Volume 13, Issue 4, April 2001
 ARTICLES


Instabilities of thermocapillary convection in a halfzone at intermediate Prandtl numbers
View Description Hide DescriptionThe stability of thermocapillary convection inside a cylindrical liquid bridge is studied using both a direct numerical simulation of the threedimensional problem and linear stability analysis of the axisymmetric basic state. Previously this has been studied extensively for low and high Prandtl numbers. However, the intermediate range of Prandtl numbers between approximately 0.07 and 0.8 which joins the low and high ranges is quite complicated and has not been studied to the same extent. One striking feature is that the axisymmetric base state is much more stable in this intermediate range than at high or low Prandtl numbers. We identify four different oscillatory modes in this range, which have different qualitative features. Direct numerical simulations have been carried out for representative parameter values, and show that the bifurcations are supercritical.

Migration of a van der Waals bubble: Lattice Boltzmann formulation
View Description Hide DescriptionA numerical study of the isothermal migration of a twodimensional bubble in Poiseuille flow is reported here for vapor–liquid density and dynamic viscosity ratios of 1/8, and A lattice Boltzmann model with a van der Waals equation of state is employed to simulate the diffuse interface for three interface thickness to bubble diameter ratios, 1/5, 1/10, and 1/20. Pointbypoint comparisons with the sharpinterface incompressible counterpart (reported in the literature) reveal velocity discrepancies which are more evident on the vapor side. These differences are a manifestation of a finite mass flux through the interface, associated with driven finite–thickness interfaces. An analytical study of the onedimensional analog of the traveling diffuse interface problem explains this phenomenon and shows that this flux vanishes as a result of viscous dissipation as the interface thickness tends to zero. This trend is corroborated by the twodimensional lattice Boltzmann results.

HighReynoldsnumber flow in a narrow gap driven by solidification. I. Theory
View Description Hide DescriptionFlow driven by solidification is studied in a limit when fluid dynamics and heat transfer decouple. The governing equations are derived by integrating the mass and momentum equations over a narrow gap of any shape within which rapid solidification occurs. These equations are then simplified by exploiting the highReynoldsnumber characteristics of the flow. The modeling leads to fluid dynamics with a suctiontype boundary layer established at the solid/liquid interface due to solid pulling and phase change. In part II [Phys. Fluids 13, 834 (2001)], the method is applied to planarflow meltspinning where liquid metal, held by surface tension in the gap between the injection nozzle and spinning chillwheel, forms a long thin puddle.

HighReynoldsnumber flow in a narrow gap driven by solidification. II. Planarflow casting application
View Description Hide DescriptionFlow driven by solidification is studied in the limit when fluid dynamics and heat transfer decouple. This occurs for processes that are characterized by a long flat solidification front and/or thin molten zones. The gapaveraged governing equations, derived in part I [Phys. Fluids 13, 826 (2001)], are applied to planarflow meltspinning where liquid metal, held by surface tension in the gap between the injection nozzle and spinning chillwheel, forms a long thin puddle. A Karman–Pohlhausen approach that includes mean flow and shear flow components leads to velocity and pressure distributions consistent with observation from experiment. A pressure rise along the flow path due to solidification suction competes with the pressure decrease due to convergent Bernoulliflow.

