulet flow along the lower side of an inclined cylinder. The influence of the inclination angle of the cylinder, mode of irrigation and flow rate on the hydrodynamical characteristics is investigated. It is shown that the rivulet flow is unstable. As a consequence the nonlinear waves appear on a rivulet surface. The different types of surface waves are described. For the purpose of a detailed study of the wave characteristics, a method of superimposed oscillations is applied. New regularities of wave motion were found in comparison to the case of liquid filmflow. The critical conditions for liquid ejection from the rivulet are determined.

On the three‐dimensional instability of a swirling, annular, inviscid liquid sheet subject to unequal gas velocities
View Description Hide DescriptionA linear model describing the instability behavior of annular, swirling, inviscid sheets subject to inner and outer gas flows of differing velocities is presented. The model considers three‐dimensional disturbances and contains previous flat sheet, cylindrical jet, and annular jet analyses as limiting cases. Model predictions show that, in the absence of swirl, (i) an increase in axial Weber number causes the range of unstable axial disturbance modes to increase, (ii) when the axial Weber numbers are small (<8), inner gas flows lead to slightly faster growing axial instability modes than outer gas flows at equivalent inner and outer Weber numbers, but inner and outer gas flows have the same effect when Weber numbers are high (≳10), (iii) the wavenumber for the axial mode having the highest growth rate decreases with a decrease in axial Weber number, (iv) an increase in the density of the atomizing gas results in a slight increase in the wavenumber of the axial disturbance mode having the highest growth rate. When swirl is present, model predictions demonstrate that (v) swirl reduces the wavenumber for the axial disturbance mode having the highest growth rate and reduces growth rates as well, (vi) an increase in the swirl Weber number beyond the stabilizing region increases the range of unstable axial and circumferential modes and increases growth rates as well for nonzero axial Weber numbers, (vii) increasing the swirl Weber number increases the axial wavenumber for the disturbance mode having the highest growth rate, but a circumferential mode number of zero is retained until the swirl Weber number exceeds about 8, at which point the axial wavenumber for the disturbance having the highest growth rate falls to zero and the circumferential wavenumber jumps to a finite value of n at which time further increases in swirl Weber number serve to increase n, (viii) up to two local nondimensional growth rate maxima can exist, and the instability domain can be simply connected or can consist of two separate regions separated by an area where disturbances are stable. The topology of the growth rate surface depends on the ratio of the annulus inner to outer radii. These findings are used to explain some observations of practical atomizer performance.

Quasi‐steady dissipative nonlinear critical layer in a stratified shear flow
View Description Hide DescriptionWhen a wave with small but finite amplitude ε propagates towards the CL, where the effects of nonlinearity and dissipation are essential, the jump of mean vorticity over the CL appears. For the dynamically stable stratified shear flow with the gradient Richardson number Ri≳1/4 the jump of vorticity has the same order as the undisturbed one [J. Fluid Mech. 233, 25 (1991)]. The process of formation of the flow with this substantial jump of vorticity (or ‘‘break’’ of the velocity profile) in the CL is studied at large time after beginning of the process. The transition region between the CL and the undisturbed flow, the dissipation boundary layer (DBL), is shown to be formed. Its thickness grows in time proportional to √t (t being time), and the CL moves towards the incident wave. When the jump of the wave momentum flux over the CL is constant in time, the flow characteristics can be found in the most simple way. The velocity profile in the DBL appears to be self‐similar, the displacement of the CL is proportional to √t and the values of vorticity at the both sides of the CL do not depend on time and they are determined only by the constant wave momentum flux. It is shown that, to provide the constant jump of the wave momentum flux the amplitude of the wave radiated by the source in the undisturbed flow region should vary in a certain complicated manner, because it reflects from the time‐dependent (broadening) velocity profile in the DBL. On the other hand, the wave momentum flux from the steady source (for example, the corrugated wall) depends on time. When the coefficients of reflection from the CL (R) and from the DBL (r) are small, this dependence is weak and the wave and flow parameters depending on time are found as series in R and r. The wave–flow interaction for this case is studied.

Short‐scale convection and long‐scale deformationally unstable Rossby wave in a rotating fluid layer heated from below
View Description Hide DescriptionA rotating fluid layer, heated from below, with a deformable upper and nondeformable lower stress free surfaces is considered in the Boussinesq approximation. The system of the differential equations that governs the long‐scale Rossby waves and short‐scale convection is obtained in the rapid‐rotation approximation. Long‐scale flows are unstable due to heating and deformation of the upper surface. The neutral stability curves for Rossby waves and convection are obtained for linearized version of the equations. In a slightly supercritical regime the amplitude equations for convection and Rossby waves are derived by the use of the method of multiscale expansions. The properties of the amplitude equations are discussed. The existence of the two weakly supercritical stationary convection regimes is shown by numerical integration of the equations in the rapid‐rotation approximation. In one of them, the amplitude of short‐scale convection is modulated due to long‐scale deformation of the upper surface associated with the excitation of the Rossby wave. In the other regime, the presence of deformation gives rise to alternating regions with and without convection.

