KOLMOGOROV LAW AND KOLMOGOROV AND TAYLOR SCALES IN ANISOTROPIC TURBULENCE. TURBULENCE AS A RESULT OF THREE-SCALE INTERACTION

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1 KOLMOGOROV LAW AND KOLMOGOROV AND TAYLOR SCALES IN ANISOTROPIC TURBULENCE. TURBULENCE AS A RESULT OF THREE-SCALE INTERACTION V. P. Maslov The interaction of waves of different scales in the development of turbulence is considered. A mechanism of occurrence of a wave whose frequency and amplitude satisfy Kolmogorov's law is described. We define by L and T the greatest fundamental scale and time scale, by L 0 and T o the least fundamental scale and time scale, by ~ the viscosity; h = Lo/L << 1, and u 0 is the velocity of the large-scale motions. The time is normalized by L/uo, the pressure by pu 2 (incompressible fluid). The Reynolds number is Re-l- -- J,/uL.,r 1. Then (in dimensionless variables) the continuity equation and Navier--Stokes equation are =0 Oz, i "t Oui Ou~ Op ~eeaui uk O~k - 0~ + (t) One says that/or=re-1/2lo and/r=re-3/4l0 are, respectively, the Taylor and Kolmogorov scales of the system. We consider a single-phase anisotropic structure [1], for example, the flow of a fluid through a comb filter, the emergence of fluid from a stratified medium, or the flow of a fluid with single-phase initial conditions [1]. Equations (9) and (3) of [1] show that the most rapid growth of perturbations will occur at the Taylor scale (i.e., for example, when the distance between the teeth of a comb filter is =Re-1/2Lo, or, in the problem with single-phase initial conditions, when h-re-1/2 [I]). The growth of an initial perturbation (in the generic case) Xo will have order xe a~/h, (~ is some constant that can be calculated. This theorem follows from Eqs. (3) and (9) (see [1]). In reality, since the growth has the form eat~hi, ~ N J~ k s (t, r) dr, h I is the scale of the oscillations, and k,,~ e -tvsl~t, i.e., c~ - IVS1,2, forh 1 =hi +7 (7>_0) VS~h -'r and (x/h I ~h ~-1, i.e., the maximum of the growth is attained for 4=0. If3' is negative, then k- 1 and ~- I, and therefore for 3' < 0 the growth of the perturbation is reduced. Accurate estimates prove this assertion. We now estimate the damping of the scale h I. The scale h I ~ VS/h gives damping k of the form e-(vs)~; if S~ h "~, then the damping has the order e -t/h2. The condition of equality of the damping of the oscillations of the scale h 1 and of the maximum growth of the Taylor scale will be 5=1/2, i.e., S~Re TM, and h 1 =S/h=Re -3/4, i.e., the Kolmogorov scale. Thus, the limiting scale that can still be "overtaken" by virtue of the instability is the Kolmogorov scale. Finer scales cannot be "sustained" by the instability and must be damped. Growth of the second phase of the Taylor scale, described in [1], leads to the appearance of a function X(r, rl, x, O, r and 77 are "fast variables," i.e., r=s/h, 7l=O/h. It is the high-frequency part of X(r, ~, x, t) with respect to ~/that "sustains" the Kolmogorov scale. Indeed, if X(r, 71, x, t) has a simple discontinuity, then its Fourier expansion with respect to ~/gives coefficients a n ~ n -t, i.e., harmonics of the form n -1 sin n~/. The Kolmogorov scale arises for n=h -l/2, nrl='~h-3/2=~, Re 3/4. Since at the same time n-l=h 1/2, the corresponding harmonic has the form h 1/2 sin ~h-3/2=h 1/3 sin Oh1-1, i.e., for the amplitude of the velocities the Kolmogorov law is satisfied. We shall consider more rigorously this "resonance" between the two scales -- Kolmogorov and Taylor -- in order to have more secure foundations for obtaining a scenario for the occurrence of turbulence. Indeed, discontinuity of the function X(r, V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 3, pp , March, Original article submitted April 27, /93/ Plenum Publishing Corporation

