Mean Enstrophy Research Articles

RETRACTED – Reynolds-number scaling and convergence time scale in two-dimensional Rayleigh–Bénard convection

An equation for the evolution of mean kinetic energy, $E$ , in a two-dimensional (2-D) or 3-D Rayleigh–Bénard system with domain height $L$ is derived. Assuming classical Nusselt-number scaling, $Nu \sim Ra^{1/3}$ , and that mean enstrophy, in the absence of a downscale energy cascade, scales as $\sim E/L^2$ , we find that the Reynolds number scales as $Re \sim Pr^{-1}Ra^{2/3}$ in the 2-D system, where $Ra$ is the Rayleigh number and $Pr$ the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that $\tilde {\tau } \gtrsim Pr^{-1/2}Ra^{1/2}$ , where $\tilde {\tau }$ is the non-dimensional convergence time scale. For the 3-D system, we make the estimate $\tilde {\tau } \gtrsim Ra^{1/6}$ for $Pr = 1$ . It is estimated that the total computational cost of reaching the high $Ra$ limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.

Enstrophy evolution during head-on wall interaction of premixed flames within turbulent boundary layers

The statistical behaviors of mean enstrophy and its evolution during head-on interaction of premixed flames propagating toward a chemically inert flat wall across the turbulent boundary layer have been analyzed using direct numerical simulations for a friction velocity-based Reynolds number of Reτ=110. The enstrophy dynamics have been analyzed for both isothermal and adiabatic thermal wall boundary conditions. The contributions of vortex-stretching and viscous dissipation are found to be leading order source and sink, respectively, to the mean enstrophy transport in both non-reacting and reacting flows irrespective of the wall boundary condition. However, the contributions due to dilatation rate and baroclinic torque play important roles in addition to the leading order contributions of the vortex-stretching and viscous dissipation terms in the enstrophy transport in turbulent premixed flames. The thermal boundary condition has been demonstrated to affect the near-wall behavior of the enstrophy transport contribution due to dilatation rate, which also affects the near-wall distribution of the enstrophy. The magnitudes of the leading order contributors to the enstrophy transport decrease with the progress of head-on interaction for both wall boundary conditions. Moreover, the overall sink contributions to the enstrophy transport dominate over the source contributions, giving rise to a drop in the mean enstrophy with the progress of head-on interaction. The enstrophy distribution changes significantly during flame-wall interaction, which gives rise to a modification of the relative proportion of the coherent structures in the reacting flow turbulent boundary layer compared to the corresponding non-reacting flow features.

Vortex stretching and enstrophy production in stationary homogeneous isotropic turbulence

On average, the length of a material line element at any given time is longer than that at the initial time. Because a vortex line is frozen into the fluid at an infinite Reynolds number, it seems that the relation between the mean enstrophies (half the squared vorticity) at two time instances can be easily derived using a method similar to that for a material line element. However, this relation has not been analytically derived because a vortex line depends on the velocity even at the initial time. In this paper, we analytically show, using the conditional average, that the mean enstrophy normalized by the initial enstrophy at any given time is larger than one. We also analytically show that the mean enstrophy production is positive in stationary homogeneous isotropic turbulence for high Reynolds numbers.

Meandering of a wing-tip vortex in a grid-generated turbulent flow

The interaction of a National Advisory Committee for Aeronautics 0012 wing-tip vortex with a grid-generated turbulent flow was experimentally investigated. The experiments were conducted in the near- and mid-wake regions at three free-stream turbulence (FST) levels, viz., 0.5%, 3%, and 6%, at a Reynolds number, based on the wing chord length, cw, of 2×105. Stereoscopic particle image velocimetry measurements were carried out at four downstream positions: x/cw = 1.25, 3.25, 6.25, and 7.75. Streamwise vorticity contours showed that the wing-tip vortex decayed with increased FST and downstream distance. Turbulent surroundings were found to affect the meandering amplitude of the vortex, which increased with the dispersed scatter pattern of the vortex center motion, resulting in a meandering-induced turbulence. Meandering-corrected turbulent kinetic energy revealed the existence of a laminar core at the center of the vortex surrounded by low turbulence levels outside the core. This was attributed to the stabilizing Coriolis effects of the strong rotational motion inside the vortex core, which tend to re-laminarize the turbulent fluid crossing the periphery of the vortex. Snapshot proper orthogonal decomposition analysis on the coherent component of the velocity field revealed two dominant modes forming a helical dipole, consistent with the helical displacement of a Kelvin wave with an azimuthal wavenumber |m=1|. An analysis of the terms balancing the rate of decay of the mean enstrophy revealed that increasing FST increases the stretching of the mean enstrophy within the vortex core while it reduces both its transport and convection terms. Nonetheless, the latter contributions were larger in all cases studied acting as the main mechanism for mean enstrophy decay.

