Volume 16, Issue 3, March 2004
 ARTICLES


Influence of subgrid scales on resolvable turbulence and mixing in Rayleigh–Taylor flow
View Description Hide DescriptionThe energy transfer process and the interaction of different scales in a flow induced by the variabledensity Rayleigh–Taylor instability in miscible fluids is investigated using a threedimensional direct numerical simulation database with a spatial resolution of The method used to study the transfer of energy between the supergrid and subgrid scales in the homogeneous planes, determined by partitioning the modes into resolved and unresolved scales defined by a twodimensional cutoff wave number in Fourier space, is applied to the kinetic energy evolution equation. The treatment of the flow inhomogeneity in the direction z parallel to the acceleration is analogous to that used in the analysis of incompressible wallbounded flows, including channel flow and Rayleigh–Bénard convection [J. A. Domaradzki et al., Phys. Fluids 6, 1583 (1994); J. A. Domaradzki and W. Liu, ibid. 7, 2025 (1995)]. Using a sharp Fourier cutoff filter, the kinetic energy transfer is decomposed into (1) the resolved part; (2) a part corresponding to the interaction between resolved and unresolved scales; and (3) a part corresponding to the interaction between unresolved scales. The sum of these last two contributions is the subgridscale kinetic energy transfer, which is studied in the present work. These zdependent spectra are computed for three different cutoff wave numbers to investigate the dependence of the transfer process on the scales contributing to the subgrid interactions. The kinetic energy transfer is further decomposed into its positive and negative components corresponding to the forward and backward cascades of energy, respectively, that arise from the nonlinear modal interactions. The decomposition into resolved and unresolved scales is used to define an effective eddyviscosity and backscatterviscosity. The principal conclusions of the analysis are (1) the transfer spectra and eddyviscosities exhibit a strong dependence on the wave number cutoff; (2) the contributions from the interaction between resolved and unresolved scales dominate the contribution to the total subgrid eddyviscosities and are responsible for the cusp at large (3) the contributions from the interaction between unresolved scales dominate the contribution to the total subgrid eddyviscosities at small and are responsible for the small, negative contribution (associated with an inverse energy transfer), and (4) backscatter is strongest in the regions near the boundaries of the mixing layer. The physical implications of these results for subgridscale modeling in a largeeddy simulation of Rayleigh–Taylor instabilityinducedturbulence are discussed.

Density waves and coherent structures in granular Couette flows
View Description Hide DescriptionDensity inhomogeneities in granular flows can dramatically influence microscopic and macroscopic properties. Here, we numerically examine dilute rapid granular flows in the Couette geometry via largescale particledynamic simulations, and characterize development of nonuniform particle distributions. For monodisperse grains we observe density waves in two and threedimensional computational domains of varying aspect ratios. Both fully developed and transient states are quantified using Fourier methods. For inelastic, planar (twodimensional) flows exceeding a minimum solids fraction, onedimensional, highdensity clusters—wellknown features of inelastic materials—align parallel to the walls. Above a critical streamwise length, these are destabilized by twodimensional antisymmetric modes with wavelength ∼100 particle diameters. We relate oscillatory behavior to an underlying physical mechanism of the slow drift of clusters towards walls and their subsequent bursting. Further streamwise or spanwise expansions permit additional wave numbers to be expressed in these directions. In “shallow” threedimensional flows, the planar wave types initially survive. As depth is increased above a critical value, crossstream invariance experiences symmetry preserving instabilities to form coherent structures resembling steady and wavy Taylor–Couette fluid vortices. Their presence strongly impacts macroscopic behavior, as regions of sustained vorticity develop, and stresses and granular temperatures deviate by up to an order of magnitude from mean values. The influence of solids fraction, particle size, material elasticity,surfacefriction, polydispersity, and gravity are considered, and instabilities are found to intensify as collisional dissipation rises. For planar flows, transient and fully developed density distributions share many parametric responses with previous continuum results using kinetic theory.

Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at
View Description Hide DescriptionA spatially developing supersonic adiabatic flat plate boundary layer flow (at and is analyzed by means of direct numerical simulation. The numerical algorithm is based on a mixed weighted essentially nonoscillatory compactdifference method for the threedimensional Navier–Stokes equations. The main objectives are to assess the validity of Morkovin’s hypothesis and Reynolds analogies, and to analyze the controlling mechanisms for turbulence production, dissipation, and transport. The results show that the essential dynamics of the investigated turbulent supersonic boundary layer flow closely resembles the incompressible pattern. The Van Driest transformed mean velocity obeys the incompressible lawofthewall, and the mean static temperature field exhibits a quadratic dependency upon the mean velocity, as predicted by the Crocco–Busemann relation. The total temperature has been found not to be precisely uniform, and total temperature fluctuations are found to be nonnegligible. Consistently, the turbulent Prandtl number is not unity, and it varies between 0.7 and 0.8 in the outer part of the boundary layer. Nonetheless, a modified strong Reynolds analogy is still verified. In agreement with the low Mach number results, the streamwise velocity component and the temperature are only weakly anticorrelated. The turbulent kinetic energy budget also shows similarities with the incompressible case provided all terms of the equation are properly scaled; indeed, the leading compressibility contributions are negligible throughout the boundary layer.

