Volume 7, Issue 7, July 1995
Index of content:

The effect of shear in selective withdrawal
View Description Hide DescriptionThe evolution of the withdrawal through a line sink of an initially quiescent, linearly stratified fluid in a semi‐infinite, horizontal duct is investigated. It is shown that due to the shear present in the withdrawal layer the previously suggested mechanism for the control of the higher mode shear fronts, which assumes that the velocity of the fronts is balanced by the oncoming flow, cannot occur. An alternative mechanism for the control of this flow is proposed based on solutions for the vertical structure of linear, long, internal waves in horizontal shear. This results in a model for unsteady selective withdrawal in agreement with steady‐state solutions.

On the breakup of viscous liquid threads
View Description Hide DescriptionA one‐dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian fluid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a finite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self‐similar pinching solutions that yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Furthermore, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one‐dimensional model gives a consistent terminal state, with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inertial forces in the momentum equations to derive an evolution system modeling the breakup of Navier–Stokes jets. The model is employed in extensive simulations to compute breakup times for different initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied.

Oscillations of a deformed liquid drop in an acoustic field
View Description Hide DescriptionThe oscillations of an axially symmetric liquid drop in an acoustic standing wave field in air have been studied using the boundary integral method. The interaction between the drop oscillation and sound field has been included in this analysis. Our computations focus on the frequency shift of small‐amplitude oscillations of an acoustically deformed drop typical of a drop levitated in air. In the presence or absence of gravity, the trend and the magnitude of the frequency shift have been given in terms of drop size, drop deformation, and the strength of the sound field. Our calculations are compared with experiments performed on the United States Microgravity Laboratory (USML‐1) and with ground‐based measurements, and are found to be in good agreement within the accuracy of the experimental data.

Binary fluid convection in a rotating cylinder
View Description Hide DescriptionThe onset of convection in binary fluid mixtures in a rotating vertical cylinder is considered. Parameter values and boundary conditions relevant to experiments on ^{3}He–^{4}He mixtures with negative separation ratio are used. The eigenfunctions take the form of rigidly precessing spirals. The azimuthal wavenumber of the first unstable mode as the Rayleigh number increases is calculated as a function of the rotation rate and the separation ratio, as are the critical Rayleigh numbers and precession frequencies. Depending on the parameters the spirals may take the form of spatially extended body modes which fill the container, or of wall modes confined to its boundary. The former typically precess in the retrograde direction, while the latter are prograde. Under appropriate circumstances the binary system with a negative separation ratio becomes unstable for lower Rayleigh numbers than a pure fluid. This property of the system is enhanced by the wall modes.

Multiple, two‐dimensional solutions in a rotating straight pipe
View Description Hide DescriptionThe multiplicity features and the secondary flow structure of the fully developed, laminar flow of a Newtonian fluid in a straight pipe that is rotating about an axis perpendicular to the pipe axis are examined. The governing equations of motion are solved numerically using the control volume method and the SIMPLE algorithm. The solution structure is governed by two dynamical parameters, Ekman number, Ek=ν/D ^{2}Ω and Rossby number, Ro=U/DΩ, where D is the pipe diameter, ν is kinematicviscosity, Ω is rotational speed, and U is velocity scale. Results are presented for a fixed Ekman number of Ek=0.01 and a range of Rossby numbers between 0 to 20. The primary solution branch begins as a unique solution at low Rossby numbers. Its secondary flow structure consists of two‐cells. At higher values of Ro a hitherto unknown solution with a four‐cell flow structure appears, which coexists with the two‐cell flow structure over a range of Ro up to 20. Transient, two‐dimensional simulations were carried out to determine the stability of the solutions to two‐dimensional perturbations. The two‐cell flow structure is stable to both symmetric and asymmetric perturbations. Four‐cell flow structure is stable to symmetric perturbations and unstable to asymmetric perturbations, where it breaks down to a two‐cell flow structure.

