Probability distribution functions for small scale turbulence

The exact expression for the probability distribution function (pdf),P(Δu_r), of a velocity difference Δu_r, over a distancer, in incompressible fluid turbulence, obtained from the Navier-Stokes equations, is used as a basis for deriving approximate profiles forP(Δu_r). These approximate forms are deduced from an approximate factorisation of the underlying functional probability distribution of the flow field, in which the individual factors capture different physical effects.P(Δu_r) is represented as the integral, with respect to the spatially averaged dissipation rateε _r, of the product of the conditionalpdf of Δu_r givenε _r, and thepdf ofε _r. The approximation yields the latter as a log-Poissonpdf, while the conditionalpdf is found to be a Gaussian for a transverse increment, and the product of a Gaussian and a cubic polynomial for a longitudinal increment. This approximation is equivalent to the refined similarity hypothesis coupled with the log-Poisson distribution, and it possesses the characteristic features ofP(Δu_r), including symmetric profiles for transverse increments, asymmetric profiles for longitudinal increments, and the development of pronounced non-Gaussian features at small separations. The associated scaling exponents for longitudinal and transverse structure functions are shown to be identical, in this approximation, and to assume the log-Poisson form.     

The velocity fluctuation spectrum of a gas with particles in the case of uniform isotropic turbulence

In the present paper, an analysis of the influence of particles on the fluctuation spectrum of a gas is based on equations for the two-point moments of the velocity component fluctuations of the solid and carrier phases. For an approximation of the terms describing the transfer of fluctuation energy through the spectrum, we use Heisenberg's hypothesis [8], modified to take into account the influence of the particles on the rate of redistribution of turbulent energy between vortices of different dimensions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 69–74, January–February, 1987.     

PROBABILITY DISTRIBUTION FUNCTION OF NEAR-WALL TURBULENT VELOCITY FLUCTUATIONS

IntroductionNear-wall turbulence plays an important role in environmental engineering due to itsgreat impact on the transport of pollutants,nutrients,sediments,etc.Therefore,peoplehave long been studying the wall-bounded turbulence[1,2],in which the proba…     

Measurement of inertial particle clustering and relative velocity statistics in isotropic turbulence using holographic imaging

We present the first measurements of relative velocity statistics of inertial particles in a homogeneous isotropic turbulent flow with three-dimensional holographic particle image velocimetry (holographic PIV). From the measurements we are able to obtain the radial relative velocity probability density function (PDF) conditioned on the interparticle separation distance, for distances on the order of the Kolmogorov length scale. Together with measurements of the three-dimensional radial distribution function (RDF) in our turbulence chamber, these statistics, in principle, can be used to determine interparticle collision rates via the formula derived by Sundaram and Collins (1997). In addition, we show temporal development of the RDF, which reveals the existence of an extended quasi-steady-state regime in our facility. Over this regime the measured two-particle statistics are compared to direct numerical simulations (DNS) with encouraging qualitative agreement. Statistics at the same Reynolds number but different Stokes numbers demonstrate the ability of the experiment to correctly capture the trends associated with particles of different inertia. Our results further indicate that even at moderate Stokes numbers turbulence may enhance collision rates significantly. Such experimental investigations may prove valuable in validating, guiding and refining numerical models of particle dynamics in turbulent flows.     

Weak turbulence in media with a decay spectrum

The theory of weak turbulence of a plasma has been investigated in many papers [1–5]. It has been established that weak turbulence may be described by means of the kinetic wave equations. Here the collision term in the kinetic equation is the sum of two substantially different components. The first of these has the character of nonlinear wave damping and differs from zero in those cases where interaction between waves and particles is significant. It has a comparatively simple mathematical nature and can be analyzed. The second component is specifically a collision term, it depends closely on the form of the spectrum in the medium and describes the exchange of energy between different groups of waves. The case when the second component plays the principal role in the collision term has scarcely been studied. The present paper is devoted to a study of this case.The analysis is carried out for a simple isotropic model of a medium with an almost linear dispersion law, but with a positive second derivative; we shall call such a spectrum a decay spectrum. This model is much closer to reality than the model considered in [6]. The results obtained from this model are evidently fairly general in character and express substantially the regularity of behavior of weak turbulence in media with a weak decay spectrum. The basic result of the paper is as follows: apart from the Rayleigh-Jeans solution, there exists another solution which reduces the collision term to zero. This solution corresponds to a process which is substantially nonequilibrium, and may be realized in actual problems, where there are always wave sources or transfer terms playing the same part, only in cases where there is wave damping in the medium with a coefficient which increases fairly rapidly into the region of large k. Here the universal character, as it were, of the nonequilibrium process is realized.Notation k wave vector - a parameter characteristic of the dispersion - _k wave frequency - _k density of wave sources - V _kkkk matrix element describing the wave interactions - (s) gamma function - u a variable describing the medium - k ₀ boundary of instability region - a _k complex wave amplitude - k ^a damping decrement - n _k wave density in k space - k ₁ boundary of the region of transparency - n _k wave density in spherical coordinates - v instability increment In conclusion the author thanks R. Z. Sagdeev for discussing the paper.     