Dynamics of axisymmetric coreannular flow in a straight tube. I. The more viscous fluid in the core, bamboo waves
View Description Hide DescriptionNonlinear dynamics of the concentric, twophase flow of two immiscible fluids in a circular tube is studied. The viscosity of the fluid around the axis of symmetry of the tube is larger than the viscosity of the fluid that surrounds it and gravity acts against the applied pressure gradient. A pseudospectral numerical method is coupled with an implicit second order timeintegration scheme to solve the complete mass and momentum conservationequations as an initial value problem. The simulations originate with the analytical solution for the pressure driven, steady, coreannular flow(CAF) in a tube. In order to replicate as closely as possible the experimental conditions reported by Bai, Chen and Joseph (1992), the volume fraction of each fluid in the tube and the total flow rate of both fluids are imposed. Furthermore, the length of the tube is taken to be as long as computationally possible in order to allow for multiple waves of different lengths to develop and interact as reported in the experiments and in earlier weakly nonlinear analyses. Having performed simulations of CAF for conditions under which the reported flow charts indicate that both phases retain their integrity but the original steady flow is unstable, it was found that indeed traveling waves develop with slightly sharper crests (pointing towards the annular fluid) than troughs, the socalled “bamboo waves.” Despite the uneven interface, the flow in the core fluid closely resembles Poiseuille flow, but in the annular fluid small recirculation zones develop at the level of each crest. As the Reynolds number or the flow rate of the core fluid increase, the average wavelength and the amplitude of these waves decrease, whereas the holdup ratio of the core to the annular fluid approaches two. Their specific values for each examined case are in closer agreement with the experiments than in earlier theoretical reports. For large values of interfacial tension, waves with even different wavelength move with the same velocity, whereas for small values, they attain variable velocities and approach or repel each other but no wave merging or splitting is observed.

Physics of oscillatory flow and its effect on the mass transfer and separation of species
View Description Hide DescriptionThe mass transport of species in the presence of oscillating flow is calculated for a flow generated by an oscillating pressure drop. Calculations have been done for several binary systems where two dilute species are transported in the presence of a carrier. It is observed that there are multiple frequencies of the imposed oscillation at which the total mass transport of each species is the same. These frequencies, called crossover frequencies, are obtained by way of a graphical intersection when the total mass transport is plotted against the imposed frequency. Explanations on the occurrence of these crossover frequencies are put forth in terms of time constants due to the kinematicviscosity,diffusion coefficients, and the periodicity of the flow. Consequently, these explanations give an idea of the separation of species in pulsatile flow.

Modification of a vortex street by a polymer additive
View Description Hide DescriptionA Kármán vortex street is created in a flowing soap film by a rod penetrating the film. The velocity field generated by this rod is modified by the addition of the polymer polyethylene oxide having a molecular weight of and a concentration of 30 wppm. The rms velocity fluctuations behind the rod are strongly suppressed by the polymer additive and the power spectrum of the velocity fluctuations is modified as well. The experiments show that the polymer additive decreases the rate at which energy is injected into the flow. The measurements further indicate that the polymer introduces an elongational viscosity term into the Navier–Stokes equation.

The linear stability of a ridge of fluid subject to a jet of air
View Description Hide DescriptionIn this paper we investigate the linear stability of an initially symmetric twodimensional thin ridge of Newtonian fluid of finite width on a horizontal planar substrate acting under the influence of a symmetric twodimensional jet of air normal to the substrate. Ridges both with and without a dry patch at their center are considered. For both problems we examine both the special case of quasistatic motion (corresponding to zero capillary number) analytically and the general case of nonzero capillary number numerically. In all cases the ridge is found to be unconditionally unstable, but the nature and location of the most unstable mode are found to depend on the details of the specific problem considered.

The interaction of two vortices on a betaplane
View Description Hide DescriptionTwo motion of two interactingvortices in the presence of a background gradient of potential vorticity due to the βeffect is considered. The vortices, and their interaction, are modeled using a twolayer, quasigeostrophic model. The paper extends that of McDonald [Philos. Trans. R. Soc. London, Ser. A 456, 1029 (2000)] who, using a similar model, considered only steady configurations of zonally aligned vortices, i.e., lying along the same parallel of latitude. Here Green’s function techniques are used to calculate the velocity of one vortex due to the presence of the Rossby wave wake of the other. This enables consideration of the more general case of vortices having arbitrary relative location, the only requirement being that the vortices are wellseparated (i.e., several Rossby radii apart). An additional quasisteady approximation enables the time evolution of vortices to be studied. Of particular interest is the existence of equilibrium configurations in which the vortices do not move relative to each other and maintain their orientation with respect to the direction defined by the gradient of the background potential vorticity field. It is shown that necessary conditions for equilibrium are that the vortices are (a) of equal strength, or (b) aligned zonally. The stability of equilibria to small perturbations in the relative location of the vortices is investigated and the equilibria found to be weakly unstable in that the vortex trajectories drift slowly (but not exponentially fast) from their equilibrium paths. This is verified by numerical computation of the trajectories near equilibrium. For equal strength vortices not in equilibrium the distance between the vortices remains constant, but the line joining the two vortices rotates with constant angular velocity. For this case, examples of the trajectories of the individual vortices and the center of mass are given. For vortices of differing strength and arbitrary initial configurations, the motion is, in general, complicated. Some examples are presented.

