Volume 11, Issue 7, July 1999
 LETTERS


An approximate deconvolution procedure for largeeddy simulation
View Description Hide DescriptionAn alternative approach to largeeddy simulation based on approximate deconvolution(ADM) is developed. The main ingredient is an approximation of the nonfiltered field by truncated series expansion of the inverse filter operator. A posteriori tests for decaying compressible isotropic turbulence show excellent agreement with direct numerical simulation. The computational overhead of ADM is similar to that of a scalesimilarity model and considerably less than for dynamic models.

Qualitative flow visualization using colored lights and reflective flakes
View Description Hide DescriptionWe present a novel flowvisualization technique utilizing reflective flakes in combination with color illumination. Three differently colored columated light beams are used to illuminate the flow, each color being directed from a separate direction. In this way, the color of the light reflected from the flakes gives an indication of the local flake orientation. The flake orientation in complex threedimensional (3D) flow is in general a complicated function of the local velocity gradient tensor, but can be calculated if the underlying velocity field is known. This has recently been demonstrated by Gauthier et al. [Phys. Fluids. 10, 2147 (1998)] using monochome light. In complex flow fields the distribution of flakes may, however, be rearranged by the motion, thus making the local intensity of reflection depend on both orientation and flake concentration. The color is, however, immune to the local number density of flakes inside the flow, making quantitative information possible. This technique is demonstrated by visualizing the finer details of vortices in a Taylor–Couette device.
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 ARTICLES


Miscible quarter fivespot displacements in a HeleShaw cell and the role of flowinduced dispersion
View Description Hide DescriptionMiscible quarter fivespot displacements in a HeleShaw cell are investigated by means of experimental measurements and numerical simulations. The experiments record both the volumetric as well as the surface efficiency at breakthrough as a function of the dimensionless flow rate in the form of a Peclet number and the viscosity contrast. For small flow rates, both of these efficiency measures decrease uniformly with increasing Peclet numbers. At large flow rates, an asymptotic state is reached where the efficiencies no longer depend on the Peclet number. Up to Atwood numbers of approximately 0.5, the less viscous fluid occupies close to of the gap width, which indicates a nearparabolic velocity profile across the gap. Consequently, in this parameter range a Taylor dispersion approach should be well suited to account for flowinduced dispersion effects. For larger viscosity contrasts, accompanying twodimensional numerical simulations based on Taylor dispersion predict an increased stabilization for high flow rates, which is not confirmed by the experiments. This suggests that, in order to extend the range of applicability to larger viscosity contrasts, the components of the dispersion tensor will have to be amended in order to account for the presence of quasisteady fingers.

Iterated stretching of viscoelastic jets
View Description Hide DescriptionWe examine, with asymptotic analysis and numerical simulation, the iterated stretching dynamics of FENE and OldroydB jets of initial radius shear viscosity ν, Weissenberg number We, retardation number S, and capillary number Ca. The usual Rayleigh instability stretches the local uniaxial extensional flow region near a minimum in jet radius into a primary filament of radius between two beads. The strainrate within the filament remains constant while its radius (elastic stress) decreases (increases) exponentially in time with a long elasticrelaxation timeInstabilities convected from the bead relieve the tension at the necks during this slow elastic drainage and trigger a filament recoil. Secondary filaments then form at the necks from the resulting stretching. This iterated stretching is predicted to occur successively to generate highgeneration filaments of radius until finiteextensibility effects set in.

A theory to describe heat transfer during laminar incompressible flow of a fluid in periodic porous media
View Description Hide DescriptionIn this paper, a theory is formulated that describes the heat transfer during the laminar flow of an incompressible fluid within a rigid periodic porous medium. To describe the macroscopic temperature distribution in porous media, the volume average concept is employed to derive the volume averaged energy equation. Previous researchers have used either the oneequation model with the local thermal equilibrium assumption between the various phases or the complex twoequation model to deal with the average temperatures of the solid phase and the fluid phase. Both types of models are derived and applied in a stationary observation frame. In this paper, we propose a generalized oneequation model by relaxing the local thermal equilibrium assumption between the phases and invoking the frame invariant principle. The three new terms arise due to the microscopic transient, convective, and conductive effects. By taking advantage of the objectivity of the generalized model, it is shown that the microscopic temperature distribution in a periodic unit cell results in a periodic heat flux boundary condition. The direct temperature solution is obtained in the periodic unit cell by solving the generalized volume averaged energy equation. A modified effective thermal conductivitytensor is calculated which encompasses the microconvective effects encountered in porous media. An analytic study of a nonisothermal flow through two parallel thick plates is performed to explore the role of each new term in the generalized model.

