Volume 23, Issue 3, March 2011
Index of content:

Rinsing flows are common processes where a jet of one liquid impinges upon a layer of a second liquid for the purpose of removing the second liquid. An imaging setup has been developed to obtain both qualitative and quantitative data on the rinsing flow of a jet of water impinging on either layers of Newtonian or elastic fluids. Three classes of test fluids have been investigated: a Newtonian glycerolwater solution, a semidilute aqueous solution of high molecular weight polyacrylamide solution displaying both elasticity and shear thinning, and an elastic but nonshear thinning Boger fluid. The fluids were designed to have approximately equal zeroshear viscosities. For all cases, a circular hydraulic jump occurs and Saffman–Taylor instabilities were observed at the interface between the low viscosity jet and the higher viscosity coating liquids. Results show that the elasticity (extensional viscosity) of the samples influences the pattern of the instabilities and contributes to dampening surface disturbances in the vicinity of the hydraulic jump. Quantitative measurements of liquid layer thicknesses were obtained using a laser triangulation technique. We observed that shear thinning contributes to increasing the velocity of the hydraulic jump circle growth, and the growth profile appears to be linear instead of logarithmiclike as in the Newtonian fluids. Shear thinning characteristics of the samples also contribute to a larger vertical height of the hydraulic jump and an undercutting phenomenon. The elasticity of the fluids contributes to a “recoil” of the hydraulic jump circle, causing the circle, after initial expansion, to shrink in size before expanding again.
 ARTICLES


Interfacial Flows

Velocity field reconstruction in gravitydriven flow over unknown topography
View Description Hide DescriptionA numerical method for reconstructing the velocity field of a viscous liquid flowing over unknown topography is presented. For a given fluid this procedure allows one to determine the velocity field as well as the topographic structure from the freesurface shape only. First, we confirm the results with previous computations in the thinfilm limit and then generalize the numerical solution to arbitrary film thicknesses and focus on the velocity field. It is documented that even smoothly corrugated freesurface shapes require strongly undulated topographies to maintain the flow structure. Finally, we discuss details of the implementation in applications, solvability in general, and sensitivity of the solution.

Nonlinear dynamics of confined thin liquidvapor bilayer systems with phase change
View Description Hide DescriptionWe numerically investigate the nonlinear evolution of the interface of a thin liquidvapor bilayer system confined by rigid horizontal walls from both below and above. The lateral variation of the vapor pressure arising from phase change is taken into account in the present analysis. When the liquid (vapor) is heated (cooled) and gravity acts toward the liquid, the deflection of the interface monotonically grows, leading to a rupture of the vapor layer, whereas nonruptured stationary states are found when the liquid (vapor) is cooled (heated) and gravity acts toward the vapor. In the latter case, vaporflowdriven convective cells are found in the liquid phase in the stationary state. The average vapor pressure and interface temperature deviate from their equilibrium values once the interface departs from the flat equilibrium state. Thermocapillarity does not have a significant effect near the thermodynamic equilibrium, but becomes important if the system significantly deviates from it.

Viscous and NonNewtonian Flows

Role of fluid elasticity on the dynamics of rinsing flow by an impinging jet
View Description Hide DescriptionRinsing flows are common processes where a jet of one liquid impinges upon a layer of a second liquid for the purpose of removing the second liquid. An imaging setup has been developed to obtain both qualitative and quantitative data on the rinsing flow of a jet of water impinging on either layers of Newtonian or elastic fluids. Three classes of test fluids have been investigated: a Newtonian glycerolwater solution, a semidilute aqueous solution of high molecular weight polyacrylamide solution displaying both elasticity and shear thinning, and an elastic but nonshear thinning Boger fluid. The fluids were designed to have approximately equal zeroshear viscosities. For all cases, a circular hydraulic jump occurs and Saffman–Taylor instabilities were observed at the interface between the low viscosity jet and the higher viscosity coating liquids. Results show that the elasticity (extensional viscosity) of the samples influences the pattern of the instabilities and contributes to dampening surface disturbances in the vicinity of the hydraulic jump. Quantitative measurements of liquid layer thicknesses were obtained using a laser triangulation technique. We observed that shear thinning contributes to increasing the velocity of the hydraulic jump circle growth, and the growth profile appears to be linear instead of logarithmiclike as in the Newtonian fluids. Shear thinning characteristics of the samples also contribute to a larger vertical height of the hydraulic jump and an undercutting phenomenon. The elasticity of the fluids contributes to a “recoil” of the hydraulic jump circle, causing the circle, after initial expansion, to shrink in size before expanding again.

