Volume 9, Issue 7, July 1997
Index of content:

Active control of Rayleigh–Bénard convection
View Description Hide DescriptionWe report on stabilizing the unstable nomotion state in a moderate aspect ratio onedimensional Rayleigh–Bénard convection experiment. A linear proportional control algorithm uses shadowgraphic convection images to determine heat flux perturbations which are applied to the lower boundary by a network of local heaters. We show that simple linear control stabilizes the otherwise unstable nomotion (conduction) state over a substantial range of supercritical Rayleigh numbers.

Depth of penetration of bubbles entrained by a plunging water jet
View Description Hide DescriptionA model is proposed to predict the depth of penetration of the air bubbles entrained by a round water jet impacting into a flat, liquid pool. This depth is shown to be determined only by the initial jet momentum and by the nonmonotonic nature of the bubble terminal velocities as a function of their size. The model is shown to be in excellent agreement with measurements of the depth and width of penetration of the bubbles performed over a wide range of jet diameters, velocities, and plunging angles.

System identification and control of a turbulent boundary layer
View Description Hide DescriptionAn experimental investigation is made into the active control of the nearwall region of a turbulent boundary layer using a linear feedforward control algorithm. A wallbased detection scheme is described which effectively detects coherent structures and predicts downstream flow behavior. A simple demonstration, using three wallbased sensors and a single actuator, achieves a maximum of 31% reduction in and 17% reduction in

A diffuseinterface description of internal waves in a nearcritical fluid
View Description Hide DescriptionWe present a diffuseinterface treatment of the internal gravity waves which have been observed experimentally by Berg et al. in xenon near its thermodynamic critical point. The results are compared with theoretical predictions by Berg et al. that were obtained using separate models above and below the critical temperature The diffuseinterface model applies both above and below and is formulated by using the density as an order parameter. The diffuse interface is represented as a transition zone of rapid but smooth density variation in the model, and density gradients appear in a capillary tensor, or Korteweg stress term, in the momentum equation. We obtain static density profiles, compute internal wave frequencies and compare with the experimental data and theoretical results of Berg et al. both above and below the critical temperature. The results reveal a singularity in the diffuseinterface model in the limit of incompressible perturbations.

Stability of an isorotating liquid bridge in an axial gravity field
View Description Hide DescriptionThe stability problem for the axisymmetric equilibrium states of an isorotating liquid bridge between equidimensional circular disks in a constant axial gravity field is considered. In particular, we examine the stability of bridges satisfying two constraints that are typical for the floating zone method used for materials purification and single crystal growth. First we consider the constraint that the relative volume of the bridge, is equal to 1. Here, is the ratio of the actual bridge volume to that of a cylinder pinned to the edges of disks. For this case, the critical values of the slenderness (Λ) (ratio of the disk separation to the diameter) and of the free surface slopes at both disks have been determined for a wide range of the Bond and Weber numbers. The second constraint is that the surface slope at one of the disks is prescribed. The chosen values are 90° and 75° and correspond to extremes in growth angle values encountered in floating zone crystal growth. For this case, the dependencies of critical Λ and values and have been calculated. In addition, both axial gravity directions are considered separately and the values of the slope angle, at the other disk are also analyzed for critical states. The solution of the stability problem for any liquid bridge is discussed in detail using the case for as an example. In particular, the relationship between the general boundary of the stability region and the stability of bridges subject to the constraints outlined above is examined.

Bifurcation of the equilibrium states of a weightless liquid bridge
View Description Hide DescriptionThe bifurcation of the solutions of the nonlinear equilibrium problem of a weightless liquid bridge with a free surface pinned to the edges of two coaxial equidimensional circular disks is examined. The bifurcation is studied in the neighborhood of the stability boundary for axisymmetric equilibrium states with emphasis on the boundary segment corresponding to nonaxisymmetric critical perturbations. The first approximations for the shapes of the bifurcated equilibrium surfaces are obtained. The stability of the bifurcated states is then determined from the bifurcation structure. Along the maximum volume stability limit, depending on values of the system parameters, loss of stability with respect to nonaxisymmetric perturbations results in either a jump or a continuous transition to stable nonaxisymmetric shapes. The value of the slenderness at which a change in the type of transition occurs is found to be Experimental investigation based on a neutral buoyancy technique agrees with this prediction. It shows that, for the jump is finite and that a critical bridge undergoes a finite deformation to a stable nonaxisymmetric state.

