Kolmogorov's Similarity Hypothesis Research Articles

Kolmogorov's similarity hypotheses application in research of the turbulent swirl flows in pipe generated by the various axial fan geometries

AbstractKolmogorov has formulated his two hypotheses of similarity in 1941 considering local‐isotropic turbulence, and later has refined his theory [1, 2]. In this paper are discussed three turbulent swirl flows generated by the three various axial fan types in‐built in the pipe. The spectral characteristics of the generated turbulent swirl flows are determined on the basis on the LDA (laser Doppler anemometry) experimental data for the circumferential velocity. In the inertial subranges these curves have the same tangent and obey the “‐ 5/3” Kolmogorov law. It is shown that in the generated turbulent swirl flows occur eddies of various scales, encompassing very wide spectra of wave numbers, i.e. frequencies.

Turbulence cascade model for viscous vortex ring-tube reconnection

The reconnection of a vortex ring and a vortex tube in a viscous fluid with the effects of two vortex core sizes (σ0=0.12r0 and 0.24r0, where r0 are initial ring radius) and three initial flow configurations (left-offset, center, and right-offset) at Reynolds number (ReΓ) of 10 000 was investigated using a high-order vortex-in-cell method combined with a large-eddy simulation model. For the left-offset case, a large part of the ring, slipping over the tube, associates with a small part of the tube to establish a new vortex ring, whereas the rest of the tube is reconnected by another part of the ring. For the center case, half of the ring joins with a part of the tube to construct an elliptical vortex ring while the rest connects because of viscosity. The reconnected ring and tube become more stable and are like the initial ones in the ultimate stage. For the right-offset case, both the ring and tube's reconnection occurs, and the reconnected elliptical vortex ring is rapidly distorted. The proportion of reconnected ring increases, and then this ring section loses its integrity, decaying into a complex cluster of various-scales vortex structures in different shapes. At σ0=0.12r0, the secondary vortex structures surrounding the tube and ring appear in three cases, while they are only observed for the center case at σ0=0.24r0. For three flow configurations and two vortex core sizes, after the reconnection, the energy cascade of the flow approaches a k−5/3 slope of Kolmogorov's similarity hypotheses and a k−3 slope in the ranges of wavenumbers (k) from 3 to 10 and from 10 to 40, respectively. The highest population of small-scale coherent vortex structures is observed for the right-offset, followed by the center and left-offset. In addition, a larger number of these structures was observed for a smaller core size. This validates that the mixing performance is the best at a small vortex core and in the right-offset configuration.

Turbulent swirl flow analysis by Kolmogorov's similarity hypotheses

AbstractTurbulent swirl flow generated by the axial fans in pipe is investigated in this paper by using Kolmogorov's similarity hypotheses. It is shown that energy in the inertial subrange is divided among eddies after the “−5/3” law, i.e. Φww ∝ f−5/3, where Φww is spectral density for circumferential velocity and f is the frequency. This is presented for two various axial fans, of which one is carefully designed to generat Rankine vortex, while the second one has solid body circumferential velocity profile.

Self-preservation relation to the Kolmogorov similarity hypotheses

A theoretical and experimental investigation validates that the Kolmogorov first similarity hypothesis ($\ensuremath{\nu}$ and $ϵ$ are the relevant controlling parameters for small-scale turbulence at high Reynolds number) is subsumed under the more general hypothesis of self-preservation of the Navier-Stokes equations.

Modeling of Propagation of Antiplane Acoustic Waves in Multiscale Media with Lognormal Distribution of Parameters

The effective coefficients for the problem of propagation of acoustic waves in multifractal elastic media using the subgrid modeling approach are obtained. The maximum scale of heterogeneities of the medium in question is assumed to be small as compared with the wavelength. If a isotropic medium is assumed to satisfy the improved Kolmogorov similarity hypothesis, the term for the effective coefficient of the elastic stiffness coincides with the Landau–Lifshitz–Matheron formula. Both isotropic and anisotropic media are considered. The numerical testing for the wave propagating at a distance, which is of the same order as a typical wavelength of a source, illustrates the efficiency of the approach proposed.

