Enstrophy Decay Research Articles

Universal Law of Enstrophy Decay in Two-Dimensional Large-Reynolds-Number Turbulence

A theory of enstrophy decay in two-dimensional turbulence, accurately accounting for the role of the large-scale coherent structures, is presented. It is shown that in the limit $\mathrm{R}\mathrm{e}\ensuremath{\rightarrow}\ensuremath{\infty}$ the total enstrophy in the system obeys a universal asymptotic relation: $\ensuremath{\Omega}\ensuremath{\propto}{t}^{\ensuremath{-}2/3}$.

Grid turbulence in shallow flows

Results of experiments on decaying turbulence in shallow water flow bounded by a solid bottom and a free surface are presented. The evolution of the turbulence structures generated by a grid, with horizontal mesh dimensions larger than the water depth, is measured using laser Doppler velocimetry and particle image velocimetry (PIV). The vertical confinement of the flow forces the large turbulence structures to move in the horizontal plane, thus attaining strongly two-dimensional features. This two-dimensionality and its consequences for the intensities and length scales of the large structures are analysed. It is shown that the decay of the vortices that are shed from the grid is determined by the characteristic size of the grid elements rather than the grid spacing. Furthermore, during the decay process the merging of vortices is observed in combination with a slope in the energy density spectrum of the velocity fluctuations. Using the PIV data, spatial properties like divergence and enstrophy can be derived for the velocity field near the free surface. The distinct effect of water depth that is found in the velocity fluctuations is almost insignificant in the enstrophy decay.

Decaying quasi-two-dimensional turbulence in a thin liquid layer

The results of laboratory measurements of decaying quasi-two-dimensional turbulence in a thin liquid layer are considered. It is shown that the enstrophy-to-energy ratio decreases according to a power law on a certain decay interval. The exponent in the power law is a function of the Reynolds number. The enstrophy decay is found to be anomalously slow as predicted in a number of numerical studies. It is shown that the anomalous decay in the quasi-two-dimensional flow under investigation is not due to intense vortex formation as in the numerical decaying turbulence, but due to the limited range of scales on which a flow can be regarded as two-dimensional.

Experiments on free decay of quasi-two-dimensional turbulent flows.

Decaying quasi-two-dimensional turbulence in a thin-layer flow is explored in laboratory experiments. We report the presence of power-law interval in the enstrophy decay law, in agreement with earlier experiments by Cardoso et al. [Phys. Rev. E 49, 454 (1994)] and Hansen et al. [Phys. Rev. E 58, 7261 (1998)]. The decay exponent proves sensitive to the way in which the energy decay is compensated. For the range of initial microscale Reynolds numbers between 35 and 95, the decay exponent is close to -0.4 for the ratio of enstrophy to energy, and to -0.75 for the enstrophy multiplied with a compensating factor of exp(-2lambda(t)), where lambda is the bottom-drag coefficient and t the decay time. The vorticity behavior does not comply with the theory of Carnevale et al. [Phys. Rev. Lett. 66, 2735 (1991)]: robust vortices are not observed in the vorticity field and the vorticity kurtosis is less than the Gaussian value.

Evolution of energy and enstrophy containing scales in decaying, two-dimensional turbulence with friction

Decaying, two-dimensional (2D), isotropic, incompressible, turbulence with finite Newtonian (lateral) friction and linear bottom drag is examined analytically. The classical result for inviscid, 2D turbulence of an upscale transfer of energy is shown to be robust to the presence of certain types of friction. It is shown that the energy-weighted mean wave number decreases with time, regardless of the Newtonian viscosity (i.e., for all Reynolds number) or linear bottom drag coefficients. For a given flow state, the time rate of change of the energy-weighted mean wave number is shown to be independent of the bottom friction but to increase with the Newtonian viscosity and the rate of spreading of energy in wave number space. There is an implicit dependence upon bottom friction since the second time derivative of the energy-weighted mean wave number involves a term linear in bottom drag coefficient. The forward transfer of enstrophy, on the other hand, is counter-balanced by the enstrophy decay due to finite Newtonian friction. The enstrophy-weighted mean wavelength decreases for sufficiently large Reynolds number, but increases when the Reynolds number becomes small enough. For a given flow state, the first time derivative of the enstrophy-weighted mean wavelength is independent of bottom friction. However, it implicitly affects the evolution through its influence upon the rate of spreading.

Explicit-filtering large-eddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term

Large-eddy simulation (LES) with regular explicit filtering is investigated. The filtered-scale stress due to the explicit filtering is here partially reconstructed using the tensor-diffusivity model: It provides for backscatter along the stretching direction(s), and for global dissipation, both also attributes of the exact filtered-scale stress. The necessary LES truncations (grid and numerical method) are responsible for an additional subgrid-scale stress. A natural mixed model is then the tensor-diffusivity model supplemented by a dynamic Smagorinsky term. This model is reviewed, together with useful connections to other models, and is tested against direct numerical simulation (DNS) of turbulent isotropic decay starting with Reλ=90 (thus moderate Reynolds number): LES started from a 2563 DNS truncated to 643 and Gaussian filtered. The tensor-diffusivity part is first tested alone; the mixed model is tested next. Diagnostics include energy decay, enstrophy decay, and energy spectra. After an initial transient of the dynamic procedure (observed with all models), the mixed model is found to produce good results. However, despite expectations based on favorable a priori tests, the results are similar to those obtained when using the dynamic Smagorinsky model alone in LES without explicit filtering. Nevertheless, the dynamic mixed model appears as a good compromise between partial reconstruction of the filtered-scale stress and modeling of the truncations effects (incomplete reconstruction and subgrid-scale effects). More challenging 483 LES are also done: Again, the results of both approaches are found to be similar. The dynamic mixed model is also tested on the turbulent channel flow at Reτ=395. The tensor-diffusivity part must be damped close to the wall in order to avoid instabilities. Diagnostics are mean profiles of velocity, stress, dissipation, and reconstructed Reynolds stresses. The velocity profile obtained using the damped dynamic mixed model is slightly better than that obtained using the dynamic Smagorinsky model without explicit filtering. The damping used so far is however crude, and this calls for further work.

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.

Entropy maximization tendency in topographic turbulence

Numerical simulations of geostrophic turbulence above topography are used to compare (a) nonlinear generation of system entropy, S, (b) selective damping of enstrophy and (c) development of vorticity–topography correlation. In the damped cases, S initially increases, approaching a quasi-equilibrium (maximum S subject to the instantaneous, though decaying, energy and enstrophy). When strongly scale-selective damping is applied, onset of the vorticity–topography correlation follows the timescales for enstrophy decay. During the period of decay, it is shown that nonlinear interaction continues to generate S, offsetting in part the loss of S to explicit damping.

Selective Decay of Enstrophy and the Excitation of Barotropic Waves in a Channel

Abstract If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, β-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave. Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective ...

Minimum enstrophy vortices

Two kinds of minimum enstrophy vortices in two-dimensional flows are found by variational analysis. Each represents the ideal limit of a selective decay of enstrophy with energy and angular momentum or circulation, respectively, remaining fixed. For each, the vorticity is confined to a disk whose radius is determined by the fixed integrals. The definition of the vortices assures a large degree of stability, but many questions about their formation and destruction have not yet been answered.