Eddy Turnover Time Research Articles

Temporal intermittency and cascades in shell models of turbulence.

The two-dimensional- (2D) and three-dimensional- (3D) like Gletzer, Okhitani, and Yamada shell models are examined. The 2D-like model shows a transition from statistical quasiequilibrium to cascade of enstrophy as a function of the spectral ratio of energy to enstrophy. The transition is related to the ratio of time scales, corresponding to eddy turnover times, between shells. The anomalous scaling, giving rise to nonlinear scaling functions, is also connected to the ratio of eddy turnover times. This is illustrated in a simple stochastic model, where the structure function \ensuremath{\zeta}(q) becomes independent of q. In the 3D-like model the multiscaling is also influenced by the existence of a second nonpositive-definite inviscid invariant, the helicity. \textcopyright{} 1996 The American Physical Society.

Elements of the Microphysical Structure of Numerically Simulated Nonprecipitating Stratocumulus

A set of 500 simulated trajectories and a simple parcel model are used to (i) evaluate the performance of a large eddy simulation model coupled to a detailed representation of the droplet spectrum (the LES-BM model) and (ii) gain insight into the microphysical structure of numerically simulated nonprecipitating stratocumulus. The LES-BM model reasonably reproduces many observed features of stratocumulus. The largest sources of error appear to be associated with limited vertical resolution, the neglect of gas kinetic effects and the inability of the model to properly represent mixing across cloud interfacial boundaries. The first two problems have simple remedies; for instance, a condensation–nucleation scheme is derived that includes gas–kinetic effects thus obviating the second problem. The third source of error poses a more vexing, and as yet unsolved, problem for models of the class described herein. Trajectories timescales are analyzed and in-cloud residence times are found to be, in the mean, on the order of the large eddy turnover time. In addition, it is shown that the length of time trajectories spend near cloud top may be an important factor in the droplet growth equation for a certain favored subset of trajectories. An analysis of the adiabatic trajectory data also indicates that (i) values of diameter dispersion are a factor of 2 to 5 smaller than commonly observed; (ii) simulated values of the dispersion in number concentration are found to be explainable solely on the basis of trajectories having different updraft velocities; (iii) diameter dispersions are not found to be equal to a third of number dispersions, nor did they relate simply to the dispersion in the cloud-base updraft velocity. Problems with coupling one- and two-dimensional models to detailed representations of the droplet spectrum are discussed. In the case of the former, the lack of an explicit representation of turbulent eddies requires that the coupling between the microphysics and the dynamics be parameterized. In the case of the latter, boundary layer eddies are represented, thus allowing for a more reasonable coupling between turbulence and microphysics. However, the resolved eddies have a different structure than their three-dimensional counterparts, one consequence of which is that timescales of in-cloud circulations are found to be shorter and have less variability.

The effect of vortex pairing on particle dispersion and kinetic energy transfer in a two-phase turbulent shear layer

The transport of heavy, polydispersed particles and the inter-phase transfer of kinetic energy due to the viscous drag forces is measured experimentally in a turbulent shear layer. To study the effect of the large-scale vortex pairing event, the shear layer is forced simultaneously with a fundarmental and subharmonic perturbation. It is shown that vortex pairing plays a homogenizing role on the particulate field, but hte amount of homogenization is strongly dependent upon the particle's viscous relaxtion time, the eddy turnover time, as well as the time the particles interact with each scale prior to a pairing event. Thus, even though the smaller size particles become well-mixed across the large eddies, the larger sizes are still dispersed in an inhormogeneous fashion. It is also found that the kinetic energy transfer between the phases occurs inhomogeneously with energy being exchanged predominantly in a sublayer just outside the region of maximum turbulence intensity. The kinetic energy transfer is shown to exhibit notable positive and negative peaks located beneath the cores and stagnation points of the large-scale eddy field, and these peaks are shown to result from the irrotational velocity perturbations created by the vortices. This energy exchange mechanism remains a prominent process as long as the Stokes number of the particles relative to the vortices is of order unity.