On a modified Taylor–Dean stability problem where the small gap between the cylinders varies in the azimuthal direction
View Description Hide DescriptionWe modify the classical Taylor–Dean stability problem to a case where the inner circular cylinder is rotating and the outer cylinder is fixed and noncircular in general. A method is given to determine some stability characteristics, in which the streamwise growth of a steady disturbance is calculated, in a smallgap approximation. Some particular examples are studied. If the gap width is increasing in the direction of flux of the basic steadystate flow, this flow is more unstable than for the case of constant gap width, and conversely when the gap width is decreasing.

Soret effect inducing subcritical and Hopf bifurcations in a shallow enclosure filled with a clear binary fluid or a saturated porous medium: A comparative study
View Description Hide DescriptionSoretdriven thermosolutal convection within a shallow porous or fluid layer subject to a vertical gradient of temperature is investigated analytically and numerically. The bridging between a clear fluid and Darcy porous media problems is conducted using the Brinkman–Hazen–Darcy model in its transient form. The analytical solution is derived on the basis of the parallel flow approximation, and validated numerically using a finite difference method by solving the full governing equations. The study is focused on the thermal diffusion effects on the flow intensity, and on the heat and mass transfer rates. In particular, a comparative study is made for the two limiting cases that emerge from the present investigation, namely the low porosity Darcy porous medium and the clear fluid medium. The flow behavior for both cases is qualitatively similar. The critical Rayleigh numbers for the onset of subcritical, oscillatory and stationary convection are determined explicitly as functions of the governing parameters for infinite and finite layers. At the onset of instabilities, the wavenumber is equal to zero and the oscillation frequency vanishes at the onset of Hopf bifurcation. For a finite aspect ratio enclosure, the frequency is finite and decreases as the aspect ratio increases. The codimension2 point exists and different flow regimes are delineated. For constant heat flux boundaries, only standing oscillatory and steady waves are found to exist. The analytical and numerical results are found to be in good agreement, within the range of the governing parameters considered in the present study. The thermal diffusion effect on the flow intensity and on the heat and mass transfer is more enhanced for Darcy medium compared to the clear fluid, for which the viscous effects are significant.

Theoretical and numerical results for spin coating of viscous liquids
View Description Hide DescriptionA mathematical model is developed for fluid flow in the spin coating process. Spin coating employs centrifugal force to produce coatings of uniform thickness. The longwave or lubrication approximation is used for the flow of thin liquid layers that are exposed to the air and lie on a spinning horizontal solid substrate. For low rotation rates, steady axisymmetric drop shapes can be found analytically. The stability of these drops is investigated, using an energy method, both with and without the longwave approximation. For industrially relevant highspeed motions, we formulate and solve a theoretical and numerical model for the threedimensional timedependent motion of the deforming drop. We pay particular attention to the formation of “fingers” at the expanding front. The model includes viscous, capillary, gravitational, centrifugal, Coriolis, and finitecontactangle effects. Both homogeneous and chemically heterogeneous substrates are considered. In agreement with published experiments, the model demonstrates that imperfect wetting behavior is the principal cause of fingering during spin coating. Features of the finger profiles are in close agreement with experimental observations.

Selfsimilar pinchoff of power law fluids
View Description Hide DescriptionPinchoff dynamics of slender liquid threads of power law fluids without inertia are studied by asymptotic analysis. Because the threads are slender, their dynamics are governed by a pair of spatially onedimensional, nonlinear evolution equations for the thread shape and axial velocity that results from a longwave asymptotic expansion of the creeping flowequations. By means of an approach that differs from those used previously in analyses of capillary pinching of threads of Newtonian fluids, a similarity transformation is derived that reduces the evolution equations to two coupled similarity equations. As in the Newtonian case, it is shown that for each value of the power law exponent n where there is a family of similarity solutions for capillary pinching of threads of power law fluids. For a given family of solutions, the radial and axial scales vary with time τ to pinchoff as and respectively, where δ is the axial scaling exponent. It is shown that for a given family of solutions characterized by a fixed value of n, each member of the family has a different scaling exponent δ. Since the viscosity of a power law fluid varies as where is the deformation rate, for each value of n a numerical method based on domain splicing is used to compute the values of the axial scaling exponent δ and the similarity solutions.

