Volume 9, Issue 10, October 1997
Index of content:

A mean field prediction of the asymptotic state of decaying 2D turbulence
View Description Hide DescriptionWe compare the quasistationary state obtained from a numerical integration of the twodimensional NavierStokes equation, performed by Matthaeus et al. [Phys. Rev. Lett. 66, 2731 (1991)] with predictions of a mean field theory based on inviscid dynamics. We find that over a relatively short initial period, the theory does not apply, whereas at later stages its prediction for the quasistationary state is very good. We relate the failure and success of the inviscid theory to the relevance of viscous effects in the dynamics.

Thermocapillary interaction between a solid particle and a liquidgas interface
View Description Hide DescriptionWhen a hot solid particle is submerged into an ambient fluid near a free liquidgas interface, a nonuniform temperature field around the particle produces surface tension gradients at the interface and generates thermocapillary flow in the surrounding fluid. This flow sets the particle itself in motion towards or away from the interface. In the first part of the paper, the interaction between a hot spherical solid particle and a plane undeformable liquidgas interface is studied. The velocity of the thermocapillary induced motion of the solid particle is proportional to the surface tension gradient at the liquidgas interface, and is calculated in the approximation of the Stokes flow and a zero Peclet number as a function of the separation distance between the particle and the interface. The asymptotic cases of both small and large separation distances are studied. In the second part of the paper, the interaction between a hot solid sphere and a gas bubble submerged into an ambient fluid is studied in the limiting case when the separation distance between them tends to zero. The velocity of a pairwise migration of a particle and a bubble in contact is calculated as a function of their radius ratio. The asymptotic values of the individual velocities of a solid particle and a gas bubble in near contact are also computed. The relative velocity of their motion towards each other is found to be proportional to the separation distance between them. In the third part of the paper, we investigate the effect of gravity on the thermocapillary driven motion of a solid particle for the two cases. It is shown that due to the thermocapillary interaction, a particle can move against the buoyancy forces.

On explicit solutions of the freesurface Euler equations in the presence of gravity
View Description Hide DescriptionThere are very few explicit solutions of the freesurface Euler equations in the presence of gravity. These include Stokes’ corner flow, Richardson’s flow, and Joukovski’s flow. In this paper, it is shown how these solutions appear as particular solutions of a more general family of solutions of the freesurface Euler equations in the presence of gravity, which can only be computed numerically. The numerical method is based on series truncation. The physical relevance of these solutions is discussed as well.

On the flux of fluctuation energy in a collisional grain flow at a flat, frictional wall
View Description Hide DescriptionWe consider a flow of colliding spheres that interacts with a flat, frictional wall and calculate the flux of fluctuation energy in two limits. In the first limit, all spheres slide upon contact with the wall. Here, we refine the calculations of Jenkins [J. Appl. Mech. 59, 120 (1992)] and show that a correlation between two orthogonal components of the fluctuation velocity of the point of contact of the grains with the wall provides a substantial correction to the flux originally predicted. In the other limit, the granular material is agitated but the mean velocity of the contact points with respect to the wall is zero and Jenkins’ earlier calculation is improved by distinguishing between those contacts that slide in a collision and those that stick. The new expressions for the flux agree well with the computer simulations of Louge [Phys. Fluids 6, 2253 (1994)]. Finally, we extend the expression for zero mean sliding to incorporate small sliding and obtain an approximate expression for the flux between the two limits.

