Kolmogorov Similarity Research Articles

On the simulation of particle trajectories in turbulent flows

A different class of stochastic model, comprising of the Langevin equation with a random time scale, for the simulation of fluid velocities along particle trajectories in high Reynolds-number turbulent flows is formulated. These velocities are neither purely Lagrangian nor purely Eulerian in character. The distribution of time scales is chosen to ensure that the modeled form of the fluid-velocity structure function and spectral functions are compatible with Kolmogorov similarity scaling and with the scaling analysis of Fung, Hunt, and Perkins [Proc. R. Soc. London, Ser. A 459, 445 (2003)]. It is shown that the model accounts naturally for the crossing trajectory effect and integral time scales are compatible with the much used parameterizations advocated by Csanady [J. Atmos. Sci. 20, 201 (1963)] and by Frenkiel [Adv. Appl. Mech. 3, 61 (1953)]. Model predictions for particle dispersion in grid generated turbulence are shown to be in close accord with the experimental data of Snyder and Lumley [J. Fluid Mech. 48, 41 (1971)].

Lagrangian statistics in turbulent channel flow

The Lagrangian dispersion of fluid particles in inhomogeneous turbulence is investigated by a direct numerical simulation of turbulent channel flow. Lagrangian velocity and acceleration along a particle trajectory are computed by employing several interpolation schemes. Among the schemes tested, the four-point Hermite interpolation in the homogeneous directions combined with Chebyshev polynomials in the wall-normal direction seems to produce most reliable Lagrangian statistics. Inhomogeneity of Lagrangian statistics in turbulent boundary layer is investigated by releasing many particles at several different wall-normal locations and tracking those particles. The fluid particle dispersion and Lagrangian structure function of velocity are investigated for the Kolmogorov similarity. The behavior of the Lagrangian integral time scales, Kolmogorov constants a0 and C0 of the velocity structure function near the wall are discussed. The intermittent behavior of the fluid particle acceleration is also examined by kurtosis of the Lagrangian structure function. Finally, the effect of the initial particle location on the dispersion is analyzed by the probability density function of particle position at several downstream locations.

Remarks concerning modified Navier-Stokes equations

We discuss some historical background concerning a modified version of the Navier-Stokes equations for the motion of an incompressible fluid. The classical (Newtonian) linear relation between the Cauchy stress tensor and the rate of strain tensor yields the Navier-Stokes equations. Certain nonlinear relations are also consistent with basic physical principals and result in equations with "stronger" dissipation. We describe a class of models that has its genesis in Kolmogorov's similarity hypothesis for 3-dimensional isotropic turbulence and was formulated by Smagorinsky in the meteorological context of rapidly rotating fluids and more generally by Ladyzhenskaya. These models also describe the motion of fluids with shear dependent viscosities and have received considerable attention. We present a dyadic model for such modified Navier-Stokes equations. This model is an example of a hierarchical shell model. Following the treatment of a (non-physically motivated) linear hyper-dissipative model given by Katz-Pavlović, we prove for the dyadic model a bound for the Hausdorff dimension of the singular set at the first time of blow up. The result interpolates between the results of solvability for sufficiently strong dissipation of Ladyzhenskaya, (later strengthened by Nečas et al) and the bound for the dimension of the singular set for the Navier-Stokes equations proved by Caffarelli, Kohn and Nirenberg. We discuss the implications of this dyadic model for the modified Navier-Stokes equation themselves.

Numerical evidence of a new similarity law in shear dominated flows

Recent theoretical results concerning the effect of shear on scaling laws in turbulence are discussed. DNS data for a turbulent channel flow and a homogeneous shear flow are used to investigate the statistical properties of turbulent fluctuations. As expected, the shear increases the level of intermittency of velocity increments. Instead, the statistical properties of the dissipation field remain similar to those of homogeneous isotropic turbulence. In fact, the increase of intermittency is found to be associated with the failure of the refined Kolmogorov similarity and the establishment of a new similarity law accounting for the prevailing role of the energy production related to the Reynolds stresses in mean shear. The relevance of these findings for eddy viscosity closures in LES of shear dominated flows is briefly outlined.

