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Volume 9, Issue 4, April 1997

Threedimensional simulation of wavy Taylor vortex flow by direct simulation Monte Carlo method
View Description Hide DescriptionWavy Taylor vortex flow between two concentric cylinders is simulated on molecular level by the direct simulation Monte Carlo method. The number of cells in circumferential direction is greatly reduced by introducing the idea of locally axisymmetrical collision. As found in experiment, the wavy Taylor vortices are seen to move in azimuthal direction. The frequency of the azimuthal waves is in agreement with the experimental observation. It is the first time that wavy Taylor vortex flow is simulated by means of a molecular approach.

The inverse diffusion time scale of velocity gradients in homogeneous isotropic turbulence
View Description Hide DescriptionThe diffusion term in the velocity gradient tensor evolution equation in homogeneous isotropic turbulence at moderate Reynolds number has been investigated using full direct numerical simulation (DNS) of the Navier–Stokes equations. This study also considered the diffusion terms associated with the strain rate tensor and the vorticity vector The statistics of the “diffusion frequency” defined as which can be thought of as an inverse of a characteristicdiffusion time scale has been studied both for diagonal and offdiagonal elements of this tensor, for and for The probability density functions (PDF’s) of the diffusion frequency for all the variables considered in this study have a distinct peak indicating a most probable positive value. It is found that the value of the frequency increases with larger absolute values of the velocity gradient. The conditional averages of and are found to be closely related by a cubic function of and respectively, which in the neighborhood of one standard deviation from the origin is very well approximated by the same linear relationship for all the variables. This latter result suggests that a linear approximation model [Dopazo et al., Ninth Symposium on Turbulent Shear Flows; J. Martin, Ph.D. dissertation] for the diffusion terms of velocity gradients may be appropriate in certain cases.

Effect of a lateral gravitational field on the nonaxisymmetric equilibrium shapes of liquid bridges held between eccentric disks and of volumes equal to those of cylinders
View Description Hide DescriptionBifurcation diagrams of nonaxisymmetric cylindrical volume liquid bridges held between nonconcentric circular disks subject to a lateral gravitational force are found by solving the YoungLaplace equation for the interface by a finite difference method. In the absence of lateral gravity, the primary family of liquid bridges that starts with the cylinder when the eccentricity of the disks, e, is zero first loses stability at a subcritical bifurcation point as e increases. Further loss of stability is experienced by the already unstable primary family as a turning point is encountered at yet higher values of the eccentricity. However, the introduction of lateral gravity changes entirely the structure of the solutions in that instability always occurs at a turning point with respect to e no matter how small the magnitude of The stability limits calculated are compared with the ones obtained using asymptotic techniques by taking as base solution the cylinder of slenderness Λ=π.

Stability of viscoelastic dynamic contact lines: An experimental study
View Description Hide DescriptionAn experimental study of the rivulet instability associated with spin coating a circular drop of fluid is conducted to examine the effect of elasticity on the onset and evolution of the instability. The spin coating experiments are conducted with viscoelasticdrops consisting of a high molecular weight polystyrene in tricresyl phosphate (TCP), as well as the Newtonian solvent TCP. Results show an unequivocal delay in the onset of the instability when the appropriate Weissenberg number is sufficiently large, resulting in a larger coated area and more finger arms relative to Newtonian results. Experiments performed with the viscoelastic fluid at low Weissenberg number exhibit similar behavior to those performed with the Newtonian solvent as expected. Additionally, the growth rate of the instability is reduced for experiments in which the elastic forces are important, in agreement with the perturbation theory of Spaid and Homsy [Phys. Fluids 8, 460 (1996)], demonstrating that elastic forces have a stabilizing influence on the contact line instability.