Effect of wall slip on the stability of viscoelastic plane shear flow
View Description Hide DescriptionWe show that for shear flow of an upper convected Maxwell fluid with small but nonzero slip velocity, an increasing dependence of the slip velocity on the elastic normal stress in the flow direction leads to short wavelength flow instability at sufficiently high Weissenberg number (≳10). Pressuredependent slip can also lead to instability, but only at unrealistically large Weissenberg number. If the slip velocity depends only on shear stress, then the flow is always stable. These analytical results are valid in a specific asymptotic limit, but are independent of the specific form of the model for slip. Numerical results for specific, phenomenological slip models and the PhanThien–Tanner bulk constitutive model show that the results are robust in the presence of nonlinear viscoelasticity. The scaling of the critical shear stress for instability with modulus and molecular weight and of the distortion period with polymer relaxation time are qualitatively consistent with experimental observations of the sharkskin instability in linear polyethylenes. The results may also have some relationship to the recent experimental observation of short wavelength instability in plane Couette flow of an entangled solution with wall slip.

Pushing a nonNewtonian fluid in a HeleShaw cell: From fingers to needles
View Description Hide DescriptionWe make a theoretical study of the finger behavior of a simple fluid displacing a nonNewtonian fluid confined in a Hele–Shaw cell. We study the Saffman–Taylor instability when the viscosity of the displaced fluid changes with shear. Our results predict a decrease of the finger width that goes to zero for large values of the velocity. An analytical treatment allows the predictions of the dynamics in radial growth.

Spontaneous thermocapillary interaction of drops, bubbles and particles: Unsteady convective effects at low Peclet numbers
View Description Hide DescriptionMass and heat transfer between two adjacent droplets and the surrounding viscous fluid induce local variations in the surfaceproperties of the drops. These may result in a selfinduced surfaceflow and a subsequent motion of the droplets toward or away from each other. Previous studies of this spontaneous thermocapillary interaction were conducted under the limiting assumptions that inertia, convective effects, and interfacial deformation were negligible. In the present paper the effect of convective transport on the spontaneous interaction of droplets at small nonzero Peclet numbers is examined. It is shown that at large separation distances the motion maintains its quasisteady nature and the correction to the approach velocity is of When the droplets are at closer proximity the temporal changes of the domain are dominant. They result in the appearance of a Basset type history term in the expansion of concentration field and, hence, in the force balance equation. The correction to the approach velocity is of and it depends on the initial position and the evolution in time of the interaction process.

Structure and dynamics of the wake of bubbles and its relevance for bubble interaction
View Description Hide DescriptionThe flow in the wake of single and two interacting air bubbles freely rising in water is studied experimentally using digitalparticleimagevelocimetry in combination with highspeed recording. The experiments focus on ellipsoidal bubbles of diameter of about 0.4–0.8 cm which show spiraling, zigzagging, and rocking motion during their rise in water, which was seeded with small tracer particles for flow visualization. Under counterflow conditions in the vertical channel, the bubbles are retained in the center of the observation region, which allows the wake oscillations and bubble interaction to be observed over several successive periods. By simultaneous diffuse illumination in addition to the light sheet, we were able to record both the path and shape oscillations of the bubble, as well as the wake structure in a horizontal and vertical cross section. The results show that the zigzagging motion is coupled to a regular generation and discharge of alternate oppositely oriented hairpinlike vortex structures. Associated with the wake oscillation, the bubble experiences a strong asymmetric deformation in the equatorial plane at the inversion points of the zigzag path. The zigzag motion is superimposed on a small lateral drift of the bubble, which implies the existence of a net lift force. This is explained by the observed different strength of the hairpin vortices in the zig and zag path; a seemingly familiar phenomenon was found in recent numerical results of the sphere wake flow. For spiraling bubbles the wake is approximately steady to an observer moving with the bubble. It consists of a twisted pair of streamwise vortex filaments which are wound in a helical path and are attached to the bubble base at an asymmetrical position. The minor axis of the bubble is tilted in the tangential plane as well as in the radial plane toward the spiral center. Due to the pressure field induced by the asymmetrically attached wake two components of the lift force exist, one that causes the lateral motion and the other a centripetal force that keeps the bubble on a circular path. A mechanism is proposed to explain the reason for one bubble to spiral or to zigzag. Experiments with two simultaneous released bubbles show that bubble interaction is strongly triggered by the wake dynamics. Once a bubble is captured in the wake of a rocking bubble, it accelerates and rises via successive jumps until they collide. The jumps are explained by the upwards induction effect of the ringlike heads of the hairpin vortices being shed from the leading bubble. The final collision and repulsion thereafter abruptly enlarges the wake for a short moment, which is suggested to be one major contribution to the amplification of turbulence production in bubbly flows.

