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Volume 11, Issue 3, March 1999
 LETTERS


Particle segregation in monodisperse sheared suspensions
View Description Hide DescriptionIt has been known for a long time that many mixtures of granular materials tend to segregate when tumbled in a rotating horizontal cylinder, with the different components separating into bands of relatively pure single concentration along the rotational axis [Mixing of Solids, Advances in Chemical Eng., edited by T. B. Drew and J. W. Hoopes (Academic Press, New York, 1952), Vol. 2, p. 211]. Here we report a phenomenon that seems to be analogous, but in suspensions of monodisperse neutrally buoyant spherical particles in a Newtonian liquid medium being sheared in a partially filled horizontal Couette device in which the suspension separates itself into alternating regions of high and low particle concentration along the length of the tube. The experiment is mostly qualitative, the aim at this stage being primarily to provide photographic evidence of a curious and as yet unexplained phenomenon.

Correction to the fourfifths law due to variations of the dissipation
View Description Hide DescriptionThe full Kolmogorov equation is analyzed with respect to Galilean invariance. It is shown that the timedependent term in typical applications as jets and wakes will cause rather large deviations from the fourfifths law, even at fairly large Reynolds numbers. With the use of the model an inertial range model expression is derived for the thirdorder structure function of freely decaying turbulence at finite Reynolds number. The expression is compared with results from previously reported experiments. It is argued that experiments to test intermittency corrections to the Kolmogorov 1941 theory should be made in flows which are both stationary and homogeneous in the mean flow direction, such as pipe and channel flows.

 ARTICLES


Inverse equations
View Description Hide DescriptionFollowing previous work by Keller [“Inverse Euler equations,” Z. Angew. Math. Phys. 49, 363 (1998)], that is extended to nonconservative flows, the general timeindependent flow equations are first written in a perfectly antisymmetric form, using a pair of stream functions as the dependent variables. In a second step the equations are written in an inverse form, using the two stream functions and the natural coordinate as independent variables. The special cases of incompressible flow and inviscid axisymmetric flow are also considered. The main advantage of using these inverse equations is associated with the possibilities of using static pressure distributions, mach number distributions, geometric constraints, etc., or any combination of geometric constraints and specifications of physical quantities to define the boundary conditions. In contrast to conventional inverse methods, that are based on iterative approximations to a desired pressure distribution along the surface of a flow device, for example, the use of inverse equations offers the possibility to arrive at the solution for any kind of boundary conditions in a single step. Furthermore, there is no need for complicated grid generation procedures, because the domain of definition in inverse space is typically a cube with Cartesian coordinates.

Formation of cusp on the free surface at low Reynolds number flow
View Description Hide DescriptionFree surface deformation and cusp formation are analyzed by considering a viscousflow due to a vortex dipole immersed beneath the free surface, with the direction and strength of the vortex dipole arbitrary. In the analysis, the Stokes’ approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex variable techniques. From the solution, deformations of the free surface are shown and the formation of a cusp on the free surface are discussed. As the capillary number grows, it is known that the radius of curvature at the converging point on the free surface becomes extremely small, which means that in a real fluid a cusp should form on the free surface above some critical capillary number. For a particular range of vortex dipole inclination, however, there exists no cusp, even for infinite capillary numbers. Typically, streamline patterns for some capillary numbers of horizontal vortex dipole are also shown.

Stability of the axisymmetric buoyantcapillary flows in a laterally heated liquid bridge
View Description Hide DescriptionThe axisymmetric steadystates solutions of buoyantcapillary flows in a cylindrical liquid bridge are calculated by means of a pseudospectral method. The free surface is undeformable and laterally heated. The working fluid is a liquid metal, with a Prandtl number value Particular care was taken to preserve the physical regularity in our model, by writing appropriate flux boundary conditions. The location and nature of the bifurcations undergone by the flows are investigated in the space of the dimensionless numbers (Marangoni, Rayleigh, . Saddlenode and Hopf bifurcations are found. By analyzing the steady state structures and the energy budgets, the saddlenode bifurcations are observed to play a determinant role. Only two sets of stable steadystates, connected by saddlenodes, are allowed by the coupling of buoyancy and capillarity. Most of the solutions of the explored part of the (Ma, Ra) plane belong to these states.

Scaling laws in granular flows down rough inclined planes
View Description Hide DescriptionIn this paper, new scaling properties for granular flows down rough inclined planes are presented. In the dense steady uniform flow regime, we have systematically measured the mean velocity of the flow as a function of the inclination of the surface θ and of the thickness h of the layer. The results obtained for different systems of beads corresponding to different surface roughness conditions are shown to collapse into a single curve when properly scaled. The scaling is based on the measurement of the minimum thickness necessary to observe a steady uniform flow at inclination θ. From this experimental observation an empirical description for granular flows down inclined planes is proposed in terms of a dynamic friction coefficient.