A model for the onset of breakdown in an axisymmetric compressible vortex
View Description Hide DescriptionA simple inviscid model to predict the onset of breakdown in an axisymmetric vortex is proposed. Three problems are considered: the shock‐induced breakdown of a compressible vortex, the breakdown of a free compressible vortex, and the breakdown of a free incompressible vortex. The same physical reasoning is used in all three problems to predict the onset of breakdown. It is hypothesized that breakdown is the result of the competing effects of adverse pressure rise and streamwise momentum flux at the vortex centerline. Breakdown is assumed to occur if the pressure rise exceeds the axial momentum flux. A formula with no adjustable constants is derived for the critical swirl number in all three problems. The dependence of the critical swirl number on parameters such as upstream Mach number, excess/deficit in centerline axial velocity, and shock oblique angle is explored. The predictions for the onset of shock‐induced breakdown and free incompressible breakdown are compared to experiment and computation, and good agreement is observed. Finally, a new breakdown map is proposed. It is suggested that the adverse pressure rise at the vortex axis be plotted against the axial momentum flux to determine the onset of breakdown. The proposed map allows the simultaneous comparison of data from flows ranging from incompressible breakdown to breakdown induced by a shock wave.

Computation of aerodynamic coefficients for a flexible membrane airfoil in turbulent flow: A comparison with classical theory
View Description Hide DescriptionIn the present paper an aeroelastic model of flexible membrane wing aerodynamics which incorporates the Reynolds‐averaged Navier–Stokes equations is presented. The Reynolds stresses are prescribed by the k–ω shear‐stress transport eddy‐viscosity model recently proposed by Menter. The computed coefficients are compared with classical inviscid membrane airfoil theory and with a portion of the available experimental data for membrane wings. The results indicate that classical potential‐based membrane airfoil theory can provide a meaningful description of membrane wing aerodynamics only for a small range of incidence angles near ideal and then only for membrane airfoils with small excess length ratios. For larger excess lengths and incidence angles viscous effects dominate the aerodynamics. The agreement of the computed results with the experimental data is mixed. The current status of the available experimental data for membrane airfoils is also reviewed.

On the large eddy simulation of a turbulent channel flow with significant heat transfer
View Description Hide DescriptionA large eddy simulation of a planar channel flow with significant heat transfer at a low Mach number was performed to study effects of fluid property variations on the near‐wall turbulence structure. A compressible dynamic subgrid scale model was used to model the residual‐scale turbulence. Two low Reynolds numberchannel flows with one wall heated and one wall cooled at temperature ratios of 1.02 and 3.0 were simulated to study the effects of property variations at low Mach number. Several features of the flow were observed to vary with the heat transfer level including velocity and temperature rms values. Specifically, the temperature‐velocity correlations were found to exhibit stronger dependency on heat transfer rate. At the higher heat transfer rate, density fluctuations at levels characteristic of flows at much higher Mach numbers were observed. Heating appeared to enhance velocity fluctuations whereas density and temperature percentage fluctuations were greatest near the cooled wall when scaled by their local mean values.

On the log‐Poisson statistics of the energy dissipation field and related problems of developed turbulence
View Description Hide DescriptionAn energy cascading model of intermittency involving rare localized regions of both large and/or weak energy dissipation (dynamical intermittency) is considered and compared to the case of intermittency arising from a large number of regions with nearly equal dissipation rates (space intermittency). The latter leads to the log‐normal statistics of the dissipation rate while the first scenario leads to shifted log‐Poisson distributions either for a large or for weak energy dissipation. The only difference between these two cases is that small values of dissipation (with respect to the maximum of PDF) are more probable for intermittency of the regions with weak dissipation than for intermittency of the regions with large values of dissipation. Some consequences are derived which show that Novikov’s inequalities are valid for intermittency with rare regions of a weak dissipation only. Different experimental data of probability distributions of dissipation are presented and compared to theoretical predictions. Some experimental evidences of quasi‐two‐dimensional vortical structures with weak dissipation are discussed. They suggest that the scenario involving dynamical intermittency with holes of dissipation could apply to a real world turbulence.

Development of a two‐equation heat transfer model based on direct simulations of turbulent flows with different Prandtl numbers
View Description Hide DescriptionLow‐Reynolds‐number type k−ε and k _{ t }−ε_{ t }models have been constructed with the aid of direct numerical simulation (DNS) databases. The proposed models incorporate new velocity and time scales to represent various sizes of eddies in velocity and thermal fields with different Prandtl numbers. The validity of the present k−ε model was tested by application to basic and complex flows such as flows with injection and suction, flows with strong adverse and favorable pressure gradients, and flows with separation and reattachment, while comparing the relevant DNS and reliable experimental data. Fundamental properties of the proposed k _{ t }−ε_{ t }model were first verified in basic flows under arbitrary wall thermal boundary conditions and next in backward‐facing step flows at various Prandtl numbers through a comparison of the predictions with the DNS and measurements. These comparisons have proven that the proposed models for both velocity and thermal fields have wide applicability to science and engineering and have sufficient capability to perform highly stable computations at any Prandtl numbers, irrespective of flow configurations.

The behavior of a gas in the continuum limit in the light of kinetic theory: The case of cylindrical Couette flows with evaporation and condensation
View Description Hide DescriptionCylindrical Couette flows of a rarefied gas between two concentric circular cylinders consisting of the condensed phase of the gas, where evaporation or condensation occurs, are considered on the basis of kinetic theory, with interest in the behavior of the gas in the continuum limit. The limiting solution is obtained by asymptotic analysis of the Boltzmann equation. In some range of the parameters of the problem, neither evaporation nor condensation occurs. The limiting solution in this case is different from the continuum solution (the conventional Couette flow without evaporation and condensation on the cylinders) and is subject to the effect of the flow that is induced if the effect of gas rarefaction is taken into account. This paradoxical result is confirmed by investigating the behavior of the numerical solution of the kinetic equation as the Knudsen number approaches zero.