Particulate, Multiphase, and Granular Flows

Settling dynamics of asymmetric rigid fibers
View Description Hide DescriptionThe threedimensional motion of asymmetric rigid fibers settling under gravity in a quiescent fluid was experimentally measured using a pair of cameras located on a movable platform. The particle motion typically consisted of an initial transient after which the particle approached a steady rate of rotation about an axis parallel to the acceleration of gravity, with its center of mass following a helical trajectory. Numerical and analytical methods were used to predict translational and angular velocities as well as the evolution of the fiber orientation as a function of time. A comparison of calculated and measured values shows that it is possible to quantitatively predict complex motions of particles that have highly asymmetric shape. The relations between particle shape and settling trajectory have potential applications for hydrodynamic characterization of fiber shapes and fiber separation.

Laminar Flows

Numerical investigations of lift suppression by feedback rotary oscillation of circular cylinder at low Reynolds number
View Description Hide DescriptionThis article describes a strategy of active flow control for lift force reduction of circular cylinder subjected to uniform flow at low Reynolds numbers. The flow control is realized by rotationally oscillating the circular cylinder about its axis with , where is the dimensionless angular speed of rotation cylinder, is the control parameter and is the feedback signal of lift coefficient. The study focuses on seeking optimum for the low Reynolds numbers of 60, 80, 100, 150, and 200. The effectiveness of the proposed flow control in suppressing lift force is examined comprehensively by a numerical model based on the finite element solution of twodimensional Navier–Stokes equations. The dependence of lift reduction on the control parameter is investigated. The threshold of , denoted by , is identified for the Reynolds numbers considered in this work. The numerical results show that the present active rotary oscillation of circular cylinder is able to reduce the amplitude of lift force significantly as long as , at least 50% for the laminar flow regime. Meanwhile, the present active flow control does not result in the undesirable increase in the drag force. The Strouhal number is observed to decrease slightly with the increase of . As for a specific Reynolds number, the larger gives rise to the larger amount of lift reduction. The lift reduction reaches the maximum at . The mechanism behind the present lift reduction method is revealed by comparing the flow patterns and pressure distributions near the active rotationally oscillating circular cylinder and the stationary circular cylinder. It is found that the critical value generally increases with Reynolds number. Two types of lift shift are observed in the numerical results for the cases with . The first is characterized by the regular fluctuation of lift coefficient but with nonzero mean value, while the second is associated with the sustaining increase of lift coefficient. The phenomenon of lift shift is found to be related closely to the evolution of vortex pattern in the near wake of circular cylinder.

Low Reynolds number flow over a square cylinder with a splitter plate
View Description Hide DescriptionFlows over a square cylinder of side length with and without a splitter plate are numerically investigated at a Reynolds number of 150. The length of the splitter plate is varied systematically from to so the sensitivity of the flow structure to the inclusion of the splitter plate can be inspected. It is found that the splitter plate introduces a strong hydrodynamic interaction to the near wake of the cylinder and the length of the plate affects significantly the flow structure. The behavior of the flow can be grouped into three regimes. For short plate lengths , the free shear layers are convected further downstream before rolling up when the plate length is increased. For intermediate plate lengths , a secondary vortex is clearly visible around the trailing edge of the splitter plate and the shear layers begin to roll up closer to the trailing edge. For long plate lengths , a regime is observed in which the free shear layers reattach to the splitter plate. The study also proposes the minimum wake halfwidth as the length scale for a possible universal Strouhal number, which is found to be valid for .

Instability and Transition

Quasitwodimensional convection in a threedimensional laterally heated box in a strong magnetic field normal to main circulation
View Description Hide DescriptionConvection in a laterally heated threedimensional box affected by a strong magnetic field is considered in the quasitwodimensional (Q2D) formulation. It is assumed that the magnetic field is strong and is normal to the main convective circulation. The stability of the resulting Q2D flow is studied for two values of the Hartmann number scaled by half of the width ratio, 100 and 1000, and for either thermally insulating or perfectly conducting horizontal boundaries. The aspect lengthtoheight ratio of the box is varied continuously between 4 and 10. It is shown that the magnetic field damps the bulk flow and creates thermal and Shercliff boundary layers at the boundaries, which become the main source of instabilities. In spite of the general tendency of the flow stabilization by the magnetic field, the flow instability takes place in different ways depending on the boundary conditions and the aspect ratio. Similarities with other magnetic field directions and flows with larger Prandtl numbers are discussed.

