An Intermittency Model For Passive-Scalar Turbulence

A phenomenological model for the inertial range scaling of passive-scalar turbulence is developed based on a bivariate log-Poisson model. An analytical formula of the scaling exponent for three-dimensional passive-scalar turbulence is deduced. The predicted scaling exponents are compared with experimental measurements, showing good agreement. 47.27.-i, 47.27.Gs Typeset using REVTEX 1 Submitted to Phys. Fluids 2 The inertial range dynamics of a passive scalar advected by fluid turbulence is of great theoretical interest [1]. In particular, recent success [2] in the deduction of the anomalous scaling exponent for the passive scalar field advected by a white and Gaussian velocity has inspired renewed enthusiasm for the problem [3]. For the three dimensional turbulence diffusion, the advective velocity field is governed by the Navier-Stokes equation whose statistics is far from white and Gaussian. The extension of Kraichnan’s theory [2] for this problem is intriguing and non-trivial. In fact it has even been a difficult task to experimentally measure the scaling exponents of the passive scalar. The existing experimental data do not yield convergent results [4–6]. Direct numerical simulation (DNS) has been a useful alternative for studying scaling behaviors in fluid turbulence [7]. In a recent note, using the DNS data, we have tried to explain the discrepancies among experimental results [8]. The existing phenomenological theories, such as the β model [9] and a bivariate logNormal model [10], are based on fractal structures of turbulence [5], yielding analytical predictions of scaling exponents. Nevertheless, it has been noted that log-normal distribution leads to negative values of scaling exponents for high order passive scalar structure functions, contradicting existing numerical and experimental measurements. In this letter, we develop a phenomenological theory based on a bivariate log-Poisson model. The analytical formula for the scaling exponents is obtained. The scaling exponents from the theory are compared with those from experimental measurements, showing good agreement. According to Kolmogorov’s refined similarity hypothesis (RSH), or K62 theory [11] for the velocity field and similar RSH for the passive scalar field proposed by Obukhov [12], ∆ul = vu(lǫl) , ∆Tl = vT l ǫ −1/6 l N 1/2 l , (1) where ∆ul = u(x + l) − u(x) and ∆Tl = T (x + l) − T (x) are the longitudinal velocity increment and the scalar increment respectively, l is in the inertial range, vu and and vT are Submitted to Phys. Fluids 3 random variables whose statistics only depend on the Reynolds number. ǫl and Nl are the locally averaged velocity dissipation and scalar dissipation, respectively [4]. Equation (1) has been verified by experiments and is in general believed to be correct [13]. Suppose that the p− th order velocity and passive scalar structure functions satisfy the scaling relations in the inertial range: Sp(l) = 〈(∆ul)〉 ∼ lp, S p (l) = 〈(∆Tl)〉 ∼ lp, then to deduce scaling exponents from (1), a probability density function (PDF) of velocity field and a joint PDF for velocity and passive scalar must be modeled. Here 〈·〉 denotes an ensemble and ζp and zp are the p − th order scaling exponents for the velocity and scalar structure functions, respectively. The log-normal assumption in K62 theory [11] has been challenged recently by She and Leveque [14] who argued that a log-Poisson PDF for ǫr leads to a better description of the statistics of ǫr. A hierarchy model based on the log-Poisson PDF was derived for the inertial range scaling exponents, showing good agreement with numerical simulations [7] and experiments [15]. The distribution function of a random variable x satisfying Poisson distribution can be written as [16]: p(x = i) = pi = e a/i!, where a is the variance and the mean for the variable. The generating function of pi has the form: P (s) = ∑ i pis i = e. The generating function for a bivariate Poisson distribution pi,j = p(x = i, y = j) can be defined as P (s1, s2) = ∑ ij pijs i 1s j 2 = e 12112212, where a1, a2 are variances for x and y, respectively, and b is a constant representing the correlation between x and y. Let us assume that the velocity dissipation and passive scalar dissipation have the hierarchy structure relation: ǫl1 = Wl1l2ǫl2 and Nl1 = Vl1l2Nl2 , where l1 and l2 are two length scales; Wl1l2 and Vl1l2 are multiplicative factors depending on l1 and l2. For l1 and l2 within the inertial range, the multiplicative factors can be written in the forms of Wl1l2 = (l1/l2) β and Vl1l2 = (l1/l2) hT β T , where β, βT , h and hT are constants to be determined later; (X, Y ) are stochastic variables. In the following, we assume that (X, Y ) follow the bivariate Poisson distribution pi,j. Submitted to Phys. Fluids 4 If there is a scaling, then for l in the inertial range, 〈ǫplN q l 〉 ∼ l, where τp,q is the scaling exponent of power order p and q. Using the bivariate Poisson distribution assumption, we have: 〈W p l1l2V q l1l2 〉 = (l1/l2) ∑ i,j ββ T pij. It is easy to recognize that the summation in the right hand side of the above equation is nothing but the generating function for the bivariate Poisson distribution. It is then straightforward to write a formula for τ(p, q): τ(p, q) = −hp− hT q + (a + aT + b− aβ − aTβ T − bββ T )/ln(l1/l2), (2) where constants a, aT and b are functions of l1 and l2. The obvious physical constrains τ(1, 0) = τ(0, 1) = 0 result the following relations: a = [h/(1 − β)]ln(l1/l2 − b) and aT = [hT/(1−βT )]ln(l1/l2 − b). If we further assume the correlation between ǫl and Nl satisfying power law scaling in the inertial range, i.e. b = γbln(l1/l2) and γb is a constant, we obtain the scaling exponent relation: τ(p, q) = τp + τq + τb(p, q), (3) where τp = −hp + [h/(1 − β)](1− β), and τq = −hT q + [hT/(1− βT )](1 − β T ), are scaling exponents for moments 〈ǫpl〉 and 〈N q l 〉 respectively, and τb(p, q) = γb(1 − β − β T + ββ T ) is the contribution from the cross correlation. There have been several discussions about how to determine the model constants in τp,q [14,17]. The parameter β depends on the Submitted to Phys. Fluids 5 constant h and the intermittency parameter μ = τ2 [17], i.e. β = 1 − μ/h. It has been argued that the co-dimension, C, for the most singular structure of the velocity dissipation [14] is related to β and h: C = h/(1− β) = h/μ. (4) Wed Nov 6 17:05:55 1996