난류채널유동의 라그란지안 해석 (II) - 라그란지안 통계분석 -

The Lagrangian dispersion of fluid particles in inhomogeneous turbulence is investigated by a direct numerical simulation of turbulent channel flow. Four points Hermite interpolation in the homogeneous direction and Chebyshev polynomials in the inhomogeneous direction is adopted to simulate the fluid particle dispersion. An inhomogeneity of Lagrangian statistics in turbulent boundary layer is investigated by releasing many particles at several different wall-normal locations and tracking those particles. The fluid particle dispersions and Lagrangian structure functions of velocity are scaled by the Kolmogorov similarity. The auto-correlations of velocity and acceleration are shown at the different releasing locations. Effect of initial particle location on the dispersion is analyzed by the probability density function at the several downstreams and time instants.<br/>

Sparseness of nonlinear coupling: importance in sparse direct-interaction perturbation

It is not weakness of nonlinearity but sparseness of nonlinear couplingsthat plays a key role in the sparse direct-interaction perturbation (SDIP), which is a second-order moment closure theory for nonlinear dynamical systems. The SDIP is similar but different in procedure from Kraichnan's direct-interaction approximation (DIA). Homogeneous Navier–Stokes turbulence is an example of dynamical systems in which nonlinearity is strong in magnitude but sparse in coupling. In order to clarify the importance of sparseness of coupling, we formulate SDIP for a model equation which has three parameters: coupling density, strength of nonlinearity and the number of degrees of freedom. By the help of numerical simulations, it is shown that SDIP is applicable when the coupling density is sufficiently smaller than the square root of the number of degrees of freedom, even if the strength of nonlinearity is infinitely large. This implies that the applicability of SDIP has nothing to do with the Gaussianity of a dynamical variable, whereas DIA is often explained as a theory based on it. The SDIP is also applied to a shell model of turbulence with sparse coupling which exhibits the Kolmogorov similarity. The solution to SDIP equations is consistent with inertial range properties such as the −(5/3) power law of the energy spectrum.

Kolmogorov two-thirds law by matched asymptotic expansion

The Kolmogorov two-thirds law is derived for large Reynolds number isotropic turbulence by the method of matched asymptotic expansions. Inner and outer variables are derived from the Karman–Howarth equation by using the von Karman self-preservation hypothesis. Matching the resulting large Reynolds number asymptotic expansions yields the Kolmogorov law. The Kolmogorov similarity hypotheses are not assumed; only the Navier–Stokes equation is employed and the assumption that dissipation is finite. This indicates that the Kolmogorov results are a direct consequence of the Navier–Stokes equations.

Scaling laws and intermittency in homogeneous shear flow

In this article we discuss the dynamical features of intermittent fluctuations in homogeneous shear flow turbulence. In this flow the energy cascade is strongly modified by the production of turbulent kinetic energy related to the presence of vortical structures induced by the shear. As a consequence, the intermittency of velocity fluctuations increases with respect to homogeneous and isotropic turbulence. By using direct numerical simulations, we show that the refined Kolmogorov similarity is broken and a new form of similarity is observed, in agreement with previous results obtained in turbulent boundary layers. We find here that the statistical properties of the energy dissipation are practically unchanged with respect to homogeneous isotropic conditions, while the increased intermittency is entirely captured in terms of the new similarity law.

Relationship between the smallest scale of flame wrinkles and turbulence characteristics of high-pressure, high-temperature turbulent premixed flames