Formation producibility and fractional flow curves from radial resistivity variation caused by drilling fluid invasion
View Description Hide DescriptionIn order to characterize conductivity profiles due to drilling fluid invasion into a hydrocarbon formation, a model for radial fluid transport is presented. The model assumes a waterbased mud and accounts for the convective movement of oil, water, and salt. A mathematical analysis of the model using the method of characteristics is given. An equivalent graphical construction is also provided. Computations of radial conductivity profiles for specified formation characteristics and total filtrate loss are given. For typical fractional flow curves, it is shown that three qualitatively different profiles may occur depending on the initial water saturation. These results are compared with numerical simulations that include capillary pressure and gravity segregation. Next, the important issue of the inverse problem is analyzed. It is shown that from a single snapshot of the conductivity profile, an exact calculation of filtrate loss and formation fractional flow curve is possible. The calculation is valid provided there is a resistivity contrast between the drilling fluid filtrate and the formation water. For practical application, we show the limits of applicability of this result with complete numerical calculations. Thus, if wireline logs of resistivity variation away from the wellbore are available, formation oil producibility and water cut at native conditions may be estimated.

Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers
View Description Hide DescriptionWe present the results of numerical simulations of the threedimensional thermocapillary motion of deformable viscousdrops under the influence of a constant temperature gradient within a second liquid medium. In particular, we examine the effects of shape deformations and convective transport of momentum and energy on the migration velocity of the drop. A numerical method based on a continuum model for the fluid–fluid interface is used to account for finite drop deformations. An octtree adaptive grid refinement scheme is integrated into the numerical method in order to track the interface without the need for interface reconstruction. Interface deformations arising from the convection of energy at small Reynolds numbers are found to be negligible. On the other hand, deformations of the drop shape due to inertial effects, though small in magnitude, are found to retard the motion of the drop. The steady drop shapes are found to resemble oblate or prolate spheroids without fore and aft symmetry, with the direction of elongation of the drop depending on the value of the density ratio between the two phases. As in the case of a gas bubble, convection of energy is shown to retard the thermocapillary motion of a viscousdrop, as the isotherms get wrapped around the front surface of the drop and effectively reduce the surface temperature gradient which drives the motion. The effect of inertia on the mobility of viscousdrops is found to be weaker than that in the case of gas bubbles.

Intercluster interactions in rapid granular shear flows
View Description Hide DescriptionOne of the possible phases of a sheared system of inelastically colliding rigid smooth disks is one in which relatively dense strips aligned at 45° to the streamwise direction are interspersed among similarly aligned dilute strips. The dense strips may have secondary microstructures in the form of elongated clusters. The latter are formed by an instability, following which they are convected, stretched, and rotated by the shear field. This process causes cluster–cluster collisions, a result of which is the partial destruction of the colliding clusters, followed by the emergence of new clusters. In addition, it is demonstrated that clusteringdynamics can be responsible for hysteresis and multistability in granular systems. The studies presented in this paper involve molecular dynamics simulations complemented by theoretical analysis.

Combined flow and evaporation of fluid on a spinning disk
View Description Hide DescriptionFluid flow and fluid evaporation both contribute to the overall rate of thinning during spinning of a fluid on a disk. Laser interferometry of solvent thinning behavior on spinning silicon wafers was performed to yield plots of solvent thickness evolution. These plots of thickness versus time were then analyzed to understand the respective contributions of viscous flow and evaporation to the thinning. A technique is described for extracting both the viscosity and the evaporation rate from the interference data. Well understood solvent systems are examined as test cases for this deconvolution. It is also demonstrated that nonevaporating fluids can be analyzed, even though their thickness evolution has no easily referenced endpoint to the thinning, in contrast to the volatile solvents which are rapidly spun dry.

Inviscid and inviscidlimit behavior of a surface quasigeostrophic flow
View Description Hide DescriptionThe growth of the gradient of a scalar temperature in a quasigeostrophic flow is studied numerically in detail. We use a flow evolving from a simple initial condition which was regarded by Constantin et al. as a candidate for a singularity formation in a finite time. For the inviscid problem, we propose a completely different interpretation of the growth, that is, the temperature gradient can be fitted equally well by a doubleexponential function of time rather than an algebraic blowup. It seems impossible to distinguish whether the flow blows up or not on the basis of the inviscid computations at hand. In the viscous case, a comparison is made between a series of computations with different Reynolds numbers. The critical time at which the temperature gradient attains the first local maximum is found to depend double logarithmically on the Reynolds number, which suggests the global regularity of the inviscid flow.