Calculation of isotropic turbulence spectrum

The spectral function of isotropic turbulence is obtained on the basis of semi-empirical turbulence theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–164, November–December, 1974.     

Development of lattice turbulence in a stream with a constant velocity gradient

On the basis of the equations for the Reynolds stresses and the equation for the scale of the turbulence, an analysis is made of the development of lattice turbulence in a stream with a constant velocity gradient. The constants in the equations are determined under the assumption that, far from the lattice and with large Reynolds numbers, the structure of the turbulence tends toward a limiting state with constant values of the correlation coefficient, the degree of anisotropy, and the dimensionless velocity gradient. The constants in terms containing the viscosity are determined from a consideration of the flow beyond the lattice without a velocity gradient in the final stage of decay of the turbulence. The equations obtained were solved in an electronic computer. The calculation is in satisfactory agreement with the existing experimental data. For calculating flows with a variable velocity gradient, instead of the equation of the scale, it is proposed to use an equation for the frequency of the turbulent pulsations obtained in the present work. The computer calculations were made by S. I. Bekritskaya.     

The role of inertial energy transfer in uniformly distorted homogeneous turbulence

Homogeneous turbulence was subjected to plane strain in a distorting tunnel of maximum strain ratio 13:1. The turbulence was generated by using bi-planar grids of square bars with different solid fractions and a range of mean flow velocities, so that relative strain rates could be varied by about a factor of three; relative strain rate being here defined as the ratio of strain rate to initial eddy turnover rate. The change in anisotropies of the turbulence intensities and also of the velocity derivatives are found to depend strongly on the relative strain rate and so does the change in turbulent kinetic energy. This dependency is explained in terms of the ratio of turbulence production to dissipation rates which are related to the relative strain rate. An interesting feature is the inflexion in the anisotropy versus strain ratio curve observed for low relative strain rates. This is thought to be a result of an ‘overshoot’ of pressure strain rates.     

Inertial obukhov-bolgiano interval in shell models of convective turbulence

The shell model of developed convective turbulence of an incompressible fluid is considered. Regimes developing at high Rayleigh numbers are investigated numerically for three- and two-dimensional motion. It is shown that in the three-dimensional turbulent convection model the inertial Obukhov-Bolgiano interval is developed on large scales, but this interval is unstable and gives way to the Kolmogorov regime in which the temperature behaves as a passive admixture. In the two-dimensional turbulent convection model a finite scale interval on which the buoyancy forces determine the nature of the fluctuations but the spectral laws established differ from those that follow from dimensional considerations for the Obukhov-Bolgiano interval is detected. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 37–46, November–December, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 94-01-00951a).     

Wind tunnel measurements of the preferential concentration of inertial droplets in homogeneous isotropic turbulence

We describe an experimental setup aimed at studying turbulent-induced droplet collisions in a laboratory setting. Our goal is to reproduce conditions relevant to warm-rain formation in clouds. In these conditions, the trajectories of small inertial droplets are strongly influenced by the background air turbulence, and collisions can potentially explain the droplet growth rates and spectrum broadening observed in this type of clouds. Warm-rain formation is currently under strong scrutiny because it is an important source of uncertainty in atmospheric models. A grid at the entrance of a horizontal wind tunnel produces homogeneous isotropic turbulence at a Re _λ in the range of 400–500. Water droplets are injected from the nodes of the turbulence-inducing grid at a volume fraction (?) of 2.7?×?10^?5 and with sizes of 10–200?μm. A complex manifold-injection system was developed to obtain uniform water droplet seeding, in terms of both water content and size distribution. We characterize the resulting droplet-laden turbulent flow, and the statistics of droplet pairs are measured and analyzed. We found that the radial distribution function (RDF), a measure of preferential concentration of droplets that plays a key role in collision kernel models, has a large peak at distances below the Kolmogorov microscale of the turbulence. At very long separations, comparable with the integral length scale of the turbulence, these RDFs show a slow decay to the average probability given by the mean droplet number density. Consistent with this result, conditional analysis shows an increased local concentration of droplets within the inertial length scale (≈ 10–100 Kolmogorov lengths). These results are in good agreement with previous experiments that found clustering of inertial droplets with St?≈ 1 at scales on the order of 10η. Ultimately, our results support the hypothesis that turbulence-induced preferential concentration and enhanced settling can lead to significant increases in the collision probability for inertial droplets in the range 10–50?μm.     

