Volume 12, Issue 8, August 2000
 LETTERS


A linear process in wallbounded turbulent shear flows
View Description Hide DescriptionA linear process in wallbounded turbulentshear flows has been investigated through numerical experiments. It is shown that the linear coupling term, which enhances nonnormality of the linearized Navier–Stokes system, plays an important role in fully turbulent—and hence, nonlinear —flows. Nearwall turbulence is shown to decay without the linear coupling term. It is also shown that nearwall turbulence structures are not formed in their proper scales without the nonlinear terms in the Navier–Stokes equations, thus indicating that the formation of the commonly observed nearwall turbulence structures are essentially nonlinear, but the maintenance relies on the linear process. Other implications of the linear process are also discussed.
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 ARTICLES


Steady freesurface thin film flows over topography
View Description Hide DescriptionWe consider the slow motion of a thin viscous film flowing over a topographical feature (trench or mound) under the action of an external body force. Using the lubrication approximation, the equations of motion simplify to a single nonlinear partial differential equation for the evolution of the free surface in time and space. It is shown that the problem is governed by three dimensionless parameters corresponding to the feature depth, feature width and feature steepness. Quasisteady solutions for the free surface are reported for a wide range of these parameters. Our computations reveal that the free surface develops a ridge right before the entrance to the trench or exit from the mound and that this ridge can become large for steep substrate features of significant depth. Such capillary ridges have also been observed in the contact line motion over a planar substrate where the buildup of pressure near the contact line is responsible for the ridge. For flow over topography, the ridge formation is a manifestation of the effect of the capillary pressure gradient induced by the substrate curvature. In addition, the minimum film thickness is always found near the concave corner of the feature. Both the height of the ridge and the minimum film thickness are found to be strongly dependent on both the profile depth and steepness. Finally, it is found that either finite feature width or a significant vertical component of gravity can suppress these effects in a way that is made quantitative and which allows the operative physical mechanism to be explained.

Circulating flows inside a drop under timeperiodic nonuniform electric fields
View Description Hide DescriptionThe circulating flows formed inside a spherical drop under timeperiodic nonuniform electric fields are considered. For simplicity, it is assumed that there are axisymmetric electric fields and that the flow fields are in the Stokes flow regime. An analytical solution of the streamfunction distribution inside and outside the drop is obtained. The flow field is found to be dependent on the frequency of the timeperiodic electric field and the ratios of the material properties such as the viscosity, the electrical conductivity, and the electrical permittivity. As part of the solution, an analytical expression of the dielectrophoretic migration velocity of a drop under a timeperiodic electric field is also obtained. The result shows an interesting physics—that dielectrophoretic migration is possible in a timeperiodic electric field even in the situation where dielectrophoresis would be impossible in a static electric field. By using the analytical solution of the streamfunction, fluid mixing inside a drop is analyzed based on the Poincaré maps. The mass transfer enhancement factor due to fluid mixing has also been computed by solving the unsteady mass transfer equation numerically. The existence of an optimal frequency has been confirmed as in other mass transfer enhancement processes by timeperiodic forcing.

Motion of three vortices near collapse
View Description Hide DescriptionA system of three point vortices in an unbounded plane has a special family of selfsimilarly contracting or expanding solutions: during the motion, the vortex triangle remains similar to the original one, while its area decreases (grows) at a constant rate. A contracting configuration brings three vortices to a single point in a finite time; this phenomenon known as vortex collapse is of principal importance for manyvortex systems. Dynamics of closetocollapse vortex configurations depends on the way the collapse conditions are violated. Using an effective potential representation, a detailed quantitative analysis of all the different types of nearcollapse dynamics is performed when two of the vortices are identical. We discuss time and length scales, emerging in the problem, and their behavior as the initial vortex triangle is approaching an exact collapse configuration. Different types of critical behaviors, such as logarithmic or powerlaw divergences are exhibited, which emphasize the importance of the way the collapse is approached. Period asymptotics for all singular cases are presented as functions of the initial vortice’s configurations. Special features of passive particle mixing by nearcollapse flows are illustrated numerically.