On the Optimal Design of Wall‐to‐Wall Heat Transport

AbstractWe consider the problem of optimizing heat transport through an incompressible fluid layer. Modeling passive scalar transport by advection‐diffusion, we maximize the mean rate of total transport by a divergence‐free velocity field. Subject to various boundary conditions and intensity constraints, we prove that the maximal rate of transport scales linearly in the r.m.s. kinetic energy and, up to possible logarithmic corrections, as the one‐third power of the mean enstrophy in the advective regime. This makes rigorous a previous prediction on the near optimality of convection rolls for energy‐constrained transport. On the other hand, optimal designs for enstrophy‐constrained transport are significantly more difficult to describe: we introduce a “branching” flow design with an unbounded number of degrees of freedom and prove it achieves nearly optimal transport. The main technical tool behind these results is a variational principle for evaluating the transport of candidate designs. The principle admits dual formulations for bounding transport from above and below. While the upper bound is closely related to the “background method,” the lower bound reveals a connection between the optimal design problems considered herein and other apparently related model problems from mathematical materials science. These connections serve to motivate designs. © 2019 Wiley Periodicals, Inc.

Genesis and evolution of velocity gradients in near-field spatially developing turbulence

This paper investigates the dynamics of velocity gradients for a spatially developing flow generated by a single square element of a fractal square grid at low inlet Reynolds number through direct numerical simulation. This square grid-element is also the fundamental block of a classical grid. The flow along the grid-element centreline is initially irrotational and becomes turbulent further downstream due to the lateral excursions of vortical turbulent wakes from the grid-element bars. We study the generation and evolution of the symmetric and anti-symmetric parts of the velocity gradient tensor for this spatially developing flow using the transport equations of mean strain product and mean enstrophy respectively. The choice of low inlet Reynolds number allows for fine spatial resolution and long simulations, both of which are conducive in balancing the budget equations of the above quantities. The budget analysis is carried out along the grid-element centreline and the bar centreline. The former is observed to consist of two subregions: one in the immediate lee of the grid-element which is dominated by irrotational strain, and one further downstream where both strain and vorticity coexist. In the demarcation area between these two subregions, where the turbulence is inhomogeneous and developing, the energy spectrum exhibits the best$-5/3$power-law slope. This is the same location where the experiments at much higher inlet Reynolds number show a well-defined$-5/3$spectrum over more than a decade of frequencies. Yet, the$Q{-}R$diagram, where$Q$and$R$are the second and third invariants of the velocity gradient tensor, remains undeveloped in the near-grid-element region, and both the intermediate and extensive strain-rate eigenvectors align with the vorticity vector. Along the grid-element centreline, the strain is the first velocity gradient quantity generated by the action of pressure Hessian. This strain is then transported downstream by fluctuations and strain self-amplification is activated a little later. Further downstream, vorticity from the bar wakes is brought towards the grid-element centreline, and, through the interaction with strain, leads to the production of enstrophy. The strain-rate tensor has a statistically axial stretching form in the production region, but a statistically biaxial stretching form in the decay region. The usual signatures of velocity gradients such as the shape of$Q{-}R$diagrams and the alignment of vorticity vector with the intermediate eigenvector are detected only in the decay region even though the local Reynolds number (based on the Taylor length scale) is only between 30 and 40.

Scale-to-scale energy and enstrophy transport in two-dimensional Rayleigh–Taylor turbulence

We apply a recently developed filtering approach, i.e. filter-space technique (FST), to study the scale-to-scale transport of kinetic energy, thermal energy, and enstrophy in two-dimensional (2D) Rayleigh–Taylor (RT) turbulence. Although the scaling laws of the energy cascades in 2D RT systems follow the Bolgiano–Obukhov (BO59) scenario due to buoyancy forces, the kinetic energy is still found to be, on average, dynamically transferred to large scales by an inverse cascade, while both the mean thermal energy and the mean enstrophy move towards small scales by forward cascades. In particular, there is a reasonably extended range over which the transfer rate of thermal energy is scale-independent and equals the corresponding thermal dissipation rate at different times. This range functions similarly to the inertial range for the kinetic energy in the homogeneous and isotropic turbulence. Our results further show that at small scales the fluctuations of the three instantaneous local fluxes are highly asymmetrically distributed and there is a strong correlation between any two fluxes. These small-scale features are signatures of the mixing and dissipation of fluids with steep temperature gradients at the fluid interfaces.