Motion of a solid object through a pasty (thixotropic) fluid
View Description Hide DescriptionFor materials assumed to be simple yield stress fluids the velocity of an object should continuously increase from zero as the applied force increases from the critical value for incipient motion. We carried out experiments of fall of a sphere in a typical, thixotropic, pasty material (a laponite suspension). We either left a sphere falling in the fluid in different initial states of structure or vibrated the fluid in a given state of structure at different frequencies. In each case three analogous regimes appear either for increasing restructuring states of the fluid or decreasing frequencies: A rapid fall at an almost constant rate; a slower fall at a progressively decreasing velocity; a slow fall at a rapidly decreasing rate finally leading to apparent stoppage. These results show that the motion of an object, due to gravity in a pasty material, is a more complex dynamical process than generally assumed for simple yield stress fluids. A simple model using the basic features of the (thixotropic) rheological behavior of these pasty materials makes it possible to explain these experimental trends. The fall of an object in such a fluid thus appears to basically follow a bifurcation process: For a sufficiently large force applied onto the object its rapid motion tends to sufficiently liquify the fluid around it so that its subsequent motion is more rapid and so on until reaching a constant velocity; on the contrary if the force applied onto the object is not sufficiently large the fluid around has enough time to restructure, which slows down the motion and so on until the complete stoppage of the object.

Miscible displacements in capillary tubes: Effect of a preexisting wall film
View Description Hide DescriptionFor miscible displacements in capillary tubes, the impact of a preexisting wall film on the tip velocity of the displacing fluid finger is analyzed by means of axisymmetric Stokes simulations. The wall film is assumed to have the same viscosity as the displacing fluid, which is less viscous than the displaced fluid. The finger of the displacing fluid is seen to move in a quasisteady fashion, with a tip velocity below the centerline velocity of an equivalent Poiseuille flow. The explanation for this behavior, which is in contrast to our earlier findings for miscible displacements without wall films, lies in the lubricating effect of the wall film. The condition is established for which the displaced fluid moves in a nearly solid bodylike motion. In this limit, a closed expression is derived for the finger tip velocity. A comparison between the simulation data and the closed form results shows reasonable agreement, provided that the criterion for solid bodylike motion is satisfied. Furthermore, results are presented for the practically relevant limit of large viscosity ratios.

Impact of an oblique breaking wave on a wall
View Description Hide DescriptionThe intention of this paper is to study impact force of an obliqueangled slamming wave acting on a rigid wall. In the present study the analytical approach is pursued based on a technique proposed by Shu (Proceedings of the International Conference on Applied Mathematics & Mathematical Physics, Sylhet, Bangladesh, 2000). A nonlinear theory in the context of potential flow is presented for determining accurately the freesurface profiles immediately after an oblique breaking wave impingement on the rigid vertical wall that suddenly starts from rest. The smalltime expansion is taken as far as necessary to include the accelerating effect. The analytical solutions for the freesurface elevation are derived up to the third order. The results derived in this paper are of particular interest to the marine and offshore engineering industries, which will find the information useful for the design of ships, coastal and offshore.

Conductances between confined rough walls
View Description Hide DescriptionTwo and threedimensional creeping flows and diffusion transport through constricted and possibly rough surfaces are studied. Asymptotic expansions of conductances are derived as functions of the constriction local geometry. The validity range of the proposed theoretical approximations is explored through a comparison either with available exact results for specific twodimensional aperture fields or with direct numerical computations for general threedimensional geometries. The large validity range of the analytical expressions proposed for the hydraulic conductivity (and to a lesser extent for the electrical conductivity) opens up interesting perspectives for the simulation of flows in highly complicated geometries with a large number of constrictions.

Experimental investigation on cellular breakup of a planar liquid sheet from an airblast nozzle
View Description Hide DescriptionThe cellular breakup phenomenon is investigated experimentally for a planar liquid sheet from an airblast nozzle. The dominant sinuous wave growing spatially downstream forms complicated cellular structures of perforated thin films and surrounding ligaments. Several characteristic parameters are measured from photographic images and compared with linear temporal analysis. The dominant wavelength is proportional to the inverse square of the relative velocity between air and liquid. The estimated breakup time matches the growth time of the most unstable wave, while the breakup length corresponds to a product of breakup time and liquid velocity. Numerical simulation shows a substantially reduced mean effective velocity near flow reattachment region of the air stream. Air turbulence seems to play a major role on initial perturbations of cellular breakup in the given nozzle configuration. The measured spatial growth rates are always less than linear predictions due to deviation from the linear regime at higher amplitudes.