Maximal mixing rate in turbulent stably stratified Couette flow
View Description Hide DescriptionA rigorous upper bound on the longtimeaveraged vertical buoyancy flux is derived from the Navier–Stokes equations for a Boussinesq fluid confined between two parallel horizontal plates a distance apart, maintained at a constant statically stabilizing temperature difference and driven at a constant relative velocity The upper bound on the volume and longtimeaveraged vertical buoyancy flux is where and is some reference density. Significantly, is independent of the bulk Richardson number of the flow and is achieved by an optimal solution with a mixing efficiency (or flux Richardson number) which approaches 0.5 as the Reynolds number becomes large. The timeaveraged turbulent dissipation of kinetic energy and the timeaveraged vertical buoyancy flux are then in equipartition for the optimizing flow.

Chaotic advection by two interacting finitearea vortices
View Description Hide DescriptionThis article deals with the advection of fluid particles in the velocity field of two identical vortices with various vorticity distributions. The twodimensional velocity field is aperiodic in the range of parameters studied here, namely, the neighborhood of the critical distance for merger. Ideas and methods from the theory of transport in dynamical systems are used to describe and quantify particle advection. These methods are applied to the numerical representation of the velocity field, which is obtained by solving the Euler equations with the vortexincell method. It is found that the strongest stirring of vortexfluid occurs slightly above the critical distance for merger. In this regime the fluid located between the vortices is subjected to intense stirring, and some vortexfluid may be entrained into the chaotic region depending on the smoothness of the vorticity distribution. Initial conditions below the critical distance lead to stirring of fluid mainly before merger. In this case the flow geometry is used to quantify the efficiency of merger, which is defined as the ratio of the circulation of the resultant vortex to the total circulation of the original vortices. It is found that the vortices with the smoothest vorticity profile have the lowest efficiency. Experimental visualizations in a twodimensional rotating fluid confirm the intense stretching and folding of fluid elements that occurs before the vortices merge.

Mixing of a passive scalar near a free surface
View Description Hide DescriptionWe study numerically and analytically the mixing of a passive scalar in turbulent shear flow with a free surface. The Navier–Stokes and scalar mixing equations are solved by direct numerical simulation. We find that the mean concentration of the scalar exhibits a twolayer structure near the free surface: an outer layer, characterized by a fast reduction in the value of the turbulent diffusivity of the scalar, and a much thinner inner layer characterized by a fast reduction of the mean concentration gradient. We develop a similarity theory for the variation of the mean concentration, which agrees very well with the numerical results and provides us with scaling relations for the thickness of each layer.

