Three-dimensional Homogeneous Turbulence Research Articles

Error Growth in Three-Dimensional Homogeneous Turbulence

We study the evolution of an error field defined by the difference between two velocity fields in statistically identical three-dimensional incompressible homogeneous turbulence. We assume that the...

An alternative formulation for exact scaling relations in hydrodynamic and magnetohydrodynamic turbulence

We propose an alternative formulation for the exact relations in three-dimensional homogeneous turbulence using two-point statistics. Our finding is illustrated with incompressible hydrodynamic, standard and Hall magnetohydrodynamic turbulence. In this formulation, the cascade rate of an inviscid invariant of turbulence can be expressed simply in terms of mixed second-order structure functions. Besides the usual variables like the velocity , vorticity , magnetic field and the current , the vectors , and are also found to play a key role in the turbulent cascades. The current methodology offers a simple algebraic form which is specially interesting to study anisotropic space plasmas like the solar wind, with, a faster statistical convergence than the classical laws written in terms of third-order correlators.

Reynolds-number dependency in homogeneous, stationary two-dimensional turbulence

Turbulent solutions of the two-dimensional Navier–Stokes equations are a paradigm for the chaotic space–time patterns and equilibrium distributions of turbulent geophysical and astrophysical ‘thin’ flows on large horizontal scales. Here we investigate how homogeneous, stationary two-dimensional turbulence varies with the Reynolds number (Re) in stationary solutions with large-scale, random forcing and viscous diffusion, also including hypoviscous diffusion to limit the inverse energy cascade. This survey is made over the computationally feasible range in Re ≫ 1, approximately between 1.5 × 103 and 5.6 × 106. For increasing Re, we witness the emergence of vorticity fine structure within the filaments and vortex cores. The energy spectrum shape approaches the forward-enstrophy inertial-range form k−3 at large Re, and the velocity structure function is independent of Re. All other statistical measures investigated in this study exhibit power-law scaling with Re, including energy, enstrophy, dissipation rates and the vorticity structure function. The scaling exponents depend on the forcing properties through their influences on large-scale coherent structures, whose particular distributions are non-universal. A striking result is the Re independence of the intermittency measures of the flow, in contrast with the known behaviour for three-dimensional homogeneous turbulence of asymptotically increasing intermittency. This is a consequence of the control of the tails of the distribution functions by large-scale coherent vortices. Our analysis allows extrapolation towards the asymptotic limit of Re → ∞, fundamental to geophysical and astrophysical regimes and their large-scale simulation models where turbulent transport and dissipation must be parameterized.

Fundamentals of pair diffusion in kinematic simulations of turbulence

We demonstrate that kinematic simulation (KS) of three-dimensional homogeneous turbulence produces fluid element pair statistics in agreement with the predictions of L F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)] even though KS lacks explicit modeling of turbulent sweeping of small eddies by large ones. This scaling is most clearly evident in the turbulent diffusivity's dependence on rms pair separation and, to a lesser extent, on the pair's travel time statistics. It is also shown that kinematic simulation generates a probability density function of pair separation which is in good agreement with recent theory [S. Goto and J. C. Vassilicos, New J. Phys. 6, 65 (2004)] and with the scaling of the rms pair separation predicted by L. F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)]. Finally, the statistical persistence hypothesis (SPH) is formulated mathematically and its validity tested in KS. This formulation introduces the concept of stagnation point velocities and relates these to fluid accelerations. The scaling of accelerations found in kinematic simulation supports the SPH, even though KS does not generate a Kolmogorov scaling for the acceleration variance (except for a specific case and a limited range of outer to inner length-scale ratios). An argument is then presented that suggests that the stagnation points in homogeneous isotropic turbulence are on average long-lived.

Decaying grid-generated turbulence in a rotating tank

The decay of initially three-dimensional homogeneous turbulence in a rotating frame is experimentally investigated. Turbulence is generated by rapidly towing a grid in a rotating water tank, and the velocity field in a plane perpendicular to the rotation axis is measured by means of particle image velocimetry. During the decay, strong cyclonic coherent vortices emerge, as the result of enhanced stretching of the cyclonic vorticity by the background rotation, and the selective instability of the anticyclonic vorticity by the Coriolis force. This asymmetry towards cyclonic vorticity grows on a time scale Ω−1 (Ω is the rotation rate), until the friction from the Ekman layers becomes dominant. The energy spectrum perpendicular to the rotation axis becomes steeper as the instantaneous Rossby number Roω=ω′∕2Ω decreases below the value 2±0.5 (ω′ is the root-mean square of the vertical vorticity). The spectral exponent increases in time from its classical Kolmogorov value 5∕3 up to values larger than 2. Below the threshold Roω<2, the velocity derivative skewness decreases as ∣S∣∝Roω, reflecting the inhibition of the energy transfers by the background rotation, with a net inverse energy cascade that develops at large scales.