Fully nonlinear oscillatory convection in a rotating layer
View Description Hide DescriptionTwodimensional overstable convection in a rotating layer is studied for large Taylor numbers. In this regime, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile. Fully nonlinear solutions in the form of traveling and standing waves are obtained via an asymptotic expansion in the Taylor number. The formulation leads to a nonlinear eigenvalue problem for the vertical structure and frequency of each solution type and allows the computation of the Nusselt number and frequency as functions of the applied Rayleigh number.

The effect of slight viscosity on a nearcritical swirling flow in a pipe
View Description Hide DescriptionThe effect of slight viscosity on a nearcritical axisymmetric incompressible swirling flow in a straight pipe is studied. We demonstrate the singular behavior of a regularexpansion solution in terms of the slight viscosity around the critical swirl. This singularity infers that largeamplitude disturbances may be induced by the small viscosity when the incoming flow to the pipe has a swirl level around the critical swirl. It also provides a theoretical understanding of Hall’s boundary layer separation analogy to the vortex breakdown phenomenon. In order to understand the nature of flows in this swirl range, we develop a smalldisturbance analysis. It shows that a small but finite viscosity breaks the transcritical bifurcation of solutions of the Euler equations at the critical swirl into two branches of solutions of the Navier–Stokes equations. These branches fold at limit points near the critical swirl with a finite gap between the two branches. This means that no nearcolumnar equilibrium state can exist for an incoming flow with swirl close to the critical level and the flow must develop large disturbances in this swirl range. Beyond this range, two equilibrium states may exist under the same inlet/outlet conditions. When the flowReynolds number is decreased this special behavior uniformly changes into a branch of a single equilibrium state for each incoming swirl. We also derive a weakly nonlinear approach to study the effect of slight viscosity on standing waves in a long pipe. This special behavior of viscoussolutions shows good agreement with the numerical simulations of the axisymmetric Navier–Stokes equations by Beran and Culick and provides a theoretical understanding of these computations. The relevance of the results to the axisymmetric vortex breakdown in a pipe is also discussed.

Convective and absolute instability of a liquid jet in a longitudinal magnetic field
View Description Hide DescriptionA circular jet of perfectly conductive inviscid liquid in a uniform longitudinal magnetic field can be absolutely or convectively unstable for different values of the flow parameters. Higher intensity of the field always reduces the domain of absolute instability. For both cases of absolute and convective instability, the corresponding jet of large but finite length is globally unstable if suitable boundary conditions hold at its beginning and end. The unstable global mode is based on a pair of waves that propagate in opposite directions and reflect from one into the other at the flow boundaries.

The longtime motion of vortex sheets with surface tension
View Description Hide DescriptionWe study numerically the simplest model of two incompressible, immiscible fluids shearing past one another. The fluids are twodimensional, inviscid, irrotational, density matched, and separated by a sharp interface under a surface tension. The nonlinear growth and evolution of this interface is governed by only the competing effects of the Kelvin–Helmholtz instability and the dispersion due to surface tension. We have developed new and highly accurate numerical methods designed to treat the difficulties associated with the presence of surface tension. This allows us to accurately simulate the evolution of the interface over much longer times than has been done previously. A surprisingly rich variety of behavior is found. For small Weber numbers, where there are no unstable lengthscales, the flow is dispersively dominated and oscillatory behavior is observed. For intermediate Weber numbers, where there are only a few unstable lengthscales, the interface forms elongating and interpenetrating fingers of fluid. At larger Weber numbers, where there are many unstable scales, the interface rollsup into a “KelvinHelmholtz” spiral with its late evolution terminated by the collision of the interface with itself, forming at that instant bubbles of fluid at the core of the spiral. Using locally refined grids, this singular event (a “topological” or “pinching” singularity) is studied carefully. Our computations suggest at least a partial conformance to a local selfsimilar scaling. For fixed initial data, the pinching singularity times decrease as the surface tension is reduced, apparently towards the singularity time associated with the zero surface tension problem, as studied by Moore and others. Simulations from more complicated, multimodal initial data show the evolution as a combination of these fingers, spirals, and pinches.

Acoustic waves in unbounded shear flows
View Description Hide DescriptionThe linear evolution of acoustic waves in a fluid flow with uniform mean density and uniform shear of velocity is investigated. The process of the mean flow energy extraction by the threedimensional acoustic waves, stimulated by the nonnormal character of the linear dynamics in the shear flow, is analyzed. The thorough examination of the dynamics of different physical variables characterizing the wave evolution is presented. The physics of gaining of the shear energy by acoustic waves is described.