Boundary layer separation due to gas thermal expansion
View Description Hide DescriptionAn analysis is carried out of the viscousinviscid interaction due to the thermal expansion of the gas in the laminar boundary layer over a flat plate whose temperature presents a step elevation above the ambient temperature at a finite distance of its leading edge. Numerical solution of the tripledeck problem for infinitely large Reynolds numbers shows that the boundary layer separates for a critical value of the ratio of platetogas temperature, dependent on the Prandtl number and the temperature sensitivity of the gas viscosity (which is assumed to increase as a power of the temperature). As the temperature ratio increases above this value, the separation point continuously shifts upstream of the temperature step whereas reattachment occurs shortly downstream of the step, leading to a triangle shaped recirculation bubble. An asymptotic description is given of the flow with incipient separation. For very large values of the temperature ratio a neatly defined interface develops between the cold and hot parts of the flow and the closure of the bubble becomes very abrupt, with the fluid recirculating near its rear section in a characteristic manner that depends on the combined effects of the large decrease of density and increase of viscosity due to the strong heating.

Largescale threedimensional Langmuir circulation
View Description Hide DescriptionA reduced equation set in two space dimensions capable of describing threedimensional Langmuir circulation in a layer of unstratified water is derived. The reduced description is valid for circulations having large horizontal scales compared to vertical scales, and should be useful for pattern formation studies when this condition is met. Windrow patterns on the surface are found from numerical simulations of the reduced equation set, guided by secondary instability theory.

On the spectrum of hyperbolic flows
View Description Hide DescriptionBy using a previously developed technique [A. Lifschitz, Phys. Fluids A 7, 1626 (1995)], it is shown that the spectrum of linear hyperbolic flows in the plane occupies a strip in the complex plane along the real axis which is independent of the geometry of the basic flow and that all the points in this strip are simple eigenvalues. The relevance of this result to the stability theory is explained.

Taylor–Couette flow with buoyancy: Onset of spiral flow
View Description Hide DescriptionNumerical simulations of the effects of buoyancy on the stability and morphology of Taylor–Couette flow have been conducted. The threedimensional equations of motion are discretized using a hybrid Chebyshev collocation/Fourier spectral method. The problem geometry consists of an airfilled vertical annulus with radius ratio, (where and are the inner and outer radii, respectively), and aspect ratio, (where is the height of the annulus). The flow is generated by combined heating and rotation of the inner cylinder. Results for various values of the Reynolds number,, and Grashof number, , show several bifurcations of the system. The most notable change in flow structure with increasing rotational effects is the onset of spiral flow in certain parameter ranges. Compared to existing analytical and experimental results, the current numerical results show good agreement.

The linear stability of flatplate boundarylayer flow of fluid with temperaturedependent viscosity
View Description Hide DescriptionThe linear stability of boundarylayer flow of fluid with temperaturedependent viscosity over a heated or cooled flatplate is investigated. Decomposition of the disturbance into normal temporal modes leads to a sixthorder “modified” eigenvalue problem. Making the additional ad hoc assumption of parallel flow leads to a simpler sixthorder “parallel” eigenvalue problem which, unlike the modified problem, reduces to the classical Orr–Sommerfeld problem in the isothermal case. Two viscositymodels are considered, and for both models numericallycalculated stability results for both the modified and parallel eigenvalue problems are obtained. For both viscositymodels it is, perhaps surprisingly, found that for both eigenvalue problems a nonuniform decrease in viscosity across the layer stabilizes the flow while a nonuniform increase in viscosity across the layer destabilizes the flow. Results for the two eigenvalue problems are shown to be quantitatively similar with, however, the parallel problem always overpredicting the critical Reynolds number in comparison to the modified problem. Finally, we discuss the physical interpretation of our results in terms of velocity–profile shape and thinlayer effects.

Azimuthal and streamwise disturbances in a fluid layer flowing down a rotating cylinder
View Description Hide DescriptionIn this paper an extensive investigation is done of the linear stability of a film falling down a vertical rotating cylinder. Two cases are considered: flow in the outside and flow in the inside of the cylinder. To this end, a system of differential equations which describe the instability is solved numerically with the Runge–Kutta fourth and sixth order methods. We show that in different circumstances the azimuthal disturbances become the most unstable when the rotation is taken into account. Moreover, a splitting of the azimuthal modes appears due to the Coriolis force. In the case of flow in the inside there is a threshold in the centrifugal force above which the flow is linearly stable. Besides, new results for the nonrotating case are also obtained.

