Single-point velocity distribution in turbulence

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 lnP(v) ∝ −v4 at v ≫ vrms, L/τ . For a short-correlated forcing when τ ≪ L/vrms there is an intermediate asymptotics lnP(v) ∝ −v3 at L/τ ≫ v ≫ vrms. PACS numbers 47.10.+g, 47.27.-i, 05.40.+j Typeset using REVTEX 1 Early experimental data on skewness and flatness of the velocity field prompted one to believe that the single-point velocity PDF in developed turbulence is generally close to Gaussian [1,2]. A possible reasoning may be that large-scale motions (that give the main contribution into velocity statistics at a point) are connected to a random external forcing f then velocity v(t) = ∫ t 0 f(t )dt has to be Gaussian if t is larger then the correlation time τ of the forcing, irrespective of the statistics of f . That would be the case if the force was the only agent affecting velocity. However, there are also nonlinearity (leading to instability and break-up of large-scale motions) and viscosity (that dissipates small-scale modes appeared as a result of the instability). Let us first explain the simple physics involved and formulate the predictions following from physical arguments, then we develop the formalism which gives the predicted PDF tails. Qualitatively, one may describe the interplay between external force and nonlinearity in the following way. Force f pumps velocity v ∼ ft until the time t∗ ∼ L/v when nonlinearity restricts the growth. The relation between velocity and forcing can thus be suggested as follows: v ∼ fL. Therefore, velocity’s PDF can be obtained by substituting f ∼ v/L into force’s PDF Pf : P(v) ∼ Pf (v /L). Those arguments presume that t∗ is less than the correlation time τ of forcing. If opposite is the case t∗ ≫ τ then the law of velocity growth is different v ∼ f tτ , so that the velocity increases up to v ∼ f Lτ ; the short-correlated pumping is effectively Gaussian Pf ∼ exp(−f ) and the velocity’s PDF is P(v) ∼ exp[−(v/vrms) ]. We see that velocity PDF is expected to be dependent on the statistics of the force. Actual mechanism of restriction for the Navier-Stokes equation (instability of a largescale flow leading to a cascade that provides for a viscous dissipation) is irrelevant for the above arguments. What matters is that we deal with the system of the hydrodynamic type so that nonlinear time t∗ can be estimated as L/v. For example, the same argument goes for Burgers equation where t∗ is a breaking time and viscous dissipation at a shock prevents further growth [5]. It is interesting that viscosity does not enter above estimates yet it’s existence is crucial for the whole picture to be valid. Let us stress that the above arguments can be only applied to rare events with velocity 2 and force being much larger than their root-mean-square values when the influence of background fluctuations can be neglected. The above predictions are thus made for PDF tails. For a a nonlinear dissipative system, it is generally difficult to relate an output statistics to the statistics of the input (be it initial conditions or external force). Our aim here is to show that it is possible, nevertheless, to relate the probabilities of rare fluctuations that is to relate the tails of the PDFs of the force and the field that is forced respectively. A rigorous way to describe rare fluctuations is the instanton formalism recently developed for turbulence [3] and employed for obtaining PDF tails in different problems [4–6]. The main idea of the method is that the tails are described by saddle-point configurations in the path integral for the correlation functions of the turbulent variable (say, velocity v). We call the configuration instanton because of a finite lifetime. One may call it also optimal fluctuation since it corresponds to the extremum of the probability. We start with the Navier-Stokes equation ∂tvα + vβ∇βvα − ν∇ vα +∇αP = fα , (1) where f is a random force (per unit mass) pumping the energy into the system and ν is the viscosity coefficient. Incompressibility is assumed so that div v = div f = 0. The field P in (1) is the pressure divided by the mass density ρ. Velocity correlation functions can be presented as path integrals which form is determined by the statistics of pumping. Let us first consider a Gaussian forcing with the correlation function 〈fα(t, r)fβ(t , r)〉 = Ξαβ(t−t , r−r) which is assumed to decay on the scale τ as a function of the first argument and on the scale L as a function of the second one. Then moments of the velocity can be written as path integrals: 〈v〉 = ∫ DpDvDP DQ exp (iI + 2n ln v) , (2) where p is an auxiliary field and the effective action has the following form [7]. I = ∫ dt dr [pα(∂tvα + vβ∇βvα − ν∇ vα +∇αP ) +Q∇αvα] + i 2 ∫ dt′dtdr′drΞαβ(t− t ′, r − r′)pαp ′ β . (3)