The mechanism determining the scale of the smallest flame wrinkles of high-pressure, high-temperatureturbulent premixed flames was investigated. The fractal inner cutoff of OH planar laser-induced fluorescence images, which is the smallest scale of flame wrinkles, was analyzed for burner-stabilized flames in a high-pressure chamber. Precise measurement of the energy spectrum of turbulence was also performed and the relationship between the intrinsic flame instability and flow turbulence was examined. Experiments were performed for CH4/air mixtures of 300 and 573 K at 0.1, 0.5, and 1.0 MPa. The experimental results clearly showed the Kolmogorov's similarity law for non-dimensional energy spectra of flow turbulence at high pressure and high temperature. A characteristics scale equivalent to the average vortex-tube diameter, lv, which is about 10 times larger than the Kolmogorov scale revealed by recent direct numerical simulation, was used as a scale corresponding to the largest wave number of initial flow disturbances in an unburned mixture. At high pressure, the fractal inner cutoff, ∈i, and lv decrease with turbulence Reynolds number basedon Taylor microscale, Rλ, regardless of mixture temperature. The magnitude, of ∈i is close to lv when lv is larger than the characteristic instability scale corresponding to the maximum growth rate of flame instability, lv. When Rλ increases further and lv becomes smaller than li, ∈i becomes almost constant. At atmospheric pressure, the relationship between lv, li, and ∈i was not obvious, but the correlation of ∈i with the integral scale, lg, was rather significant. These characteristics of ∈i variation for high-pressure, high-temperature turbulent premixed flames can be explained using the scale-relation model based on lg, lv, and li for Rλ variation proposed in this study.

LAGRANGIANINVESTIGATIONS OFTURBULENCE

▪ Abstract A Lagrangian description of turbulence has unique physical advantages that are especially important in studies of mixing and dispersion. We focus on fundamental aspects, using mainly data from direct numerical simulations capable of great detail and precision when specific accuracy requirements are met. Differences between time evolution in Eulerian and Lagrangian frames illustrate the dominance of advective transport. We examine basic results in Kolmogorov similarity, giving an estimate of an inertial-range universal constant and the grid resolution and Reynolds number needed to attain the requisite scaling range of time lags. The Lagrangian statistics of passive scalars are discussed in view of current efforts in model development, with differential diffusion between multiple scalars being characterized by shorter timescales. We also note the need for new data in more complex flows and in other applications where a Lagrangian viewpoint is especially useful.

TURBULENT RELATIVE DISPERSION

▪ Abstract This review begins with the classical foundations of relative dispersion in Kolmogorov's similarity scaling. Analysis of the special cases of isotropic and homogeneous scalar fields is then used to establish most simply the connection with turbulent mixing. The importance of the two-particle acceleration covariance in relative dispersion is demonstrated from the kinematics of the motion of particle-pairs. A summary of the development of two-particle Lagrangian stochastic models is given, with emphasis on the assumptions and constraints involved, and on predictions of the scalar variance field for inhomogeneous sources. Two-point closures and kinematic simulation are also reviewed in the context of their prediction of the Richardson constant and other fundamental constants. In the absence of reliable field data, direct numerical simulations and laboratory measurements seem most likely to provide suitable data with which to test the assumptions and predictions of these theories.

Pair dispersion in synthetic fully developed turbulence.

The Lagrangian statistics of relative dispersion in fully developed turbulence is numerically investigated. A scaling range spanning many decades is achieved by generating a two-dimensional velocity field by means of a stochastic process with prescribed statistics and of a dynamical model (shell model) with fluctuating characteristic times. When the velocity field obeys Kolmogorov similarity, the Lagrangian statistics is self similar and agrees with Richardson's predictions [Proc. R. Soc. London Ser. A 110, 709 (1926)]. For intermittent velocity fields the scaling laws for the Lagrangian statistics are found to depend on the Eulerian intermittency in agreement with the multifractal description. As a consequence of the Kolmogorov law the Richardson law for the variance of pair separation is, however, not affected by intermittency corrections. Moreover, Lagrangian exponents do not depend on the particular Eulerian dynamics. A method of data analysis, based on fixed scale statistics rather than usual fixed time statistics, is shown to give much wider scaling range, and should be preferred for the analysis of experimental data.

Intermittency and scaling laws for wall bounded turbulence

Well defined scaling laws clearly appear in wall bounded turbulence, very close to the wall, where a distinct violation of the refined Kolmogorov similarity hypothesis (RKSH) occurs together with the simultaneous persistence of scaling laws. A new form of RKSH for the wall region is here proposed in terms of the structure functions of order two which, in physical terms, confirms the prevailing role of the momentum transfer towards the wall in the near wall dynamics.