The role of capillary waves in twofluid atomization
View Description Hide DescriptionA mechanistic study of twofluid atomization has been carried out using a new spray technique called ultrasoundmodulated twofluid (UMTF) atomization. This technique is based on resonance between the liquid capillary waves generated by ultrasound and those generated by highvelocity air. Specifically, capillary waves are established on the surface of a liquid jet as it issues from a coaxial twofluid atomizer, the nozzle tip of which vibrates at the same frequency as the ultrasound while the frequency of the capillary waves is only half of the ultrasound frequency. As these capillary waves travel downstream in the direction of air flow, their amplitude is further amplified by the air flowing around them. Atomization occurs when the wave amplitude becomes too great to maintain wave stability; the resulting drop sizes are proportional to the wavelength of the resonant capillary waves which is determined by the harmonic frequency of the ultrasound in accordance with the Kelvin equation. Theoretical calculations of the amplitude growth rate are based on two models of temporal instability of windgenerated capillary waves: Taylor’s dispersion relation and Jeffreys’ oneparameter (sheltering factor) model. Good agreements between the theoretical predictions by these models and the experimental results of how dropsize and size distributions are influenced by air velocity and surface tension led to the conclusion that Taylormode breakup of capillary waves plays a very important role in twofluid atomization. Furthermore, all peak drop diameters can be accounted for by the harmonic frequencies of the ultrasound. Hence, it is further concluded that secondary atomization is negligible in coflow twofluid atomization of a water jet at air velocities up to 170 m/s and airtowater mass ratio up to 5.6. In addition, uniform drops with diameters predetermined by the ultrasound frequency can be accomplished by adjusting the air velocity.

Comparison of flow characteristics in the near field of two parallel plane jets and an offset plane jet
View Description Hide DescriptionTwo parallel plane air jets and offset air jets have several common features such as the existence of a subatmospheric pressure region and the formation of a flow recirculation zone adjacent to the nozzle plate. In particular, the symmetry plane that exists between two parallel plane jets may appear to affect the flow field in much the same way as a solid wall does in a reattaching offset jet. It is obvious, however, that there are significant differences far downstream from the nozzles because the two parallel plane jets will combine to form a single free jet while the offset jet will develop into a wall jet. Differences in the near field between these two jet configurations with a separation ratio of 2.125 are examined here under identical initial flow exit conditions through laser Doppler anemometer measurements of the mean velocity components, turbulence intensities, and Reynolds shear stress. The results indicate that the wall exerts significant retarding and turbulence suppression effects on the offset jet in the flow development region. Through the use of comparisons, the interaction of two inner shear layers on both sides of the symmetry plane in two parallel plane jets results in a much more turbulent near field than that of the offset jet.

Model dynamics and vertical collapse in decaying strongly stratified flows
View Description Hide DescriptionRecent laboratory experiments with decaying strongly stratified grid turbulence at moderate Reynolds numbers reveal remarkable behavior. These experiments document the evolution of an initial sea of columnar dipole vortex pairs with dominant vertical vorticity to stratified “pancake” vortex sheets with dominant horizontal vorticity, together with a concurrent dominance of vertical dissipation of kinetic energy as compared with horizontal dissipation. Here we build exact solutions of the equations for low Froude number limiting dynamics, which capture basic qualitative features observed in the above experiments. Unlike the actual turbulent experiments, these exact solutions are laminar and do not involve a cascade of many scales in the horizontal. The exact solutions of the limiting dynamics involve a periodic array of dipole vortices in a weakly vertically sheared horizontal flow. The effect of finite Rossby numbers on the collapse of these exact solutions is also described here. At moderately large Rossby numbers, the effect of rotation is to inhibit the vertical collapse process.