Liquid mixture convection during phase separation in a temperature gradient
View Description Hide DescriptionWe simulate the phase separation of a lowviscosity binary mixture, assuming that the fluid system is confined between two walls that are cooled down to different temperatures below the critical point of the mixture, corresponding to quenches within the unstable range of its phase diagram. Spinodal decomposition patterns for offcritical mixtures are studied numerically in two dimensions in the creeping flow limit and for a large Lewis number, together with their dependence on the fluidity coefficient. Our numerical results reproduce the largescale unidirectional migration of phaseseparating droplets that was observed experimentally by Califano et al. [“Largescale, unidirectional convection during phase separation of a densitymatched liquid mixture,” Phys. Fluids17, 094109 (2005)], who measured typical speeds that are quite larger than the Marangoni velocity. To understand this finding, we then studied the temperaturegradientinduced motion of an isolated droplet of the minority phase embedded in a continuous phase, showing that when the drop is near local equilibrium, its speed is of the same order as the Marangoni velocity, i.e., it is proportional to the unperturbed temperature gradient and the fluidity coefficient. However, far from local equilibrium, i.e., for very large unperturbed temperature gradients, the drop first accelerates to a speed that is larger than the Marangoni velocity, then, later, it decelerates, exhibiting an increasedecrease behavior, as described by Yin et al. [“Thermocapillary migration of nondeformable drops,” Phys. Fluids20, 082101 (2008)]. Such behavior is due to the large nonequilibrium, Kortewegdriven convection, which at first accelerates the droplets to relatively large velocities, and then tends to induce an approximately uniform inside temperature distribution so that the drop experiences an effective temperature gradient that is much smaller than the unperturbed one and, consequently, decelerates.

The influence of shear layer thickness on the stability of confined twodimensional wakes
View Description Hide DescriptionThe influence of confinement onto the inviscid and incompressible linear stability of the family of wakes introduced by Monkewitz [Phys. Fluids31, 999 (1988)] is examined. The nondimensional parameters of the model, the velocity ratio , defined as ratio of the velocity gap to the mean velocity, the profile shape parameter, which controls the shear layer thickness , and the confinement parameter, are varied and their effect onto temporal and spatiotemporal stability properties is considered. Particularly, the limit between absolute (A) and convective (C) instability is investigated as a function of the different parameters. For a given confinement, there exists an optimal value of the shear layer thickness for which the absolute instability is maximal. The absolute frequency and complex wavenumber of the mode at the C/A transition are discussed. Furthermore, the continuous profiles are approximated by means of piecewise brokenline profile, with similar spatiotemporal properties. As a typical application, a few nonparallel base flows, computed by direct numerical simulation at , are analyzed on a weakly nonparallel basis by plotting the locus of the local velocity profile in the plane. The absolute frequency at the C/A transition point is seen to predict accurately the frequency prevailing in the nonlinear direct numerical simulations. These results further help interpreting the influence of confinement on the Strouhal number measured experimentally in the wake of confined cylinders.

Experimental control of vortex breakdown by density effects
View Description Hide DescriptionThe vortex breakdown inside a cylinder with a rotating top lid is controlled experimentally by injecting at the bottom a fluid with a small density difference. The density difference is obtained by mixing a heavy dye or alcohol with water in order to create a jet denser or lighter than water. The injection of a heavy fluid creates a buoyancy force downward, which counteracts the meridional recirculation in the cylinder and thus enhances the formation of a vortex breakdown bubble. The stability diagram shows that even a very small density difference of 0.02% is able to decrease by a factor of 2 the critical Reynolds number of appearance of the breakdown. On the other hand, the injection of a lighter fluid does not destroy the vortex breakdown. However, for large enough density differences (larger than 0.03%), the lighter fluid is able to pierce through the bubble and leads to a new structure of the vortex breakdown. Finally, a parallel is drawn between a light jet and a vortex ring generated at the bottom of the cylinder: strong vortex rings are able to pierce through the bubble, whereas weak vortex rings are simply advected around the bubble.