A direct numerical simulation study of vorticity transformation in weakly turbulent premixed flames

Database obtained earlier in 3D Direct Numerical Simulations (DNS) of statistically stationary, 1D, planar turbulent flames characterized by three different density ratios σ is processed in order to investigate vorticity transformation in premixed combustion under conditions of moderately weak turbulence (rms turbulent velocity and laminar flame speed are roughly equal to one another). In cases H and M characterized by σ = 7.53 and 5.0, respectively, anisotropic generation of vorticity within the flame brush is reported. In order to study physical mechanisms that control this phenomenon, various terms in vorticity and enstrophy balance equations are analyzed, with both mean terms and terms conditioned on a particular value c of the combustion progress variable being addressed. Results indicate an important role played by baroclinic torque and dilatation in transformation of average vorticity and enstrophy within both flamelets and flame brush. Besides these widely recognized physical mechanisms, two other effects are documented. First, viscous stresses redistribute enstrophy within flamelets, but play a minor role in the balance of the mean enstrophy \documentclass[12pt]{minimal}\begin{document}$\overline{\Omega }$\end{document}Ω¯ within turbulent flame brush. Second, negative correlation \documentclass[12pt]{minimal}\begin{document}$\overline{\mathbf {u}^{\prime } \cdot \nabla \Omega ^{\prime }}$\end{document}u′·∇Ω′¯ between fluctuations in velocity u and enstrophy gradient contributes substantially to an increase in the mean \documentclass[12pt]{minimal}\begin{document}$\overline{\Omega }$\end{document}Ω¯ within turbulent flame brush. This negative correlation is mainly controlled by the positive correlation between fluctuations in the enstrophy and dilatation and, therefore, dilatation fluctuations substantially reduce the damping effect of the mean dilatation on the vorticity and enstrophy fields. In case L characterized by σ = 2.5, these effects are weakly pronounced and \documentclass[12pt]{minimal}\begin{document}$\overline{\Omega }$\end{document}Ω¯ is reduced mainly due to viscosity. Under conditions of the present DNS, vortex stretching plays a minor role in the balance of vorticity and enstrophy within turbulent flame brush in all three cases.

Three-dimensional direct numerical simulation study of conditioned moments associated with front propagation in turbulent flows

In order to gain further insight into (i) the use of conditioned quantities for characterizing turbulence within a premixed flame brush and (ii) the influence of front propagation on turbulent scalar transport, a 3D Direct Numerical Simulation (DNS) study of an infinitely thin front that self-propagates in statistically stationary, homogeneous, isotropic, forced turbulence was performed by numerically integrating Navier-Stokes and level set equations. While this study was motivated by issues relevant to premixed combustion, the density was assumed to be constant in order (i) to avoid the influence of the front on the flow and, therefore, to know the true turbulence characteristics as reference quantities for assessment of conditioned moments and (ii) to separate the influence of front propagation on turbulent transport from the influence of pressure gradient induced by heat release. Numerical simulations were performed for two turbulence Reynolds numbers (50 and 100) and four ratios (1, 2, 5, and 10) of the rms turbulent velocity to the front speed. Obtained results show that, first, the mean front thickness is decreased when a ratio of the rms turbulent velocity to the front speed is decreased. Second, although the gradient diffusion closure yields the right direction of turbulent scalar flux obtained in the DNS, the diffusion coefficient Dt determined using the DNS data depends on the mean progress variable. Moreover, Dt is decreased when the front speed is increased, thus, indicating that the front propagation affects turbulent scalar transport even in a constant-density case. Third, conditioned moments of the velocity field differ from counterpart mean moments, thus, disputing the use of conditioned velocity moments for characterizing turbulence when modeling premixed turbulent combustion. Fourth, computed conditioned enstrophies are close to the mean enstrophy in all studied cases, thus, suggesting the use of conditioned enstrophy for characterizing turbulence within a premixed flame brush.