Numerical investigation of the scalar probability density function distribution in neutral and stably stratified mixing layers
View Description Hide DescriptionDirect numerical simulations of a temporallygrowing mixing layer are performed to examine how the resulting concentration probability density function (PDF) of an advected, nearly nondiffusive, passive scalar (numerical dye) varies under the combined effects of stable stratification, thermal conductivity, and perturbation of the initial velocity field. In stablystratified mixing layers convective instabilities are responsible for the development of streamwise vortices or ribs. These exhibit shorter spanwise separation than in nonstratified shear layers and lead to differences in the subsequent threedimensionalization of the flow field. Furthermore, vortex development is very efficiently suppressed by stratification in highthermallyconducting fluids, because only small temperature gradients arise and thus negligible baroclinic vorticity reinforcement is available to counterbalance the stabilizing effects of buoyancy. This is reflected in smaller mixedfluid total PDF areas with decreasing Prandtl number. The effect of coherent structures on the PDF distribution is seen to be significant. The main rolls are the cause of “globalconcentration” mixing, i.e., mixing of fluid lumps with vastly different species concentration, reflected in nonmarching PDF peaks, where the peak of the PDF is located at a constant concentration value. Ribs, on the other hand, having shorter spatial extent and engendering mixing on a narrower concentration range, cause “localconcentration” mixing, which translates into marching PDF peaks. The combined action of the spanwise vortices rolling up, or pairing, and the ribs, may then give rise to tilted PDF distributions, which are intermediate between nonmarching and marching. The “pairing parameter,” used to predict the transition from nonmarching to marching PDFs, was found not to be reliable, small values being sufficient to allow the PDF to be marching in stratified flow. The scalar mean, and the scalar meanmixedfluid, concentrations are also investigated and are seen to deviate considerably from each other, depending on the strength and coherence of the vortical structures, the imposed stratification, and the thermal conductivity.

A numerical study of dynamics of a temporally evolving swirling jet
View Description Hide DescriptionDirect numerical simulation (DNS) of a swirling jet near the outlet of a nozzle with axisymmetric and nonaxisymmetric disturbances is performed to investigate the dynamic characteristics of the flow. The early (linear) stage of the jet evolution agrees well with the predictions of linear stability theory. In the nonlinear stage, the axisymmetric DNS shows that the interaction between the primary vortex ring and the streamwise columnar vortex creates a secondary vortex structure with opposite azimuthal vorticity near the columnar vortex. Then a vortex pair consisting of the primary and secondary vortices forms and travels radially away from the symmetry axis, causing a rapid increase of the thickness of mixing layer. The nonaxisymmetric DNS shows that the streamwise vortex layer developed in the early stage of evolution due to azimuthal instability breakdowns into small eddies under the joint stretch of the axial and azimuthal shear. The results reveal several mechanisms of mixing enhancement by swirl, i.e., the radial motion of vortex ring pairs, the rapid growth of streamwise vorticity, and the creation of threedimensional small eddies. They are all favorable for fluid entrainment in swirling jets.

Nonlinear threemode interactions in a developing mixing layer
View Description Hide DescriptionA formulation based on the integral energy method is presented for the nonlinear interaction problem between three largescale coherent modes in a developing, laminar, free shear layer. Both binary and threemode nonlinear, modal energy and phase interactions are included. The modal evolution and the development of the mean flow are very sensitive to the initial energy contents and phases of the participating modes. A strong nonlinear coupling of the modal phase variations with the variations of the modal energies is shown to exist and is responsible for the rapid reversal of the intermodal energy interactions in favor of the occasionally declining mode. This nonlinear mechanism preserves the higher frequency modes far downstream and is shown to contribute to the appearance of mean flow contraction observed in experiments through local collective return of energy to the mean by all modes. Depending on initial conditions, our results support the arguments of R. A. Petersen and R. C. Clough [“The influence of higher harmonics on vortex pairing in an axisymmetric mixing layer,” J. Fluid Mech. 239, 81 (1992)] and M. R. Hajj, R. W. Miksad, and E. J. Powers [“Subharmonic growth by parametric resonance,” ibid. 236, 35 (1992)] according to which threemode resonances are more efficient than the binary ones. Two cases relevant to situations where the first and second subharmonics of the most amplified mode are forced involving the 3/2 harmonic and 2/3 subharmonic, respectively, are examined. Comparisons of the calculations with experimental results indicate good qualitative agreement.