A phenomenological treatment of rotating turbulence

In this Brief Communication, we first note the strong similarity between the magnetohydrodynamic (MHD) turbulence and initially isotropic turbulence subject to rotation. We then applied the MHD phenomenologies of Kraichnan [Phys. Fluids 8, 1385 (1965)] and Matthaeus and Zhou [Phys. Fluids B 1, 1929 (1989)] to rotating turbulence. We deduced a ‘‘rule’’ that relates spectral transfer time to the eddy turnover time and the time scale for decay of the triple correlations. Our hypothesis on the triple correlation decay rate leads to the spectral law, which varies between the ‘‘−5/3’’ (without rotation) and ‘‘−2’’ laws (with strong rotation). For intermediate rotation rates, the spectrum varies according to the value of a dimensionless parameter that measures the strength of the rotation wave number kΩ=(Ω3/ε)1/2 relative to the wave number k. The eddy viscosity is derived with an explicitly dependence on the rotation rate.

The nonlinear galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows

Using results of DNS in the case of two-dimensional homogeneous isotropic flows, we first analyze in detail the behavior of the small and large scales of Kolmogorov like flows at moderate Reynolds numbers. We derive several estimates on the time variations of the small eddies and the nonlinear interaction terms; those terms play the role of the Reynolds stress tensor in the case of LES. Since the time step of a numerical scheme is determined as a function of the energy-containing eddies of the flow, the variations of the small scales and of the nonlinear interaction terms over one iteration can become negligible by comparison with the accuracy of the computation. Based on this remark, we propose a multilevel scheme which treats differently the small and the large eddies. Using mathematical developments, we derive estimates of all the parameters involved in the algorithm, which then becomes a completely self-adaptive procedure. Finally, we perform realistic simulations of (Kolmorov like) flows over several eddy-turnover times. The results are analyzed in detail and a parametric study of the nonlinear Galerkin method is performed.

Passive scalar dispersion and mixing in a turbulent jet

The dispersion and mixing of passive scalar (temperature) fluctuations is studied in a turbulent jet. The temperature fluctuations were produced by heated fine wire rings placed axisymmetrically in the flow. Typically the ring diameters were of the same order as the jet,Dj, and they were placed in the self-similar region. However, other initial conditions were studied, including a very small diameter ring used to approximate a point source. Using a single ring to study dispersion (which is analogous to placing a line source in a planar flow such as grid turbulence), it was found that the intense local thermal field close to the ring disperses and fills the whole jet in approximately 1.5 eddy turnover times. Thereafter the thermal field evolves in the same way as for the traditional heated jet experiments. Two heated rings were used to study the mixing of two independently introduced scalar fields. Here an inference method (invoking the principle of superposition) was used to determine the evolution of the cross-correlation coefficient, ρ, and the segregation parameter, α, as well as the coherence and co-spectrum. While initially strongly dependent on ring locations and spacing, ρ and α reached asymptotic values of 1 and 0.04, respectively, also in about 1.5 eddy turnover times. These results are contrasted with mixing and dispersion in grid turbulence where the evolution is slower. Measurements in the far field of the jet (where ρ = 1) of the square of the scalar derivative conditioned on the scalar fluctuation itself, as well as other conditional statistics, showed strong dependence on the measurement location, as well as the direction in which the derivative was determined. The cross-correlation between the square of the scalar derivative and the signal showed a clear Reynolds-number trend, decreasing as the jet Reynolds number was varied from 2800 to 18 000. The far-field measurements, using the heated rings, were corroborated by new heated jet experiments.