Application of spectral forcing in lattice-Boltzmann simulations of homogeneous turbulence

An efficient numerical method for the direct simulation of homogeneous turbulent flow has been obtained by combining a spectral forcing algorithm for homogeneous turbulence with a lattice-Boltzmann scheme for solution of the continuity and Navier–Stokes equations. The spectral forcing scheme of Alvelius [Alvelius K. Random forcing of three-dimensional homogeneous turbulence. Phys Fluids 1999;11(7):1880–89] is used which allows control of the power input by eliminating the force–velocity correlation in the Fourier domain and enables anisotropic forcing. A priori chosen properties such as the Kolmogorov length scale, the integral length scale and the integral time scale are recovered. This demonstrates that the scheme works accurately with the lattice-Boltzmann method and that all specific features of the forcing scheme are recovered in the lattice-Boltzmann implementation.

Coherent vortex extraction in three-dimensional homogeneous turbulence: Comparison between CVS-wavelet and POD-Fourier decompositions

The coherent vortex simulation (CVS) decomposes each realization of a turbulent flow into two orthogonal components: An organized coherent flow and a random incoherent flow. They both contribute to all scales in the inertial range, but exhibit different statistical behaviors. The CVS decomposition is based on the nonlinear filtering of the vorticity field, projected onto an orthonormal wavelet basis made of compactly supported functions, and the computation of the induced velocity field using Biot–Savart’s relation. We apply it to a three-dimensional homogeneous isotropic turbulent flow with a Taylor microscale Reynolds number Rλ=168, computed by direct numerical simulation at resolution N=2563. Only 2.9%N wavelet modes correspond to the coherent flow made of vortex tubes, which contribute 99% of energy and 79% of enstrophy, and exhibit the same k−5/3 energy spectrum as the total flow. The remaining 97.1%N wavelet modes correspond to a incoherent random flow which is structureless, has an equipartition energy spectrum, and a Gaussian velocity probability distribution function (PDF). For the same flow and the same compression rate, the proper orthogonal decomposition (POD), which in this statistically homogeneous case degenerates into the Fourier basis, decomposes each flow realization into large scale and small scale flows, in a way similar to large eddy simulation (LES) filtering. It is shown that the large scale flow thus obtained does not extract the vortex tubes equally well as the coherent flow resulting from the CVS decomposition. Moreover, the small scale flow still contains coherent structures, and its velocity PDF is stretched exponential, while the incoherent flow is structureless, decorrelated, and its velocity PDF is Gaussian. Thus, modeling the effect of the incoherent flow discarded by CVS-wavelet shall be easier than modeling the effect of the small scale flow discarded by POD-Fourier or LES.

Comparison of a Spectral Method with a Higher-Order Finite Difference Method in Direct Numerical Simulations of Three-Dimensional Homogeneous Turbulence.

A finite difference method (FDM) is easier and cheaper to use than a spectral method (SM) in flows with complicated geometry, but is inferior to a SM in both accuracy and resolution. In this paper, effects of both accuracy and resolution in a FDM on the quality of incompressible turbulent flow description were studied by comparing a SM with a FDM in direct numerical simulations (DNS) of the three-dimensional freely decaying homogeneous turbulence with 1283 and 2563 grids. We adopted the higher-order method of lines and Fourier spectral method as a FDM and a SM, respectively. The value of the kinematic viscosity was determined so that the minimum value of Kmaxη in time was about 2 in the SM with 2563 grids. Here, Kmax and η are the maximum resolved wavenumber and the Kolmogorov length scale, respectively. The statistical quantities, such as the total energy, the energy dissipation ratio, obtained by the higher-order method of lines were in good agreement with those by the SM. Furthermore, comparing their DNS fields, we found that the performance of the 4th-order method of lines with 2563 grids was comparable to that of the SM with 1283 grids in which the minimum value of Kmaxη in time was about 1.

Random forcing of three-dimensional homogeneous turbulence

A method of using a fully random force to numerically generate statistically stationary homogeneous turbulence has been developed. The forcing is implemented in spectral space where it is concentrated at small wave numbers. Hence, the power input is introduced into the flow at large scales. The randomness in time makes the force neutral, in the sense that it does not directly correlate with any of the time scales of the turbulent flow, and it also makes the power input determined solely by the force-force correlation. This means that it is possible to generate different desirable turbulence states, such as axisymmetric turbulence, where the degree of anisotropy of the forcing can be chosen a priori through forcing parameters. In particular, the total amount of power input from the forcing can be set to balance a desired dissipation at a statistically stationary state. In order to only get a contribution from the force-force correlation to the input power in the discrete equations, the force is determined so that the velocity-force correlation vanishes for each Fourier mode. In direct numerical simulations (DNS) of forced isotropic turbulence, universality of the small scales is shown for the kinetic energy spectrum at different Reynolds numbers and the velocity derivative skewness obtains the value −0.5. The forcing method is used in a large eddy simulation (LES), where it is compared with a simulation of decaying turbulence to show the importance of having a statistically stationary flow if well known inertial laws are to be recovered at moderate Reynolds numbers.

