Volume 9, Issue 8, August 1997
Index of content:

Inertial effects on contact line instability in the coating of a dry inclined plate
View Description Hide DescriptionIt has been observed that when a thin liquid film coats an initially dry inclined plane, a spanwise instability occurs at the leading edge. Here we develop a model for the evolution of this coating film which includes inertia, gravity, surface tension and the contact angle at the leading edge of the film. A Kármán–Pohlhausen method is used to include inertia. We determine steady state profiles of the film and investigate their stability. The predictions of the model are compared to some recent experiments and we find good agreement. This theory gives improvement over a lubricationtheory in experiments where Reynolds numbers are significantly larger than one.

The flow of a liquid film on the inside of a rotating cylinder, and some related problems
View Description Hide DescriptionA liquid filmflows on the inner surface of a rotating horizontal cylinder. The simplest lubrication model assumes a force balance between viscosity and the streamwise component of gravity, and the equation for the film thickness admits a continuous solution only when the average thickness is less than a certain critical value. Above this value, a discontinuous solution is possible but the details are not accessible by means of the simple theory. This behavior can be traced to the gradual periodic variation of the streamwise component of gravity in the streamwise direction. We consider also two related problems in which this variation occurs more or less abruptly: (i) when the moving wall comprises two straight segments inclined at different angles, and (ii) when the dragged film emerges through the free surface of a second, overlying liquid. These problems are approached by introducing a smoothing parameter, namely surface tension, and solving a suitable initial value problem. We use the method of lines for this purpose because of the availability of robust ODE software which can exploit the structure of the problem; however, the periodic conditions of the cylinder problem necessitate a special approach to the discretization.

Local similarity solutions for the stress field of an OldroydB fluid in the partialslip/slip flow
View Description Hide DescriptionLocal similarity solutions are presented for the stress field of a fluid described by the OldroydB viscoelasticconstitutive equation near the singularity caused by the intersection of a planar free surface and a solid surface along which Navier’s slip law holds, the partialslip/slip problem. For the case where the velocity field is given by Newtonian kinematics, the elastic stress field is predicted to have a logarithmic singularity as the point of attachment of the free surface is approached. Asymptotic analysis for the fullycoupled flow, where the stress and flow fields are determined simultaneously, results in a local form for the flow and elastic stress fields that is similar in form to that for the decoupled case. For both the coupled and decoupled flow problems, the strength of the singularity depends on the dimensionless solventviscosity and the slip coefficient, but not upon the Deborah number. The asymptotic results for the coupled flow differ from the predictions with Newtonian kinematics in that the strength of the singularity in the rateofstrain and elastic stress fields scales with the inverse of the dimensionless solventviscosity, and suggest that calculations with decreasing solventviscosity become increasingly difficult. The fullycoupled analysis also suggests that the asymptotic behavior in the limit of vanishing solventviscosity, the UCM limit, is qualitatively different from that for finite values of the solventviscosity. The structure of the flow and stress fields for both the coupled and decoupled flow problems is reproduced by finite element calculations.

Wetting of heterogeneous surfaces: Influence of defect interactions
View Description Hide DescriptionWe investigate the motion of a fluid interface through a thin cell, used as a model for porous media with heterogeneous wettability properties. We analyze how the distribution of wettability defects modifies the shape of the advancing interface. Solid substrates are prepared with a photolithography technique to create wettability defects. Several distributions of defects are generated, with different surface densities and different position ordering. The amplitude of fluctuation of the fluid interface is measured for the different defect distributions and with or without gravity stabilization of the interface. When the interface is stabilized by gravity, the capillary length sets the distance below which defects strongly cooperate to pin the interface and trap air bubbles. Trapping of the advancing interface by the defects is dependent on the spatial distribution of defects. The interface fluctuation is much larger for strongly disordered distributions and is related to a correlation length of the defect pattern.