Coupled Korteweg–de Vries equations describing, to highorder, resonant flow of a fluid over topography
View Description Hide DescriptionThe nearresonant flow of a fluid over a localized topography is examined. The flow is considered in the weakly nonlinear longwave limit and is governed by the forced Korteweg–de Vries (fKdV) equation at first order. It is shown that the unidirectional assumption, of rightmoving waves only, is incompatible with mass conservation at second order. To resolve this incompatibility, a forced coupled KdV system, which allows leftmoving waves, is derived to third order (two orders beyond the fKdV approximation). The secondorder fKdV equation is reformulated as an asymptotically equivalent forced Benjamin–Bona–Mahony (fBBM) equation, as its numerical scheme has superior stability. First and secondorder predictions for the resonant flow of surface waterwaves are compared and the mass flux between the right and leftmoving waves is found. An analytical estimate for the mass flux between the right and leftmoving waves is also derived and good agreement with the numerical solution is obtained.

Inviscid instability of the Batchelor vortex: Absoluteconvective transition and spatial branches
View Description Hide DescriptionThe main objective of the study is to examine the spatiotemporal instability properties of the Batchelor qvortex, as a function of swirl ratio q and external axial flow parameter a. The inviscid dispersion relation between complex axial wave number and frequency is determined by numerical integration of the Howard–Gupta ordinary differential equation. The absoluteconvective nature of the instability is then ascertained by application of the Briggs–Bers zerogroupvelocity criterion. A moderate amount of swirl is found to promote the onset of absolute instability. In the case of wakes, transition from convective to absolute instability always takes place via the helical mode of azimuthal wave number For sufficiently large swirl, coflowing wakes become absolutely unstable. In the case of jets, transition from absolute to convective instability occurs through various helical modes, the transitional azimuthal wave number m being negative but sensitive to increasing swirl. For sufficiently large swirl, weakly coflowing jets become absolutely unstable. These results are in good qualitative and quantitative agreement with those obtained by Delbende et al. through a direct numerical simulation of the linear response. Finally, the spatial (complex axial wave number, real frequency) instability characteristics are illustrated for the case of zeroexternal flow swirling jets.

Nonaxisymmetric flow in a finitelength cylinder with a rotating magnetic field
View Description Hide DescriptionThis paper treats the flow of an electrically conducting liquid in an insulating cylinder with a spatially uniform, transverse, rotating magnetic field. The frequency of the externally applied magnetic field is sufficiently low that this field penetrates throughout the liquid. Previous researchers have treated the steady, axisymmetric flow driven by the azimuthal average of the electromagnetic body force. This paper presents an analytical solution for the nonaxisymmetric flowperturbation driven by the deviation of the body force from its azimuthal average. For the magnetic field strengths and frequencies currently used in crystalgrowth processes, the values of the interaction parameter N and the Reynolds number Re, both based on the frequency of the magnetic field, are small and large, respectively. The results of the appropriate asymptotic analysis show that the ratio of the nonaxisymmetric flow to the axisymmetric one is extremely small, thus validating the common neglect of the nonaxisymmetric flow. Therefore a rotating magnetic field may provide beneficial stirring without disturbing the desirable axisymmetric distribution of additives in the liquid.