Vorticity constraints on a fluid/fluid interface
View Description Hide DescriptionGeneral relations among the components of the strain rate tensors and those of the tangential vorticities on the two sides of a liquid/gas interface are derived; kinematic constraints as well as the tangentialstress balance at the interface are used. For small gas to liquid dynamic viscosity ratios compared to unity simple expressions relating the liquid tangential vorticity components to the tangential velocity component perpendicular to them, the interface curvatures and the normal velocity surfacegradient components are obtained. Starting from the customary Eulerian vorticity equation, a transport equation for the vortex sheet strength is obtained.

A vorticity dynamics theory of threedimensional flow separation
View Description Hide DescriptionA theory of threedimensional incompressible flow separation is presented in terms of the onwall signatures of the flow. Some longstanding controversial issues are revisited and answers given, such as the inconsistency of the separation criteria based on the topological theory and “open separation,” and whether a separation line is an asymptote or envelope of neighboring skinfriction lines. General criteria for identifying an “open” or “closed” flow separation zone and separation line (including the initial point of the latter), steady and unsteady, are obtained, which apply to a generic smooth curved wall at any Reynolds numbers. The criteria are found to be most clearly given in terms of onwall signatures of vorticitydynamics. These are then specified to steady boundary layer separation at large Reynolds numbers. A scale analysis under mild assumptions leads to a threedimensional tripledeck structure near a generic boundary layer separation line. Criteria are presented for “separation watch,” which tells that a boundarylayer may soon break away, and for “separation warning,” which identifies the vorticity characteristics in an already formed boundarylayer separation zone and along a boundarylayer separation line. Flow behavior and its dependence on outer flow conditions are examined qualitatively. A numerical example is given which confirms the predictions of the theory.

Energy dissipation in a shear layer with suction
View Description Hide DescriptionThe rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. The injection and suction rates are maintained constant and equal and this leads to solutions with constant throughput. For strong enough suction, expressed in terms of the entry angle between the injection velocity and the boundaries, a steady laminar flow is nonlinearly stable for all Reynolds numbers. For a narrow range of small but nonzero angles, the laminar flow is linearly unstable at high Reynolds numbers. The upper bound on the energy dissipation rate—valid even for turbulent solutions of the Navier–Stokes equations—scales with viscosity in the same way as the laminar dissipation in the vanishing viscosity limit. For both the laminar and turbulent flows, the energy dissipation rate becomes independent of the viscosity for high Reynolds numbers. Hence the laminar energy dissipation rate and the largest possible turbulent energy dissipation rate for flows in this geometry differ by only a prefactor that depends only on the angle of entry.

Mode coalescence in a twofluid boundarylayer stability problem
View Description Hide DescriptionThe tripledeck analysis of a twofluid boundary layer stability at high Reynolds numbers in Timoshin [J. Fluid Mech. 353, 163 (1997)] is extended to include broader parameter variations, most notably variations in the scaled film thickness. The two underlying instabilities, the Tollmien–Schlichting and interfacial waves, are shown to have points of mode coalescence located in the stable or unstable part of the spectrum. An interpretation of the instability mechanisms operational in twofluid tripledeck flows is proposed in terms of the disturbance vorticity equation.

Zeroabsolutevorticity state in a rotating turbulent shear flow
View Description Hide DescriptionStability characteristics of a rotating simple shear flow modulated periodically in the shear direction is investigated by direct numerical simulation. The temporal evolution and the final profile of the mean flow are examined, starting with a random perturbation imposed on it under the assumption that the velocity and pressure fields are uniform in the mean flow direction. It is found that the mean velocity profile changes due to the shearCoriolis instability. If there is a locally unstable region in a stable rotating simple shear, the mean velocity profile there is deformed into a linear one with nearlyzero absolute vorticity. An analogy holds between rotating uniformly sheared turbulence and thermally convective turbulence that the region of nearlyzero absolute vorticity in the former corresponds to that of nearlyuniform temperature in the latter.

Spinup in a rectangular container with an internal cylindrical obstacle
View Description Hide DescriptionThis paper describes a study of the spinup of a freesurface fluid in a rectangular container in which an internal cylindrical obstacle is mounted. Laboratory experiments have been carried out for a variety of obstacle positions. It was found that the flow evolution during the adjustment process leading to the final state of solidbody rotation is crucially dependent on the obstacle position. As found in previous studies, in the absence of any obstacle the relative flow becomes organized in a domainfilling regular cellular pattern, upon which it decays according to the wellknown spinup mechanism provided by the Ekman layer at the tank bottom. Since the obstacle acts both as a barrier and as a source of viscously produced wall vorticity, the formation of the cell pattern is in most cases drastically influenced (either impeded or promoted) by the solid obstacle. Theoretical predictions of the flow pattern in the starting stage agree very well with the laboratory observations.