On a selfsustaining process in shear flows
View Description Hide DescriptionA selfsustaining process conjectured to be generic for wallbounded shear flows is investigated. The selfsustaining process consists of streamwise rolls that redistribute the mean shear to create streaks that wiggle to maintain the rolls. The process is analyzed and shown to be remarkably insensitive to whether there is noslip or freeslip at the walls. A loworder model of the process is derived from the Navier–Stokes equations for a sinusoidal shear flow. The model has two unstable steady solutions above a critical Reynolds number, in addition to the stable laminar flow. For some parameter values, there is a second critical Reynolds number at which a homoclinic bifurcation gives rise to a stable periodic solution. This suggests a direct link between unstable steady solutions and almost periodic solutions that have been computed in plane Couette flow. It is argued that this selfsustaining process is responsible for the bifurcation of shear flows at low Reynolds numbers and perhaps also for controlling the nearwall region of turbulentshear flows at higher Reynolds numbers.

Angular dependence and growth of vorticity in the threedimensional Euler equations
View Description Hide DescriptionAn investigation of lower bounds on the quantities for for the incompressible threedimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted “eigenvalues,” and , of the strain matrix and the Hessian matrix of the pressure , respectively. It is shown that these obey the simple inequality,, where The are spatially averaged weighted angles between the vorticity vector ω and the vortex stretching vector σ=ω⋅∇u. The weighting in the averaging process highlights regions of large vorticity. This is the angle considered by Tsinober, Kit, and Dracos in their analysis of data from turbulent grid flow experiments in which they noted a tendency toward alignment between ω and σ. The Burgers vortex turns out to be a sharp solution of this inequality with a corresponding angle , giving rise to exponential growth in Some special solutions for cases where moves between and are displayed. The work of Ohkitani and Kishiba on the alignment in 3D Euler flows between ω and the third eigenvector of at maximum enstrophy is also particularly relevant and is applied to the modified pressure matrix in the limit . The finite time blowup problem is discussed in this context. In an Appendix it is shown that an identical inequality holds for the barotropic compressible Euler equations where ζ=ω/ρ and replace ω and respectively.

Convective instability boundary of Couette flow between rotating porous cylinders with axial and radial flows
View Description Hide DescriptionThe convective instability boundary of a circular Couette flow in the annular region bounded by two co or counterrotating coaxial cylinders with angular velocities and , respectively, is studied in the presence of an axial flow due to a constant axial pressure gradient and a radial flow through the permeable walls of the cylinders. A linear stability analysis is carried out for positive and negative radial Reynolds numbers corresponding to outward and inward radial flows, respectively. Axisymmetric and nonaxisymmetric disturbances are considered. In the particular case of no axial flow, the Couette flow is stabilized by an inward, or a strong outward, radial flow, but destabilized by a weak outward radial flow. Nonaxisymmetric disturbances lead to instability for some negative values of . Bifurcation diagrams for combined radial and axial flows are more complicated. For particular values of the parameters of the problem, the Couette flow has regions of stabilization and destabilization in the parameter space. Computational results are compared with experimental data.