The Dynamics of Flux Tubes in a High- beta Plasma. I. A General Description

view Abstract Citations (39) References (29) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Dynamics of Flux Tubes in a High- beta Plasma. I. A General Description Vishniac, Ethan T. Abstract We suggest a new model for the structure of a magnetic field embedded in a plasma whose average turbulent and magnetic energy densities are both much less than the gas pressure. This model is based on the popular notion that the magnetic field will tend to separate into individual flux tubes. We point out that interactions between the flux tubes will be dominated by coherent effects stemming from the turbulent wakes created as the fluid streams by the flux tubes. Balancing the attraction caused by shielding effects with turbulent diffusion we find that flux tubes have typical radii comparable to the local Mach number squared times the large-scale eddy length, are arranged in a one-dimensional fractal pattern, have a radius of curvature comparable to the largest scale eddies in the turbulence, and have an internal magnetic pressure comparable to the ambient pressure. When the average magnetic energy density is much less than the turbulent energy density, the radius and curvature scale of the flux tubes will be smaller than these estimates. Allowing for resistivity changes these properties but does not alter the macroscopic properties of the fluid or the large-scale magnetic field. In either case we show that the Sweet-Parker reconnection rate is at least as fast as an eddy turnover time. Realistic stellar plasmas are expected to either be in the ideal limit (e.g., the solar photosphere) or the resistive limit (the bulk of the solar convection zone). Allowing for significant viscosity drastically changes the macroscopic properties of the magnetic field. We find that all current numerical simulations of three-dimensional MHD turbulence are in the viscous regime and are inapplicable to stars or accretion disks. However, these simulations are in good quantitative agreement with our model in the viscous limit. Publication: The Astrophysical Journal Pub Date: June 1995 DOI: 10.1086/175830 Bibcode: 1995ApJ...446..724V Keywords: MAGNETOHYDRODYNAMICS: MHD; PLASMAS; TURBULENCE full text sources ADS |

Three-dimensional multimode simulations of the ablative Rayleigh–Taylor instability

Multimode simulations of the evolution of the laser-driven, ablative Rayleigh–Taylor instability on planar, plastic targets are performed in three dimensions, with FAST3D–CM. The initial mass density target perturbations are random, with a power law dependence of k−2, a RMS surface finish of 0.1 μm, and perturbation wave numbers ranging from 2π/dmax to √2×(12π/dmax), for dmax=128 μm. At early nonlinear times, the perturbations grow to tile the target with approximately hexagonal bubbles that are of the shortest, initially seeded wavelengths not stabilized by density gradients. This tiling occurs on a time scale that is comparable to the eddy turnover time of the dominant bubble wavelength. When the target thickness is large compared to the dominant, short wavelengths, the bubbles continue to burn into the target and to evolve to progressively longer spatial scales. Predictions from second-order mode coupling and saturation models are found to be consistent with the simulation results.

Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and Fourier-space views

As discussed in a recent paper by Brasseur & Wei (1994), scale interactions in fully developed turbulence are of two basic types in the Fourier-spectral view. The cascade of energy from large to small scales is embedded within ‘local-to-non-local’ triadic interactions separated in scale by a decade or less. ‘Distant’ triadic interactions between widely disparate scales transfer negligible energy between the largest and smallest scales, but directly modify the structure of the smallest scales in relationship to the structure of the energy-dominated large scales. Whereas cascading interactions tend to isotropize the small scales as energy moves through spectral shells from low to high wavenumbers, distant interactions redistribute energy within spectral shells in a manner that leads to anisotropic redistributions of small-scale energy and phase in response to anisotropic structure in the large scales. To study the role of long-range interactions in small-scale dynamics, Yeung & Brasseur (1991) carried out a numerical experiment in which the marginally distant triads were purposely stimulated through a coherent narrow-band anisotropic forcing at the large scales readily interpretable in both the Fourier- and physical-space views. It was found that, after one eddy turnover time, the smallest scales rapidly became anisotropic as a direct consequence of the marginally distant triadic group in a manner consistent with the distant triadic equations. Because these asymptotic equations apply in the infinite Reynolds number limit, Yeung & Brasseur argued that the observed long-range effects should be applicable also at high Reynolds numbers.We continue the analysis of forced simulations in this study, focusing (i) on the detailed three-dimensional restructuring of the small scales as predicted by the asymptotic triadic equations, and (ii) on the relationship between Fourier- and physical-space evolution during forcing. We show that the three-dimensional restructuring of small-scale energy and vorticity in Fourier space from large-scale forcing is predicted in some detail by the distant triadic equations. We find that during forcing the distant interactions alter small-scale structure in two ways: energy is redistributed anisotropically within high-wavenumber spectral shells, and phase correlations are established at the small scales by the distant interactions. In the numerical experiments, the long-range interactions create two pairs of localized volumes of concentrated energy in three-dimensional Fourier space at high wavenumbers in which the Fourier modes are phase coupled. Each pair of locally phase-correlated volumes of Fourier modes separately corresponds to aligned vortex tubes in physical space in two orthogonal directions. We show that the dynamics of distant interactions in creating small-scale anisotropy may be described in physical space by differential advection and distortion of small-scale vorticity by the coherent large-scale energy-containing eddies, producing anisotropic alignment of small-scale vortex tubes.Scaling arguments indicate a disparity in timescale between distant triadic interactions and energy-cascading local-to-non-local interactions which increases with scale separation. Consequently, the small scales respond to forcing initially through the distant interactions. However, as energy cascades from the large-scale to the small-scale Fourier modes, the stimulated distant interactions become embedded within a sea of local-to-non-local energy cascading interactions which reduce (but do not eliminate) small-scale anisotropy at later times. We find that whereas the small-scale structure is still anisotropic at these later times, the second-order velocity moment tensor is insensitive to this anisotropy. Third-order moments, on the other hand, do detect the anisotropy. We conclude that whereas a single statistical measure of anisotropy can be used to indicate the presence of anisotropy, a null result in that measure does not necessarily imply that the signal is isotropic. The results indicate that non-equilibrium non-stationary turbulence is particularly sensitive to long-range interactions and deviations from local isotropy.

On the non-linear stability of the 1:1:1 ABC flow

ABC flows which can be considered as prototypes for the study of the onset of three-dimensional spatio-temporal turbulence are known both analytically and numerically to be linearly unstable. We analyze the nonlinear evolution of the ABC flow A 1 with A = B = C = 1 and with characteristic wavenumber k 0 = 1 in the interval of Reynolds number 13 ≤ R ≤ 50. We solve numerically the forced Navier-Stokes equations with periodic boundary conditions for up to 9.9 × 10 4 eddy turnover times. Bifurcations towards progressively more complex flows obtain, with a relaminarization window, loss of symmetries, and chaotic oscillations probably revealing an underlying heteroclinic structure. In the chaotic regime, only three steady solutions emerge besides A 1; they consist of a perturbed ABC flow A 2 with A = B ≠ C plus cyclic permutations. At 23 ≤ R ≤ 50 an unstructured temporal chaos is observed with the flow still dominated by the largest scales.

On the Lagrangian nature of the turbulence energy cascade

The spatial and temporal evolution of turbulence kinetic energy at different scales is studied using direct numerical simulations of isotropic turbulence. To explicitly follow the energy during the cascade process in physical space, a Lagrangian correlation coefficient between local kinetic energy at different scales is computed. This correlation is found to peak only after a Lagrangian time delay that is an increasing function of the scale separation. It is shown that a characteristic length reduction of a factor of 2 is achieved approximately after the local eddy-turnover time scale. The results show that the view of spatially localized eddy structures transferring their kinetic energy to smaller scales appears to be, on average, quite realistic.