Kolmogorov's refined similarity hypothesis for hyperviscous turbulence.

Kolmogorov's refined similarity hypothesis (RSH) is tested in high resolution numerical simulations of forced three-dimensional homogeneous turbulence. High Reynolds numbers are achieved by using hyperviscous dissipation (-1${)}^{\mathit{h}+1}$${\mathrm{\ensuremath{\Delta}}}^{\mathit{h}}$ (h=8) instead of Newtonian (h=1) dissipation. It is found that, in the inertial range, the RSH is reasonably well satisfied for low order moments with noticeable systematic corrections for higher order moments. Within the constraints imposed by the use of hyperviscosity our data nearly eliminate trivial kinematic dependencies between longitudinal velocity differences and the energy dissipation rate thus helping to reveal the true dynamical nature of the RSH. \textcopyright{} 1996 The American Physical Society.

Self-similar decay of three-dimensional homogeneous turbulence with hyperviscosity.

Numerical simulations of the Navier-Stokes equations with hyperviscosity (-1${)}^{\mathit{h}+1}$${\mathrm{\ensuremath{\Delta}}}^{\mathit{h}}$ (h=8) show that periodic-box turbulence exhibits self-similar decay. The inertial-range energy spectrum has the scaling law E(t${)}^{2/3}$/${\mathit{k}}^{5/3}$, where E(t) is the energy dissipation rate at time t. The total energy of the system decreases at 1/${\mathit{t}}^{2}$. The concept of constant Reynolds number decay is introduced, enabling us to perform long time averages and reliably measure higher-order correlation functions. Comparisons are made with the case of forced turbulence reported earlier.

Line dispersion in homogeneous turbulence: Stretching, fractal dimensions, and micromixing.

Motivated by the understanding of the turbulent mixing mechanisms at small scale, the time evolution of an initially regular passive scalar pattern in three-dimensional homogeneous turbulence is investigated in a line-dispresion experiment. The major observation is the increasing space-fillingness of the line as time elapses. It is shown that the evolution of the fractal dimension ${d}_{f}(t)$ of the support of the line can be related to the mean-field Kolmogorov scaling of velocity differences incorporating dissipative corrections and an expression for ${d}_{f}(t)$ is derived that compares very well with the experiments for several distinct Reynolds numbers. Following, a criterion of micromixing is discussed.

Statistics of Energy-Dissipation Clusters in Three-Dimensional Homogeneous Turbulence

The concept of energy-dissipation clusters is introduced for investigating an intermittent turbulence field, Which is obtained by a direct numerical simulation of the three-dimensional incompressible Navier-Stokes equation. The statistics of the clusters characterize some intermittent properties, such as the degrees of concentration and connectivity of highly energy-dissipated regions. The concentration may not depend on the Reynolds number, while the connectivity becomes weaker as the number increases: The higher Reynolds number field appears more spotty. Moreover the cluster size suggests the power-law distribution with a universal exponent.

Statistics of Energy-Dissipation Clusters in Three-Dimensional Homogeneous Turbulence

The concept of energy-dissipation clusters is introduced for investigating an intermittent turbu­ lence field, which is obtained by a direct numerical simulation of the three-dimensional incompressible Navier-Stokes equation. The statistics of the clusters .characterize some intermittent properties, such as the degrees of concentration and connectivity of highly energy-dissipated regions. The concentration may not depend on the Reynolds number, while the connectivity becomes weaker as the number increases: The higher Reynolds number field appears more spotty. Moreover the cluster size suggests the power-law distribution with a universal exponent.

Dynamical behavior of the fundamental triad-interaction system in three-dimensional homogeneous turbulence

The fundamental triad-interaction system is the building block of the nonlinear interaction of the three-dimensional Navier–Stokes equations. It satisfies incompressibility, conserves energy and helicity, preserves measure under the time evolution, and involves an arbitrary parameter taking any value in [0,2π]. Under several values of this parameter, it is found that the fundamental triad-interaction system cannot be ergodic on the constant energy-helicity surface because of the extraneous constants of motion other than energy and helicity. This at once implies that isotropy is not possible for the spectral tensor, for isotropy calls for mixing which is a stronger requirement than ergodicity. Although the dynamics of a single triad-interaction system is not directly relevant to real homogeneous turbulence, a theoretical framework to investigate ergodicity and mixing and isotropy of the three-dimensional turbulence model by consistently including many fundamental triad-interaction systems is provided.