Direct numerical simulation of a turbulent channel flow by an improved vortex in cell method

Purpose – This study is concerned with the direct numerical simulation (DNS) of a turbulent channel flow by an improved vortex in cell (VIC) method. The paper aims to discuss these issues. Design/methodology/approach – First, two improvements for VIC method are proposed to heighten the numerical accuracy and efficiency. A discretization method employing a staggered grid is presented to ensure the consistency among the discretized equations as well as to prevent the numerical oscillation of the solution. A correction method for vorticity is also proposed to compute the vorticity field satisfying the solenoidal condition. Second, the DNS for a turbulent channel flow is conducted by the improved VIC method. The Reynolds number based on the friction velocity and the channel half width is 180. Findings – It is highlighted that the simulated turbulence statistics, such as the mean velocity, the Reynolds shear stress and the budget of the mean enstrophy, agree well with the existing DNS results. It is also shown that the organized flow structures in the near-wall region, such as the streaks and the streamwise vortices, are favourably captured. These demonstrate the high applicability of the improved VIC method to the DNS for wall turbulent flows. Originality/value – This study enables the VIC method to perform the DNS for wall turbulent flows.

Rayleigh–Bénard convection in a supercritical fluid along its critical isochore in a shallow cavity

We perform a numerical investigation of the Rayleigh–Bénard convection in supercritical nitrogen in a shallow enclosure with an aspect ratio of 4. The transient and steady-state fluid behaviors over a wide range of initial distances to the critical point along the critical isochore are obtained and analyzed in response to modest homogeneous bottom heating. On account of the fluid layer being extremely thin, density stratification is notably excluded from consideration herein, which leads to the dominating role of the Rayleigh criterion in the onset of convection. Following the Boussinesq approximation, we find the power law scaling relationships over five decades of the Rayleigh number (Ra) for various transient quantities including the exponential growth rate of the mean enstrophy in the cavity and the characteristic times of the development of convective motion. The correlation of the Nusselt number versus the Rayleigh number shows asymptotic features at the two ends of the Ra spectrum, which incidentally correspond to different convection patterns. Under the regime of high Ra, the heat transfer through the fluid cavity is enhanced by the turbulent bursts of thermal plumes from the boundary layers. On the other hand, under the regime of low Ra, it is the orderly multicellular flow that moves heat from the bottom of the layer to the top, which includes a transition from a four-cell structure to a six-cell structure with decreasing Ra.

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.

On the effects of microbubbles on Taylor–Green vortex flow

The paper describes a numerical study of the effects of microbubbles on the vorticity dynamics in a Taylor–Green vortex flow (TGV) using the two-fluid approach. The results show that bubbles with a volume fraction ∼10−2 enhance the decay rate of the vorticity at the centre of the vortex. Analysis of the vorticity equation of the bubble-laden flow shows that the local positive velocity divergence of the fluid velocity, ∇·U, created in the vortex core by bubble clustering, is responsible for the vorticity decay. At the centre of the vortex, the vorticity ωc(t) decreases nearly linearly with the bubble concentration Cm(t). Similarly, the enstrophy in the core of the vortex, ω2(t), decays nearly linearly with C2(t). The approximate mean-enstrophy equation shows that bubble accumulation in the high-enstrophy core regions produces a positive correlation between ω2 and ∇·U, which enhances the decay rate of the mean enstrophy.

Structural studies of decaying fluid turbulence: Effect of initial conditions

We present results from a systematic numerical study of structural properties of an unforced, incompressible, homogeneous, and isotropic three-dimensional turbulent fluid with an initial energy spectrum that develops a cascade of kinetic energy to large wave numbers. The results are compared with those from a recently studied set of power-law initial energy spectra [C. Kalelkar and R. Pandit, Phys. Rev. E 69, 046304 (2004)] which do not exhibit such a cascade. Differences are exhibited in plots of vorticity isosurfaces, the temporal evolution of the kinetic energy-dissipation rate, and the rates of production of the mean enstrophy along the principal axes of the strain-rate tensor. A crossover between "non-cascade-type" and "cascade-type" behavior is shown numerically for a specific set of initial energy spectra.

Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence

Some rigorous results connected with the conventional statistical theory of turbulence in both the two- and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular forcing term and for arbitrary stationary statistical solutions.