Influence of highfrequency vibration on the Rayleigh–Marangoni instability in a twolayer system
View Description Hide DescriptionThe influences of highfrequency vibrations on the Rayleigh–Marangoni instability in a twolayer system are investigated theoretically in the framework of the averaging method. We focus on the effects of vertical and horizontal vibrations on the stability of different convection modes. The results show that vertical vibrations significantly stabilize the system, while horizontal vibrations significantly destabilize it. In the presence of vertical vibrations, instability only occurs in a system heated from below. However, in the presence of horizontal vibrations, instability can also occur in a system cooled from below. When Marangoni effect is dominant at the interface, it is found that there are four types of coupling modes. The oscillatory convection is the result of the competition between different modes. In the presence of Marangoni effect at the interface, the structure of the interfacial flow is complicated. In some cases, small counterrolls may develop to preserve the nonslip condition of fluids in either the upper layer or the lower layer.

Simulations of Richtmyer–Meshkov instabilities in planar shocktube experiments
View Description Hide DescriptionIn the large eddy simulation(LES) approach, largescale energycontaining structures are resolved, smaller structures are filtered out, and unresolved subgrid effects are modeled. Extensive recent work has demonstrated that predictive underresolved simulations of the velocity fields in turbulent flows are possible without resorting to explicit subgrid models when using a class of physicscapturing highresolution finitevolume numerical algorithms. This strategy is denoted as implicit LES (ILES). Tests in fundamental applications ranging from canonical to complex flows indicate that ILES is competitive with conventional LES in the LES realm proper—flows driven by largescale features. The performance of ILES in the substantially more difficult problem of underresolved material mixing driven by underresolved velocity fields and initial conditions is a focus of the present work. Progress in addressing relevant resolution issues in studies of mixing driven by Richtmyer–Meshkov instabilities in planar shocktube laboratory experiments is reported. Our particular focus is devoted to the initial material interface characterization and modeling difficulties, and effects of initial condition specifics (resolved spectral content) on transitional and latetime turbulent mixing—which were not previously addressed.

Boundary layer receptivity to freestream turbulence and surface roughness over a swept flat plate
View Description Hide DescriptionAn experimental study of the receptivity of disturbances and their subsequent development into a threedimensional boundary layer has been carried out. The threedimensional boundary layer was set up using a flat plate with a swept leading edge and a pressure gradient using a displacement body at the ceiling of the test section. Low level freestream turbulence was generated with five different screens and was shown to generate traveling crossflow modes for all but the lowest turbulence level, i.e., for , where instead a stationary crossflow disturbance dominated. Stationary crossflow disturbances were triggered by small cylindrical roughness elements arranged in an array. For high enough roughnessReynolds number () stationary disturbances growing exponentially were seen and their amplitude seems to scale with .

Turbulent Flows

Lagrangian properties of turbulent diffusion with passive chemical reaction in the framework of the premixed combustion theory
View Description Hide DescriptionIn this paper, we analyze two effects caused by the Lagrangian nature of turbulent transfer which are usually ignored in the theory of turbulent premixed combustion. These effects are (i) the nonequilibrium behavior of the turbulent diffusion coefficient, which is important for modeling the initial stage of combustion (for example, in the spark ignition engine), and (ii) the existence of a traveling front of turbulent diffusion with finite speed, which controls the velocity of the steady state flame in strong turbulence. However, in order to derive simple and exact results, the hydrodynamical and the combustion subproblems are stated to be independent by assuming a constant density so that a passive chemical reaction is actually considered. First, we derive a parabolic diffusionequation with both diffusion and chemical source terms expressed by Lagrangian characteristics of turbulence. We show that, in general, the diffusivity of product particles is not zero in the moment of their generation by chemical transformation and this result is important for combustiontheories that relate the formation of the initial flame with the development of the diffusion coefficient. Afterward, a hyperbolic diffusionequation based on hydrodynamics is derived with turbulent diffusion front velocity , where is the root mean square of turbulent velocity fluctuations, and we analyze the relationships between and the speed of the steady state premixed flame. In particular, for the flamelet combustion mechanism, we obtain , where is the normal laminar flame speed. This result shows that, in moderate turbulence, the usually assumed relation is not consistent with an accurate statistical analysis and more when gives a percent error around 40%, which cannot be neglected in applications. In strong turbulence case , the value of the flame speed is very close to that of the diffusion front velocity.

