Enstrophy Dissipation Rate Research Articles

Vanishing of Long Time average p-enstrophy Dissipation Rate in the Inviscid Limit of the 2D Damped Navier–Stokes Equations

In Constantin and Ramos (Commun Math Phys 275(2), 529–551, 2007), Constantin and Ramos proved a result on the vanishing long time average enstrophy dissipation rate in the inviscid limit of the 2D damped Navier–Stokes equations. In this work, we prove a generalization of this for the p-enstrophy, 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1< p < \\infty $$\\end{document}, sequences of distributions of initial data and sequences of strongly converging right-hand sides. We simplify their approach by working with invariant measures on the global attractors which can be characterized via bounded complete solution trajectories. Then, working on the level of trajectories allows us to directly employ some recent results on strong convergence of the vorticity in the inviscid limit.

Investigation on leakage vortex cavitation and corresponding enstrophy characteristics in a liquid nitrogen inducer

A modified thermal cavitation model was employed to study the thermodynamic cavitation and effects of tip leakage vortex (TLV) on hydraulic losses in a three-blade inducer. Comparative analyses for TLV cavitation under different temperatures were presented. The tip leakage flow was spirally distributed, and its shape was negatively correlated with the nitrogen temperature. Thermodynamic effects reduced the vapor volume in the cavitation region and vorticity primarily at the leading edge, thus inhibiting the development of TLV. Enstrophy analysis was proposed to reveal the energy losses on TLV, and the enstrophy dissipation rate showed good predictability for TLV evolution. With the increasing temperature, the intensity of energy losses was reduced due to the thermal effects combined with weak cavitation intensity. Moreover, the change of temperature mainly influenced the primary tip leakage vortex (PTLV) but had minimal impact of the secondary tip leakage vortex. The energy losses were positive correlated with TLV, and PTLV was offset to the hub of the inducer due to the thermodynamic effects. Thus, the tip leakage can be considered as a key parameter to minimize the occurrence of thermodynamic cavitation in a liquid nitrogen inducer toprovide a reference for the design of the inducer.

Intermittency of turbulent velocity and scalar fields using three-dimensional local averaging

An efficient approach for extracting 3D local averages in spherical subdomains is proposed and applied to study the intermittency of small-scale velocity and scalar fields in direct numerical simulations of isotropic turbulence. We focus on the inertial-range scaling exponents of locally averaged energy dissipation rate, enstrophy and scalar dissipation rate corresponding to the mixing of a passive scalar $\theta$ in the presence of a uniform mean gradient. The Taylor-scale Reynolds number $R_\lambda$ goes up to $1300$, and the Schmidt number $Sc$ up to $512$ (albeit at smaller $R_\lambda$). The intermittency exponent of the energy dissipation rate is $\mu \approx 0.23$, whereas that of enstrophy is slightly larger; trends with $R_\lambda$ suggest that this will be the case even at extremely large $R_\lambda$. The intermittency exponent of the scalar dissipation rate is $\mu_\theta \approx 0.35$ for $Sc=1$. These findings are in essential agreement with previously reported results in the literature. We further show that $\mu_\theta$ decreases monotonically with increasing $Sc$, either as $1/\log Sc$ or a weak power law, suggesting that $\mu_\theta \to 0$ as $Sc \to \infty$, reaffirming recent results on the breakdown of scalar dissipation anomaly in this limit.

Enstrophy dissipation of the tip leakage vortex in a multiphase pump

Tip clearance is the distance required between the blade tip and the pump body wall of the impeller in the multiphase pump, of which the resulting tip leakage vortex (TLV) causes unstable flow, leading to energy dissipation. However, only a few studies have been made on the energy dissipation caused by the TLV, and the internal mechanism of energy dissipation has not been revealed. In the present work, enstrophy dissipation theory is innovatively applied to quantitatively study the energy dissipation of the TLV in impeller, to provide guidelines for controlling energy dissipation associated with the TLV, and to optimize the design of the multiphase pump. The location, mode, and energy dissipation rate caused by the TLV are analyzed. The relationship between vorticity and the enstrophy dissipation rate is summarized, and the energy dissipation law is revealed during the inception, development, and dissipation of the TLV. The present analysis indicates that the vorticity is highest at the core of the TLV and gradually weakens along the radial direction with the vortex core at the center; the enstrophy dissipation, however, presented the opposite distribution law. The enstrophy dissipation rate changes as the spatialtemporal evolution of the TLV. The gas phase significantly deteriorates the flow pattern of the TLV, enhances the volume enstrophy dissipation rate, and reduces the wall enstrophy dissipation rate. The volume enstrophy dissipation power increases by 45.33% with an inlet gas void fraction of 10%, the wall enstrophy dissipation power decreases by 23.90%, and the total enstrophy dissipation power increases by 17.21%.