Double diffusive instability in an inclined cavity
View Description Hide DescriptionDouble diffusive convection in a rectangular twodimensional cavity with imposed temperatures and concentrations along two opposite sidewalls is considered. The study is performed for twodimensional cavities in which the thermal and solutal buoyancy forces have the same magnitude, but are of opposite sign. The influence on the convective instability of the aspect ratio A (height/length) of the cavity and the cavity inclination α with respect to gravity is discussed. The onset of convection is computed for an infinite layer and compared to that for bounded boxes. The study is completed by the continuation of bifurcating solutions. It is found that, due to centrosymmetry, steady bifurcations are either pitchfork or transcritical depending on A and α. However, a primary pitchfork bifurcation is found to create unstable steady solutions, even if it is the first bifurcation. For the aspect ratios we studied, and close to the onset of convection, the stable solutions are mainly oneroll structures that can be destabilized by further interactions with asymmetric solutions created at primary pitchfork bifurcations. For large aspect ratios, additional asymmetric oneroll solutions are created via more complex branch interactions.

Threedimensional Floquet instability of the wake of square cylinder
View Description Hide DescriptionIn this study we investigate the onset of threedimensionality in an otherwise twodimensional periodic wake of a square cylinder. Floquet stability analysis is employed to extract the different modes of threedimensional instabilities. It is observed that the threedimensional transition process for a square cylinder is similar to that of a circular cylinder. Most notably, there is a longwavelength (mode A) threedimensional disturbance that becomes unstable first at a Reynolds number of about 161, followed by a shortwavelength (mode B) threedimensional disturbance that becomes unstable at a Reynolds number of about 190. In addition, a third intermediatewavelength mode is also observed to become unstable at around Re=200. Unlike modes A and B, the intermediatewavelength mode is subharmonic with a period of twice the shedding period of the twodimensional base state. This mode also breaks the reflection translation symmetry observed in the other two modes and as a result appears with multiplicity two. The space–time symmetries of the three modes are explored in detail.

Stability characteristics of wavy walled channel flows
View Description Hide DescriptionThe linear temporal stability characteristics of converging–diverging, symmetric wavy walled channel flows are numerically investigated in this paper. The basic flow in the problem is a superposition of plane channel flow and periodic flow components arising due to the small amplitude sinusoidal waviness of the channel walls. The disturbance equations are derived within the frame work of Floquet theory and solved using the spectral collocation method. Twodimensional stability calculations indicate the presence of fast growing unstable modes that arise due to the waviness of the walls. Neutral stability calculations are performed in the disturbance wavenumber–Reynolds number plane, for the wavy channel with wavenumber and the wall amplitude to semichannel height ratio, up to 0.1. It is also shown that the twodimensional wavy channel flows can be modulated by a suitable frequency of wall excitation thereby stabilizing the flow.

Interactions of three components and subcritical selfsustained amplification of disturbances in plane Poiseuille flow
View Description Hide DescriptionA lowdimensional nonlinear model for the normal velocity and normal vorticity (η) disturbance development in plane Poiseuille flow is studied. The study is restricted to the interactions of a pair of oblique components of the form and the component of the form where α and β are streamwise and spanwise wave numbers, respectively. The disturbances considered are also assumed to be highly elongated in the streamwise direction. Owing to the nonnormal properties of the basic equations, the η disturbance is first transiently amplified. Then, if the Reynolds number (R) and the initial disturbance are sufficiently large, the nonlinear interactions lead to a selfsustained process of disturbance amplification at subcritical R. For large R the threshold disturbance amplitude scales like The results also strongly indicate that the nonlinear feedback from η to is crucial for the establishment of the selfsustained process.