In Memoriam: Sergei A. Kitaigorodskii, 1934—2014

Professor Sergei A. Kitaigorodskii, one of the pioneers of air-sea interaction studies, passed away on 4 December 2014 in Helsinki, at the age of 80. He had a clear vision of the fundamental problems of marine boundary layer physics. His bold ideas opened new paths more than a half century ago, and they continue to stimulate active research. In particular, he made significant progress on the difficult problem of estimating gas transfer rates through gas-liquid interfaces and contributed to many of the Gas Transfer at Water Surfaces conferences.Kitaigorodskii was born in Moscow on 13 September 1934. His father, Professor Alexander Kitaigorodskii, was a well-known physicist. Sergei Kitaigorodskii completed his undergraduate studies at Moscow State University in 1956, and he continued his postgraduate studies on the theory of turbulent mixing at the Academy of Sciences, Moscow, where he was awarded his Ph.D. from the Institute of Physics of the Atmosphere in 1960.As a post-doctoral fellow at the Institute of Oceanology of the Academy of Sciences, he formulated the Kitaigorodskii similarity hypothesis for the wind-generated wave spectrum in 1961. He later recounted how, as a young student, he had been impressed by the simple and at the same time fundamental results of O. M. Phillips, and he had wanted to follow that path by combining it with the Russian tradition, the Kolmogorov similarity hypothesis of turbulence. The Kitaigorodskii hypothesis immediately showed its value when W. J. Pierson Jr. and L. Moskowitz were able in 1964 to collapse their empirical data into a single dimensionless curve, and to formulate the Pierson-Moskowitz spectrum for fully developed seas. At the same time Kitaigorodskii continued his studies of the atmospheric side and showed how swell could influence the marine boundary layer, a topic upon which his ideas remain important.In 1962 Kitaigorodskii became a Senior Scientist at the institute and turned his main interest back to turbulence of the oceanic mixed layer. Here he again made fundamental contributions. These, combined with his earlier studies of waves and the air-sea boundary layer, were summarized in his thesis for the Russian degree of Doctor of Science in 1968. This work was the basis of a monograph, Physics of Air- Sea Interaction, published in 1970 and translated into English in 1973; it was used widely as a textbook. In the same year, 1973, he received the Rosenstiel gold medal, and in 1978 the Liege University Award.Between 1968 and 1977 Kitaigorodskii was head of the Laboratory of the Physics of Ocean- Atmosphere Interaction in the Institute of Oceanology.During the 1970s Kitaigorodskii established contacts with the Finnish geophysics community. His Finnish wife had completed her term as a press correspondent in Moscow, and she returned to Helsinki with their twin daughters. The Soviet system allowed Kitaigorodskii to visit his family every other year, and he used this opportunity to lecture and conduct research at several Finnish institutes.

Breakdown of Kolmogorov's first similarity hypothesis in grid turbulence

Kolmogorov's first similarity hypothesis (or KSH1) stipulates that two-point statistics have a universal form which depends on two parameters, the kinematic viscosity ν and the mean energy dissipation rate ⟨ε⟩. KSH1 is underpinned by two assumptions: the Reynolds number is very large and local isotropy holds. To disentangle the intricacies of these two requirements, we assess the validity of KSH1 in a flow where local isotropy is a priori tenable, i.e. decaying grid turbulence. The main question we address is how large should the Reynolds number be for KSH1 to be valid over a range of scales wider than, say, five Kolmogorov scales. To this end, direct numerical simulations based on the lattice Boltzmann method are carried out in low Reynolds number grid turbulence. The results show that when the Taylor microscale Reynolds number Rλ drops below about 20, the Kolmogorov normalised spectra deviate from those at higher Rλ; the deviation increases with decreasing Rλ. It is shown that at Rλ ≃ 20, the contribution of the energy transfer in the scale-by-scale energy budget becomes smaller than the contributions from the viscous and (large-scale) non-homogeneous terms at all scales, but never vanishes, at least for the range of Reynolds investigated here. A phenomenological argument based on the ratio N between the energy-containing timescale and the dissipative range timescale leads to the condition for KSH1 to hold. The numerical data indicate that N = 5, yielding Rλ ≃ 20, thus confirming our numerical finding. The present results show that KSH1, unlike the second Kolmogorov similarity hypothesis (KSH2,) does not require the existence of an inertial range. While it may seem remarkable that KSH1 is validated at much lower Reynolds numbers than required for KSH2 in grid turbulence (Rλ ≥ 1000,), KSH1 applies to small scales which include both dissipative scales and inertial range (if it exists). One can expect that, as the Reynolds number increases, the dissipative scales should satisfy KSH1 first; then, as the Reynolds number attains very high values, the inertial range is established in conformity with KSH2.

Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence

A scale-similarity model for Lagrangian two-point, two-time velocity correlations (LVCs) in isotropic turbulence is developed from the Kolmogorov similarity hypothesis. It is a second approximation to the isocontours of LVCs, while the Smith-Hay model is only a first approximation. This model expresses the LVC by its space correlation and a dispersion velocity. We derive the analytical expression for the dispersion velocity from the Navier-Stokes equations using the quasinormality assumption. The dispersion velocity is dependent on enstrophy spectra and shown to be smaller than the sweeping velocity for the Eulerian velocity correlation. Therefore, the Lagrangian decorrelation process is slower than the Eulerian decorrelation process. The data from direct numerical simulation of isotropic turbulence support the scale-similarity model: the LVCs for different space separations collapse into a universal form when plotted against the separation axis defined by the model.