Motion and arrest of a molten contact line on a cold surface: An experimental study
View Description Hide DescriptionAn experimental study is presented of the behavior of a molten contact line under conditions which simulate what happens when a molten droplet touches a subcooled solid, spreads partly over it, and freezes. We restrict our attention to the case where the solid and melt are of the same material and have approximately the same thermal properties, and reach two conclusions. First, we show that an advancing molten contact line is arrested at an apparent dynamic contact angle, which for a given material depends primarily on the Stefan number based on the temperature difference between the fusion point and the temperature of the solid over which the melt spreads. Second, during much of the spreading prior to contactline arrest, the relationship between the melt’s apparent dynamic contact angle and the contactline speed appears to obey the Hoffman–Tanner–Voinov law with the equilibrium contact angle taken as zero.

On the theory for the arrest of an advancing molten contact line on a cold solid of the same material
View Description Hide DescriptionWe show that a conventional continuum formulation of the equations and boundary conditions for the spreading of a pure molten material over a cold, solid substrate of its own kind has no meaningful solution for the angle of attack of the fusion front at the contact line, which is the quantity that determines contactline arrest. is determined by the heat flux just behind the contact line, and the heat flux in the mathematical model is singular at the contact line. The scale of the physical mechanism which limits the heat flux at the contact line and removes the singularity is estimated by computing the point where the continuum model must be cut off in order to bring it into agreement with the experimental data for a microcrystalline wax. The cutoff scale is in the range 0.1–1 μm, that is, much larger than molecular dimensions, but of order times the convective thermal length scale

Drop formation in viscous flows at a vertical capillary tube
View Description Hide DescriptionDrop formation at the tip of a vertical, circular capillary tube immersed in a second immiscible fluid is studied numerically for lowReynoldsnumber flows using the boundary integral method. The evolution and breakup of the drop fluid is considered to assess the influences of the viscosity ratio , the Bond number B, and the capillary number C for , , and . For very small , breakup occurs at shorter times, there is no detectable thread between the detaching drop and the remaining pendant fluid column, and thus no large satellite drops are formed. The distance to detachment increases monotonically with and changes substantially for , but the volume of the primary drop varies only slightly with . An additional application of the numerical investigation is to consider the effect of imposing a uniform flow in the ambient fluid [e.g., Oguz and Prosperetti, J. Fluid Mech. 257, 111 (1993)], which is shown to lead to a smaller primary drop volume and a longer detachment length, as has been previously demonstrated primarily for highReynoldsnumber flows.

Dispersion in twodimensional periodic porous media. Part I. Hydrodynamics
View Description Hide DescriptionIn this paper we investigate the hydrodynamics of twodimensional spatially periodic porous media. The finite volume method is used to compute the flow field in these media with the Navier–Stokes equations. Two particular points are described: how to deal with the periodic boundary conditions on the surface of a unit cell; and how to avoid numerical dispersion. The fluid flow is computed for the Stokes regime and for moderate values of the particle Reynolds number up to . Four types of media are studied. Three of them are “ordered,” with inline or staggered square cylinders and zigzag medium. The fourth is “disordered,” with randomly distributed square cylinders. The effect of the direction of the average flow velocity is analysed in all these cases. In the Stokes regime, the results for the pressure drop agree with the usual evaluation in terms of permeability. For higher particle Reynolds numbers, the nonlinear correction to Darcy’s law is discussed. The correction term for moderate Reynolds number is found to be cubic in the average flow velocity for isotropic media.

Dispersion in twodimensional periodic porous media. Part II. Dispersion tensor
View Description Hide DescriptionThe dispersion tensor of twodimensional periodic porous media is investigated numerically. The theory is first briefly reviewed using the volume averaging method. Then, with the help of the hydrodynamics determined in Part I of this study, the dispersion tensor is calculated both for ordered (inline or staggered square cylinders) and disordered (randomly distributed square cylinders) varying both the particle Péclet number , the particle Reynolds number, from the Stokes flow to the laminar–inertial regime , and the direction of the average flow with the axes of the unit cell. The influence of order and spatial periodicity is discussed. Lastly the results are compared with those for “real” porous media.