The role of self-consistent Lagrangian chaos in Bénard convection in an annulus

The nonlinear behavior of the two-dimensional Bénard problem with periodic boundary conditions in the horizontal direction is studied with particular emphasis on the role of self-consistent chaotic advection. The results show a complex interplay between vortices driven by the Bénard (Rayleigh–Taylor) instability and shear flow, which is driven by the vortices [J. Drake et al., Phys. Fluids B 4, 4881 (1992)] and which causes their decay. Chaotic advection occurs in the transition from the low Rayleigh number (Ra) regime to the high Ra regime [J. Finn, Phys. Fluids B 5, 415 (1993)]. For the former, vortex flow and shear flow coexist, possibly with slow relaxation oscillations. In the high Ra regime there are vortices localized near the upper and lower boundaries with a shear flow in between. As Ra is decreased from the high Ra regime, these vortices broaden, eventually overlapping, causing self-consistent Lagrangian chaos. This onset of chaos is responsible for several properties of the transition state between the low Ra and the high Ra regimes, most notably the damping of the relaxation oscillations involving vortex and shear flow. It is also observed that the Nusselt number Nu has a peak with respect to Ra in this transition regime characterized by Lagrangian chaos. In the low Ra regime, on the other hand, the relaxation oscillations are on a much slower time scale than the eddy turnover time and the Lagrangian behavior is described by separatrix crossing.

Generation of Mean Flows and Jets on a Beta Plane and over Topography

Abstract This paper proposes and discusses mechanisms whereby mean flows and jets are produced by differential rotation and by topographic effects. It is shown that, in general, a mean gradient of potential vorticity not only inhibits the cascade of energy to large scales but directly produces anisotropic structures. Scalings for this are examined on the β plane using ideas from classical phenomenology. The scalings are naturally anisotropic and predict the formation of zonal flows directly through a turbulent cascade. Numerical simulations and two-point closure calculations qualitatively confirm the predictions. Also, simulations of barotropic flow on the β plane can produce zonal jet structures of exceptional persistence over many eddy turnover times. Unsteady flow over topography generally produces a mean flow with a correlation between streamfunction and topography, with anticyclonic motion over humps. If the topography is shallow (or the flow sufficiently energetic) the mean streamfunction will be of...

Optimal perturbations and streak spacing in wall-bounded turbulent shear flow

The mean streak spacing of approximately 100 wall units that is observed in wall-bounded turbulent shear flow is shown to be consistent with near-wall streamwise vortices optimally configured to gain the most energy over an appropriate turbulent eddy turnover time. The streak spacing arising from the optimal perturbation increases with distance from the wall and is nearly independent of Reynolds number, in agreement with experiment.

Intermittency and predictability in turbulence.

We discuss the relation between predictability and the sensitive dependence on initial conditions in turblent flows. We find that the maxium Lyapunov exponent \ensuremath{\lambda} and the variance of the effective Lyapunov exponent diverge as a power of the Reynolds number with scaling exponents which can be calculated from the eddy turnover time at the Kolmogorov length using the multifractal approach. The intermittency leads to long tails in the distribution of the predictability time. The typical predictability time T is related to the value of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$. We provide numerical evidence of this picture within the framework of a cascade model for three-dimensional fully developed turbulence.

Spatiotemporal intermittency and instability of a forced turbulence

The time development of an infinitesimal disturbance and the relation to the spatiotemporal intermittency of a developed turbulence is studied by solving numerically the Navier–Stokes equation and its linearized form. A high-symmetric flow is integrated by the use of the spectral method with 3403 effective modes to realize a developed turbulent field with Reynolds number of about 186 which is in statistically stationary state. Energy transfer in the wave-number space is not uniform in time but quiescent and active periods of energy transfer and of the energy dissipation rate occur repeatedly in a large-scale eddy-turnover time. Vorticity tends to be concentrated in long thin tubelike regions and the energy dissipation field forms double peaks around a tube, which resembles a flow around a nonaxisymmetric Burgers vortex. Both the energy and the enstrophy of the disturbance field increase exponentially in time with growth rate comparable to the reciprocal of the time scale of the peak wave number of the energy dissipation spectrum. The energy spectrum of the disturbance field shows an equipartition of energy at small wave numbers including the inertial range. The disturbance vorticity field has a dipole structure at the position of a vorticity concentrated tube. The direction of this dipole structure is orthogonal to that of the energy dissipation field.