Vortex sinks with axial flow: Solution and applications
View Description Hide DescriptionIn this paper we develop a new class of analytical solutions of the Navier–Stokes equations and suggest ways to predict and control complex swirling flows. We consider vortex sinks on curved axisymmetric surfaces with an axial flow and obtain a fiveparameter solution family that describes a large variety of flowpatterns and modelsfluid motion in a cylindrical can, whirlpools, tornadoes, and cosmic swirling jets. The singularity of these solutions on the flow axis is removed by matching them with swirling jets. The resulting composite solutions describe flows, consisting of up to seven separation regions (recirculatory “bubbles” and vortex rings), and modelflows in the Ranque–Hilsch tube, in the meniscus of electrosprays, in vortex breakdown, and in an industrial vortex burner. The analytical solutions allow a clear understanding of how different control parameters affect the flow and guide selection of optimal parameter values for desired flow features. The approach permits extension to swirling flows with heat transfer and chemical reaction, and have the potential of being significantly useful for further detailed investigation by direct or largeeddy numerical simulations as well as laboratory experimentation.

Unsteady nature of leading edge vortices
View Description Hide DescriptionFlow visualization and velocity measurements were carried out in leading edge vortices over a delta wing. Very large velocity fluctuations were observed well upstream of breakdown and also in the absence of breakdown. The maximum rms swirl velocity can exceed the free stream velocity depending on the angle of attack. Examination of the probability density functions and spectra suggest that the core of the vortex is characterized by large amplitude, broad band random velocity fluctuations. It was suggested that these large velocity fluctuations are due to apparently random displacements of the vortex core. A simple model of the flow that produced the essential features of the experimental results was presented.

The threedimensional interaction of a vortex pair with a wall
View Description Hide DescriptionThe interaction of vortices passing near a solid surface has been examined using direct numerical simulation. The configuration studied is a counterrotating vortex pair approaching a wall in an otherwise quiescent fluid. The focus of these simulations is on the threedimensional effects, of which little is known. To the authors’ knowledge, this is the first threedimensional simulation that lends support to the shortwavelength instability of the secondary vortex. It has been shown how this Crowtype instability leads to three dimensionality after the rebound of a vortex pair. The growth of the instability of the secondary vortex in the presence of the stronger primary vortex leads to the turning and intense stretching of the secondary vortex. As the instability grows the secondary vortex is bent, stretched, and wrapped around the stronger primary. During this process reconnection was observed between the two secondary vortices. Reconnection also begins between the primary and secondary vortices but the weaker secondary vortex dissipates before the primary, leaving reconnection incomplete. Evidence is presented for a new type of energy cascade based on the shortwavelength instability and the formation of continual smaller vortices at the wall. Ultimately the secondary vortex is destroyed by stretching and dissipation leaving the primary vortex with a permanently distorted shape but relatively unaffected strength compared to an isolated vortex.

One and twoparticle Lagrangian acceleration correlations in numerically simulated homogeneous turbulence
View Description Hide DescriptionLagrangian statistics of the fluid particle acceleration are studied by direct numerical simulation, in stationary isotropic turbulence (at three different Reynolds numbers) and homogeneous shear flow with uniform mean shear rate. The oneparticle acceleration autocorrelation decays rapidly with time, with a zero crossing just over two Kolmogorov time scales. In contrast, twoparticle correlations are relatively persistent, especially for particle pairs of small initial separation distance. Results for intermediate times at a Taylorscale Reynolds number of 140 resemble a inertialrange scaling suggested in the literature, but even higher Reynolds numbers are needed for more definitive comparisons. Use of a theoretical argument and conditional sampling indicates that the twoparticle correlation is determined by a coupling between a correlation localized in space and a particlepair separation probability density of positive skewness. The scenario which emerges is that whereas some fluid particles accelerate rapidly away from each other, the majority of particle pairs can still be relatively close together and hence help maintain the twoparticle correlation at significant levels. Acceleration correlations in homogeneous shear flow are found to display a tendency toward local isotropy.