A posteriori analysis of numerical errors in subfilter scalar variance modeling for large eddy simulation
View Description Hide DescriptionSubfilter scalar variance is a critical indicator of small scale mixing in large eddy simulation(LES) of turbulent combustion and is an important parameter of conserved scalar based combustionmodels. Realistic combustionmodels have a highly nonlinear dependence on the conserved scalar, making the prediction of flow thermochemistry sensitive to errors in subfilter variance modeling, including errors due to numerical discretization. Large numerical errors can result from the use of gridbased filtering and the resulting underresolution of the smallest filtered scales, which are a key to variance modeling. Hence, the development of variance models should take into account this sensitivity to numerical discretization. In this work, a novel coupled direct numerical simulation (DNS)LES a posteriori method is used to study the role of discretization errors in variance prediction for the two most widely used types of models: algebraic dynamic models and transport equationbased models. Algebraic models are found to be illsuited to discretization due to their dependence on filtered scalar gradient values. Additionally, the use of dynamic modeling procedures enhances their sensitivity to filtered scalar errors. The accuracy of transport equationmodels primarily rests on the accuracy of the scalar dissipation rate closure with numerical error having a secondary effect. The influence of dissipation rate modeling error is investigated using the unique information provided by the combined DNSLES simulation method. Overall, transport equationmodels are found to offer a more powerful approach to variance modeling due to more complete model physics and reduced effects of discretization error.

Interaction of a Taylor blast wave with isotropic turbulence
View Description Hide DescriptionSimulations of the Taylor blast wave through a region of compressible isotropic turbulence are carried out. The turbulent fluctuations are either significantly attenuated or unchanged depending on the initial strength of the shock wave. It is shown through Eulerian simulations and Lagrangian tracking of particles that both these effects are primarily related to the vorticitydilatation term in the vorticity transport equation. The turbulence length scales associated with this problem are defined and the effect on them quantified. Turbulence also distorts the shock, which can lead to substantial local variations in shock strength and asphericity. Transverse vorticity amplification is compared with linear planar shockturbulence theory. Aspects that distinguish spherical shockturbulence interaction from the planar case are stressed.

Largeeddy simulation of the flow and acoustic fields of a Reynolds number subsonic jet with tripped exit boundary layers
View Description Hide DescriptionLargeeddy simulations (LESs) of isothermal round jets at a Mach number of 0.9 and a diameterbased Reynolds number of originating from a pipe are performed using lowdissipation schemes in combination with relaxation filtering. The aim is to carefully examine the capability of LES to compute the flow and acoustic fields of initially nominally turbulent jets. As in experiments on laboratoryscale jets, the boundary layers inside the pipe are tripped in order to obtain laminar mean exit velocity profiles with high perturbation levels. At the pipe outlet, their momentum thickness is times the jet radius, yielding a Reynolds number, and peak turbulence intensities are around 9% of the jet velocity. Two methods of boundarylayer tripping and five grids are considered. The results are found to vary negligibly with the tripping procedure but appreciably with the grid resolution. Based on analyses of the LES quality and on comparisons with measurements at high Reynolds numbers, fine discretizations appear necessary in the three coordinate directions over the entire jet flow. The final LES carried out using points with minimum radial, azimuthal, and axial mesh spacings, respectively, of 0.20, 0.34, and is also shown to provide shearlayer solutions that are practically grid converged and, more generally, results that can be regarded as numerically accurate as well as physically relevant. They suggest that the mixinglayer development in the present tripped jet, while exhibiting a wide range of turbulent scales, is characterized by persistent coherent vortex pairings.

Flow structure and momentum transport for buoyancy driven mixing flows in long tubes at different tilt angles
View Description Hide DescriptionBuoyancy driven mixing of fluids of different densities ( and ) in a long circular tube is studied experimentally at the local scale as a function of the tilt angle from vertical and of the Atwood number . Particle Image Velocimetry (PIV) and Laser Induced Fluorescence(LIF)measurements in a vertical diametral plane provide the velocity and the relative concentration (and, hence, density) fields. A map of the different flow regimes observed as a function of At and has been determined: as At increases and is reduced, the regime varies from laminar to intermittent destabilizations and, finally, to developed turbulence. In the laminar regime, three parallel stable layers of different densities are observed; the velocity profile is linear and well predicted from the density profile. The thickness of the intermediate layer can be estimated from the values of At and . In the turbulent regime, the density varies slowly with in the core of the flow: there, transverse turbulent momentum transfer is dominant. As At decreases and increases, the density gradient in the core (and, hence, the buoyancy forces) becomes larger, resulting in higher extremal velocities and indicating a less efficient mixing. While the mean concentration varies with time in the turbulent regime, the mean velocity remains constant. In the strong turbulent regime (highest At and lowest values), the transverse gradient of the mean concentration and the fluctuations of concentration and velocity remain stationary, whereas they gradually decay with time when turbulence is weaker.