A MODEL OF TWO-DIMENSIONAL TURBULENCE USING RANDOM MATRIX THEORY

We derive a formula for the entropy of two-dimensional incompressible inviscid flow, by determining the volume of the space of vorticity distributions with fixed values for the moments Qk = ∫ ω(x)kd2x. This space is approximated by a sequence of spaces of finite volume, by using a regularization of the system that is geometrically natural and connected with the theory of random matrices. By taking the limit we get a simple formula for the entropy of a vortex field. We predict vorticity distributions of maximum entropy with given mean vorticity and enstrophy; we also predict the cylindrically symmetric vortex field with maximum entropy. This could be an approximate description of a hurricane.

Soap film flows: Statistics of two-dimensional turbulence

Soap film flows provide a very convenient laboratory model for studies of two-dimensional (2-D) hydrodynamics including turbulence. For a gravity-driven soap film channel with a grid of equally spaced cylinders inserted in the flow, we have measured the simultaneous velocity and thickness fields in the irregular flow downstream from the cylinders. The velocity field is determined by a modified digital particle image velocimetry method and the thickness from the light scattered by the particles in the film. From these measurements, we compute the decay of mean energy, enstrophy, and thickness fluctuations with downstream distance, and the structure functions of velocity, vorticity, thickness fluctuation, and vorticity flux. From these quantities we determine the microscale Reynolds number of the flow Rλ≈100 and the integral and dissipation scales of 2D turbulence. We also obtain quantitative measures of the degree to which our flow can be considered incompressible and isotropic as a function of downstream distance. We find coarsening of characteristic spatial scales, qualitative correspondence of the decay of energy and enstrophy with the Batchelor model, scaling of energy in k space consistent with the k−3 spectrum of the Kraichnan–Batchelor enstrophy-scaling picture, and power-law scalings of the structure functions of velocity, vorticity, vorticity flux, and thickness. These results are compared with models of 2-D turbulence and with numerical simulations.

Noise-sustained currents in quasigeostrophic turbulence over topography

We study the development of directed mean currents in a nonlinear model of fluid flow appropriate for large-scale ocean dynamics in the presence of bottom topography and dissipation, and forced with a noise term. We show that the presence of noise leads to persistent average currents directed along isobaths. The effect requires the presence of both the nonlinear and the random terms, and can be though of as a rectification of the stochastic energy input by the combined effect of nonlinearity and topography. At variance with previous works we use a scale-unselective dissipation, so that the phenomenon can not be explained in terms of minimum enstrophy states. Despite the fact that the system is out of equilibrium the large-scales are well described by a generalized canonical equilibrium with mean energy and enstrophy determined by a balance between random forcing and dissipation. This result allows predicting the strenght of the noise-sustained currents. Finally we discuss the relevance that these noise-induced currents could have on real ocean circulation.

Small-scale characteristics of a turbulent boundary layer over a rough wall

Measurements of the spanwise and wall-normal components of vorticity and their constituent velocity derivative fluctuations have been made in a turbulent boundary layer over a mesh-screen rough wall using a four-hot-wire vorticity probe. The measured spectra and variances of vorticity and velocity derivatives have been corrected for the effect of spatial resolution. The high-wavenumber behaviour of the spectra conforms closely with isotropy. Over most of the outer layer, the normalized magnitudes of the velocity derivative variances differ significantly from those over a smooth wall layer. The differences are such that the variances are much more nearly isotropic over the rough wall than on the smooth wall. This behaviour is consistent with earlier observations that the large-scale structure in this rough wall layer is more isotropic than that in a smooth wall layer. Isotropy-based approximations for the mean energy dissipation rate and mean enstrophy are consequently more reliable in this rough wall layer than in a smooth wall layer. In the outer layer, the vorticity variances are slightly larger than those over a smooth wall; reflecting structural differences between the two flows.

Statistical solutions of the Navier–Stokes equations on the phase space of vorticity and the inviscid limits

Using the methods of Foias [Sem. Math. Univ. Padova 48, 219–343 (1972); 49, 9–123 (1973)] and Vishik–Fursikov [Mathematical Problems of Statistical Hydromechanics (Kluwer, Dordrecht, 1988)], we prove the existence and uniqueness of both spatial and space–time statistical solutions of the Navier–Stokes equations on the phase space of vorticity. Here the initial vorticity is in Yudovich space and the initial measure has finite mean enstrophy. We show under further assumptions on the initial vorticity that the statistical solutions of the Navier–Stokes equations converge weakly and the inviscid limits are the corresponding statistical solutions of the Euler equations.