一种用于无陀螺捷联惯导系统的角速度融合算法

无陀螺捷联惯导系统(GFSINS)是用加速度计的合理空间组合解算出载体的角速度。载体角速度的解算精度是GFSINS的技术关键。在分析GFSINS九加速度计配置方案的基础上,提出一种新的角速度融合算法,消除该方案解算过程中开方计算及符号判断带来的误差。该算法还明显抵消了加速度计输出中包含的常值零点偏移误差和温度漂移误差等,具有实时性好、计算量小、通用性强的优点。仿真计算表明该算法可行,并能在一定程度上提高系统解算精度。     

Reynolds-number-dependence of the maximum in the streamwise velocity fluctuations in wall turbulence

A survey is made of the standard deviation of the streamwise velocity fluctuations in near-wall turbulence and in particular of the Reynolds-number-dependency of its peak value. The following canonical flow geometries are considered: an incompressible turbulent boundary layer under zero pressure gradient, a fully developed two-dimensional channel and a cylindrical pipe flow. Data were collected from 47 independent experimental and numerical studies, which cover a Reynolds number range of R _θ=U _∞ θ/v=300−20,920 for the boundary layer with θ the momentum thickness and R ⁺=u _*R/v=100-4,300 for the internal flows with R the pipe radius or the channel half-width. It is found that the peak value of the rms-value normalised by the friction velocity, u _*, is within statistical errors independent of the Reynolds number. The most probable value for this parameter was found to be 2.71±0.14 and 2.70±0.09 for the case of a boundary layer and an internal flow, respectively. The present survey also includes some data of the streamwise velocity fluctuations measured over a riblet surface. We find no significant difference in magnitude of the normalised peak value between the riblet and smooth surfaces and this property of the normalised peak value may for instance be exploited to estimate the wall shear stress from the streamwise velocity fluctuations. We also consider the skewness of the streamwise velocity fluctuations and find its value to be close to zero at the position where the variance has its peak value. This is explained with help of the equations of the third-order moment of velocity fluctuations. These results for the peak value of the rms of the streamwise velocity fluctuations and also the coincidence of this peak with the zero value of the third moment can be interpreted as confirmation of local equilibrium in the near-wall layer, which is the basis of inner-layer scaling. Furthermore, these results can be also used as a requirement which turbulence models for the second and triple velocity correlations should satisfy. The authors are indebted to Prof. P. Bradshaw for making available his list of references on this topic and for his remarks on “active” and “inactive” motions. We also gratefully acknowledge discussions with Prof. I. Castro regarding the value of σ _u ⁺ above rough walls.     

Modelling of inhomogeneous turbulence in the absence of mean velocity gradients

In this paper we propose a new rational model for turbulent transport of Reynolds stresses and dissipation. For this purpose we first analyse some properties of diffused turbulence i.e. turbulence where dissipation is balanced by turbulent diffusion arising from inhomogeneity. Then we use some of these results to deduce some mathematical requirements that must be satisfied by any rational model and we show that they are not verified by available models. Consequently we derive a more general model taking into account the various properties of diffused turbulence.     

Reconstruction of turbulence spectrum from transient characteristics of a shadow-instrument signal

With the investigation of turbulence using a shadow instrument with photoelectric recording, the statistical characteristics of the signal taken off from the instrument are used to obtain information on the statistics of the investigated medium [1, 2]. In situations where the investigated medium is moving perpendicular to the instrument axis (for example, with experiments in hydro- and aerodynamic tubes), it is convenient to use the transient characteristics of the signal. In the present article an investigation is made of the connection of the transient correlation function and the frequency spectrum of a shadow-instrument signal with the energy spectrum of the optical inhomogeneities in the medium; a method is given for reconstructing the spectrum of the inhomogeneities from the correlation function or the transient spectrum of the signal.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 73–77, January–February, 1978.