On the decay rate of isotropic turbulence laden with microparticles
View Description Hide DescriptionThis paper is concerned with the twoway coupling effects on the decay rate of isotropic turbulence laden with solid spherical microparticles whose response time, is much smaller than the Kolmogorov time scale The particles volumetric concentration, is small enough to neglect particle–particle interactions, and the material density of the particle is much larger than the fluid density We obtain asymptotic analytical solutions for the instantaneous particle velocity and kinetic energy spectrum, in the limit which indicate that the twoway coupling increases the fluid inertia term in the fluid momentum equation by the factor Consequently, the highwave number components of the spectra of turbulence energy and dissipation develop in time as The net result is a reduction of the decay rate of turbulence energy compared to that of particlefree turbulence (i.e., the oneway coupling case where . We also perform direct numerical simulation (DNS) of isotropic turbulence laden with microparticles, using the twofluid (TF or Eulerian–Eulerian) approach developed in an earlier study [Druzhinin and Elghobashi, Phys. Fluids 3 (1998)]. Excellent agreement is achieved between the DNS results and the analytical solution for the particle kinetic energy spectrum. The DNS results show that the twoway coupling reduces the decay rate of turbulence energy compared to that of oneway coupling. In addition, we compare the temporal developments of the turbulence kinetic energy and its dissipation rate, obtained from DNS using TF, with those from DNS using the trajectory (Eulerian–Lagrangian) approach. Satisfactory agreement is achieved between the two approaches, with TF requiring considerably less computational time.

Decaying twodimensional turbulence in square containers with noslip or stressfree boundaries
View Description Hide DescriptionWe report results of direct numerical simulations of decaying twodimensional (2D) turbulence inside a square container with rigid boundaries. It is shown that the type of boundary condition (noslip or stressfree) determines the flow evolution essentially. During the initial and intermediate stages of decaying 2D turbulence is comparable with an eddy turnover time, Re is the Reynolds number of the flow), the decay scenario for simulations with noslip boundary conditions can be understood from turbulent spectral transfer and selective decay. A third mechanism can be recognized for A decay stage where diffusion dominates over nonlinear advection, i.e., spectral transfer is then absent in favor of selfsimilar decay. The present results show that at presently accessible Reynolds numbers and computation times, laboratory experiments cannot be accurately compared with quasistationary states from ideal maximumentropy theories or with computed solutions of flows in containers with stressfree boundaries. The decay which results in rectangular containers with noslip boundaries does not yet yield anything that is meaningfully comparable with these formulations. The evolution of the number of vortices the average vortex radius a, the ratio of enstrophy Ω over energy E, and the extremum of vorticity (normalized by ) have been computed based on ensemble averaging of the noslip runs. An algebraic regime has been observed with and Finally, quantities such as a measure of the viscous stresses near the boundaries have been computed in order to analyze the decay of 2D turbulence in containers with rigid boundaries.

Turbulent force as a diffusive field with vortical sources
View Description Hide DescriptionIn Reynoldsaverage Navier–Stokes equation it is the divergence of Reynolds stresstensor, i.e., the turbulent force, rather than the tensor itself, is to be simulated and partially modeled. Thus, directly working on turbulent force could bring significant simplification. In this paper a novel exact equation for incompressible turbulent force f is derived: where ν is the molecular viscosity and all source terms in tensorS to be modeled are vortical. The dominant mechanism is the advection and stretching (with an opposite sign) of a “pseudoLamb vector” by fluctuating velocity field. No coupling with pressure is involved. The equation follows from a study of the mean fluctuating Lamb vector and kinetic energy, which constitute the turbulent force. Both constituents are governed by the same kind of equations as f. This innovative turbulentforce equation is similar to Lighthill’s acoustic analogy and naturally calls one’s attention to studying the vortical sources of turbulent force. The methodology described here may lead to turbulencemodels which provide more complete treatment than that of twoequation models, but relatively easier computation than that of secondorder closures.

Some new features of the passive scalar mixing in a turbulent flow
View Description Hide DescriptionWe analyze experimentally the statistical properties of a turbulent mixing created in the gap between two counterrotating disks at a Taylor Reynolds number. Local isotropy is investigated for the inertial and dissipative scales , using two tests, one applied on , the correlation coefficient between temperature increments and velocity increments, and the other one on , the temperature increment skewness factor. When heating one of the disks and cooling the other one, either positive, negative or almost null values of C and S can be obtained at small scales as a direct result of the presence of several temperature sources. In particular, we emphasize the fact that null or small values for these quantities in the inertial range are an evidence of local isotropy of the temperature field. In these cases, we use the Vaienti et al. equation [Physica D 73, 99 (1994)] for the evolution of the temperature increments probability density functions(PDFs) to predict the inertial and dissipative range PDFs, using an initial PDF, and two measurable closure functions. The intermittent behavior quantified through these statistics is well reproduced by the numerical integration of this evolution equation.