On the decay of homogeneous isotropic turbulence
View Description Hide DescriptionDecaying homogeneous, isotropic turbulence is investigated using a phenomenological model based on the threedimensional turbulentenergy spectra. We generalize the approach first used by ComteBellot and Corrsin [J. Fluid Mech. 25, 657 (1966)] and revised by Saffman [J. Fluid Mech. 27, 581 (1967); Phys. Fluids 10, 1349 (1967)]. At small wave numbers we assume the spectral energy is proportional to the wave number to an arbitrary power. The specific case of power 2, which follows from the Saffman invariant, is discussed in detail and is later shown to best describe experimental data. For the spectral energy density in the inertial range we apply both the Kolmogorov −5/3 law, and the refined Kolmogorov law by taking into account intermittency. We show that intermittency affects the energy decay mainly by shifting the position of the virtual origin rather than altering the power law of the energy decay. Additionally, the spectrum is naturally truncated due to the size of the wind tunnel test section, as eddies larger than the physical size of the system cannot exist. We discuss effects associated with the energycontaining length scale saturating at the size of the test section and predict a change in the power law decay of both energy and vorticity. To incorporate viscous corrections to the model, we truncate the spectrum at an effective Kolmogorov wave number where γ is a dimensionless parameter of order unity. We show that as the turbulence decays, viscous corrections gradually become more important and a simple power law can no longer describe the decay. We discuss the final period of decay within the framework of our model, and show that care must be taken to distinguish between the final period of decay and the change of the character of decay due to the saturation of the energy containing length scale. The model is applied to a number of experiments on decaying turbulence. These include the downstream decay of turbulence in wind tunnels and a water channel, the temporal decay of turbulence created by an oscillating grid in water and the decay of energy and vorticity created by a towed grid in a stationary sample of water. We also analyze decaying vorticity data we obtained in superfluid helium and show that decaying superfluidturbulence can be described classically. This paper offers a unified investigation of decaying isotropic, homogeneous turbulence that is based on accepted forms of the threedimensional turbulent spectra and a variety of experimental decay data obtained in air, water, and superfluid helium.

Long time mass fraction statistics in stationary compressible isotropic turbulence at supercritical pressure
View Description Hide DescriptionDirect numerical simulations are used to investigate the “long time” distribution of mass fraction fluctuations in stationary compressible isotropic turbulent binary nitrogen–hydrocarbon mixtures under supercritical pressure conditions. The governing equations are the compressible Navier–Stokes equations together with the cubic Peng–Robinson real gas state equation, and generalized heat and mass diffusion derived from nonequilibrium thermodynamics. A highly efficient procedure is presented which allows for the solution of all thermodynamic quantities without iterations or interpolation tables. The simulations consider equal mass binary mixtures of various combinations of nitrogen, heptane, dodecane, and 3methylhexane, having molecular weight ratios in the range It is shown that temperature and pressuregradientinduced Soret mass diffusion results in statistically stationary mass fraction distributions at long times. The results reveal that the mass diffusion term due to the pressure gradient acts as a production mechanism in the Favre averaged scalar variance transport equation, and is balanced by Fickian dissipation to produce the stationary states. The resulting scalar probability density function is characterized by a larger than Gaussian flatness factor, and is asymmetric due to the mass fraction dependence of the partial molar volume. The stationary scalar variance amplitude increases both with increasing turbulenceMach number and molecular weight ratio of the species, but is inversely related to the turbulence Reynolds number. The scalar energy spectra exhibit peak values at wave numbers corresponding to the peak in the velocity dissipation spectra.