Potential/complexlamellar descriptions of incompressible viscous flow
View Description Hide DescriptionIn this manuscript a taxonomy of potential/complexlamellar decompositions will be developed for describing the flow of an incompressible viscous fluid. Since these decompositions are not unique, they are best defined by their governing equations and not by their individual components. Within this framework we will construct a number of decompositions, both classical and contemporary, inviscid and viscous. Finally, we will use these results to derive the conditions under which an unsteady flow is circulation preserving. First, using a decomposition based on the Lighthill–Darwin–Hawthorne drift function, we will show that a flow ceases to be circulation preserving when the cross product of its circulation and drift function gradients are not harmonic. Then we will show that an unsteady flow can be circulation preserving only when it is an irrotational one as well.

The effect of small pipe divergence on near critical swirling flows
View Description Hide DescriptionThe effect of small pipe divergence on an inviscid, incompressible, near critical axisymmetric swirling flow is investigated. The singular behavior of a regular expansion solution, in terms of the pipe divergence parameter, around the critical swirl of a flow in a straight pipe is demonstrated. This singularity infers that largeamplitude disturbances may be induced by the small pipe divergence when incoming flows have a swirl level near the critical swirl. In order to gain insight to the behavior of flows in this swirl range, a smalldisturbance analysis is developed. It is found that a small but finite pipe divergence breaks the transcritical bifurcation of solutions of a flow in a straight pipe into two equilibrium solution branches. These branches fold at limit swirl levels near the critical swirl resulting in a finite gap of swirl that separates the two branches. This suggests that no nearcolumnar axisymmetric state can exist within this range of incoming swirl around the critical level; the flow must develop large disturbances in this swirl range. Beyond this range, two steady states may exist under the same inlet/outlet conditions. However, when the pipe divergence is increased, this special behavior uniformly changes into a branch of solutions with no fold. A weakly nonlinear approach to study the effect of slight pipe divergence on standing waves in a long pipe is also derived. The behavior of the asymptotic solutions match the bifurcation diagrams from previous theoretical and numerical studies and extends their results. The relevance of the results to axisymmetric vortex breakdown in a diverging pipe is discussed.

Response of the Blasius boundary layer to freestream vorticity
View Description Hide DescriptionTwo and threedimensional vortical modes that solve the linearized NavierStokes equations in the free stream are used in the present theory to represent some of the key features of lowlevel turbulence. Excluding the leading edge, the effect of these modes on the Blasius boundary layer is investigated using the parabolized stabilityequations (PSE). When the vortical modes are steady, or have low frequencies, the PSE analysis is started at a location from the solution to a new set of ordinary differential equations. This solution is able to satisfy the linearized NavierStokes equations in a rather large neighborhood of When the vortical modes have frequencies equal to those of unstable TollmienSchlichting waves, the scattering of the vortical modes by surface undulation produces only a weak response in the boundary layer, in agreement with other investigations. However, when steady and lowfrequency vortical modes are considered, the analysis yields results that successfully reproduce a number of the experimental measurements of Kendall [AIAA Paper 901504 (1990)] on streaky structures, known as Klebanoff modes, that cause a periodic spanwise modulation of the streamwise velocity.