Enstrophy dissipation and vortex thinning for the incompressible 2D Navier–Stokes equations

Vortex thinning is one of the main mechanisms of two-dimensional turbulence. By direct numerical simulation to the two-dimensional Navier–Stokes equations with small-scale forcing and large-scale damping, Xiao et al (2009 J. Fluid Mech. 619 1–44) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. On the other hand, Alexakis and Doering (2006 Phys. Lett. A 359 652–657) calculated the upper bound of the bulk averaged enstrophy dissipation rate of the steady-state two dimensional turbulence. In this paper, we show that vortex thinning induces enhanced dissipation with strictly slower vanishing order of the enstrophy dissipation than Re−1.

Relative particle dispersion in two-dimensional and quasi-geostrophic turbulence

In this paper, phenomenological developments are used to explore relative particle dispersion (RPD) in two-dimensional (2D) and quasi-geostrophic (QG) fully-developed turbulence (FDT). The role played by the 2D and QG FDT cascade physics underlying this process is given special attention. Prevalence of spatial intermittency effects in 2D FDT enstrophy cascade, however small, is shown to lead to a structural change in the RPD growth law, more specifically the development of power-law scaling of RPD; this corroborates the difficulty in observing Lin’s (1972) exponential scaling law in laboratory experiments (Jullien (2003)). QG effects are found to lead to an enhanced RPD in the baroclinic regime of the energy cascade, which seems to be traceable to a negative eddy-viscosity characterizing the latter regime. They are also found to lead to particle clumping in the baroclinic regime of the enstrophy cascade (this aspect appears to be associated with the tendency of divorticity sheets to occur near the vortex nulls where particle clumping is known, as per numerical simulations and laboratory experiments, to be favored to occur). These results are developed from the established scaling relations for 2D and QG FDT and are validated further via alternative dimensional/scaling developments for 2D and QG FDT similar to the one given for 3D FDT by Batchelor and Townsend (1956). The feasibility of spatial intermittency effects is underscored via the nonlinear scaling dependence of RPD on the enstrophy (or energy) dissipation rate.

Enstrophy spectrum in freely decaying two-dimensional self-similar turbulent flow

We consider freely decaying two-dimensional homogeneous and isotropic turbulent motion in a self-similar limit that is achieved at large Reynolds numbers based on time and the mean kinetic energy of the flow provided that initial average enstrophy tends to infinity as fluid viscosity tens to zero. In this case, the enstrophy dissipation rate has a nonzero finite limit. The vorticity correlation function and the spectral enstrophy density are investigated in an inertial range of distances and wave numbers where these functions are free from the influence of viscosity and large-scale flow parameters. It turns out that in freely decaying two-dimensional self-similar turbulence, the inertial range exists in real space but is absent in the space of wave numbers. This means that turbulent eddies of the appropriate size do not contribute to the spectral density and the known ${k}^{\ensuremath{-}1}$ law does not hold. The spectral enstrophy density at large wave numbers behaves nonmonotonically: it first decreases faster than in accordance with the ${k}^{\ensuremath{-}1}$ law and then, in the dissipation region, has a growth portion and a second maximum. The enstrophy spectral flux at the boundary of the dissipation region is only a fraction of the enstrophy dissipation rate.

Statistical features of rapidly rotating decaying turbulence: Enstrophy and energy spectra and coherent structures

In this paper, we investigate the properties of rapidly rotating decaying turbulence using numerical simulations and phenomenological modeling. We find that as the turbulent flow evolves in time, the Rossby number decreases to ∼10−3, and the flow becomes quasi-two-dimensional with strong coherent columnar structures arising due to the inverse cascade of energy. We establish that a major fraction of energy is confined in Fourier modes (±1, 0, 0) and (0, ±1, 0) that correspond to the largest columnar structure in the flow. For wavenumbers (k) greater than the enstrophy dissipation wavenumber (kd), our phenomenological arguments and numerical study show that the enstrophy flux and spectrum of a horizontal cross section perpendicular to the axis of rotation are given by ϵω⁡exp(−C(k/kd)2) and Cϵω2/3k−1⁡exp(−C(k/kd)2), respectively; for this 2D flow, ϵω is the enstrophy dissipation rate, and C is a constant. Using these results, we propose a new form for the energy spectrum of rapidly rotating decaying turbulence: E(k)=Cϵω2/3k−3⁡exp(−C(k/kd)2). This model of the energy spectrum is based on wavenumber-dependent enstrophy flux, and it deviates significantly from power law energy spectrum reported earlier.