Advected Scalar Version of the Differences between the True Dissipation and Its One-Dimensional Surrogate in Isotropic Turbulence

The one-dimensional (1D) surrogates of scalar dissipation rate as well as energy dissipation rate are often used in place of the true ones in investigations relevant to the Obukhof-Corrsin spectral form and the joint-multifractal nature of isotropic turbulence with an advected scalar. As a result of direct numerical simulation, it turns out that the surrogates cause a similar fundamental change in statistics to that caused by the 1D surrogate of energy dissipation rate in the treatment of the Kolmogorov similarity variable. The conditional and unconditional probability densites of the scalar similarity variable built from the true energy and scalar dissipations are never bimodal and always nearly Gaussian.

On the onset of high-Reynolds-number grid-generated wind tunnel turbulence

Using an active grid devised by Makita (1991), shearless decaying turbulence is studied for the Taylor-microscale Reynolds number, Rλ, varying from 50 to 473 in a small (40 × 40 cm2 cross-section) wind tunnel. The turbulence generator consists of grid bars with triangular wings that rotate and flap in a random way. The value of Rλ is determined by the mean speed of the air (varied from 3 to 14 m s–1) as it passes the rotating grid, and to a lesser extent by the randomness and rotation rate of the grid bars. Our main findings are as follows. A weak, not particularly well-defined scaling range (i.e. a power-law dependence of both the longitudinal (u) and transverse (v) spectra, F11(k1) and F22(k1) respectively, on wavenumber k1) first appears at Rλ ∼ 50, with a slope, n1, (for the u spectrum) of approximately 1.3. As Rλ was increased, n1 increased rapidly until Rλ ∼ 200 where n ∼ 1.5. From there on the increase in n1 was slow, and even by Rλ = 473 it was still significantly below the Kolmogorov value of 1.67. Over the entire range, 50 [les ] Rλ [les ] 473, the data were well described by the empirical fit: . Using a modified form of the Kolmogorov similarity law: F11(k1) = C1*e2/3k1–5/3(k 1η)5/3–n1 where e is the turbulence energy dissipation rate and η is the Kolmogorov microscale, we determined a linear dependence between n1 and C1*: C1* = 4.5 – 2.4n1. Thus for n1 = 5/3 (which extrapolation of our results suggests will occur in this flow for Rλ ∼ 104), C1* = 0.5, the accepted high-Reynolds-number value of the Kolmogorov constant. Analysis of the p.d.f. of velocity differences Δu(r) and Δv(r) where r is an inertial subrange interval, conditional dissipation, and other statistics showed that there was a qualitative difference between the turbulence for Rλ 200 (strong turbulence). For the latter, the p.d.f.s of Δu(r) and Δv(r) had super Gaussian tails and the dissipation (both of the u and v components) conditioned on Δu(r) and Δv(r) was a strong function of the velocity difference. For Rλ 200 are consistent with the predictions of the Kolmogorov refined similarity hypothesis (and make a distinction between the dynamical and kinematical contributions to the conditional statistics). They have much in common with similar statistics done in shear flows at much higher Rλ, with which they are compared.

Experimental study of the Kolmogorov refined similarity variable

Recently Thoroddsen and Van Atta (1992, Phys. Fluids A4, 2592) showed that Kolmogorov's refined similarity hypothesis (1962, J. Fluid Mech. 13, 77; 82) is supported by experimental data from a wind-tunnel study of a cylinder wake, at Reλ of 550. We show here that the probability density of the Kolmogorov similarity variable approaches a Gaussian distribution. The data also suggest that it may obey an even more demanding conditional similarity, which leads to important conclusions regarding the scaling exponents.

Energy and spectral dynamics in decaying compressible turbulence

The statistical properties of decaying compressible turbulence are investigated by direct numerical simulations of flow in a periodic cube. Starting with fully developed turbulence for various microscale Reynolds numbersR λ , rms Mach numbersM, and small- and large-scale compressive ratiosr CS andr CL , we find that the kinetic energy exhibits an exponential decay in time. Interactions between the compressive and rotational components of kinetic energy are weaker than self-interactions of the respective components. The energy spectrum of rotational component obeys the same Kolmogorov similarity law as for incompressible turbulence and forced compressible turbulence. The form of the energy spectrum of the compressive component, on the other hand, depends strongly onM. As the turbulence decays freely,R λ andM decrease in time butr CS andr CL tend to some universal values.