Investigation of dissipation elements in a fully developed turbulent channel flow by tomographic particleimage velocimetry
View Description Hide DescriptionA new method to describe statistical information from passive scalar fields has been proposed by Wang and Peters [“The lengthscale distribution function of the distance between extremal points in passive scalar turbulence,”J. Fluid Mech.554, 457 (2006)]. They used direct numerical simulations (DNS) of homogeneous shear flow to introduce the innovative concept. This novel method determines the local minimum and maximum points of the fluctuatingscalar field via gradient trajectories, starting from every grid point in the direction of the steepest ascending and descending scalar gradients. Relying on gradient trajectories, a dissipation element is defined as the region of all the grid points, the trajectories of which share the same pair of maximum and minimum points. The procedure has also been successfully applied to various DNS fields of homogeneous shear turbulence using the three velocity components and the kinetic energy as scalar fields [L. Wang and N. Peters, “Lengthscale distribution functions and conditional means for various fields in turbulence,”J. Fluid Mech.608, 113 (2008)]. In this spirit, dissipation elements are, for the first time, determined from experimental data of a fully developed turbulent channel flow. The dissipation elements are deduced from the gradients of the instantaneous fluctuation of the three velocity components , , and and the instantaneous kinetic energy , respectively. The measurements are conducted at a Reynolds number of based on the channel halfheight and the bulk velocity. The required threedimensional velocity data are obtained investigating a test volume using tomographic particleimage velocimetry. Detection and analysis of dissipation elements from the experimental velocity data are discussed in detail. The statistical results are compared to the DNS data from Wang and Peters [“The lengthscale distribution function of the distance between extremal points in passive scalar turbulence,”J. Fluid Mech.554, 457 (2006); “Lengthscale distribution functions and conditional means for various fields in turbulence,”J. Fluid Mech.608, 113 (2008)]. Similar characteristics have been found especially for the pdf’s of the large dissipation element length regarding the exponential decay. In agreement with the DNS results, over 99% of the experimental dissipation elements possess a length that is smaller than three times the average element length.

Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence
View Description Hide DescriptionThe stretched spiral vortex [T. S. Lundgren, “Strained spiral vortexmodel for turbulent structures,” Phys. Fluids25, 2193 (1982)] is identified in turbulence in homogeneous shear flow and the spectral properties of this flow are studied using directnumerical simulation data. The effects of mean shear on the genesis, growth, and annihilation processes of the spiral vortex are elucidated, and the role of the spiral vortex in the generation of turbulence is shown. As in homogeneous isotropic turbulence [K. Horiuti and T. Fujisawa, “The multi mode stretched spiral vortex in homogeneous isotropic turbulence,”J. Fluid Mech.595, 341 (2008)], multimodes of the spiral vortex are extracted. Two symmetric modes of configurations with regard to the vorticity alignment along the vortex tube in the core region and dual vortex sheets spiraling around the tube are often educed. One of the two symmetric modes is created by a conventional rollingup of a single spanwise shear layer. Another one is created by the convergence of the recirculating flow or streamwise roll [F. Waleffe, “Homotopy of exact coherent structures in plane shear flows,” Phys. Fluids15, 1517 (2003)] caused by the upward and downward motions associated with the streaks. The vortex tube is formed by axial straining and lowering of pressure in the recirculating region. The spanwise shear layers are entrained by the tube and they form spiral turns. The latter symmetric mode tends to be transformed into the former mode with lapse of time due to the action of the pressure Hessian term. The power law in the inertial subrange energy spectrum is studied. The base steady spectrum fits the equilibrium Kolmogorov −5/3 spectrum, to which a nonequilibrium component induced by the fluctuation of the dissipation rate is added. This component is extracted using the conditional sampling on , and it is shown that it fits the −7/3 power in accordance with the statistical theory. The correlation between these spectra and the appearance and diminution of the streaks and the two modes of the spiral vortex is discussed. The temporal variations of the spectrum are divided into two regimes, Phases 1 and 2. Large energy contained in the lowwavenumber range in Phase 1 is cascaded to the small scales in Phase 2. This energy transfer is accomplished by the reversal in the sign of −7/3 power component.