Linear stability analysis of plane quadratic flows in a rotating frame with applications to modeling
View Description Hide DescriptionThe linear response of turbulence to a distortion and simple rotation is investigated in this paper from a fundamental theoretical standpoint. Quadratic flows are a special case of planar flows with constant mean velocity gradients, which can be characterized by a constant rate of strain and a constant spanwise (normal to the plane) vorticity component , with arbitrary values (here, we can take and without any loss of generality). According to the sign of , streamlines are hyperbolic, rectilinear (pure shear flow) or elliptic. Since these flows can also be considered as meanflows, when superimposing a threedimensional disturbance (or fluctuatingturbulent) field which satisfies statistical homogeneity, the linearized analysis of the disturbance field is of interest both from the point of view of hydrodynamicstability (e.g., the elliptical flow instability) and from the point of view of homogeneous rapid distortion theory (RDT) including applications to the basic statistics. The case of quadratic flow in a rotating frame (with angular velocity in the direction normal to the plane of the basic flow) is revisited in this paper, in order to complete—with the three parameters , and —previous works on linear theory by Cambon et al. [J. Fluid Mech. 278, 175 (1994)] and Speziale, Abid, and Blaisdell [Phys. Fluids 8, 781 (1996)]. From a simplified “pressureless” linear approach, a general stability criterion is derived based on the value of the modified Bradshaw number , which coincides with the rotational Richardson number introduced by Bradshaw [J. Fluid Mech. 36, 177 (1969)] and denoted by (in particular, for the case of pure shear flow where . It is shown that this criterion gives results identical to the “true” linear stabilityanalysis (including the effect of the fluctuating pressure) if the absolute vorticity has a zero value. In addition, the relevance of this criterion is checked with respect to the true linear approach in distorted wave space and related RDT applications. For all of the cases, the maximum amplification for the threedimensional disturbance field is found for zero tilting vorticity and for pure spanwise modes (with wave vector normal to the plane of the quadratic flow), in accordance with the generalized Bradshaw criterion and other results in hydrodynamicstability. For other spectral directions, the agreement is not as complete except for the pure shear case, and this is particularly discussed looking at statistical RDT solutions, which involve a summation over all directions of the wave vector. Finally, the impact of the whole analysis on secondorder, onepoint modeling is discussed.

Hamiltonian moment reduction for describing vortices in shear
View Description Hide DescriptionThis paper discusses a general method for approximating twodimensional and quasigeostrophic threedimensional fluid flows that are dominated by coherent lumps of vorticity. The method is based upon the noncanonical Hamiltonian structure of the ideal fluid and uses special functionals of the vorticity as dynamical variables. It permits the extraction of exact or approximate finite degreeoffreedom Hamiltonian systems from the partial differential equations that describe vortex dynamics. We give examples in which the functionals are chosen to be spatial moments of the vorticity. The method gives rise to constants of motion known as Casimir invariants and provides a classification scheme for the global phase space structure of the reduced finite systems, based upon Lie algebra theory. The method is illustrated by application to the Kida vortex [S. Kida, J. Phys. Soc. Jpn. 50, 3517 (1981)] and to the problem of the quasigeostrophic evolution of an ellipsoid of uniform vorticity, embedded in a background flow containing horizontal and vertical shear [Meacham et al., Dyn. Atmos. Oceans14, 333 (1994)]. The approach provides a simple way of visualizing the structure of the phase space of the Kida problem that allows one to easily classify the types of physical behavior that the vortex may undergo. The dynamics of the ellipsoidal vortex in shear are shown to be Hamiltonian and are represented, without further approximation beyond the assumption of quasigeostrophy, by a finite degreeoffreedom system in canonical variables. The derivation presented here is simpler and more complete than the previous derivation which led to a finite degreeoffreedom system that governs the semiaxes and orientation of the ellipsoid. Using the reduced Hamiltonian description, it is shown that one of the possible modes of evolution of the ellipsoidal vortex is chaotic. These chaotic solutions are noteworthy in that they are exact chaotic solutions of a continuum fluid governing equation, the quasigeostrophic potential vorticityequation.

Quasisteady monopole and tripole attractors for relaxing vortices
View Description Hide DescriptionUsing fully nonlinear simulations of the twodimensional Navier–Stokes equations at large Reynolds number (Re), we bracket a threshold amplitude above which a perturbed Gaussian monopole will relax to a quasisteady, rotating tripole, and below which will relax to an axisymmetric monopole. The resulting quasisteady structures are robust to small perturbations. We propose a means of measuring the decay rate of disturbances to asymptotic vortical structures wherein streamlines and lines of constant vorticity correspond in some rotating or translating frame. These experiments support the hypothesis that small or moderate deviations from asymptotic structures decay through inviscid and viscous mixing.