Rimming flows with an axially varying viscosity
View Description Hide DescriptionWe consider rimming flows in the presence of an axially varying viscosity but with inertia and surface tension effects being negligible. First, we find that a modified lubrication analysis (MLA) presented earlier [M. Tirumkudulu and A. Acrivos, Phys. Fluids 13, 14 (2001)] can predict accurately the thickness of the film profile over the whole range of Ω, the angular velocity of the rotating cylinder, even when the fill fraction F is as large as 0.36, where the film is far from thin. This is also the case with the analysis due to Benjamin et al. [T. B. Benjamin, W. G. Pritchard, and S. J. Tavener (preprint, 1993)] except that, here, F cannot exceed 0.29. On the basis of this MLA, we propose a model to describe the threedimensional free surface shape of rimming flows with an axially varying viscosity and show that the free surface profiles thereby obtained agree with those determined by solving numerically the threedimensional Stokes equations. In the accompanying article, this model will be used as the basis of a stability analysis which will explain the origin of the observed particle band formation in rimming flows of suspensions containing neutrally buoyant particles [M. Tirumkudulu, A. Mileo, and A. Acrivos, Phys. Fluids 12, 1615 (2000)].

Theory of particle segregation in rimming flows of suspensions containing neutrally buoyant particles
View Description Hide DescriptionIt has recently been reported that an initially uniform suspension of neutrally buoyant particles within a partially filled horizontal rotating cylinder can, under certain conditions, segregate into bands of particles separated by regions of low particle concentration or even particlefree liquid [M. Tirumkudulu, A. Mileo, and A. Acrivos, Phys. Fluids 12, 1615 (2000)]. An explanation for this phenomenon is proposed on the basis of a model of rimming flows with an axially varying viscosity plus the experimental observation that, when the liquid contains a recirculating region (puddle), the particles segregate radially by migrating out of the puddle into the unidirectional circumferential flow. A linear stability analysis for dilute suspensions shows that such a particle distribution is unstable to axial perturbations with the surface tension being responsible for the selection of the wavelength of the most rapidly amplified disturbance. The calculated and measured spacings between the bands are in good agreement. In addition, since, in the absence of a puddle, the particle concentration appears to remain uniform throughout the cross section of the film, no axial particle segregation is predicted to occur nor has it ever been seen experimentally, even when an axial viscosity variation is imposed on the flow by cooling a preselected portion of the cylinder.

Inertialess instability of a twolayer liquid film flow
View Description Hide DescriptionThe physical mechanism of instability in a superposed twolayer liquid filmflow down an incline plane is analyzed. If the layer adjacent to the wall is sufficiently thin and less viscous in certain twolayer parallel Newtonian liquid flows of the same density with an interface but without a free surface, the flows are stable with respect to long waves. This is the socalled “thin lubrication layer effect.” However, when a free surface exists in the twolayered flow, the flow becomes unstable even when the Reynolds number approaches zero. Thus the thinlayer lubrication effect is lost due to the presence of the free surface, and inertialess instability occurs. The reason for the loss of the lubrication effect and the mechanism of inertialess instability are explained by use of the energy budget in the mechanical energy equation. Contrary to the case of twolayer flows without a free surface, the flow with a free surface is stable if the layer adjacent to the solid wall is more viscous. The stabilization is achieved even without help from surface or interfacial tension. The mechanism of stabilization is also elucidated from an energy consideration. Navier–Stokes simulations are then performed to determine the effect of the layer viscosity ratio when nonlinear effects are included.

Singularity method for oblate and prolate spheroids in Stokes and linearized oscillatory flow
View Description Hide DescriptionThis paper applies the singularity method to obtain analytic solutions for oblate spheroids in Stokes flow, and to obtain numerical results for prolate and oblate spheroids undergoing oscillatory translation, and oscillatory rotation. To apply the method to oblate spheroids, singularities are placed along an imaginary focal length. A novel method is used to determine the hydrodynamictorque by deriving Green’s functions for torque for the unsteady rotlet, stresslet, and potential quadrupole. The results agree with analytic solutions for high and low frequencies, the results of previous studies, and results calculated using the boundary element method.