Strain, vortices, and the enstrophy inertial range in twodimensional turbulence
View Description Hide DescriptionThe properties of vortices in a strain field are used to construct a phenomenological theory of the enstrophy inertial range in twodimensional incompressible turbulence. The theory, based in part on the results and behavior of numerical simulations, attempts to combine spectral inertial range theories of the Kolmogorov type with the dynamics of vortex interactions in physical space. It is based on the assumptions that coherent vortices can survive in a turbulent flow if of sufficient strength compared to the background straining field, and that coherent structures feel a mean strain field, independent of their scale. The first assumption is suggested by a result in the theory of uniform elliptic vortices, while the second comes from numerical simulations. The theory employs a single nondimensional parameter, essentially the ratio between the enstrophy flux and the mean strain, which then characterizes flows from extremely intermittent decaying turbulence to nearly Gaussian passive scalar dynamics. The theory predicts that in forced twodimensional turbulence, coherent structures reside in a “background” straining field. The coherent vortices will dominate the flow at a sufficiently large scale, with a fairly abrupt transition at a small scale to a flow in which the classical enstrophy spectrum holds. In this classical region small amplitude vortices do not survive because the (largescale) straining field is of larger amplitude than the (smallscale) vorticity. The vorticity itself is passively advected in this regime. If the enstrophy flux is very small compared to the enstrophy itself, then the dynamics will be highly intermittent, with a spectrum determined by the spectrum of the vortices themselves, rather than by the dynamics of the enstrophy flux. The theory predicts that at small scales in forceddissipative twodimensional turbulence the energy spectrum will obey the classical enstrophy inertial range predictions even though the nonlinear interactions remain spectrally nonlocal. Passive scalar dynamics are predicted to be similar to vortex dynamics, at small scales. Available numerical simulations are consistent with these suggestions.

Dissipation due to particle/turbulence interaction in a twophase, turbulent, shear layer
View Description Hide DescriptionExperimental measurements of particle velocity, size, concentration and gas velocity have enabled the calculation of additional carrier phase dissipation due to the Stokes disturbance flowgenerated by small, heavy droplets interacting with the coherent largescale eddies of a turbulent shear layer. The flow field was generated by mixing a homogeneous, dropletladen (volume fraction highspeed air stream with the ambient atmosphere. Ensemble averaged measurements of the largescale spanwise vortices through the first pairing event show that the additional dissipation is primarily concentrated into intense regions located beneath the core of the vortex and extends into the mixing layer close to the free stagnation point. The magnitude of the dissipation is typically on the order of 10% of the rate at which kinetic energy is transferred between the gas and the particles. A simple model based on the steadystate response of heavy particles to an oscillatory forcing qualitatively illustrate the evolution of the dissipation and kinetic energy transfer within the freestream outside the mixing layer. The comparison also indicates that improved results might be attained by accounting for the unsteady growth of the gas phase velocity fluctuation resulting from the evolution of the coherent structures. Estimates of the singlephase turbulent dissipation indicate that the additional dissipation due to the presence of the particles is approximately 1% of the singlephase dissipation. This is the same order of magnitude as the mass loading and is in agreement with numerical simulation estimates of the increased dissipation in homogeneous turbulent flows.

Modeling of compressible effects on the Reynolds stress using a Markovianized twoscale method
View Description Hide DescriptionCompressibility effects on the Reynolds stress are modeled using a Markovianized twoscale method. These effects occur twofold. One is the effect on the turbulentviscosity, which is expressed in terms of the ratio of the normalized density variance to the squared turbulentMach number. Another comes from the deviation of the Reynolds stress from a turbulentviscosity representation, which is written using the Langrange derivative of the mean velocity and the spatial derivatives of the mean density and internal energy. A simple model with the former compressibility effect incorporated is applied to fully developed freeshear layers and is shown to capture the steep decrease in the growth rate with the increasing convective Mach number.