Rayleigh convection in a closed cylinder—Experiments and a three‐dimensional model with temperature‐dependent viscosity effects
View Description Hide DescriptionIn this paper our most recent research results on natural convection in a closed cylinder, where our interest focuses on pattern structure dependence on aspect ratio and on temperature‐dependent viscosity, are summarized. The main results are (a) the experiments on the onset pattern and conditions for pure Rayleigh convection in circular cylinders compare favorably with linearized stability results of Hardin et al. [Int. J. Num. Methods Fluids 10, 79 (1990)], as well as three‐dimensional nonlinear calculations made by us; and (b) experiments and nonlinear calculations indicate a variation of the patterns at and near the codimension two points when large temperature differences are introduced, so as to cause a substantial change in viscosity.

Dynamics of vortex rings in crossflow
View Description Hide DescriptionThe spread and mixing of a fluid jet into an ambient stream occurs at a rate which deserves further study to improve efficient mixing.Mixing enhancement techniques, such as introduction of periodic disturbances into the jet flow, are used to increase mixing between a jet and the surrounding fluid. Pulsations were generated by the periodic closing and opening of a jet flow. The dynamics and trajectories of vortex rings, formed by the pulsation of the jet in a uniform crossflow, are studied. In particular, the effects of pulsation on the development of vortex rings and their penetration in a crossflow were investigated. Detailed measurements were made using flow visualization techniques including laser‐induced fluorescence and hot‐film anemometry. Vortex rings generated in a crossflow at one specific frequency (1 Hz) were measured using a hot‐film probe. Measurements indicated that vortex rings were fully‐formed at a distance of three times the jet exit diameter. To simulate the dynamics of vortex rings in crossflows, a numerical experiment was performed based on a Lagrangian, grid‐free, three‐dimensional vortex element method. At low frequencies, the fluid in the vortex rings penetrated into the crossflow to a height much greater than that for either high frequency pulsation or for a steady jet. At low frequencies, interaction between sequentially generated vortex rings was negligible; therefore, each ring behaved as a single discrete vortex ring. The vortex rings moved into the flow occasionally tilting up to about 30°, depending on the ring’s strength. Numerical simulation indicated that the tilting of the ring was due to the combined effects of viscosity and the crossflow. It is postulated that the increased penetration combined with the discretization of a jet into vortex rings results in a more efficient mixing rate.

A closure model for intermittency in three‐dimensional incompressible turbulence
View Description Hide DescriptionA simplified Lagrangian closure for the Navier–Stokes equation is used to study the production of intermittency in the inertial range of three‐dimensional turbulence. This is done using localized wave packets following the fluid rather than a standard Fourier basis. In this formulation, the equation for the energy transfer acquires a noise term coming from the fluctuations in the energy content of the different wave packets. Assuming smallness of the intermittency correction to scaling allows the adoption of a quasi‐Gaussian approximation for the velocity field, provided a cutoff on small scales is imposed and a finite region of space is considered. In these approximations, the amplitude of the local energy transfer fluctuations can be calculated self‐consistently in the model. Definite predictions on anomalous scaling are obtained in terms of the modified structure functions: 〈〈E(l,a)〉^{ q } _{ R }〉, where 〈E(l,a,r,t)〉_{ R } is the part of the turbulent energy coming from Fourier components in a band (a−1)k around k∼l ^{−1}, spatially averaged over a volume of size R∼l/(a−1) around r.

Effect of cross‐flow on Görtler instability in incompressible boundary layers
View Description Hide DescriptionLinear stability theory is used to study the effect of cross‐flow on Görtler instability in incompressible boundary layers. The results cover a wide range of sweep angle, pressure gradient, and wall curvature parameters. It is shown that the cross‐flow stabilizes Görtler disturbances by reducing the maximum growth rate and shrinking the unstable band of spanwise wave numbers. On the other hand, the effect of concave wall curvature on cross‐flow instability is destabilizing. Calculations show that the changeover from Görtler to cross‐flow instabilities is a function of Görtler number, pressure gradient, and sweep angle. The results demonstrate that Görtler instability may still be relevant in the transition process on swept wings even at large angles of sweep if the pressure gradient is sufficiently small. The influence of pressure gradient and sweep can be combined by defining a cross‐flow Reynolds number. Thus, the changeover from Görtler to cross‐flow instability takes place at some critical cross‐flow Reynolds number whose value increases with Görtler number.

