60 1: (a) Navier-Stokes (21) kl) Fourier 2 $\tilde{u}(k_{1})$ $\tilde{u}(k_{4})$ $\tilde{u}(-k_{1}-k_{4})$ 2 (b) (a) 2 $C_{ijk}$ 2 $\tilde{u}(k_{1})$

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そよあいしま
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1 Reynolds (SUSUMU GOTO) (SHIGEO KIDA) Navier-Stokes $\langle$ Reynolds 2 1 (direct-interaction approximation DIA) Kraichnan [1] (\S 31 ) Navier-Stokes Navier-Stokes [2] 2 Navier-Stokes $L$ Fourier ( $k$ ) $[ \frac{\partial}{\partial t}+\nu k^{2}]\tilde{u}_{i}(k t)=-\frac{\mathrm{i}}{2}(\frac{2\pi}{l})^{3}\tilde{p}_{ijm}(k)$ $\sum_{p}\sum_{q}$ $\tilde{u}_{j}(-p t)\tilde{u}m(-q t)$ (21) $(k+p+q=\mathit{0})$ Navier-Stokes $\tilde{p}_{ijm}(k)=km\tilde{p}_{i}j(k)+k_{j}\overline{p}_{im}(k)$ $\tilde{p}_{ij}(k)=\delta_{ijj/}-k_{i}kk^{2}$ $\nu$ (21) $\tilde{u}_{j}(p)\tilde{u}_{m}(q)$ $P$ $q$ $p$ $q$ $k+p+q=\mathit{0}$

2 60 1: (a) Navier-Stokes (21) kl) Fourier 2 $\tilde{u}(k_{1})$ $\tilde{u}(k_{4})$ $\tilde{u}(-k_{1}-k_{4})$ 2 (b) (a) 2 $C_{ijk}$ 2 $\tilde{u}(k_{1})$ $\overline{u}(k_{2})$ $\tilde{u}(-k_{1}-k_{2})$ ( ) 1(a) 2 (21) ( - Fourier ) Navier-Stokes (21) $[ \frac{\mathrm{d}}{\mathrm{d}t}+\nu]x_{i}(t)=\sum_{j}\sum_{k}$ Oijk $xj(t)xk(t)+f_{i}(t)$ $(i=12 \cdots N)$ (22) $\nu$ $X_{i}(i=12 \cdots N)$ (21) $F_{i}$ $2\nu/(N\Delta t)$ ) (\Delta k $C_{ijk}=cikj$ (23) $C_{ijk}+ojki+Ckij=0$ (24) 2 - $\{X_{i} X_{j} X_{k}\}$

3 61 3 $1(\mathrm{b})$ $X_{1}$ $X_{1}$ $X_{4}$ $X_{41}$ 2 $\{X_{i} X_{j}\}$ Cijk 3 s $\{X_{i}\}$ $C_{ijk}$ ( [2] ) $X_{i}^{2}+X_{j}^{2}+X_{k}^{2}$ 3 31 (22) $X_{i}$ $V_{ij}(t t) =\overline{x_{i}(t)x_{j(t })}$ $(t\geq t )$ (31) ( ) (22) $V_{ij}$ $\frac{\partial}{\partial t}+\nu\rfloor V_{in}(t t )=\sum_{j}\sum_{k}c_{ijkj}\overline{x(t)x_{k(}t)x_{n}(t )}$ $(t>t )$ (32) $[ \frac{\mathrm{d}}{\mathrm{d}t}+2\nu]v_{in}(t t)=\sum_{j}\sum_{k}c_{ijkj}\overline{x(t)x_{k(}t)x_{n}(t)}+\overline{f_{i}(t)x_{n}(t)}+(irightarrow n)$ (33) Navier-Stokes 2 2 Kolmogorov [3] [4] - 32 DIA 2 1 $\{X_{i} X_{j}\}$