Linear stability of laminar plane Poiseuille flow of helium II under a nonuniform mutual friction forcing
View Description Hide DescriptionThe calculation of the stability characteristics of superfluidhelium II is complicated by the fact that we have to consider the interaction of two velocity fields: the normal fluid and the superfluid. We consider 2D channel flow and concentrate on the linear stability of the normal fluid under a mutual friction forcing from the superfluid. The linear stability of the superfluid component is also briefly discussed. We consider nonuniform distributions of superfluidvorticity, leading to a nonuniform mutual friction forcing. We derive a modified Orr–Sommerfeld equation for the stability of the normal fluid component and find the neutral stability curves for the normal fluid. We identify a new branch to the neutral stability curve, which can significantly lower the critical Reynolds number of the normal fluid flow.

Spatial theory of optimal disturbances in a circular pipe flow
View Description Hide DescriptionA spatial theory of linear transient growth for disturbances in a circular pipe is presented. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes. Therefore, the problem of optimal disturbances in the pipe flow may be considered as an initial value problem on the subset of the downstream decaying eigenmodes, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. Examples are presented for spatial transient growth of axisymmetric and nonaxisymmetric disturbances. It is shown that stationary disturbances may achieve more significant transient growth than nonstationary ones. The maximum of the transient growth exists at azimuthal index for stationary perturbations, whereas nonstationary perturbations may achieve their maxima at higher azimuthal indices.

An approximate deconvolution model for largeeddy simulation with application to incompressible wallbounded flows
View Description Hide DescriptionThe approximate deconvolutionmodel(ADM) for the largeeddy simulation of incompressible flows is detailed and applied to turbulent channel flow. With this approach an approximation of the unfiltered solution is obtained by repeated filtering. Given a good approximation of the unfiltered solution, the nonlinear terms of the filtered Navier–Stokes equations can be computed directly. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation. Largeeddy simulations are performed for incompressible channel flow at Reynolds numbers based on the friction velocity and the channel halfwidth of and Both simulations compare well with direct numerical simulation (DNS) data and show a significant improvement over results obtained with classical subgrid scale models such as the standard or the dynamic Smagorinsky model. The computational cost of ADM is lower than that of dynamic models or the velocity estimation model.

Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow
View Description Hide DescriptionThe budgets of the Reynolds stress, turbulent kinetic energy and streamwise enstrophy are evaluated through direct numerical simulations for the turbulent channel flow of a viscoelasticpolymer solution modeled with the Finitely Extensible Nonlinear Elastic with the Peterlin approximation (FENEP) constitutive equation. The influence of viscoelasticity on the budgets is examined through a comparison of the Newtonian and the viscoelastic budgets obtained for the same constant pressure drop across the channel. It is observed that as the extensional viscosity of the polymer solution increases there is a consistent decrease in the production of Reynolds stress in all components, as well as in the other terms in the budgets. In particular, the effect of the flowelasticity, which is associated with the reduction in the intensity of the velocitypressure gradient correlations, potentially leads to a redistribution of the turbulent kinetic energy among the streamwise, the wallnormal and the spanwise directions. In this work, we also show that in the presence of viscoelasticity there is a significant reduction in all components of the production of streamwise enstrophy. This is consistent with a proposed mechanism for polymerinduced drag reduction through the inhibition of vortex stretching by the high extensional viscosity of the polymer solution.

Effect of wall boundary condition on scalar transfer in a fully developed turbulent flume
View Description Hide DescriptionWe performed direct numerical simulation of fully developed turbulent velocity and temperature fields in a flume, for Reynolds number, based on the wall shear velocity and the height of the flume, and Prandtl numbers and To elucidate exactly the role of the wall boundary condition for passive scalar, the system considered was the flow at constant properties of the fluid. Two types of thermal wall boundary conditions (BCs) for the dimensionless temperature equation were studied: isothermal wall boundary condition—H1, and isoflux wall boundary condition—H2. The profile of the mean temperature was not affected by the type of BC. However, the type of BC has a profound effect on the statistics of the temperature fluctuations in the nearwall region Comparison of nearwall statistics of temperature fluctuations shows that at the buffer part of the turbulent boundary layer significantly influences the scalar transfer in the conductive sublayer, whereas at the nearwall temperature field may be associated with predominant motion in the viscous sublayer.
