S. Goto

S. Kida

National Institute for Fusion Science (NIFS-Repository)

§26. Lagrangian Direct-interaction Approximation for a Passive Scalar Field Goto, S., Kida, S. It is interesting to investigate statistical property of a scalar field O(x, t) such as temperatu~e, particle concentration, etc., which is diffused and passively ad-vected by turbulent flow as 80 80 8 20 - +Ui- =K,---. 8t 8Xi 8Xi8xi (1) Here, K, is the diffusion coefficient and Ui (x, t)" (8Ui/8xi = 0) is an incompressible turbulent veloc-ity field governed by the Navier-Stokes equation, 8Ui 8Ui 1 8p 82 Ui -8 + Uj -8 = -- -8 + v 8 ,(2) t Xj P Xi x j8xj where v is the kinematic viscosity, and p and pare the pressure and the constant density of the fluid, re-spectively. Micro-scale statistics of a passive scalar field has been studied by many researchers from phe-nomenological viewpoints (see §VI-IO of Ref. 1 for a review). They have pointed out that statistics of the passive scalar field strongly depends upon the Schmidt number s = v / K,. For example, it has been proposed that the passive scalar spectrum function E>(k), which is defined by the Fourier transform of the two-point correlation function of O(x), obeys different scaling laws depending upon the Schmidt number s: (i) k-5 / 3 law in the inertial-advective range (irrespective of s), (ii) k-1 law in the viscous-advective range (for s » 1, see Fig.I) and (iii) k-17/ 3 law in the inertial-diffusive range (for s « 1). Although these scaling laws are supported by experiments or numerical simulations, the phenomenologies can predict neither the univer-sal coefficients of the scaling functions nor the func-tional forms of the spectrum in the whole universal wavenumber range. A purpose of this study is to predict functional forms of the passive scalar spectrum from the basic equations (1)(2) by overcoming the closure problem of moment hierarchy. We have developed a closure theory so-called the Lagrangian direct-interaction ap-proximation (DIA) for velocity correlation functions 2) baSed upon the weak dependence principle proposed by Kraichnan 3). Note that the Lagrangian DIA formu-lation is different from those of the other DIAs. The Lagrangian DIA is formulated for the Lagrangian cor-relation and Lagrangian response functions in contrast with the original Eulerian DIA 3), and its formulation is much simpler than that of Kraichnan's Lagrangian 300 DIA 4). Since the Lagrangian DIA for the velocity cor-relation function was shown to be quite successful 2), we apply it to a passive scalar field to derive from (1)(2) a closed set of equations for the passive scalar spectrum function and the velocity correlation func-tion. Then, we can analytically show that the Schmidt number dependence of solutions to the resultant clo-sure equations is completely consistent with the phe-nomenologies; the three scaling functions (k- 5 / 3 , k- 1 , k- 17/ 3 ) are solutions to them. (See Ref.5 for a deriva-tion of the Lagrangian DIA equations and analyses of the solutions.) Furthermore, we can evaluate func-tional forms of the spectrum function for arbitrary Schmidt numbers in the entire universal wavenumber range by solving the closure equations numerically. Fig.1 is an example of the results, which shows the universal form of the passive scalar spectrum in the large Schmidt number limit. We can confirm that the k-5/ 3 and the k-1 scaling functions are solutions to the Lagrangian DIA equations. (a) 1015 ,....--..,--.......------,----, 10-6 104 10-2 102 k/kB Fig.I Non-dimensional passive scalar spectrum func-tion 8(k) around (a) kK = k./v3)1/4 and (b) kB = kK s l / 2 (f is the energy dissipation rate) in the large Schmidt number limit. The line denotes a solution to the Lagrangian DIA equations. We call the wavenum-ber ranges k « kK and kK « k « kB the inertial-advective and viscous-advective ranges, respectively. References 1) Lesieur, M., "Turbulence in Fluids", Kluwer (1997). 2) Kida, S. and Goto, S., J. Fluid Mech. 345 (1997) 307." 3) Kraichnan, R. H., J. Fluid Mech. 5 (1959) 497. 4) Kraichnan, R. H., Phys. Fluids 8 (1965) 575. 5) Goto, S. and Kida, S., Phys. Fluids (in press). 

Lagrangian Direct-interaction Approximation for a Passive Scalar Field

Abstract

Similar works

Full text

Available Versions

National Institute for Fusion Science (NIFS-Repository)