Kolmogorov’s refined similarity hypotheses

Abstract

Using velocity data obtained in the atmospheric surface layer, we examine Kolmogorov's refined hypotheses. In particular, we focus on the properties of the stochastic variable K=Aw(r)/(re r ) ,/3 , where Auir) is the velocity increment over a distance r, and e r is the dissipation rate averaged over linear intervals of size r. We show that V has an approximately universal probability density function for r in the inertial range and discuss its properties; we also examine the properties of V for r outside the inertial range.PACS numbers: 05.45.+b, 02.50,+s, 03.40.Gc, A fruitful idea in high-Reynolds-number turbulence is the phenomenological picture of local structure introduced by Kolmogorov [1]. In its simplified version, the picture relates to the probability density function (PDF) of the velocity increment Aw (/*) = u (x + r) -u (x), where u (x+r) and uix) are velocities along the x axis at two points x and x +r separated by a distance r<^L, L being the integral scale of turbulence. According to the first hypothesis, the PDF of Auir) depends (besides on r itself) only on the average rate of energy dissipation (e) and the fluid viscosity v. The second hypothesis is that if the Reynolds number is very large, there exists a range of scales (the so-called inertial range) for which v becomes irrelevant, so that the only important external parameter is (s). The consequences of these universality arguments are well known [2] and need no repetition. Reference [2] also describes the circumstances which led to the modification of these hypotheses.Kolmogorov [3] himself introduced refinements of his 1941 theory, which abandoned local universality and introduced more restrictive alternatives to the 1941 hypotheses. The first refined hypothesis postulated a certain relation between Au(r) and the local average of the energy dissipation rate e r (x,t), taken over an interval of size r centered at x. Define, following Kolmogorov [3], a velocity scale at the point (x,t) by U r = (re r ) l/3 and form the local Reynolds number as Re r =t/ r r/v.(1)The first refined hypothesis then takes the form of the statement that, for r<£L,where V is a stochastic variable whose PDF depends only on Re r . The second refined hypothesis states that, if Re r^> 1, the PDF of V becomes independent also of Re r and thus universal. Unlike the original hypotheses, these refinements require a supplementary statement on the PDF of Au(r) or of e r . Kolmogorov's third hypothesis is that the PDF of e r is log-normal with a specific form for its variance. The consequences of Eq.(2) have been tested in vari-ous ways, but mainly by viewing it as a dimensional relation |Aw(r)|~(re r ) 1/3 . The stochastic function V is usually disregarded, often leading to misinterpretations of Eq.(2). Indeed, V contains much of the inertial range physics. For example, if Kolmogorov's j law [4]is understood to express the cascade of turbulent kinetic energy from large to small scales, the nonzero value of (K 3 > should account for this process. Similarly, the knowledge of t...

On conditional scalar increment and joint velocity–scalar increment statistics

“…According to equation (1), the normalized conditional probability density function (PDF) of u(r) conditional on r and that of φ(r) conditional on r and χ r (specifically 1/3 r and χ r −1/6 r , which provide velocity and scalar scales at scale r) are universal. Previous experimental and numerical results support these predictions and show that the conditional PDFs are usually quasi-Gaussian [3,11,12], i.e. with kurtosis close to three but with non-zero skewness (only when the separation is comparable to the dissipation scales do the PDFs deviate significantly from Gaussian and become bimodal [3,11,12]).…”

“…Previous experimental and numerical results support these predictions and show that the conditional PDFs are usually quasi-Gaussian [3,11,12], i.e. with kurtosis close to three but with non-zero skewness (only when the separation is comparable to the dissipation scales do the PDFs deviate significantly from Gaussian and become bimodal [3,11,12]). Therefore, for an inertial-range separation, conditional velocity and scalar increments, statistics conditional on their respective scales are generally considered to be universal.…”

“…The change from Gaussian to bimodal conditional PDF shows that, when conditioned on the scalar scale given by the local scalar variance ( φ 2 1/2 r ), the scalar increments have non-universal PDFs. This is in contrast with the universal, quasi-Gaussian conditional-increment PDFs conditional on χ r −1/3 r previously observed [3,9] as hypothesized in K62. Note that, in K62, the parameters are χ r and r ; therefore the mean scalar and energy dissipation rates do not influence the conditional local turbulence.…”

“…Previous studies have shown that the conditional velocity-increment PDF for an inertial range separation is close to Gaussian when conditioned on the locally averaged energy dissipation rate [3] but when conditioned on the local kinetic energy it can be close to Gaussian or bimodal [9]. Here we examine the conditional JPDF of velocity and scalar increments.…”

“…The increments are often analysed in the context of Kolmogorov's refined similarity hypotheses (hereafter referred to as K62) [10]. According to the hypotheses the velocity and scalar increments for an inertial-range separation r, u(r) = u(x + r) − u(x), and φ(r) = φ(x + r) − φ(x) can be written as [3,11] …”