Experiments on capillarygravity waves of solitary type on deep water
View Description Hide DescriptionIt is shown that capillarygravity waves of solitary type on deep water can be generated by the resonant excitation of a water surface at a speed close to the phasespeed of free waves. This speed depends upon the wave amplitude. When the source of excitation is removed, the waves are shown to propagate as free solitary waves, damped by viscosity.

A model of wave dynamics in the far wake of a cylinder
View Description Hide DescriptionInteresting interactions in the far wake behind a cylinder, leading in particular to the appearance of a strong secondary oblique wave, have recently been discovered experimentally by Williamson and Prasad [J. Fluid Mech. 256, 269, 315 (1993)]. They are induced from a very small amount of noise, added to the decaying primary wave. The problem is investigated theoretically with simple amplitude equations, based on temporal instability of the small amplitude waves. The symmetry of the wake flow requires that quadratic interactions arise only among triads of wave numbers involving one varicose and two sinuous waves, or three varicose waves. As the primary wave, corresponding to vortex shedding, is sinuous, the interacting secondary waves must be of opposite parities. In this case, it is found that the sinuous wave will always prevail downstream. The preferential appearance of the secondary oblique wave in the far wake can be reproduced by letting an initially very small varicose parallel wave interact with the primary wave. The secondary oblique wave results from a classical threewave, quadratic nonlinear interaction between the waves. In addition, our model reproduces the observation that upon increasing the noise amplitude the oblique wave appears sooner (further upstream) in the wake. The occurrence of parallel waves very far downstream, which depends on the frequency relationship between the waves, can be understood by considering the interactions between parallel and oblique secondary waves of both parities. We propose a reasonable scenario explaining the experimental observations of Williamson and Prasad.

Thermal inertial waves in a rotating fluid layer: Exact and asymptotic solutions
View Description Hide DescriptionThe problem of the onset of convective instability in a rapidly rotating fluid layer heated from below with various velocity boundary conditions is investigated by constructing exact solutions and by asymptotic analysis. It is shown that convective motions at sufficiently small Prandtl numbers are described in leading order by a thermal inertial wave. It is at the next order that buoyancy forces drive the wave against the weak effect of viscous dissipation. On the basis of the perturbation of the thermal inertial wave, asymptotic convectionsolutions for rigid boundaries can be expressed in simple analytic form. A new asymptotic power law between the critical Rayleigh number and the Ekman number is derived. In addition, solutions for both stationary and timedependent convection are calculated numerically. Comparison between the numerical and asymptotic solutions is then made to show that a satisfactory quantitative agreement has been achieved.

A waveletpacket census algorithm for calculating vortex statistics
View Description Hide DescriptionA generalized waveletpacket based technique for decomposing signals into coherent and noncoherent parts is presented. The method is tested on the vorticity field of numerical simulations of weakly decaying twodimensional turbulence. The easily recognizable coherent vortex structures that emerge are systematically filtered from the solution. Once extracted, various properties of the vortices, such as their number, size, circulation, and peak value are computed. The results compare well with a similar study [J. Fluid Mech. 219, 361 (1990); Phys. Fluids 5, 608 (1993)], which employs a complex pattern recognition technique based exclusively on a priori knowledge of the properties of the solution—that is, the features typical of the resulting vortex structures. The similarity of the results is encouraging, suggesting that the wavelet packet technique, by absorbing much of the complexity into the mathematical features of the transform itself, can provide an efficient, standardized tool that is readily extendible to more complex problems in two and three dimensions.

Azimuthal mode measurements of elliptic jets
View Description Hide DescriptionA procedure to quantitatively measure the relative amplitudes of azimuthal modes in the acoustic field of an elliptic jet is presented. The work describes how the azimuthal modes in an elliptic jet can be represented by a linear combination of Mathieu function modes and how the amplitude coefficients of each individual mode can be determined through an orthogonal decomposition based on Mathieu functions. The modal decomposition is performed in an elliptic cylindrical coordinate system natural to the elliptic jet geometry. The procedure is first tested on an artificially excited, perfectly expanded Mach 1.5 elliptic jet with preferential varicose and flapping mode excitation of discrete frequencies. The excitation was provided with a four electrodeglow dischargesystem with phase control of the individual electrodes. Following that, the procedure was applied to naturally excited Mach 1.5 jets with both air and a helium/air mixture as the jet working gas. The helium/air jets simulate the higher jet velocity and lower jet density of heated jets. The modal decomposition technique is verified by experiment, allowing significant differences to be identified in the azimuthal modal content as the jet simulated temperature is increased.

