Use of single-point velocity probability distributions in describing turbulent flows (original) (raw)

Using single-point velocity probability distributions for characterizing turbulent flows

Journal of Applied Mechanics and Technical Physics, 1982

A semiempirical equation for a single-point velocity probability distribution in turbulent flow is proposed and analyzed. A general solution to the equation is presented, and its properties are analyzed for the case of homogeneous turbulence. A boundary-value problem is formulated and a numerical solution is obtained for steady flow in a flat duct (including the case where the friction forces

Single-Point Velocity Distribution in Turbulence

Physical Review Letters, 1997

We show that the tails of the single-point velocity probability distribution function (PDF) are generally non-Gaussian in developed turbulence. By using instanton formalism for the Navier-Stokes equation, we establish the relation between the PDF tails of the velocity and those of the external forcing. In particular, we show that a Gaussian random force having correlation scale L and correlation time τ produces velocity PDF tails ln P(v) ∝ −v 4 at v ≫ v rms , L/τ. For a short-correlated forcing when τ ≪ L/v rms there is an intermediate asymptotics ln P(v) ∝ −v 3 at L/τ ≫ v ≫ v rms .

Model equation for the probability distributions of the velocity and concentration during turbulent mixing and diffusion combustion of gases

Combustion, Explosion, and Shock Waves, 1983

It is known that under certain assumptions of a general nature, the solution of the problem of subsonic turbulent combustion of previously unmixed gases with combustion limited by the process of fuel and oxidizer mixing requires knowledge about the combined probability distribution of the velocity and concentration of an inert (i.e., not participating in reactions) impurity (see, e.g., [1]). A derivation of the exact equation forthe mentioned probability distribution is given in [2-7]. This equation is not closed, and the main problem isto close it. A semiempirical closure which is an extension of the closure of the velocity probability distribution [7, S] is proposed in this paper. ]~ is shown in [9-13] that in the general case miscibility exerts substantial influence on the concentration probability distribution. As is seen from [9-12], a theoretical analysis of miscibility is fraught with significant difficulties to be overcome; hence, in this paper miscibility is not taken into account as a first approximation (~he deduction made in [14, 15] that taking account of miscibility has a slight effect on the results of averaging the density, temperature, and other thermodynamic characteristics in a number of cases is a certain justification of such an approximation).t FUNDAMENTAL EQUATIONS. CLOSING RELATIONSHIPS. SELECTION OF EMPIRICAL CONSTANTS An equation for the density of the combined velocity and concentration probability distribution during diffusion combustion or mixing of gases can be obtained by using the methods developed in [16-19] (see also the surveys in [7, 20]). After utilizing the hypothesis of statistical independence of the micro-and macrocharacteristics of turbulence at high Reynolds numbers [18, 19], as well as the generalization of the closing relationships from [7, S], the equation mentioned is reduced to the following: apP.4.

Universality of probability distributions among two-dimensional turbulent flows

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

We study statistical properties of two-dimensional turbulent flows. Three systems are considered: the Navier-Stokes equation, surface quasigeostrophic flow, and a model equation for thermal convection in the Earth's mantle. Direct numerical simulations are used to determine one-point fluctuation properties. Comparative study shows universality of probability density functions (PDFs) across different types of flow. For instance, the PDFs for derivatives of the advected quantity are the same for the three flows, once normalized by the average size of fluctuations. The single-point statistics is surprisingly robust with respect to the nature of the nonlinearity.

Mean Velocity Equation for Turbulent Fluid Flow: An Approach via Classical Statistical Mechanics

2003

The possibility to derive an equation for the mean velocity field in turbulent flow by using classical statistical mechanics is investigated. An application of projection operator technique available in the literature is used for this purpose. It is argued that the hydrodynamic velocity defined there, in situations where the fluid is turbulent, is to be interpreted as the mean velocity field; in that case, the momentum component of the generalized transport equation derived there is the mean velocity equation. In this paper, stationary incompressible flow for constant mass density and temperature is considered. The stress tensor is obtained as a nonlinear functional of the mean velocity field, the linear part of which is the Stokes tensor. The formula contains a time correlation function in local equilibrium. Presently, there exists a microscopic theory for time correlations in total equilibrium only. For this reason and as a preliminary measure, the formula has been expanded into a power series in the mean velocity; though this limits the applicability to low Reynolds number flow. The second order term has been evaluated in a former paper of the author. For the third order term, the form of the kernel function is derived. Its calculation with the aid of the mode-coupling theory is completed; it will be reported in an separate paper. An numerical application with the data of the circular jet is under way.

Taylor's hypothesis and the probability density functions of temporal velocity and temperature derivatives in a turbulent flow

Journal of Fluid Mechanics, 1980

Equations for the instantaneous velocity and temperature fluctuations in a turbulent flow are used to assess the effect of a fluctuating convection velocity on Taylor's hypothesis when certain simplifying assumptions are made. The probability density function of the velocity or temperature derivative is calculated, with an assumed Gaussian probability density function of the spatial derivative, for two cases of the fluctuating convection velocity. I n the first case, the convection velocity is the instantaneous longitudinal velocity, assumed to be Gaussian. I n the second, the magnitude of the convection velocity is equal to that of the total velocity vector whose components are Gaussian. The calculated probability density function shows a significant departure, in both cases, from the Gaussian distribution for relatively large amplitudes of the derivative, a t only moderate values of the turbulence intensity level. The fluctuating convection velocity affects normalized moments of measured velocity and temperature derivatives in the atmospheric surface layer. The effect increases with increasing order of the moment and is more significant for odd-order moments than even-order moments.

Probability distribution functions in turbulent flows and shell models

In the multifractal framework some recent results [Phys. Rev. Lett. 67, 2295 (1991)] on the probability distribution functions (pdf) in fully developed turbulence are reviewed, for the increments of the velocity field in the inertial range and for transversal gradients. New comparisons are also produced, in the same scenario, with clusters of isogradients and with pdf evaluated from a numerical simulation of a shell model. The Reynolds dependence of the flatness in the multifractal model is discussed.