Relaxation in two dimensions and the ‘‘sinh-Poisson’’ equation

Long-time states of a turbulent, decaying, two-dimensional, Navier–Stokes flow are shown numerically to relax toward maximum-entropy configurations, as defined by the ‘‘sinh-Poisson’’ equation. The large-scale Reynolds number is about 14 000, the spatial resolution is (512)2, the boundary conditions are spatially periodic, and the evolution takes place over nearly 400 large-scale eddy-turnover times.

Two-dimensional hydrodynamic calculations of the nonlinear development of the Goldreich-Schubert-Fricke instability in a rotating annulus

Two-dimensional nonlinear hydrodynamic calculations are presented which may help assess the effectiveness of the instability in transporting angular momentum in the equatorial zones of stars and planets which are stably stratified with respect to convection. The calculations were made by numerically integrating the 2D axisymmetric Navier-Stokes equations, including viscosity and heat conduction. The instability was followed into the nonlinear regime. The maximum rms velocity amplitude was found to correlate well with the product of the linear growth rate and radial length scale of the instability, consistent with the idea that the instability grows to an amplitude such that an eddy turnover time becomes equal to the growth time defined by the inverse of the growth rate. The time scale for angular momentum to be redistributed to a state of marginal stability was consistent with this picture. The results suggest that in physical situations a state of marginal stability will be maintained, since departures from such a state will be rapidly corrected.

Decaying, two-dimensional, Navier-Stokes turbulence at very long times

Two-dimensional Navier-Stokes turbulent decay has been followed numerically for very long times. The code is spectral, satisfies periodic boundary conditions, and does not make use of hyperviscosity or any small-scale smoothing. The resolution is (512) 2, the initial Reynolds number based on the box dimension is about 14000, and the run continues for over 190 initial eddy turnover times. Isolated vortices form, and the turbulence has become highly intermittent in the manner seen by McWilliams and Brachet et al., for times greater than about 30. However, it is not the case that merger of like-signed vortices stops; it only slows down. By t = 210, the final merger is complete, and the vorticity distribution is dominated by one large vortex of either sign. The negative (positive) vortices each occupy about two percent of the box area, and together account for over 98 percent of the total enstrophy. Their alignment suggests the formation of an Ewald lattice with a basic cell containing two point vortices. The ratio of enstrophy to energy continues to decrease monotonically, and the picture is consistent with a “selective decay” process, as described some time ago. A not-entirely-understood phenomenon is the concentration of the vorticity into two cores, suggestive of a negative-temperature state of the discrete line vortex model.

The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales

The nonlinear interscale couplings in a turbulent flow are studied through direct numerical simulations of the response of isotropic turbulence to isotropic and anisotropic forcing applied at the large scales. Specifically, forcing is applied to the energy-containing wave-number range for about two eddy turnover times to fully developed isotropic turbulence at Taylor-scale Reynolds number 32 on an 1283 grid. When forced isotropically, the initially isotropic turbulence remains isotropic at all wave numbers. However, anisotropic forcing applied through an array of counter-rotating rectilinear vortices induces high levels of anisotropy at the small scales. At low wave numbers the force term feeds energy directly into two velocity components in the plane of the forced vortices. In contrast, at high wave numbers the third (spanwise) component receives the most energy, producing small-scale anisotropy very different from that at the large scales. Detailed analysis shows that the development of small-scale anisotropy is caused primarily by nonlocal wave-vector triads with one leg in the forced low-wave-number range. This latter result is particularly significant because asymptotic analysis of the Fourier-transformed Navier–Stokes equations shows that distant triadic interactions coupling the energy-containing and dissipative scales persist at asymptotically high Reynolds numbers, suggesting that the structural couplings between large and small scales in these moderate Reynolds number simulations would also exist in high Reynolds number forced turbulence. The results therefore imply a departure from the classical hypothesis of statistical independence between large- and small-scale structure and local isotropy.