Stabilization mechanisms of short waves in stratified gas–liquid flow
View Description Hide DescriptionInterfacial waves grow in a cocurrent, stratified gas–liquid flow by extracting energy from the main flow. The most unstable mode typically has a wavelength comparable to or less than the liquid depth. Experiments show that these short waves can saturate at small amplitude with no generation of longwave or transverse modes. By decomposing the typical Stuart–Landau analysis into three components, it is found that saturation usually occurs by cubic selfinteraction of the fundamental mode but quadratic resonant interaction with the first overtone is also possible. Interaction with mean flow modes is usually much less important. Experiments confirm the predictions of weakly nonlinear theory. The measured overtone is found to be and is phaselocked with the fundamental except near a 1–2 resonance point where the fundamental and the overtone have comparable speeds. Near this resonance, the amplitudes are of the same order and the phase angle between them is observed to jump irregularly as predicted by modern dynamical systems theory for intermittent chaos near a heteroclinic cycle. The phase and magnitude of the overtone interaction specify the shape, chaotic dynamics and symmetry of the waves across resonance which are analyzed and confirmed experimentally.

Ship waves on a viscous fluid of finite depth
View Description Hide DescriptionIn this paper we are concerned with the wave generation by a singular forcelet in a viscous fluid of finite depth, where the singularity is located far from the bottom and not very near the free surface. In the first part of this work, the image system of an Oseenlet bounded by a noslip wall, is considered. It is found that the resultant velocity field can be described by a planar distribution of vertical Oseen doublets and a negative Oseenlet located at the mirror point of the singularity with respect to the plane wall. In the second part of the work we deal with the generation of waves by these solutions. By imposing the linearized freesurface conditions on the solutions obtained from the first part, the wave generated is shown to exhibit the Kelvin shipwave pattern that agrees with observation. The effects of water depth and of submergence on the wave amplitude are also investigated.

Chaotic mixing by internal inertiagravity waves
View Description Hide DescriptionThe Lagrangian transport of “passive” particles advected by inertiagravity waves is investigated. We consider two classes of waves, namely, vertically trapped, horizontally propagating waves, and those propagating in three dimensions (3D). In the former case, it is shown that the superposition of at least two waves is necessary to produce chaotic particle paths; whereas for the latter case, at least three waves are required to initiate chaotic mixing. Liapounov exponents are used to quantify the predictability of particle trajectories in the chaotic region. Whether the chaotic mixing process is temporally uniform or intermittent is deduced from the local deviation from the Liapounov exponent. Typical estimates of Liapounov exponents give errordoubling times of the order of a few hours which roughly decreases as the amplitude of the perturbing wave (ε) increases. For waves propagating only in the horizontal, the chaotic mixing process tends to be more uniform as ε increases, while the reverse is the case for waves propagating in 3D with more intermittent mixing for larger values of ε. The chaos induced transport process is characterized from a relation of the form , for large , where is the mean square distance traveled by a cloud of particles. For lower values of ε, the horizontally propagating case gives values of α greater than 2 and is nearly 2 for a larger value of ε. The value of α is nearly 2 for chaotically dispersing particle clouds in the 3D propagating case. Also, correlation dimensions are used to learn about the geometry of the cloud evolution. The results show that clouds originating in the chaotic zone initially spread more than like a filament, subsequently become area filling, and then proceed toward space filling behavior. This sequence of transition has been found to be faster for the 3D propagating waves than for the vertically trapped case. The implications of the results to the waveinduced mixing phenomena in geophysical flows are discussed.

Turbulent travelingwave convection in a twolayer system
View Description Hide DescriptionWhen layers of salt and sugar solution are separated by a “diffusive” interface, interfacial waves are spontaneously generated by the turbulent convection once the system evolves to a critical value of the density–anomaly ratio (Stamp et al., to appear in J. Fluid Mech). The waves modulate the interfacial fluxes by modifying the interface thickness and thereby organize the otherwise random convective motions into coherent largescale circulations. In narrow rectangular channels a wide range of conditions give rise to a single wave which propagates backandforth, resulting in quasiperiodic reversals of tankscale circulations. Here it is shown that in annular and equant rectangular cavities this same coupling phenomenon produces turbulent convection cells of a travelingwave nature, coupled to largeamplitude solitary waves on the interface.