Single-Interface Richtmyer–Meshkov Turbulent Mixing at the Los Alamos Vertical Shock Tube

Mach number and initial conditions effects on Richtmyer–Meshkov (RM) mixing are studied by the vertical shock tube (VST) at Los Alamos National Laboratory (LANL). At the VST, a perturbed stable light-to-heavy (air–SF6, A = 0.64) interface is impulsively accelerated with a shock wave to induce RM mixing. We investigate changes to both large and small scales of mixing caused by changing the incident Mach number (Ma = 1.3 and 1.45) and the three-dimensional (3D) perturbations on the interface. Simultaneous density (quantitative planar laser-induced fluorescence (PLIF)) and velocity (particle image velocimetry (PIV)) measurements are used to characterize preshock initial conditions and the dynamic shocked interface. Initial conditions and fluid properties are characterized before shock. Using two types of dynamic measurements, time series (N = 5 realizations at ten locations) and statistics (N = 100 realizations at a single location) of the density and velocity fields, we calculate several mixing quantities. Mix width, density-specific volume correlations, density–vorticity correlations, vorticity, enstrophy, strain, and instantaneous dissipation rate are examined at one downstream location. Results indicate that large-scale mixing, such as the mix width, is strongly dependent on Mach number, whereas small scales are strongly influenced by initial conditions. The enstrophy and strain show focused mixing activity in the spike regions.

Stably stratified turbulence in the presence of large-scale forcing.

We perform two high-resolution direct numerical simulations of stratified turbulence for Reynolds number equal to Re≈25000 and Froude number, respectively, of Fr≈0.1 and Fr≈0.03. The flows are forced at large scale and discretized on an isotropic grid of 2048(3) points. Stratification makes the flow anisotropic and introduces two extra characteristic scales with respect to homogeneous isotropic turbulence: the buoyancy scale, L(B), and the Ozmidov scale, ℓ(oz). The former is related to the number of layers that the flow develops in the direction of gravity, and the latter is regarded as the scale at which isotropy is recovered. The values of L(B) and ℓ(oz) depend on the Froude number, and their absolute and relative amplitudes affect the repartition of energy among Fourier modes in ways that are not easy to predict. By contrasting the behavior of the two simulated flows we identify some surprising similarities: After an initial transient the two flows evolve towards comparable values of the kinetic and potential enstrophy and energy dissipation rate. This is the result of the Reynolds number being large enough in both flows for the Ozmidov scale to be resolved. When properly dimensionalized, the energy dissipation rate is compatible with atmospheric observations. Further similarities emerge at large scales: The same ratio between potential and total energy (≈0.1) is spontaneously selected by the flows, and slow modes grow monotonically in both regimes, causing a slow increase of the total energy in time. The axisymmetric total energy spectrum shows a wide variety of spectral slopes as a function of the angle between the imposed stratification and the wave vector. One-dimensional energy spectra computed in the direction parallel to gravity are flat from the forcing up to buoyancy scale. At intermediate scales a ∼k(-3) parallel spectrum develops for the Fr≈0.03 run, whereas for weaker stratification, the saturation spectrum does not have enough scales to develop and instead one observes a power law compatible with Kolmogorov scaling. Finally, the spectrum of helicity is flat until L(B), as observed in the nocturnal planetary boundary layer.