Exact description of the spectrum of elliptical vortices in hydrodynamics and magnetohydrodynamics
View Description Hide DescriptionA method for studying natural oscillations of fluids and plasmas in the neighborhood of two‐dimensional elliptical flows is presented. The method uses scaling combined with the Fourier transformation to reduce the spectral stability problem for such flows to a spectral problem for an ordinary differential operator. This reduction is used to obtain a complete description of the spectrum for fluid flows and a qualitative description of the spectrum (including bounds for the complex part of the spectrum) for plasma flows. It is shown that a steady planar fluid flow with elliptical streamlines is spectrally unstable. It is also shown that all planar magnetized plasma flows with elliptical streamlines are spectrally unstable, except for the case when the magnitudes of the fluid velocity and the Alfvén velocity are exactly equal to each other.

Refined similarity hypotheses for turbulent velocity and temperature fields
View Description Hide DescriptionThe refined similarity hypothesis of Kolmogorov [J. Fluid Mech. 13, 82 (1962)] is extended to a scalar field. These hypotheses are tested using measurements in a circular jet and the atmospheric surface layer. Over a significant part of the inertial range, statistics of the normalized stochastic variables for velocity and temperature indicate a dependence on the separation r. This dependence is also quantified through the probability density functions of the stochastic variables and the correlation between the velocity (or temperature) increment and the local energy (or temperature) dissipation rates. Probability density functions of the stochastic variables are conditioned on the local Reynolds number Re_{ r } based on r and the local energy dissipation rate. These functions depend on Re_{ r } when the latter is small and are approximately universal when Re_{ r } is very large. This behaviour is consistent with the refined similarity hypothesis. There is however a slight difference between the shapes of the conditional probability density functions in the two flows, implying a weak dependence on the turbulenceReynolds numberR _{λ} and flow conditions.

A numerical study of free‐surface turbulence in channel flow
View Description Hide DescriptionDirect numerical simulations of open‐channel flow indicate that turbulence at the free surface contains large‐scale persistent structures. They are ‘‘upwellings’’ caused by impingement of bursts emanating from the bottom boundary; ‘‘downdrafts’’ in regions where adjacent upwellings interact, and whirlpool‐like ‘‘attached vortices’’ which form at the edge of upwellings. The attached vortices are particularly long‐lived in the sense that once formed, unless destroyed by other upwellings, they tend to interact with each other and dissipate only slowly. If turbulence generation at the bottom wall is turned off by changing the boundary condition to free slip, then the upwellings (related to bursts) and downdrafts no longer form. The dominant structures at the free surface become the attached vortices which pair, merge, and slowly dissipate. In the central regions, as expected, the structure remains three dimensional throughout the decay process. Near the free surface, the structure appears to be quasi‐ two dimensional, as indicated by quantitative measures such as energy spectra, interwave number energy transfer, invariants of the anisotropy tensor, and length scales. In the decaying case, the quasi‐two‐dimensional region increases in thickness, with decay time, though the structure in the central regions of the flow remains three dimensional.

Joint statistics between temperature and its dissipation rate components in a round jet
View Description Hide DescriptionThe joint statistics between the temperature fluctuation θ and all three components of the temperature dissipation rate ε_{θ} are investigated in the self‐preserving region of a slightly heated turbulent round jet. The main factors which determine the correlation between θ and ε_{θ} are the asymmetry of p(θ), the probability density function (PDF) of θ, and the anisotropy of the small‐scale turbulence. The assumption of statistical independence between θ and ε_{θ} appears to be more closely approximated in this flow than in a turbulent plane jet. Relatedly, the assumption of local isotropy is also more closely satisfied in the round jet than in the plane jet. When θ is in the range ±2 standard deviations, the expectations of all components of ε_{θ}, conditioned on θ, are approximately equal in the fully turbulent part of the flow; the magnitude of the conditional expectation is consistent with the independence assumption.