4 \mathrm{d}\mathrm{i}_{\gamma \mathrm{e}\mathrm{c}\mathrm{t}}-\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ ) 62 $X_{i}$ NDI $X_{i/i\mathrm{o}j_{0}k\text{ }^{}(0)}$ $X_{i_{\text{ }}}$ 2: $X_{j_{0}}$ $X_{j_{0}}$ t 3 NDI 1 $X_{i/0}^{(0)}ij0k_{0}$ 3 \sim ( 2) $X_{i}$ $t_{0}$ $X_{i}(t)=x_{i/00}^{()}(t0_{ijk0} t\mathrm{o})+x_{i/i\mathrm{o}j\mathrm{o}0}^{(1)}(kt t_{0})$ $(t\geq t_{0})$ (34) $x_{i/00}^{(0)}(ijk_{0}t t_{0})$ $X_{i_{\text{ }}}$ 3 $X_{j_{0}}$ $X_{k_{0}}$ $\mathrm{n}\mathrm{o}\mathrm{n}- ( NDI) $X^{(1)}$ $\mathrm{d}\mathrm{i}$ $-$ (direct-interaction $(22)i/i0k\text{ }$ NDI $[ \frac{\mathrm{d}}{\mathrm{d}t}+\nu]x_{i/i\mathrm{o}j\mathrm{o}0}^{(0)}(k tt\mathrm{o})=$ $\sum_{j}\sum_{k}$ $C_{ijk}X^{()}(j/i_{0}j\mathrm{o}k0t t_{0})xk/0(0)(i\mathrm{o}j0k_{0} tt\mathrm{o})+f_{i}(t)$ (35) $\{ijk\}\neq\{i_{0\dot{o}_{0}}k\mathrm{o}\}$ 2 $\{X_{i_{0}/i_{0}}^{(0)}j\mathrm{o}k0 [DIA 1] X_{j_{0}}^{(0)}/i_{0}j\mathrm{o}k0 X_{k_{0/0}}^{(0)}ij\mathrm{o}k0\}$ 3 $X_{i/i\mathrm{o}j\mathrm{o}k0}^{(1)}$ [DIA 2] $X_{i/i\mathrm{o}jk}^{()}000$ + 2 t t

5 $X_{i}$ 63 DIA t 33 DIA-RRE DIA 2 (32) (33) 3 $G_{ij}(t t) = \frac{\delta X_{i}(t)}{\delta X_{j}(t)}$ $(t\geq t )$ (36) $\delta$ $G_{ij}$ (22) $X_{j}$ DIA $[\mathrm{a}]x_{i}$ $G_{ij}$ $[\mathrm{b}]\mathrm{d}\mathrm{i}$ $X_{i}^{(1)}$ $G_{ij}^{(1)}$ NDI [cl $X_{i}^{(1)}$ (32) (33) (34) 1 [DIA 2] $[\mathrm{d}]$ $[\mathrm{b}]$ [C] [DIA 1] $V_{ij}$ $\overline{g}_{ij}$ [el $G_{ij}$ $[\mathrm{d}]$ [C] $V_{ij}$ $\overline{g}_{i}$ $X_{i}$ ( s ) $V_{ii}$ $V_{ii}(t t )=\mathcal{v}(t-t )$ (37) $[\mathrm{a}]\sim$ [el $[ \frac{\mathrm{d}}{\mathrm{d}\tau}+\nu]\mathcal{v}(\tau)=-\frac{2c_{1}}{\mathcal{v}(0)}\int_{0}^{\tau_{\mathrm{d}\mathcal{t} }}[v(\mathcal{t}^{j})]2\tau \mathcal{v}(-\mathcal{t} )$ (38) $c_{1}= \sum_{j}\sum_{k}$ (Cijk)2 ) ( DIA-RRE 4 Reynolds (22) Reynolds (Reynoldsnumber reversed expansion RRE) $\sim t=\nu t$ (41) $\tilde{x}_{i}(^{\wedge}t)=xi(t)$ $\frac{\mathrm{d}}{\mathrm{d}t\sim}\tilde{x}_{i}(^{\sim}t)=\lambda\sum_{j}\sum_{k}$ cijk $\tilde{x}j(^{\sim}t)\overline{x}k(^{\sim}t)-\tilde{x}i(t\sim\sim)+^{\tilde{p}_{i(t)}}$ (42)

6 64 $\lambda=1/\nu$ Navier-Stokes $\tilde{x}_{i}$ Reynolds $\lambda<<1$ $\tilde{x}_{i}(t)=\tilde{x}_{i}^{(0)}(t)+\lambda\tilde{x}_{i}^{(1)}(t)+o(\lambda^{2})$ (43) $\lambda$ $\tilde{x}_{i}$ (42) $\tilde{x}_{i}^{(0)}$ $\lambda$ $\frac{\mathrm{d}}{\mathrm{d}t}\tilde{x}_{i}^{(0)}(t)=-\tilde{x}_{i}^{(0)}(t)+\tilde{f}_{i}(t)$ (44) $\tilde{x}_{i}^{(1)}$ $\text{ }$ $\tilde{g}_{ij}=\delta\tilde{x}_{i}/\delta\tilde{x}_{j}$ $\tilde{x}_{i}$ $\tilde{g}_{ij}^{(0)}\text{ }$ $\tilde{g}_{ij}^{(1)}$ Reynolds $\tilde{x}_{i}^{(0)}$ [2] (44) 4 2 RRE DIA RRE DIA DIA DIA $\lambda=1$ $\lambda$ (22) ( [5]) Reynolds Kraichnan [1] DIA ([67] ) DIA RRE Reynolds Kraichnan [6] reversion Reynolds (1 ) Reynolds reversion (renormalization) 5 DIA DIA 2 $N$ 1 2 Reynolds Reynolds Reynolds