2 y) at t=t 0 is none other than frequency multiplication, subdivision of vortices, etc., which Richardson already saw as part of the turbulence scenario. Our scheme is as follows: 1) subdivision of "single-phase" frequencies of the Taylor scale; 2) growth of the second phase of the Taylor scale as a consequence of strong instability; 3) subdivision of waves of the two-phase Taylor scale and appearance of the Kolmogorov scale; 4) developed turbulence as a process of interaction of three scales (large "common" scale and the Taylor and Kolmogorov scales). This scenario requires the presence in the original situation of the one-dimensional Taylor scale, for example, the existence of boundary layers (boundary-layer width -Re-1/2) or the formation of random one-dimensional Taylor structures, recently investigated by Majda (see his plenary talk at the Mathematicians Congress in Kyoto). We shall consider periodic waves. However, everything can be extended to the almost periodic case and to random (in a certain sense) waves. According to Landau, frequency doubling is essentially due to nonlinearity and resonance. We shall determine this resonance, taking as basis [2,3]. We consider an initial wave of the form h~ x exp(is(x)/hl). Let v(x, t) be the large-scale velocity. The formal perturbation theory that arises from solution of the equation (in the given ease, the Navier--Stokes equation, for other examples see [2]) must satisfy the following conditions: I) the dissipative (viscous) term of the Navier--Stokes equation, applied to the n-th term of the series, has the same order in h as the term (v, V)v; 2) the general term of the formal series has the same order in h i as the initial term (i.e., ha). In this case, we shall say that the series is in limiting resonance, i.e., admits frequency doubling, while a finer scale is already damped. We shall call h i the resonance scale, and hc~ the value of the resonance velocity. The following assumptions are made: 1) if the initial term is a single-phase term of the form ~ exp(is/h), then under the condition <~, VS)=0 the limiting resonance frequency has the form h=re 1/2, and the value of the resonance velocity is -1 (i.e., a=0). Up to a shift, this formal series satisfies Eq. (3) in [1], i.e., the solution of this equation describes frequency multiplication; 2) the obtained solution is unstable, and by virtue of Eq. (9) (see [1]) the second phase grows. If the "noise" is small =exp(-h-1), then the interval of time during which the second phase grows is t~- 1, and the third phase will grow during a time (to) to a finite value different from the value - 1 to which the second has grown, and it will remain small at t=t o. Thus, we can again regard the third phase as a perturbation to the equation that has been established for the two phases. In the leading term in h, this last equation will again be an Euler equation with respect to the fast variables. If we take into account the viscous term, it will be of order h. We consider the last equation: 0o(0 0) h-~+ X-~r+x-~ w+h(uo,v)w+h(w,v}uo+ (w, VS)~-~+(w, Vg2} uo + ((w, VS)~ + (w, Vcb)~-~+h{w,V))w+ (VSO-~ + VfO~ + hv)ii =h(vs~---~+~7,~ff-~+h~7)2w,, (2) T = s(~, Olh,,7 = '~(~, L)lh, ~: = & + (Uo, VS), ~ = r + (Uo, ~7,:I,), and u 0 is the two-phase "Taylor" solution. We consider an initial perturbation wit=0 that depends on the third phase ~=~/h and has the form Ao(Z, ~1, 0exp(igo(z, ~?, 0/e), v--~0. If A 0 is sufficiently small, the leading (in h) term w satisfies the linear equations 261

3 (3) v=(uo, V~). The solution of these equations has the form (vs, + o. / + or =o, we = A(r, rh~,t,h)exp( ig(r'o'~'t'h) hi/~ } +(terms of higher order in e), ~ =0, da d'-t + GA - (P, GA) = O, d 9 O 0 0 One 5uo =h=o, + x~ + ~ + ~-~, a = VS ~ + W ~ a--~-' (4) Og Og ~. Og Numerical investigation of the last equations shows that the solutions (4) grow exponentially (see also [4]), and the wave vector P/c linearly. The maximum value ofw is of order A 0 exp{e/x/h}. Therefore, ifa0_<exp{- 1/v~h}, then w remains small. With growth of the "noise" A 0 there appears a new wave w; the wave number ]P t e-~ of this wave has order h_ ~/2, i.e., w = f(g(r'rhr ] Since r=s/h, ~l=r ~=~/h, we obtain the Kolmogorov scale of the oscillations: h3/2~re -3/4, The amplitude of this last wave is determined by the nonlinear terms in (2) and has order h 1/2 =Re-1/4, i.e., w satisfies Kolmogorov's law. Thus, in the considered situation (limiting noise level) there are waves of two scales -- Ko!mogorov and Taylor; intermediate scales are absent. This "gap in the spectrum" can be filled by lack of smoothness of the Taylor wave; as we have already noted, a simple discontinuity of it leads to the appearance of oscillations of scale h t +~ with amplitude h B. For the large-scale velocity, we shall have the Reynolds equation; while the high-frequency oscillations of the Kelmogorov scale will satisfy an equation of the form (3.2) in [2]: uo =uo + hll~ lc(y, r, rt, (, x, t), y =gh -~ /~, o ={o1(t,~,r ~,t),o~(~, ~,4, ~,t)}, ~ =0, (5) de d---[+gic + (K:, D)I(. =- -DTr + D~I(., (D, E) =0, Hence, they satisfy the Navier--Stokes equation. Thus, one can say that on the "anisotropic" motion of the vortices of the Taylor scale "isotropic" turbulence of the Kolmogorov scale is formed. The back reaction of the Kolmogorov scale on the Taylor scale is described by an equation that contains derivatives with respect to the slow variables x. We note that the Navier--Stokes type equation (5) no longer contains the small parameter Re -I, and therefore for this equation it is indeed natural to consider the behavior as t---,~, i.e., the RueUe--Takens scenario. If the "noise" level is greater (for example, of order ha, 0 </~ < oo), there will be growth of harmonics of intermediate scales -~ Re-% 1/2 < c~ < 3/4. The 262