Double diffusive convection in a vertical rectangular cavity
View Description Hide DescriptionIn the present work, we study the onset of double diffusive convection in vertical enclosures with equal and opposing buoyancy forces due to horizontal thermal and concentration gradients (in the case , where and are, respectively, the solutal and thermal Grashof numbers). We demonstrate that the equilibrium solution is linearly stable until the parameter reaches a critical value, which depends on the aspect ratio of the cell, . For the square cavity we find a critical value of while previous numerical results give a value close to 6000. When increases, the stability parameter decreases regularly to reach the value , and the wave number reaches a value , for . These theoretical results are in good agreement with our direct simulation. We numerically verify that the onset of double diffusive convection corresponds to a transcritical bifurcation point. The subcritical solutions are strong attractors, which explains that authors who have worked previously on this problem were not able to preserve the equilibrium solution beyond a particular value of the thermal Rayleigh number, . This value has been confused with the critical Rayleigh number, while it corresponds in fact to the location of the turning point.

Eulerian analysis of concentration fluctuations in dispersing plumes and puffs
View Description Hide DescriptionA recent analysis of the secondorder moments of concentration fluctuations in dispersing plumes and puffs has been reexamined in order to obtain a picture of the budget for the secondorder moments. We have compared this picture with the parametrizations of the transport and dissipation of concentration variance in a secondorder closure model. The comparison lends support to the form of the secondorder closure parametrizations and also provides a constraint on the constants used in the parametrization of the dissipation of concentration variance, thereby providing the means to reduce the number of tunable constants in the model by one. The constraint implies that the constants must differ for area, line and point sources and implies that finding a good parametrization of the dissipation of concentration variance for arbitrary source geometries may be difficult.

(1+1)dimensional turbulence
View Description Hide DescriptionA class of dynamical models of turbulence living on a onedimensional dyadictree structure is introduced and studied. The models are obtained as a natural generalization of the popular GOY shell model of turbulence. These models are found to be chaotic and intermittent. They represent the first example of (1+1)dimensional dynamical systems possessing non trivial multifractal properties. The dyadic structure allows us to study spatial and temporal fluctuations. Energy dissipation statistics and its scaling properties are studied. The refined Kolmogorov hypothesis is found to hold.

The Lagrangian spectral relaxation model of the scalar dissipation in homogeneous turbulence
View Description Hide DescriptionLagrangian pdf methods are employed to extend the spectral relaxation (SR) model of the scalar dissipation of an inert, passive scalar in homogeneous turbulence. The Lagrangian spectral relaxation (LSR) model divides wavenumber space into a finite number (the total number depending on the Taylorscale Reynolds number and the Schmidt number of wavenumber bands whose time constants are determined from the mean turbulent kinetic energy and instantaneous turbulent energy dissipation rate. The LSR model accounts for the evolution of the scalar spectrum (viz., pdf) from an arbitrary initial shape to its fully developed form. The effect of turbulentfrequencyfluctuations on the instantaneous scalar dissipation rate following a Kolmogorovscale fluid particle is incorporated into the LSR model through a Lagrangian pdf model for the turbulent frequency. Model results are compared with DNS data for passive scalar mixing in stationary, isotropic turbulence. Two distinct causes of nonGaussian scalar statistics are investigated: smallscale intermittency due to scalardissipation fluctuations at scales near the Kolmogorov scale, and transient largescale inhomogeneities due to the form of the initial scalar spectrum at scales near the integral scale. Despite the absence of fitting parameters, the LSR model shows satisfactory agreement with available DNS data for both types of flows.

Turbulent cascades: Limitations and a statistical test of the lognormal hypothesis
View Description Hide DescriptionMultiplicative random cascade models were introduced in the 1970s to explain the intermittency of turbulent energy dissipation. The rigorous results in the multifractality of cascade measures recently derived by this author are used in two ways. (1) The statistical test for the Kolmogorov–Obukhov lognormal hypothesis (K62) is revised. Contrary to what is generally believed, we show that the K62 theoretical prediction is in good agreement with experimental data in the range (1,18) of the parameter (the order of velocity structure functions). This revised conclusion was necessitated by violations in previous comparisons of the “ergodic hypothesis” for large . (2) Physical limitations on cascade models are analyzed. We show that cascade measures demonstrate a strict dependence on the scaling parameter. This circumstance affects interpretations of statistics of multipliers, shows that the models used in practice are not really superior to others, and indicates the necessity to study cascades with a random scaling parameter.