On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface

AbstractWe investigate the shock-induced turbulent mixing between a light and a heavy gas, where a Richtmyer–Meshkov instability (RMI) is initiated by a shock wave with Mach number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ma}= 1.5$. The prescribed initial conditions define a deterministic multimode interface perturbation between the gases, which can be imposed exactly for different simulation codes and resolutions to allow for quantitative comparison. Well-resolved large-eddy simulations are performed using two different and independently developed numerical methods with the objective of assessing turbulence structures, prediction uncertainties and convergence behaviour. The two numerical methods differ fundamentally with respect to the employed subgrid-scale regularisation, each representing state-of-the-art approaches to RMI. Unlike previous studies, the focus of the present investigation is to quantify the uncertainties introduced by the numerical method, as there is strong evidence that subgrid-scale regularisation and truncation errors may have a significant effect on the linear and nonlinear stages of the RMI evolution. Fourier diagnostics reveal that the larger energy-containing scales converge rapidly with increasing mesh resolution and thus are in excellent agreement for the two numerical methods. Spectra of gradient-dependent quantities, such as enstrophy and scalar dissipation rate, show stronger dependences on the small-scale flow field structures as a consequence of truncation error effects, which for one numerical method are dominantly dissipative and for the other dominantly dispersive. Additionally, the study reveals details of various stages of RMI, as the flow transitions from large-scale nonlinear entrainment to fully developed turbulent mixing. The growth rates of the mixing zone widths as obtained by the two numerical methods are ${\sim } t^{7/12}$ before re-shock and ${\sim } (t-t_0)^{2/7}$ long after re-shock. The decay rate of turbulence kinetic energy is consistently ${\sim } (t-t_0)^{-10/7}$ at late times, where the molecular mixing fraction approaches an asymptotic limit $\varTheta \approx 0.85$. The anisotropy measure $\langle a \rangle _{xyz}$ approaches an asymptotic limit of ${\approx }0.04$, implying that no full recovery of isotropy within the mixing zone is obtained, even after re-shock. Spectra of density, turbulence kinetic energy, scalar dissipation rate and enstrophy are presented and show excellent agreement for the resolved scales. The probability density function of the heavy-gas mass fraction and vorticity reveal that the light–heavy gas composition within the mixing zone is accurately predicted, whereas it is more difficult to capture the long-term behaviour of the vorticity.

Energy and potential enstrophy flux constraints in quasi-geostrophic models

We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to become large enough to yield a mixed energy spectrum where the dominant k−3 scaling is overtaken by a subdominant k−5/3 contribution beyond a transition wavenumber kt situated in the inertial range. The validity of the flux inequality implies that this scaling transition cannot occur within the inertial range, whereas a violation of the flux inequality beyond some wavenumber kt implies the existence of a scaling transition near that wavenumber. This flux inequality holds unconditionally in two-dimensional Navier–Stokes turbulence, however, it is far from obvious that it continues to hold in multi-layer quasi-geostrophic models, because the dissipation rate spectra for energy and potential enstrophy no longer relate in a trivial way, as in two-dimensional Navier–Stokes. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic model, where the dissipation terms for each layer consist of the same Fourier-diagonal linear operator applied on the streamfunction field of only the same layer, the flux inequality continues to hold. It follows that a necessary condition to violate the flux inequality is the use of asymmetric dissipation where different operators are used on different layers. We explore dissipation asymmetry further in the context of a two-layer quasi-geostrophic model and derive upper bounds on the asymmetry that will allow the flux inequality to continue to hold. Asymmetry is introduced both via an extrapolated Ekman term, based on a 1980 model by Salmon, and via differential small-scale dissipation. The results given are mathematically rigorous and require no phenomenological assumptions about the inertial range. Sufficient conditions for violating the flux inequality, on the other hand, require phenomenological hypotheses, and will be explored in future work.

Intermittency in premixed turbulent reacting flows

Intermittency in premixed reacting flows is studied using numerical simulations of premixed flames at a range of turbulence intensities. The flames are modeled using a simplified reaction mechanism that represents a stoichiometric H2-air mixture. Intermittency is associated with high probabilities of large fluctuations in flow quantities, and these fluctuations can have substantial effects on the evolution and structure of premixed flames. Intermittency is characterized here using probability density functions (pdfs) and moments of the local enstrophy, pseudo-dissipation rate (strain rate magnitude), and scalar (reactant mass fraction) dissipation rate. Simulations of homogeneous isotropic turbulence with a nonreacting passive scalar are also carried out in order to provide a baseline for analyzing the reacting flow results. In the reacting flow simulations, conditional analyses based on local, instantaneous values of the scalar are used to study variations in the pdfs, moments, and intermittency through the flame. For low intensities, pdfs of the local enstrophy vary substantially through the flame, with greater intermittency near the products. Changes in the pseudo-dissipation pdfs are, however, less pronounced. As the intensity increases, both the enstrophy and pseudo-dissipation pdfs become increasingly independent of position in the flame and are similar to results from the nonreacting simulations. The scalar dissipation intermittency is largest near the reactants and increases at all flame locations with increasing turbulence intensity. For low intensities and in the reaction zone, however, scalar dissipation pdfs approximately follow a Gaussian distribution, indicative of substantially reduced intermittency. Deviations from log-normality are observed in the pdfs of all quantities, even for intensities and flame locations characterized by strong intermittency. The implications of these results for the internal structure of the flame are discussed, and we also propose a connection between reacting flow intermittency and anisotropic vorticity suppression by the flame.