A spectral model applied to homogeneous turbulence
View Description Hide DescriptionBecause a spectral model describes distributions of turbulent energy and stress in wave‐number space or, equivalently, in terms of a distribution of length scales, it can account for the variation of evolution rates with length scale. A spectral turbulencemodel adapted from a model introduced by Besnard, Rauenzahn, Harlow, and Zemach is applied here to homogeneous turbulent flows driven by constant mean‐flow gradients and to free decay of such flows. To the extent permitted by the experimental data, initial turbulent spectra are inferred, and their evolutions in time are computed to obtain detailed quantitative predictions of the spectra, relaxation times to self‐similarity, self‐similar spectrum shapes, growth rates, and power‐law time dependence of turbulent energies and dominant‐eddy sizes, and integral data, such as the components of the Reynolds stresstensor and the Reynolds stressanisotropytensor. The match to experimental data, within the limits of experimental uncertainties, is good. Some qualifications on the limits of validity of the model are noted. Among phenomena encountered for which the spectral description provides quantitative understanding are the convergence of the anisotropytensor to a nonzero limit under conditions of free decay (i.e., incomplete return to isotropy, implying a Rotta constant of unity) and the apparent ‘‘return to anisotropy,’’ observed after an anisotropytensor vanishes due to a temporary cancellation of positive and negative parts of a spectrum, which evolve at different rates.

On the interpretation of vortex breakdown
View Description Hide DescriptionStudying the numerous papers that have appeared in the recent past that address ‘‘vortex breakdown,’’ it may be difficult for a reader to avoid getting rather confused. It appears that various authors or even schools have conflicting views on the correct interpretation of the physics of vortex breakdown. Following the investigation by Keller et al. [Z. Angew. Math. Phys. 36, 854 (1985)], in this paper, axisymmetric forms of vortex breakdown, as originally defined by Benjamin [J. Fluid Mech. 14, 593 (1962)] are addressed. It is argued that at least some of the previous investigations have been concerned with different aspects of the same phenomena and may, in fact, not disagree. One of the most fundamental questions in this context concerns the properties of the distributions of total head and circulation on the downstream side of vortex breakdown transitions. Some previous investigators have suggested that the downstream flow would exhibit properties that are similar to those of a wake. For this reason the phenomenon of vortex breakdown is investigated for a class of distributions of total head and circulation in the domain of flow reversal that is substantially more general than in previous investigations. Finally, a variety of problems are discussed that are crucial for a more complete theory of vortex breakdown, but have not yet been solved. It is shown that for the typically small flow speeds in a domain of flow reversal produced by a vortex breakdown wave, the departures of both vortex core size and swirl number, with respect to the case of uniform total pressure in the zone of flow reversal, as discussed by Keller et al. [Z. Angew. Math. Phys. 36, 854 (1985)], remain surprisingly small. As a consequence, the possible appearance of large departures from a Kirchhoff‐type wake must be due to viscousdiffusion at low and due to shear‐layer instabilities at high Reynolds numbers.

A numerical turbulence model for multiphase flows in the protoplanetary nebula
View Description Hide DescriptionIt is thought that planets form from solid particles in a flattened, rotating, 99% gaseous nebula. These grains gradually coagulate into millimeter‐to‐meter sized aggregates which settle toward the midplane of the nebula. It is widely believed that the resulting dense layer eventually becomes gravitationally unstable and collapses into ‘‘planetesimals.’’ A new numerical model is presented to simulate the predominant processes (gravitation, vertical convection, and shear‐driven turbulence) during the stage while the particulate material is still dispersed about the midplane of the nebula. In our previous work, particles were assumed to be spheres of a single radius; in the present work, particles are spheres of different radii. Results indicate that neither a broad nor a narrow distribution of particle sizes is likely to become gravitationally unstable.