Asymptotic analysis of a threedimensional Bridgman furnace at large Rayleigh number
View Description Hide DescriptionWe now know that the operation of a vertical Bridgman furnace, used for solidifying a dilute binary alloy, may be optimized, in the sense of minimizing radial material segregation, by a proper choice of the geometrical parameters of the insulation zone that surrounds the crystalmelt interface. We have found [Phys. Fluids 9, 683 (1997)], as an extension of the work of Tanveer [Phys. Fluids 6, 2270 (1994)], that small surface tension reduces the radial segregation but does not modify that optimization condition. Most previous analyses have been done for axisymmetric ampoules. However, in a laboratory setting, it may not be possible to assure axisymmetry, due to, say, azimuthal variations in the heat transfer or a small tilt of the ampoule axis. We find that nonaxisymmetric effects lead to optimization conditions similar to their axisymmetric counterparts. We include in the analysis geometric (Gibbs–Thomson) and concentration effects in melt temperature; the former effect decreases the segregation, and the latter increases segregation—for either axisymmetric or asymmetric thermal loading. We identify the mechanism for the trend noted by other investigators in prior numerical work: even very small asymmetry causes large increases in crystalline segregation.

On the mode development in the developing region of a plane jet
View Description Hide DescriptionThe development of the mode arrangement in the developing region of an acoustically excited plane jet is extensively studied using hotwire measurements. Two singlewire probes are located within the two shear layers of the jet to detect the flow structure patterns simultaneously. The jet is excited at its fundamental frequency in either varicose mode or sinuous mode condition. Experimental results show that different excitation mode conditions lead to different spreading rates and velocity fluctuation distributions in the developing flow field. The frequency spectral distributions for the excitation exhibit double peaks of and rather than a single subharmonic for the excitation. These three components constitute a relationship of The mode switching phenomenon is found to be prominent under the case, while less pronounced for the case. The mode development is mainly governed by the evolution of the primary instabilities in the jet shear layers.

A PDF model for dispersed particles with inelastic particle–wall collisions
View Description Hide DescriptionA simple model is proposed for the phasespace distribution of dispersed, highinertia particles in a turbulent boundary layer. This model includes a boundary condition describing inelastic particle–wall collisions. Two models for the normal coefficient of restitution are considered. The simpler model treats this coefficient as a constant, and this is shown to lead to a singular distribution of particle velocities at the boundary. A numerical scheme for treating this model is presented, and results from this are compared with those obtained from particle randomwalk simulations. An asymptotic analysis is given for this singular model; the asymptotic behavior of the phasespace distribution shows that the onedimensional steady state distribution exists only for relatively high values of the coefficient of restitution, The more detailed model for the coefficient of restitution treats this coefficient as a function of the normal impact velocity. Numerical results from this model are also given and compared with those obtained by simulation. For both models the particle number density and particle fluctuation energy at the wall, required to formulate boundary conditions for the “macroscopic” twofluid models, are calculated as functions of model parameters. The results also illustrate the phenomenon of particle segregation towards the wall in turbulent gasparticulate suspensions, i.e., the formation of the nearwall dense layer of particles; the thickness of this layer is also determined as a function of model parameters.

Scaling laws for the turbulent mixing of a passive scalar in the wake of a cylinder
View Description Hide DescriptionTurbulence is triggered by an air flow passing around a cylinder and heated by an array of hot wires parallel to the cylinder axis. Simultaneous velocity and passive temperature fluctuations are measured locally on the centerline of the cylinder wake. The focus is on the scaling properties of the mixed scalarvelocity structure functions It is observed that displays the same scaling behavior as over an extended range of scales. This yields accurate measurements for the relative scaling exponents It is further observed that the moments of the coarsegrained scalar energy diffusion rate satisfy a closure condition as proposed by She and Lévêque [Phys. Rev. Lett. 72, 336 (1994)]. Otherwise it is proposed that scales as This generalized similarity relation is well supported by our measurements over a range of scales which greatly exceeds the inertial range. A theoretical prediction is finally proposed, in good agreement with our experimental data.