7 $\overline{\simeq>}$ $\overline{\backslash _{\frac{\backslash []}{>}}}$ $\overline{\backslash _{\frac{\backslash []}{>}}}$ $\overline{\frac{*}{>}}$ $\overline{\backslash _{\frac{\backslash []}{>}}}$ $\overline{\simeq>}$ 65 (a) (b) (c) (d) $(\mathrm{e})$ (f) $t$ $t$ 3: DIA-RRE (38) - (22) $(\mathrm{a})(\mathrm{b})$ (32) (a) $(N \nu)=(710)$ $(\mathrm{b})(71)$ $(\mathrm{c})(70)$ $(\mathrm{d})(100)$ $(\mathrm{e})(200)$ $(\mathrm{f})(400)$ $(\mathrm{a})(\mathrm{b})$ (c) $(\mathrm{e})(\mathrm{f})$ DIA-RRE $\nu>>1$ $N>>1$

8 66 $(\lambda\ll 1\Leftrightarrow\nu>>1)$ $\nu>>1$ DIA-RRE $N>>1$ DIA-RRE (38) (22) $N$ $\nu$ ( 3) $\nu>>1$ (RRE $N>>1$ (DIA - ) ) DIA \S 32 2 DIA $N>>1$ 1 $\{X_{i} X_{j} X_{k}\}$ 3 $R_{ijk}(t-t )= \frac{\overline{x_{i}(t)x_{j(}t)xk(t )}}{\sqrt{\overline{x_{i}(t)^{2}}\overline{x_{j}(t)2}\overline{x_{k}(t)2}}}$ (51) $X_{i}$ $\{X_{i} X_{j} X_{k}\}$ NDI $X_{i/}^{(0)}ijk$ $R_{ijk}\approx \mathrm{o}$ 1 4 $R_{ijk}$ - NDI $N=7$ Rijk NDI - $N=20$ NDI Rijk $\langle$ DIA 1 $\mathrm{d}\mathrm{i}$ 2 $D(t-t \mathrm{o})=\langle\sum_{i}[x_{i/}^{(1)}i\mathrm{o}j0k_{0}(t t_{0})]^{2}\rangle$ (52) $\langle$ $)$ $N$ $5(\mathrm{a})$ ( DIA-RRE (38) ) $5(\mathrm{b})$ $N$ $5(\mathrm{a})$ $N$ $D$ DIA Navier-Sotkes DIA DIA \S 32 2 \S 4 Reynolds (RRE) DIA RRE DIA DIA DIA DIA [2]

9 $\{\mathrm{a}_{-}11$ 67 (b-1) (a-2) (b-2) (a-qt (b-3) 4: DIA 1 (51) 3 $R_{ijk}$ $(\{ij k\}=\{124\})$ $X_{i}$ $\{x_{1} x_{2} x_{4}\}$ ) NDI (a) $X_{i/4}^{(0_{12}}$ $(N \nu)=(70)$ (b) $(N \nu)=(200)$ $R\approx \mathrm{o}$ NDI DIA 1 $N=7$ $N=20$

10 68 $\tau/t(n)$ $\tau/\prime \mathit{1}^{\cdot}(n)$ $\mathrm{d}\mathrm{i}$ 5: DIA 2 (a) (52) $\mathrm{d}\mathrm{i}$ $\tau=t-t_{0}$ NDI ( 1) $N$ (b) (a) $(N= )$ (a) (b) $\nu=0$ Kraichnan [1] Navier-Stokes Eulerian Kolmogorov - DIA Lagrangian Navier-Stokes [8-10] Eulerian Lagrangian (22) [1] R H Kraichnan The structure of isotropic turbulence at very high Reynolds number J Fluid Mech (1959) [2] S Goto and S Kida Direct-interaction approximation and Reynolds-number reversed expansion for a dynamical system Physica (1998) in $\mathrm{d}$ press [3] A N Kolmogorov The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers Dokl Akad Nauk SSSR (1941) English translation in Proc R Soc London SerA (1991) [4] K R Sreenivasan On the universality of the Kolmogorov constant Phys Fluids (1995)

11 $\mathrm{n}\mathrm{y}$ 69 [5] D C Leslie Developments in the theory of turbulence (Clarendon Press Oxford 1973) [6] R H Kraichnan Eulerian and Lagrangian renormalization in turbulent theory J Fluid Mech (1977) [7] J H W Wyld Formulation of the theory of turbulence in an incompressible fluid Ann Phys $14$ (1961) [8] R H Kraichnan Lagrangian-history closure approximation for turbulence Phys Fluids (1965) (erratum 1966 ibid 1884) [9] Y Kaneda Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function J Fluid Mech (1981) [10] S Kida and S Goto A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence J Fluid Mech (1997)

,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), Mellor and Yamada 1974) 4 2 3, 2 4,

Mellor and Yamada1974) The Turbulence Closure Model of Mellor and Yamada 1974) Kitamori Taichi 2004/01/30 ,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), 4 1 4 Mellor and Yamada 1974) 4 2 3, 2