On the motion of slender vortex filaments
View Description Hide DescriptionSeveral approaches for slender vortex motion (the local induction equation, the Klein–Majda equation, and the Klein–Knio equation) are compared on a specific example of sideband instability of Kelvin waves on a vortex. Numerical experiments on this model problem indicate that all these equations yield qualitatively similar behavior, and this behavior is different from the behavior of a nonslender vortex with variable crosssection. It is found that the boundaries between stable, recurrent, and chaotic regimes in the parameter space of the model problem depend on the equation used. The boundaries of these domains in the parameter space for the Klein–Majda equation and for the Klein–Knio equation are closely related to the core size. When the core size is large enough, the Klein–Majda equation always exhibits stable solutions for our model problem. Various conclusions are drawn; in particular, the behavior of turbulent vortices cannot be captured by these approximations, and probably cannot be captured by any slender vortexmodel with constant vortex crosssection. Speculations about the differences between classical and superfluidhydrodynamics are also offered.

Formation and temporal evolution of the Lambdipole
View Description Hide DescriptionThe formation and dynamics of dipolar vortex structures in twodimensional flows are studied. Localized initial structures possessing a finite linear momentum are found to develop into dipoles by direct numerical solutions of the twodimensional NavierStokes equations. The detailed structure of the evolving dipoles depend on the initial condition. However, the gross properties of their evolution are only weakly dependent on the detailed structure and can be welldescribed by the socalled Lambdipole solution. The viscous decay of the Lambdipole, leading to an expansion and a decreasing velocity, is well described by an adiabatic theory. During the expansion the dipole is found to trap fluid as it evolves.

The role of nonunique axisymmetric solutions in 3D vortex breakdown
View Description Hide DescriptionThe threedimensional, compressible Navier–Stokes equations in primitive variables are solved numerically to simulate vortex breakdown in a constricted tube. Time integration is performed with an implicit BeamWarming algorithm using fourthorder compact operators to discretize spatial derivatives. Initial conditions are obtained by solving the steady, compressible, and axisymmetric form of the Navier–Stokes equations with Newton’s method. The effects of threedimensionality on flows that are initially axisymmetric and stable to 2D disturbances are examined. Stability of the axisymmetric base flow is assessed through 3D time integration. Axisymmetric solutions at a Mach number of 0.3 and a Reynolds number of 1000 contain a region of nonuniqueness. Within this region, 3D time integration reveals only unique solutions, with nonunique axisymmetric initial conditions converging to a unique solution that is steady and axisymmetric. Past the primary limit point, which approximately identifies the appearance of critical flow (a flow that can support an axisymmetric standing wave), the solutions bifurcate into 3D timeperiodic flows. Thus this numerical study shows that the vortex strength associated with the loss of stability to 3D disturbances and that of the primary limit point are in close proximity. Additional numerical and theoretical studies of 3D swirling flows are needed to determine the impact of various parameters on dynamic behavior. For example, it is possible that a different flow behavior, leading to a nearly axisymmetric vortex breakdown state, may develop with other inlet profiles and tube geometries.

Instability of vortical and acoustic modes in supersonic round jets
View Description Hide DescriptionThe stability of “tophat” and fully developed jet profiles is investigated by an inviscid linear stability theory for compressible flow. The study covers a wide range of the Mach number and the temperature ratio. Two types of instabilities are found: vortical and acoustic, each of which can be subdivided into nonradiating (subsonic) and radiating (supersonic) modes. The vortical mode is the continuation of the KelvinHelmholtz instability from incompressible flow. The acoustic mode is a compressible flow phenomenon, which becomes important at large Mach numbers. Temperatureratio effects can be destabilizing or stabilizing, depending on the Mach number and mode of instability. A spectrum of unstable acoustic modes, including axisymmetric ones, are found to exist in the fully developed jet. For this jet, acoustic axisymmetric waves become more unstable than both vortical and acoustic helical waves at Mach numbers over about 3. Strong evidence of a resonance mechanism for acoustic modes is seen in the growth rate curves at high Mach numbers, where a spectrum of local peaks and valleys appears at regularly distributed frequencies.