Space-time correlations of fluctuating velocities in turbulent shear flows

Space-time correlations or Eulerian two-point two-time correlations of fluctuating velocities are analytically and numerically investigated in turbulent shear flows. An elliptic model for the space-time correlations in the inertial range is developed from the similarity assumptions on the isocorrelation contours: they share a uniform preference direction and a constant aspect ratio. The similarity assumptions are justified using the Kolmogorov similarity hypotheses and verified using the direct numerical simulation (DNS) of turbulent channel flows. The model relates the space-time correlations to the space correlations via the convection and sweeping characteristic velocities. The analytical expressions for the convection and sweeping velocities are derived from the Navier-Stokes equations for homogeneous turbulent shear flows, where the convection velocity is represented by the mean velocity and the sweeping velocity is the sum of the random sweeping velocity and the shear-induced velocity. This suggests that unlike Taylor's model where the convection velocity is dominating and Kraichnan and Tennekes' model where the random sweeping velocity is dominating, the decorrelation time scales of the space-time correlations in turbulent shear flows are determined by the convection velocity, the random sweeping velocity, and the shear-induced velocity. This model predicts a universal form of the space-time correlations with the two characteristic velocities. The DNS of turbulent channel flows supports the prediction: the correlation functions exhibit a fair good collapse, when plotted against the normalized space and time separations defined by the elliptic model.

Approach to the 4/5 law in homogeneous isotropic turbulence

Kolmogorov's similarity hypotheses and his 4/5 law are valid at very large Reynolds numbers. For flows encountered in the laboratory, the effect of a finite Reynolds number and of the non-stationarity or inhomogeneity associated with the large scales can affect the behaviour of the scales in the inertial range significantly. This paper focuses on the source of inhomogeneity in two types of flows, those dominated mainly by a decay of energy in the streamwise direction and those which are forced, through a continuous injection of energy at large scales. Results based on a parameterization of the second-order velocity structure function indicate that the normalized third-order structure function approaches 4/5 much more rapidly for forced than for decaying turbulence. This trend is supported by grid turbulence measurements and numerical data in a periodic box.

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.

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.

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.

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 > 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<x> 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.

The Prediction of Clear Air Turbulence over Mountainous Terrain

Abstract Recent aircraft measurements of clear air turbulence over Australia have shown that the phenomenon of CAT in a thermally stable environment is associated with a breakdown of waves, presumably gravity waves or Helmholtz waves on a stable interface, into random turbulent eddies. The energy distribution in the wavelength range in which clear air turbulence is experienced seems to follow the “−5/3 law” postulated by Kolmogorov's similarity hypothesis. The “−5/3 law” seems to extend to much longer wavelengths than previously anticipated. Combining these results of aircraft measurements with the theory on lee waves which has been derived by the use of perturbation equations, one finds that the energy involved in standing lee waves over mountains may “cascade” down from a wavelength range of approximately 10 km to a range near 100 m which then would be experienced as clear air turbulence, provided that the energy levels are high enough to cause any responses in an aircraft. This physical model of turbul...

Irreversible Statistical Mechanics of Incompressible Hydromagnetic Turbulence

The irreversible statistical mechanics of incompressible hydromagnetic turbulence driven by external forces is treated by methods which do not require that the system be close to a state of detailed balance. The equations of motion are expressed in terms of linearly independent modes formed from the wave-vector components of velocity and magnetic fields, and the nonlinear interaction is exhibited as the sum of individually conservative three-mode interactions. A fundamental statistical equation is constructed giving necessary and sufficient conditions for all members of a distribution of time-functions to satisfy the equations of motion; it involves only second-, third-, and fourth-order distribution moments. A variational criterion is proposed for specifying a distribution consistent with the fundamental equation under physically appropriate constraints. It leads to a complete formal solution of the statistical problem. This solution is not exploited. Instead, two statistical hypotheses based on the assumption of high mode density are advanced. With their aid, each three-mode interaction is treated as a small perturbation on the motion due to all the three-mode interactions and the external forces. The moments in the fundamental equation for the stationary case thereby are expressed in terms of the diagonal elements of the time-covariance matrix and distribution-averaged infinitesimal-impulse-response matrix of the system. Closed equations are obtained which fix these matrix elements in terms of the covariance matrix of the external forces. If the statistical hypotheses are sound, this provides a theory of unbounded turbulence (infinite mode density) driven by Gaussian-distributed homogeneous forces which is exact at all Reynolds numbers based on rms velocity and the macroscale determined by the driving forces. The general theory is specialized to obtain integro-differential equations determining the covariance scalars and modal impulse-response functions for stationary, isotropic hydromagnetic turbulence. In the nonmagnetic case, the asymptotic inertial-range solution yields the wave-number spectrum $E(k)=2\ensuremath{\pi}c{(\ensuremath{\epsilon}{v}_{0})}^{\frac{1}{2}}{k}^{\ensuremath{-}\frac{3}{2}}$ and the modal time-autocorrelation function $\frac{{J}_{1}(2{v}_{0}k\ensuremath{\tau})}{({v}_{0}k\ensuremath{\tau})}$, where ${v}_{0}$ is the rms velocity in any direction, $\ensuremath{\epsilon}$ is the mean rate of energy-cascade/unit-mass, and $c$ is a universal number fixed by the theory. This contradicts the Kolmogorov similarity hypotheses; independent arguments are advanced against the latter.