On the local topology evolution of a high‐symmetry flow
View Description Hide DescriptionThe local topology evolution of a high‐symmetry, high resolution (effective maximum resolution of 1024^{3} grid points, maximum wave number of 341) incompressible flow simulation having a Reynolds number (=1/ν) of 1000 is investigated. The Q–R invariants of the velocity gradient tensorA _{ ij }, the enstrophy, Ω_{ ij }Ω_{ ij } and the mean‐square strain rate S _{ ij } S _{ ij } are computed at an interval when the local maximum vorticity increases drastically. All the analysis of the computations are done on the z=0 plane, where the maximum vorticity and strain are located during the evolution. In the Q–R plane, most of the collocation points evolve towards the lower right corner, a region where strain dominates over vorticity. The pressure Hessian tensor components are computed in the 0 planes. Points with very large strain and no vorticity, which are located along the boundaries separating oppositely signed vortices, are found to have a diagonal pressure Hessian tensor. It is discussed how such a Hessian tensor form can result in a singularity formation in strain (and Q invariant). Relevance of the results to turbulence is discussed. Results are compared to the predictions of a singular model by Léorat (Ph.D. thesis, Université de Paris VII, pp. 125–129, 1975), Vieillefosse [J. Phys. 43, 837 (1982)], and Cantwell [Phys. Fluids A 5, 2008 (1992)].

Acoustic energy exchange in compressible turbulence
View Description Hide DescriptionThe mechanism of exchange of kinetic energy and internal energy in a three‐dimensional forced compressible turbulence is investigated by the analysis of a direct numerical simulation data. The Helmholtz decomposition of ρrlx u (ρ is the density of a fluid and u is the velocity), which yields positive–definite spectra of the compressive and rotational kinetic energies, is employed for exploring the relative importance of various terms in the kinetic energy equation. It is shown that the pressure–dilatation term makes a dominant contribution to the exchange of compressive kinetic energy and internal energy. This exchange occurs periodically at all wave numbers, the period of which is inversely proportional to the wave number. This indicates that acoustic waves take part in the exchange of energy.

Thermophoresis of a spherical particle in a rarefied gas: Numerical analysis based on the model kinetic equations
View Description Hide DescriptionA kinetic theory for the thermophoretic force and velocity of a spherical particle in a rarefied gas is presented. The analysis is carried out on the basis of the linearized Bhatnagar–Gross–Krook (BGK) and Smodel [Fluid Dyn. 3, 95 (1968)] kinetic equations. The integral‐moment method of solution for arbitrary values of Knudsen number is employed. The set of integral moment equations was solved by the Bubnov–Galerkin method. The possibility of arbitrary energy and tangential momentum accommodation of gas molecules on the particle surface is taken into account in the boundary condition. The particle–gas heat conductivity ratio Λ is assumed to be arbitrary. The results obtained are compared to the available theoretical and experimental data.

Dissociation modeling in low density hypersonic flows of air
View Description Hide DescriptionVibration–dissociation coupling in low‐density, hypersonic flows of air is investigated. Radiative emission data for nitric oxide and for atomic oxygen measured by a reentry flight experiment are used to assess the modeling of this phenomenon. Flow field computations are performed using the direct simulation Monte Carlo method. Due to the relatively small number of collisions under high‐altitude, low‐density flow conditions, an overlay approach is used to simulate changes in chemical composition of trace species, including both nitric oxide and atomic oxygen. Radiative emission is calculated using a nonequilibrium radiation method. It is found that the strong degree of thermal nonequilibrium that occurs in high‐altitude, hypersonic flows makes the chemistry very sensitive to the vibration–dissociation coupling model. A number of such models based on continuum and particle representations of the flow are assessed. A variation in dissociation rate of up to nine orders of magnitude among these models is found for the lowest‐density flight conditions. By using a sophisticated dissociationmodel, the emission calculated at the highest altitude for which measurements are available is improved from a factor of 220 too low to within a factor of 4 too low. With the same model, improvement by a factor of 50 is also obtained for the computation of emission from atomic oxygen. This is the first time that the observed dependence of the flight data on the free‐stream density has been predicted correctly.