4 developed wave [ g(,',,1) _ q) again satisfies a Navier--Stokes equation containing the small parameter h3-4u=re 2~-3/2 multiplying the viscous terms. Therefore, in accordance with (2)--(3) perturbations of higher frequency will develop on the background of this wave; the scale of the oscillations will change until the Kolmogorov scale is reached. Remark. In writing down Eqs. (5), we assumed implicitly that the two phases (gi, g2) of the wave of the Kolmogorov scale grow faster. This assumption is confirmed by numerical calculation: The amplitude A [see (4)] increases fastest for two values of P[ t=0 E S 2 [note that (4) does not depend on the absolute value of P]. If more than two growth maxima arise, the developed wave will depend on a larger number of phases. In this case, the dimension of the problem (5) (the number of variables) will be larger. Items 1 and 3 of the scheme we outlined earlier can be formulated with mathematical rigor, and the corresponding theorems can be proved (at the level of formal series). In essence, the scheme of the proof has been given here. As regards item 2, the scheme of the proof is also based on the construction of a formal perturbation series and the construction of the equation that it satisfies. The formal perturbation series is constructed, like the corresponding series in Lyapunov theory, without proof of the corresponding convergence theorems. As zeroth term of the series, one takes a "singlephase" function u(r, x, 0 (r=so/h, h=re-1/2) that satisfies Eq. (3) of [1]. We construct the solution in the form u(~, ~, t, h) + ~ ~(~, ~, ~, ~, h) ~xp h ' k=l p=~(x, t)/h, r=s(x, t)/h, uk(r, 77, x, t, h) are trigonometric polynomials of degree k in 7/. The analogous series determines the pressure II + IIk(r, rl, z, t, h) exp h " The function X=(Ul, Vr r +ir satisfies [1] the Rayleigh equation k----i -~ + (~, re) (vs) ~ ~ + (w), rj - (vs) ~(~, w)x: 0, vr = w - ~sr, ~ = ~(r ~, t, 0) - v(~, t), v = f0 ~ ~(~, ~, t, 0) ar is the large-scale component of the velocity. The complex eigenvalue of d~/dt with positive imaginary part determines the growth of the initial "noise" (numerical calculation shows there is one such eigenvalue). The functions u 1 and H 1 can be expressed in terms of the solution of the Rayleigh equation in accordance with the formulas Here, n is the unit vector orthogonal to the plane (VS, V~). The following terms of the series (in the zeroth order in h) are determined by the operator s II~ \ dt '. ' j=l q) has the form s vl(v--~3rs~ -r i~-t or - ~4/"r''~ a+(,,q~ in which 263

5 P= 0 1 ' 0 0 and the function ~1 satisfies the inhomogeneous Rayleigh equation -(~+(~,q)) (vs) ~ +q -~-~, j +(vs)~(~-,q>~ =~, = (~TS) ~(PF,, q> - (VS) ~ (~, q) - (VS) ~ (a + <~, ~>) ~ of, - iq'(pf, VS>. By means of the Fredholm alternative we obtain from the condition for the existence of the following approximation in h an equation that determines the dependence of the mean values of u n on x and t. The formal series of exponentials obtained in this manner (Dirichlet series) satisfy Eqs. (3.17), (3.24), (3.25), and (3.26) in [1]. We now assume that the constructed functions have a limit as ~2--,oo. Then this limit (in the leading term in h) satisfies a two-dimensional stationary Euler equation for the unknowns (u, VS) and (u, V~I>. Note, however, that these equations are not, in general, identical to the "nonlinear Rayleigh equations," since the mean values (with respect to r/) of the limits of the constructed functions need not satisfy F~s. (3) of [~]. The effect of the three-scale interaction is obtained because the single-phase solution of the Taylor scale is unstable but the viscosity for it is not small. It is damped over a long time that does not depend on the parameter Re. When the ~'o-phase solution grows, its viscosity is of order Re- 1/2, i.e., it is damped only over a time T- Re 1/2, allowing an additional "resonance" and this resonance is of Kolmogorov scale with velocity satisfying Kolmogorov's law. 1. V.P. Maslov, Dokl. Akad. Nauk SSSR, 310, 795 (1990). 2. V.P. Maslov, Usp. Mat. Nauk, 41, 19 (1986). 3. V.P. Maslov, Teor. Mat. Fiz., 65, 448 (1985). 4. B.J. Bayly, Phys. Fluids, 31, 56 (1988) REFERENCES 264

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