Eulerian acceleration statistics as a discriminator between Lagrangian stochastic models in uniform shear flow
View Description Hide DescriptionDirect numerical simulation (DNS) calculations of Eulerian acceleration statistics for homogeneous turbulence in uniform shear flow are used to test the closures implied by two different Lagrangianstochastic models for turbulent dispersion. These different models, due to Thomson [J. Fluid Mech. 180, 529 (1987)] and Borgas [Preprints of the Eighth Symposium on Turbulence and Diffusion (American Meteorological Society, Boston, 1988), p. 96], are representative of the range of a class of models which are quadratic in the velocity and are both consistent with the Eulerian velocity statistics which characterize the flow. This is the socalled nonuniqueness problem; Eulerian velocity statistics are not sufficient to uniquely define a Lagrangianstochastic model. We show here that these two models give different Eulerian acceleration statistics, which thus serve to discriminate between the models. The drift term in these stochastic models is related to the mean of the acceleration conditioned on the velocity, enabling related joint statistics of the velocity and the acceleration, such as their covariance and their cross product, to be determined. The model closures represent these joint statistics in terms of the mean shear, the Reynolds stress tensor, and its rate of change. Differences between the two models show up in the direct contribution of the mean shear to offdiagonal components of the conditional mean acceleration and the acceleration–velocity covariance in the shear plane, and in the mean rate of rotation of the velocity vector in the shear plane. In particular, Thomson’s model allocates the direct shear contribution to the correct component of the acceleration–velocity covariance in the shear plane, whereas Borgas’s model does not. Other components are identical in the two models. Overall, Thomson’s model represents the DNS results very well. However, the relatively small deviations from Thomson’s model are real and these are reflected in the fact that the mean rotation of the velocity vector has a nonzero contribution from terms that are not closed in terms of the mean flow, the Reynolds stress, and its rate of change.

The inviscid impingement of a jet with arbitrary velocity profile
View Description Hide DescriptionAccurate determination of wall shear stress and heat and mass transfer rates under an impinging jet requires careful analysis of the boundary layer at the impingement surface due to the large pressure gradients near the stagnation point. Modeling the inviscid flow just outside the boundary layer provides the boundary conditions necessary for such an analysis. Previous inviscid models have considered only a small subset of possible jet velocity profiles and with limited spatial resolution. In the present work, analytical solutions to the streamvorticity equation for twodimensional and axisymmetric impingement flow with arbitrary velocity profile are found in terms of a surface integral involving the vorticity function, allowing an iterative determination of the stream function throughout the impingement region. Surface pressure distributions and streamline plots are calculated for various impinging jet configurations, including plane, round, and annular jet nozzles. The calculations show excellent agreement with previous experimental and numerical results, while requiring relatively short computation times. Flow predictions are also made for impinging jet configurations for which no previous data or calculations exist.

Eulerian space–time correlations in turbulent shear flows
View Description Hide DescriptionThis paper is concerned with the generic form of space–time correlations of instantaneous velocity fluctuations in turbulentshear flows. The study has as its basis the Kovasznay–Corrsin conjecture modified to account for inhomogeneities. Details of the modifications were dictated by comparison with direct numerical simulations of turbulent channel flow. For example, analysis of the simulations shows that the space–time correlations at optimal delay depict, when appropriately normalized, a form independent of turbulence component. It is also evident that the half width of the correlation grows, in the outer region of the layer, like while the Lagrangian time scale grows like where and is the mean convection velocity. The resulting generic form for the space–time correlation captures these effects well and brings to light a Reynolds number effect that is most evident in the tail of the correlations at nonoptimal delay.

Porescale simulation of dispersion
View Description Hide DescriptionTracer dispersion has been simulated in threedimensional models of regular and random sphere packings for a range of Peclet numbers. A randomwalk particletracking (PT) method was used to simulate tracer movement within porescale flow fields computed with the latticeBoltzmann (LB) method. The simulation results illustrate the time evolution of dispersion, and they corroborate a number of theoretical and empirical results for the scaling of asymptotic longitudinal and transverse dispersion with Peclet number. Comparisons with nuclear magnetic resonance(NMR)spectroscopy experiments show agreement on transient, as well as asymptotic, dispersion rates. These results support both NMR findings that longitudinal dispersion rates are significantly lower than reported in earlier experimental literature, as well as the fact that asymptotic rates are observed in relatively short times by techniques that employ a uniform initial distribution of tracers, like NMR.