Probability density function in the loglaw region of low Reynolds number turbulent boundary layer
View Description Hide DescriptionThe logarithmic velocity region is considered in zeropressuregradient turbulent boundary layers. We propose a new definition of the loglaw region using the probability profiles of streamwise velocity fluctuation. The loglaw profile is extracted readily from the experimental data as well as from the probability density function (pdf) equation. The measure called Kullback Leibler divergence is applied for distinguishing the probability profiles. If the logarithmic profile, , is a good representation of experimental data, our results show that A is independent of the Reynolds number while B depends on it. The ratio of boundary layer thickness to the upper end of loglaw extent, is not constant but approaches the value as the Reynolds number increases.

A dynamical model for turbulence. VII. Complete system of five orthogonal tensors for sheardriven flows
View Description Hide DescriptionThe onepoint Reynolds stresses have been traditionally expressed in terms of ten tensors. It is, however, known that the independent tensors are only five. We construct a complete set of five orthogonal, traceless, and symmetric second rank tensors in terms of mean strain and vorticity. The system is used to express the onepoint Reynolds stresses. The coefficients of the expansion are evaluated in papers VIII and IX.

A dynamical model for turbulence. VIII. IR and UV Reynolds stress spectra for sheardriven flows
View Description Hide DescriptionThe basic equations for the twopoint Reynolds stresses derived in paper II are solved analytically in two regimes: the UV (ultraviolet) region corresponding to the inertial range and the IR (infrared) region corresponding to The analytic treatment is possible due to the existence of two smallness parameters: in the UV region and kL in the IR region; is the mean velocity gradient, is the turbulentviscosity, and L is the integral length scale. For an arbitrary flow, the Reynolds stressspectrum in the UV region is given by Eqs. (53545556575859). In the IR region, and in the firstorder approximation in kL, the spectra coincide with those of the rapid distortion theory. Since they are flow dependent, we shall discuss a few representative cases. The resulting Reynolds stressspectra, which are shown to reproduce existing data, are the basis for the calculation of the onepoint Reynolds stresses to be presented in paper IX. The model has no free parameters.

A dynamical model for turbulence. IX. Reynolds stresses for sheardriven flows
View Description Hide DescriptionWe present a new expression for the onepoint Reynolds stress in terms of the strain and vorticity of the large scales. The are expressed in terms of only five basic orthogonal tensors rather than the traditional ten tensors. The expression for Eq. (24), contains no adjustable parameters. The derivation of is based on the twopoint closure dynamic equations for the spectral Reynolds stresses that were developed earlier and the results of which were validated on a wide variety of data comprising shear, buoyancy, twodimensional (2D) turbulence, rotation, etc. For the case of homogeneous turbulence, we also derive an expression for the empirical coefficients of the ε equation that depend on the invariants of the flow, the turbulent kinetic energy K and the production P. Examples for special flows are given. The new expressions for are shown to reproduce well data from Tavoularis and Corrsin, DNS data, stationary data (pipe flow,channel flow, and homogeneous flow), and the Smagorinsky–Lilly constant, which is shown to be a dynamical variable since it depends on the ratio and on the invariant

Shockinduced flow resonance in supersonic jets of complex geometry
View Description Hide DescriptionJets with complex shockcell structures exist in numerous technological applications. This paper describes a fundamental study of shockinduced flow resonance (commonly referred to as “jet screech”) in supersonic jets with spanwise nonuniform shockcell structures. Experiments that involve flow visualization and detailed mapping of the near field reveal unsteady aspects of shockinduced flow resonances, mode transitions, and directivity of the radiated noise. The following important results about the role of spanwise nonuniform shockcells emerged: (1) It is possible to have two coexisting, independent feedback loops at nonharmonically related frequencies and different spanwise modes. (2) The same type of spanwise asymmetric mode was produced by two entirely different source configurations. (3) Nozzle geometry significantly altered the intensity and directivity of screech and broadband shock noise. The results presented here provide considerable insight into the fluid dynamics and acoustics of jets with spanwise oblique shockcell structures and provide grounds for believing that shockinduced noise can be controlled by tailoring nozzle geometry.

A mechanistic model for shock initiation of solid explosives
View Description Hide DescriptionThis paper is devoted to the building of a model for the ignition and growth of a detonation in pressed solid explosives. The ignition model describes the various phenomena occurring at the microscopic scale during viscoplastic pore collapse. The growth stage is represented by a model combining inner combustion inside the pores and outer combustion on the surface of the grains. These microscopic models are incorporated into a macroscopic one. The macroscopic model reproduces waves propagation and takes into account the various couplings between the microscopic and macroscopic scales. Pores and grain size distributions are also considered. The governing equations are solved using a shock tracking high resolution scheme, in order to avoid numerical smearing of the shock front. The role of microscopic topology of the explosive is investigated. Results are validated on pressure gauge records and shock to detonation transition distance (Popplots).