Relative Diffusion in Turbulent Flow by Usingthe Effective Hamiltonian Method

Richardson's four-thirds law is both analytically and numerically confirmed by applying the effective Hamiltonian method to relative diffusion of a pair of fluid particles subject to convection in steady, incompressible, statistically isotropic turbulent flow with a joint Gaussian distribution for velocity field when the spectrum satisfies Kolmogorov's similarity law and the scaling law. In addition, the interesting phenomenon that the mean square value of the relative distance of a pair of particles increases faster than the cube of the time of diffusion at the part which connects with the range satisfying this law when the initial distance between them is small enough in comparison with the typical length of the flow is reported. This method is, further, applied to relative diffusion in the turbulent flow with multi-range spectrum. There it is qualitatively tried to reproduce the experimental data of Kellogg (1956). The possibility that two separate ranges, which satisfy Richardson's law respectively, a...

Kolmogorov similarity hypotheses for scalar fields: sampling intermittent turbulent mixing in the ocean and galaxy

Kolmogorov’s three universal similarity hypotheses are extrapolated to describe scalar fields like temperature mixed by turbulence. The analogous first and second hypotheses for scalars include the effects of Prandtl number and rate-of-strain mixing. Application of velocity and scalar similarity hypotheses to the ocean must take into account the damping of active turbulence by density stratification and the Earth’s rotation to form fossil turbulence. By the analogous Kolmogorov third hypothesis for scalars, temperature dissipation rates X averaged over lengths r &gt; L K should be lognormally distributed with intermittency factors σ 2 that increase with increasing turbulence energy length scales L O as σ 2 ln&lt;x&gt; r ≈ μ θ ln( L O / r ). Tests of kolmogorovian velocity and scalar universal similarity hypotheses for very large ranges of turbulence length and timescales are provided by data from the ocean and the galactic interstellar medium. These ranges are from 1 to 9 decades in the ocean, and over 12 decades in the interstellar medium. The universal constant for turbulent mixing intermittency μ θ is estimated from oceanic data to be 0.44 ± 0.01, which is remarkably close to estimates for Kolmogorov’s turbulence intermittency constant μ of 0.45 ± 0.05 from galactic as well as atmospheric data. Extreme intermittency complicates the oceanic sampling problem, and may lead to quantitative and qualitative undersampling errors in estimates of mean oceanic dissipation rates and fluxes. Intermittency of turbulence and mixing in the interstellar medium may be a factor in the formation of stars.

Reynolds number effects in Lagrangian stochastic models of turbulent dispersion

A second-order autoregressive equation is used to model the acceleration of fluid particles in turbulence in order to study the effect of Reynolds number on Lagrangian turbulence statistics. It is shown that this approach provides a good representation of dissipation subrange structure of Lagrangian velocity and acceleration statistics. The parameters of the model, two time scales representing the energy-containing and dissipation scales, are determined by matching the model velocity autocorrelation function to Kolmogorov similarity forms in the inertial subrange and the dissipation subrange. The model is tested against the Lagrangian statistics obtained by Yeung and Pope [J. Fluid Mech. 207, 531 (1989)] from direct numerical simulations of turbulence. Agreement between the model predictions and simulation data for second-order Lagrangian statistics such as the velocity structure function, the acceleration correlation function, and the dispersion of fluid particles is excellent, indicating that the main departures from Kolmogorov’s theory of local isotropy shown by the simulation data are due to low Reynolds number. For Reynolds numbers typical of laboratory experiments and direct numerical simulations of turbulence the root-mean-square dispersion of marked particles is changed from the Langevin equation (i.e., infinite Reynolds number) prediction by up to about 50% at large times. Most of this change can be accounted for by the change in the Lagrangian integral time scale. It is also shown that Reynolds number effects in laboratory dispersion or Lagrangian turbulence measurements can cause significant errors (typically of order 50%) when the value of the Kolmogorov Lagrangian structure function constant C0 is estimated by fitting the predictions of the Langevin equation to these data. A value C0 = 7 is obtained by fitting the new model to the direct simulation data.