GRANULAR TURBULENCE IN TWO DIMENSIONS: MICROSCALE REYNOLDS NUMBER AND FINAL CONDENSED STATES

Granular gases from the viewpoint of "two-dimensional turbulence" are investigated. In the quasi-elastic and thermodynamic limit, we obtained clear evidence for an enstrophy (square of vorticity) cascade and -3 exponent in the Kraichnan–Leith–Bachelor energy spectrum by performing large-scale (N ~ 16.8 million number of disks) event-driven molecular dynamics simulations. In these calculations, the enstrophy dissipation rate showed a strong relationship with the evolution of the exponent in the energy spectrum. The growth of the Reynolds number based on the microscale confirmed that the enstrophy cascade regime was that of fully developed turbulence. Moreover, a condensed state resembling Bose–Einstein condensation in decaying two-dimensional Navier–Stokes turbulence also appeared as the final attractor of the evolving granular gas in the long time limit.

Testing Batchelor’s similarity hypotheses for decaying two-dimensional turbulence

We carry out three high resolution direct numerical simulations of the two-dimensional Navier–Stokes equation to test Batchelor’s similarity hypotheses of an equilibrium spectral range and an inertial subrange where the enstrophy wave number spectrum has the form Φ(k)=Cχ2/3k−1, where χ is the mean enstrophy dissipation rate and C is a constant. We use very different initial conditions in the three simulations and find that Batchelor’s hypotheses are well satisfied in each simulation. However, there is a small but significant difference between the equilibrium range spectrum of one of the simulations as compared to the spectra of the other two. We suggest that the difference is linked to the stronger degree of large scale variation of the enstrophy dissipation which is observed in this simulation as compared to the other two.

Freely decaying two-dimensional turbulence

High-resolution direct numerical simulations are used to investigate freely decaying two-dimensional turbulence. We focus on the interplay between coherent vortices and vortex filaments, the second of which give rise to an inertial range. We find that Batchelor's prediction for the inertial-range enstrophy spectrum Eω(k, t) ~ β2/3k−1, where β is the enstrophy dissipation rate, is reasonably well satisfied once the turbulence is fully developed, but that the assumptions which underpin the usual interpretation of his theory are not valid. For example, the lack of a quasi-equilibrium cascade means the enstrophy flux Πω(k) is highly non-uniform throughout the inertial range, thus the common assumption that β can act as a surrogate for Πω(k) becomes questionable. We present a variant of Batchelor's theory which accounts for the wavenumber-dependence of Πω; in particular we propose Eω(k, t) ~ Πω(k1)2/3k−1, where k1 is the wavenumber marking the start of the observed k−1 region of the enstrophy spectrum. This provides a better collapse of the data and, unlike Batchelor's original theory, can be justified on theoretical grounds. The basis for our proposal is the observation that the straining of the vortex filaments, which fuels the enstrophy flux through the inertial range, comes almost exclusively from the strain field of the coherent vortices, and this can be characterized by Πω(k1)1/3. Thus Eω(k) is a function of only k and Πω(k1) in the inertial range, and dimensional analysis then yields Eω ~ Πω(k1)2/3k−1. We also confirm the prediction by Davidson (Phys. Fluids, vol. 20, 2008, 025106) that in the inertial range Πω varies as Πω(k)/Πω(k1) = 1 − a−1 ln(k/k1), where a is a constant of order 1. This corresponds to ∂Eω/∂t ~ k−1. Surprisingly, the measured enstrophy fluxes imply that the dynamics of the inertial range as defined by the behaviour of Πω extend to wavenumbers much smaller than k1, but this is masked in Eω(k, t) by the presence of coherent vortices which also contribute to Eω in this region. In fact, we find that kEω(k, t) ≈ H(k) + A(t), or ∂Eω/∂t ~ k−1 in this extended low-k region, where H(k) is almost independent of time and represents the signature of the coherent vortices. In short, the inertial range defined by ∂Eω/∂t ~ k−1 or Πω(k) ~ ln(k) is much broader than the observed Eω ~ k−1 region.