Modeswitching and nonlinear effects in compressible flow over a cavity
View Description Hide DescriptionMultiple distinct peaks of comparable strength in unsteady pressure autospectra often characterize compressible flowinduced cavity oscillations. It is unclear whether these different largeamplitude tones (i.e., Rossiter modes) coexist or are the result of a modeswitching phenomenon. The cause of additional peaks in the spectrum, particularly at low frequency, is also unknown. This article describes the analyses of unsteady pressure data in a cavity using timefrequency methods, namely the shorttime Fourier transform(STFT) and the continuous Morlet wavelet transform, and higherorder spectral techniques. The STFT and waveletanalyses clearly show that the dominant mode switches between the primary Rossiter modes. This is verified by instantaneous schlieren images acquired simultaneously with the unsteady pressures. Furthermore, the Rossiter modes experience some degree of lowfrequency amplitude modulation. An estimate of the modulation frequency, obtained from the waveletanalysis, matches the lowfrequency peak seen in the autospectrum. Higherorder spectral methods were employed to investigate potential quadratic nonlinear interactions between the Rossiter modes and to determine if they are responsible for the lowfrequency mode present in the autospectrum. In turn, this lowfrequency mode could interact with the Rossiter modes to modulate their amplitude. Significant nonlinearities, in the form of sum and difference frequencies of the Rossiter modes, are present in the cavity at while nonlinear effects are much smaller in the at The bispectral analysis indicates that quadratic interactions between Rossiter modes in the nearfield pressure are not responsible for the observed lowfrequency peak in the pressure autospectrum. Furthermore, the lowfrequency mode does not exhibit a strong nonlinear coupling with the Rossiter modes.

Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk
View Description Hide DescriptionBoth direct numerical simulation and theoretical stability analysis are performed together in order to study the transition process to turbulence in a flow between a rotating and a stationary disk. This linear stability analysis considers the complete rotorstator flow and then extends the results of Lingwood [J. Fluid Mech. 299, 17 (1995); 314, 373 (1996)] obtained in a single disk case. The present linear analysis also extends the former twodisk computations of Itoh [ASME FED 114, 83 (1991)], only limited to a hydrodynamicspatialinstability analysis. Moreover, in the present work, this approach is completed by discussing the effects of buoyancydriven convection on the flow stability and by absolute/convective analysis of the flow. Coupled with accurate numerical computations based on an efficient pseudospectral Chebyshev–Fourier method, this study brings new insight on the spatiotemporal characteristics of this flow during the first stages of transition. For instance, an exchange of stability from a steady to a periodic flow with spiral structures is observed for the first time numerically in such cavity of large aspect ratio. The nature of the first bifurcation is discussed as well as the influence on it of disturbances coming from the endwall boundary layer. Annular and spiral patterns are observed in the unstable stationary disk layer with characteristic parameters agreeing very well with the present theoretical results. Then, these structures are interpreted in terms of type I and type II generic instabilities. Moreover, the absolute instability regions which are supposed to be strongly connected with the turbulent breakdown process are also identified and the critical Reynolds numbers of the convective/absolute transition in both Ekman and Bödewadt layers are given.

Rotating disk flow stability in electrochemical cells: Effect of viscosity stratification
View Description Hide DescriptionThis work is about the effect of viscosity stratification on the hydrodynamicinstability of rotating disk flow, and whether or not it can take into account experimental observations of the lowering of critical Reynolds numbers in electrochemical systems, where a viscosity stratification is assumed to result from the gradients of chemical species existing in the convective boundary layer near the disk electrode. The analysis is for temporal stability of a class of von Kármán solutions: fully threedimensional modes are considered and the neutral curves are therefore functions of not only the Reynolds number but also the wave frequency and the two wave numbers. Global minimization over wave numbers and also over the frequency gives the critical Reynolds number. The neutral curves exhibit a twomode structure and the dependence of both modes on parameters is studied. It is shown that viscosity stratification leads to an increase in the range of parameters where perturbations are amplified and to a reduction of the critical Reynolds number, in a wide range of perturbation frequencies. The results support the hypothesis that the current oscillations may originate from a hydrodynamicinstability.

Motion of a vortex sheet on a sphere with pole vortices
View Description Hide DescriptionWe consider the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices fixed on north and south poles. Analytic and numerical research revealed that a vortex sheet in twodimensional space has the following three properties. First, the vortex sheet is linearly unstable due to Kelvin–Helmholtz instability. Second, the curvature of the vortex sheet diverges in finite time. Last, the vortex sheet evolves into a rollingup doubly branched spiral, when the equation of motion is regularized by the vortex method. The purpose of this paper is to investigate how the curvature of the sphere and the presence of the pole vortices affect these three properties mathematically and numerically. We show that some low spectra of disturbance become linearly stable due to the pole vortices and thus the singularity formation tends to be delayed. On the other hand, however, the vortex sheet, which is regularized by the vortex method, acquires complex structure of many rollingup spirals.
