S. Goto

S. Kida

National Institute for Fusion Science (NIFS-Repository)

§24. Passive Material Line Statistics in Turbulence Goto, S., Kida, S. This study is intended as the investigation of statis-tics of material lines passively floating in turbulence of incompressible fluid (Fig.1). The position vector xp(t) of a point on the line is advected by the local turbulent velocity field u(xp, t), while the dynamics of u(x, t) are independent of the existence of the line, and governed by the Navier-Stokes equation and the equation of continu-ity. A passive material line can be regarded as a set of many successive line segments of sufficiently short length z(i)(t) (i = 1, 2, · · · , I). Then, the line average of a quan-tity g(i) (t) accompanied with the line is defined by I (g)!ine = L~t) E gU) ( t) zUl (t) , (1) where L(t) = 2:{=1 zCi)(t) is the total length of the line. On the other hand, the line-element average of gCi) (t) is defined by I . - 1 "'"""' (i) (g)Ime-element- I ~ g (t) · (2) i = 1 Since Batchelor's pioneer work1) it has been believed that these two averages are identical in statistically homoge-neous turbulence because stretching histories of all the line segments might become statistically equivalent after sufficiently long time. Since the line average (1) can be rewritten as ( ) (gl)Iine-element 9 line = (l) ' line-element (3) if the correlation between gC i) ( t) and l ( i) ( t) vanished as t -+ oo, the two averages would be identical. However, this is not true. The present study shows theoretically and numerically that the correlation between the length zCi)(t) of the segment and the quantity g(i)(t) cannot de-cay in general even after long time, and therefore (g)Iine f=. (g)Iine-element · (4) As a typical and important example, we consider the stretching rate r( t) of the material line, d r(t) = dt log L(t) , (5) which is nothing but the line average of the stretch-ing rate r~i)(t) = Jt logl(i)(t) of each line segment, i.e., r = (!e)Iine· It should be stressed that the line-element average (!e)Iine-element of the stretching rate underesti-mates the true value of r· This can be understood by looking at the relation between zCi)(t) and '"Y~i\t), z(i)(t) = z<'l(o) exp (fo'7l'l(t') dt') . (6) 356 It can be shown numerically that r~i) (t) has a finite auto-correlation time of 0(5Try), where Try denotes the Kol-mogorov time. This finite correlation time implies the positive correlation between zCi) (t) and ')'~i) (t) even in the limit of t -+ oo. Note that the time integral in (6) is in an exponential function. If it were not in the ex-ponent, the contribution to the correlation from a finite time period t -Try < t' < t would vanish as t -+ oo. In order to confirm the above argument, we carry out direct numerical simulations of passive material lines, and estimate both of the line and the line-element av-erages of the stretching rate of the lines. Results are plotted in Fig.2. The averages in the statistically sta-tionary state ( t 2: 20Try) are constants around (!~i) ) Iine = 0.17Try - 1 > (')'~i))Iine-element = 0.13Try - l · (7) As expected, the line-element average of the stretching rate underestimates the true value. Fig.l Passive material line in turbulence. A result of direct numerical simulation. 0.2 ....-... }!; 0.1 20 40 t/Try 60 80 Fig.2 Line average (thick curve) and line-element av-erage (thin curve) of stretching rate of passive material lines. Reference I) Bathelor, G. K., Proc. Roy. Soc. London A 213 (1952) 349. 

Passive Material Line Statistics in Turbulence

Abstract

Similar works

Full text

Available Versions

National Institute for Fusion Science (NIFS-Repository)