Analogy between velocity and scalar fields in a turbulent channel flow

The relationship between the fluctuating velocity vector and the temperature fluctuation has been examined using direct numerical simulation databases of a turbulent channel flow with passive scalar transport using a constant time-averaged heat flux at each wall for h+ = 180, 395, 640 and 1020 (where h is the channel half-width with the superscript denoting normalization by wall variables) at Prandtl number Pr=0.71. The analogy between spectra corresponding to the kinetic energy and scalar variance is reasonable in both inner and outer regions irrespective of whether the spectra are plotted in terms of kx or kz, the wavenumbers in the streamwise and spanwise directions respectively. Whereas all three velocity fluctuations contribute to the energy spectrum when kx is used, the longitudinal velocity fluctuation is the major contributor when kz is used. The quality of the analogy in the spectral domain is confirmed by visualizations in physical space and reflects differences between spatial organizations in the velocity and scalar fields. The similarity between the spectra corresponding to the enstrophy and scalar dissipation rate is not as good as that between the kinetic energy and scalar variance, emphasizing the prominence of the scalar sheets as the centre of the channel is approached. The ratio R between the characteristic time scales of the velocity and scalar fluctuations is approximately constant over a major part of the channel and independent of h+, when the latter is sufficiently large. This constancy, which is not observed in quantities such as the turbulent Prandtl number, follows from the spectral similarities discussed in this paper and has implications for turbulent heat transport models.

Cascades and fluxes in two-dimensional turbulence

We develop simple models of forced and unforced turbulence at large but finite Reynolds number. These models yield a number of predictions, such as (i) the form of the second- and third-order structure functions in the inertial range; (ii) the rate of generation of palinstrophy in forced and unforced turbulence; (iii) the rate of destruction of enstrophy in decaying turbulence; (iv) the rate of change of Loitsyansky’s integral in decaying turbulence; and, again for decaying turbulence; and (v) the existence of an inverse energy cascade, embedded within the direct enstrophy cascade. We show that these predictions are not mere artifacts of our simple models, but rather genuine features of two-dimensional turbulence. The most surprising findings relate to (iii), (iv), and (v) above, where we show that, once the turbulence is fully developed, the enstrophy dissipation rate scales as β∼⟨ω2⟩3∕2∕ln(Re), Loitsyansky’s integral grows at a rate proportional to βℓ6∕ln(Re), and the filamentation of vorticity fuels an inverse energy cascade with a flux of order βℓ2∕ln(Re). (Here, ω is the vorticity, ℓ is the integral scale, β the enstrophy dissipation rate, and Re the Reynolds number.) This suggests that, in the limit of Re→∞, the enstrophy dissipation rate tends to zero and the inverse energy cascade progressively shuts down. Loitsyansky’s integral is also an invariant in this limit, provided that ℓ remains finite as Re→∞.

Energy and enstrophy dissipation in steady state 2d turbulence

Upper bounds on the bulk energy dissipation rate $\epsilon$ and enstrophy dissipation rate $\chi$ are derived for the statistical steady state of body forced two dimensional turbulence in a periodic domain. For a broad class of externally imposed body forces it is shown that $\epsilon \le k_{f} U^3 Re^{-1/2}(C_1+C_2 Re^{-1})^{1/2}$ and $\chi \le k_{f}^{3}U^3 (C_1+C_2 Re^{-1})$ where $U$ is the root-mean-square velocity, $k_f$ is a wavenumber (inverse length scale) related with the forcing function, and $Re = U /\nu k_f$. The positive coefficients $C_1$ and $C_2$ are uniform in the the kinematic viscosity $\nu$, the amplitude of the driving force, and the system size. We compare these results with previously obtained bounds for body forces involving only a single length scale, or for velocity dependent a constant-energy-flux forces acting at finite wavenumbers. Implications of our results are discussed.

Critical exponents and universality in fully developed turbulence

Abstract A new perspective on the critical phenomena aspects of fully developed turbulence (FDT) is considered. The energy (or enstrophy) dissipation rate e (or τ) is pointed out to be the suitable quantity to be identified as the order parameter for the FDT problem. Multi-fractal model has been used to provide a detailed structure for the critical exponent σ associated with this order parameter that appears to exhibit an interesting universality covering radically different FDT systems in fluid and plasma dynamics. This result also appears to provide a consistent framework for classification of dissipation field into critical, subcritical and supercritical cases. Some FDT problems that exemplify these cases are discussed. This classification is validated further by considering the probability density functions (PDF) of relevant flow-variable gradients in the FDT problems exemplifying the various cases.