Random forcing of threedimensional homogeneous turbulence
View Description Hide DescriptionA method of using a fully random force to numerically generate statistically stationary homogeneous turbulence has been developed. The forcing is implemented in spectral space where it is concentrated at small wave numbers. Hence, the power input is introduced into the flow at large scales. The randomness in time makes the force neutral, in the sense that it does not directly correlate with any of the time scales of the turbulent flow, and it also makes the power input determined solely by the forceforce correlation. This means that it is possible to generate different desirable turbulence states, such as axisymmetric turbulence, where the degree of anisotropy of the forcing can be chosen a priori through forcing parameters. In particular, the total amount of power input from the forcing can be set to balance a desired dissipation at a statistically stationary state. In order to only get a contribution from the forceforce correlation to the input power in the discrete equations, the force is determined so that the velocityforce correlation vanishes for each Fourier mode. In direct numerical simulations (DNS) of forced isotropic turbulence, universality of the small scales is shown for the kinetic energy spectrum at different Reynolds numbers and the velocity derivative skewness obtains the value The forcing method is used in a large eddy simulation(LES), where it is compared with a simulation of decaying turbulence to show the importance of having a statistically stationary flow if well known inertial laws are to be recovered at moderate Reynolds numbers.

The nonlinear equation for curved flames applied to the problem of flames in cylindrical tubes
View Description Hide DescriptionThe nonlinear equation for curved stationary flames of realistic expansion coefficients is solved numerically for the problem of flame propagation in cylindrical tubes. Two different configurations of a flame front corresponding to convex and concave flames are obtained. The convex and concave flames propagate with different velocities that depend on the tube radius and on the expansion coefficient of the burning matter. For tubes of a moderate radius the velocity amplification for convex flames exceeds the respective velocity amplification of twodimensional flames almost twice. For tubes of a large radius unlimited increase of the curved flame velocity with increase of the tube width takes place. The obtained theoretical results are in good quantitative agreement with the results of numerical experiments on flame dynamics in cylindrical tubes.

Conditional moment closure for large eddy simulation of nonpremixed turbulent reacting flows
View Description Hide DescriptionA method for closing the chemical source terms in the filtered governing equations of motion is proposed. Conditional filtered means of quantities appearing in the chemical reaction rate expressions are approximated by assuming that these conditional filtered means are constant for some ensemble of points in the resolved flow field; for such an ensemble, integral equations can be solved for the conditional filtered means. These conditional filtered means are then used to approximate the conditional filtered mean of the chemical source term by invoking the Conditional Moment Closure hypothesis. The filtered means of the chemical source terms are obtained by integrating their conditional filtered means over the filtered density function of the conditioning variable(s). The method is applied to direct numerical simulation results to directly compare the prediction of the reaction rates with the actual filtered reaction rates. The results of this a priori test appear to show that the method is capable of predicting the filtered reaction rates with adequate accuracy—even in the presence of heat release, and local extinction phenomena. This is especially true for predictions obtained using two conditioning variables.

Dynamic molecular collision (DMC) model for rarefied gas flow simulations by the DSMC method
View Description Hide DescriptionThe Dynamic Molecular Collision (DMC) model is constructed for accurate and realistic simulations of rarefied gas flows of nonpolar diatomic molecules by the Direct Simulation Monte Carlo (DSMC) method. This model is applicable for moderate temperatures (up to a few hundred K for nitrogen), where most molecules are in the vibrational ground state and the vibrational degree of freedom can be neglected. In this range, moreover, the rotational energy can be considered as a continuous one. The collisions of diatomic molecules are simulated many times by the Molecular Dynamics (MD) method at various initial conditions. The site to site potential is used as an intermolecular one. The collision cross section is developed from the database obtained by MD simulation and kinetic theory of viscosity coefficient of diatomic molecules. The probability density function of energy after collision is also developed using the database. In order to verify the DMC model, two flow fields are simulated. First, the DMC model is applied to the simulation of the translational and rotational energy distribution at the equilibrium condition and the results are compared with the Maxwell distribution. The results agree very well with each other. Second, the DMC model is applied to the simulation of the rotational relaxation through low and high Mach number normal shock wave. These results also agree very well with the experimental results of Robben and Talbot, although the upstream rotational temperature is a little lower.