Nonlinear feedback control of the wake past a plate: From a loworder model to a higherorder model
View Description Hide DescriptionA framework to estimate, with an intelligent and systematic approach, the applicability of a controller derived for a loworder model to a higherorder model is presented. This framework is applied to the test case of nonlinear feedback control of the twodimensional unsteady separated flow past a plate perpendicular to the freestream velocity. A hierarchy of models of increasing complexity is generated by using vortex methods. A loworder point vortexmodel and a highorder vortex blob model are used to simulate the rollup of the separated shear layer. A nonlinear controller able to manipulate the wake by means of a suction point located on the downstream wall of the plate is given in closed form for the point vortexmodel. This controller is applied to the vortex blob model by defining a center of circulation. The topological equivalence of the phase spaces of the two models is verified. Finally, the two models are used to simulate the same flows using the same controller and the results are discussed.

Lowdimensional models for flows with density fluctuations
View Description Hide DescriptionA lowdimensional model, using the proper orthogonal, or Karhunen–Loève decomposition, has been remarkably successful in representing the behavior of the wall region of a turbulent boundary layer. We briefly summarize this work. We may hope for similar success in other flows in which coherent structures play an important role, in particular flows with density fluctuations. We sketch an approach to such a decomposition for flows with density fluctuations, suggesting various alternatives which weigh the available information differently. In such a lowdimensional model, obtaining the empirical eigenfunctions poses a problem, since they can usually be determined only from extensive measurements or direct numerical simulations. However, recent work with energy method stability theory (modified by use of an anisotropiceddyviscosity and feedback to the mean profile) has been remarkably successful in predicting the form of the empirical eigenfunctions in the isothermal boundary layer. We present here preliminary results for sheared Rayleigh–Bénard convection; these results do not include anisotropiceddyviscosities and feedback, and do not predict directly the form of the POD eigenfunctions; however, a very satisfactory comparison can be made with the second order moments obtained from a DNS.

Nonpremixed spontaneous ignition in the laminar wake of a splitter plate
View Description Hide DescriptionIgnition in the laminar wake that forms at the trailing edge of a thin splitter plate separating two parallel streams of fuel and oxidizer is studied in the limit of large activation energy. The analysis presented covers ignition events ocurring in the Rott–Hakkinen and Goldstein regions, where selfsimilar solutions for the different frozen flow variables are available. Because of the strong exponential sensitivity of the reaction rate on the temperature, ignition becomes a selfaccelerating phenomenon on the temperature increment that leads to a thermal runaway a finite distance downstream from the splitter plate. The dependence of the ignition distance on the values of the velocity and temperature gradients that exist at the trailing edge of the splitter plate is investigated. In particular, it is seen that, for sufficiently large values of the activation energy, the small temperature variations that exist across the wake must be taken into account in calculating the ignition distance. As the transverse gradient of temperature at the trailing edge increases, the reaction zone becomes thinner and migrates towards the hotter side of the mixing layer, where convection becomes the dominant transport mechanism, causing the thermal runaway distance to be determined in the first approximation by a convectivereactive balance across the thin reaction layer. Ignition in the wake when one of the streams is initially stagnant is also addressed. The character of the resulting solution is seen to depend strongly on the Lewis number of the reactant supplied by the colder stream, yielding three distinct ignition regimes that are analyzed separately.

Helicity fluctuations and turbulent energy production in rotating and nonrotating pipes
View Description Hide DescriptionIn this paper finitedifference secondorder accurate direct simulations have been used to investigate how the helicity density fluctuations change when a turbulent pipe rotates about its axis. In this case the rotation axis is parallel to the near wall vortical structures which play a fundamental role on the wall friction and turbulence production. The helicity density is the trace of the tensor whose elements form the components of . When the momentum equations are written in rotational form the turbulence energy production term splits into two parts, one related to the convection of the large scales and the other related to the energy cascade to the small scales. From data of direct simulations the changes of the turbulent production term profile with the rotation have been explained by the pdf of the components. The links between the changes on the pdf of the and the modifications of the vortical structures have been also investigated. The joint pdf of the dissipation with the helicity density has shown that the dissipation is highly correlated with regions of very low helicity density in the nonrotating pipe. When the pipe rotates the helicity density increases and the dissipation decreases. Since a drag reduction is one of the results of the background rotation, in this paper, it has been speculated that the alignment between velocity and vorticity could be a common feature in drag reducing flows.