On the prediction of gas–solid flows with twoway coupling using large eddy simulation
View Description Hide DescriptionThe purpose of this paper is to examine the feasibility of large eddy simulation(LES) for predicting gas–solid flows in which the carrier flowturbulence is modified by momentum exchange with particles. Several a priori tests of subgridscale (SGS) turbulencemodels are conducted utilizing results from direct numerical simulation (DNS) of a forced homogeneous isotropic turbulent flow with the back effect of the particles modeled using the pointforce approximation. Properties of the subgridscale field are computed by applying Gaussian filters to the DNS database. Similar to the behavior observed in singlephase flows,a priori test results show that, while the local energy flux is inaccurately estimated, the overall SGS dissipation is reasonably predicted using the conventional Smagorinsky model and underestimated using the Bardina scalesimilarity model. Very good agreement between model predictions and DNS results are measured using closures whose coefficients are computed using the resolved field, the socalled dynamic subgrid models, with the mixed model yielding more accurate predictions than the dynamic Smagorinsky model.A priori test results are then confirmed in actual LES calculations used to investigate the sensitivity of the predictions to mesh refinement. The LES was performed at infinite turbulentReynolds number and for a range of particle response times and mass loadings. Grid resolution in the LES was varied from to collocation points, with particle sample sizes of for each response time. LES predictions of the flow with twoway coupling are independent of mesh refinement when using the dynamic mixed model and when the particle relaxation time becomes larger than the characteristic time scale of the unresolved fluid turbulent field.

Surface topology of a buoyant turbulent nonpremixed flame
View Description Hide DescriptionThe temporal evolution of the threedimensional stoichiometric mixture fraction isosurface is investigated using direct numerical simulations of a turbulent nonpremixed flame with and without buoyancy. After an initial transient, the surface area of the isosurface increases monotonically in time. The rate of area increase of the buoyant flame is larger than that of the nonbuoyant flame. The stretch rate of the buoyant flamesurface indicates that the tangential strain rate is dominant and positive at the troughs, whereas the relative propagation velocity term, is dominant and negative at the crests of the isosurface. Thus, the local surface area of the crests decreases in time while that of the troughs increases, leading to the steep ridge topology. The strain rate field generated by the oppositely signed vortices saddling the isosurface is responsible for this topology. The reaction rate in the buoyant flame is largest at the troughs where the scalar dissipation is maximum.

Vortexdynamics model for entrainment in jets and plumes
View Description Hide DescriptionRecent experimental results of Bhat and Narasimha (1996) have revealed a dramatic difference in the entrainment between jets and plumes subjected to offsource volumetric heating and their unheated counterparts. Experimental observations show that plumes entrain more rapidly than jets; the greater entrainment by the plume is typically attributed to the presence of buoyancy in the plume. In contrast, the addition of buoyancy away from the source by volumetric heating produces the opposite effect of reduced entrainment. Apart from buoyancy, other factors such as acceleration due to pressure gradients or other body forces can also affect the rate of entrainment. In this paper, we develop a model for entrainment to explain the mechanism by which buoyancy produces contrasting effects on entrainment in volumetrically heated flows in comparison to their unheated counterparts. The model highlights the role of density stratification in the process of vortexsheet rollup in free shear flows. With this model, we are also able to explain the higher entrainment of the plume relative to the unheated jet. The model is further extended to explain entrainment behavior during acceleration due to an applied pressure gradient or other body forces.

The latetime development of the Richtmyer–Meshkov instability
View Description Hide DescriptionMeasurements have been made of the growth by the Richtmyer–Meshkov instability of nominally singlescale perturbations on an air/sulfur hexafluoride interface in a large shock tube. An approximately sinusoidal shape is given to the interface by a wire mesh which supports a polymeric membrane separating the air from the A single shock wave incident on the interface induces motion by the baroclinic mechanism of vorticity generation. The visual thickness δ of the interface is measured from schlieren photographs obtained singly in each run and in highspeed motion pictures. Data are presented for δ at times considerably larger than previously reported, and they are tested for selfsimilarity including independence of initial conditions. Four different initial amplitude/wavelength combinations at one incident shock strength are used to determine the scaling of the data. It is found that the growth rate decreases rapidly with time, (i.e., where and that a small dependence on the initial wavelength persists to large time. The larger value of the power law exponent agrees with the result of the latetimedecay similarity law of Huang and Leonard [Phys. Fluids 6, 3765–3775 (1994)]. The influence of the wire mesh and membrane on the mixing process is assessed.
