カルマン渦列の消滅と再生成 (乱流研究次の10年 : 乱流の動的構造の理解へ向けて)

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$1771 2011 34-42 34 Annihilation and reincamation of Karan s vortex street (Hiroshi Al anine) (Jiro Mizushima) (Shunsuke Ohashi) (Kakeru Sugita) 1 1 1 2 2 $h$ 100 B\ enard[1] $a$ $a/h>0.$

1 Annihilation and reincamation of Karan s vortex street (Hiroshi Al anine) (Jiro Mizushima) (Shunsuke Ohashi) (Kakeru Sugita) $h$ 100 B\ enard[1] $a$ $a/h>0.366$ Kirm$4n[2]$ Sato and Kuriki[5] ( ) 1959 Taneda[3] 2 $A\searrow$ 100 ( 2 1 $50d\sim 100d$ Durgin and $100d$ Karlsson[4] Cimbala, Taneda Nagib and Rosho[6] Cimbala et al Karasudani and Funakoshi[7] Taneda Durgin and Karlsson[4] Durgin and Karllson $\backslash$ $A$,

$$\frac{\partial^{2}\psi}{\partial x^{2}}=0$, 35 2 1 $A$ 2 1 Mat ui and Okude[8] $\backslash$ $A$. 1 2 2 2 3 2.1 $U$ [6] $w$, $d$ 1 2 $A=w/d$ $O$ $x$ $y$ (1).$

2 $\frac{\partial^{2}\psi}{\partial x^{2}}=0$, $A$ 2 1 Mat ui and Okude[8] $\backslash$ $A$ $U$ [6] $w$, $d$ 1 2 $A=w/d$ $O$ $x$ $y$ (1). 2 s Takemoto and $\psi(x,y, t)$ $\omega(x,y,t)$ $\psi$ $\omega$ Mizushima[9] $d$ $U$ $\frac{\partial\omega}{\partial t}=j(\psi,\omega)+\frac{1}{{\rm Re}}\Delta\omega$, (1) $J(f,g)$ $\Delta\psi=-\omega$, $\frac{\partial f}{\partial x}\frac{\partial g}{\partial $=$ (2) y}-\frac{\partial f\partial g}{\partial y\partial x}$, ( 100 $\Delta$ ) $=$ $( \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}})$ ${\rm Re}\equiv Ud/\nu$ Inasawa and Asai [10] $\nu$ $u=\frac{\partial\psi}{\partial $v=- \frac{\partial\psi}{\partial x}=0$ y}=0$, (3) $w$, $d$ $A=w/d$ 1 $U$ $A=0.4$ $A$ $(x$ $)$ $\frac{\partial\omega}{\partial x}=0$ (4)

$$J(\overline{\psi},\overline{\omega})$ $\Delta\overline{\psi}$ $(\overline{\psi},\overline{\omega})$ 36 2.$

3 $J(\overline{\psi},\overline{\omega})$ $\Delta\overline{\psi}$ $(\overline{\psi},\overline{\omega})$ : $(\{\psi\rangle,$ $(\omega))$ 1 2 $P_{1}((x,y)=(20,0))$ P2 $((x,y)=(100,0))$ $T_{1}$ 1 2 $T_{1}$ $T$ 5 $(\overline{\psi},\overline{\omega})$ $\overline{\psi}(x, -y)=-\overline{\psi}(x, y)$ 2.4 $\overline{\omega}(x, -y)=-\overline{\omega}(x, y)$ (1) $+$ $\frac{1}{r\epsilon}\delta\overline{\omega}=0$, (5) $\omega $ $-\overline{\omega}$ $\psi $ $=$ (6) $\omega=\overline{\omega}+\omega $ $\psi=\overline{\psi}+\psi $ 2 (1) (2) ${\rm Re}=50$ $-10\leq$ (5) (6) $(\psi,\omega )$ $x\leq 40$ $-10\leq y\leq 10$ $(\psi,\omega )$ $({\rm Re}=50)$ 3.$0d$ $\frac{\partial\omega }{\partial t}=j(\psi,\overline{\omega})+j(\overline{\psi},\omega )+\frac{1}{{\rm Re}}\Delta\omega $, (7) $(\langle\psi\rangle, \langle\omega\rangle)$ $(\langle\psi\rangle,$ $(\omega\rangle)$ $\Delta\psi =-\omega^{l}$ (8) 2: ( ). $Rae=50$. $(\overline{\psi},\overline{\omega})$ $((\psi\rangle, \langle\omega\rangle)$ $\psi =$ $\psi(x,y)\exp(\lambda t)$ $,$ $\omega =\hat{\omega}(x,y)\exp(\lambda t)$

$$(\overline{\psi},\overline{\omega})$ $\lambda_{r}$ $\lambda_{i}$ 37 $\lambda$ $\delta t=$ 0.001 $\delta x=\delta y=0.1$ ( ) $\delta x=\delta y=0.$

4 $(\overline{\psi},\overline{\omega})$ $\lambda_{r}$ $\lambda_{i}$ 37 $\lambda$ $\delta t=$ $\delta x=\delta y=0.1$ ( ) $\delta x=\delta y=0.05$ (7) (8) $\psi(x,y)$ 2% $\omega$ $(X, y)$ $\lambda\omega \text{^{}}=j(\psi,\overline{\omega})+j(\overline{\psi},\omega$ $)+ \frac{1}{{\rm Re}}\Delta\hat{\omega}$, (9) $\Delta\psi=-\omega \text{_{}}$ (10) (9) (10) $\psi =0$ $\omega =0$ $(\psi,\omega )$ $(\psi$ $)$ $,\omega$ : $u=\frac{\partial\psi^{l}}{\partial y}=0$, $v=- \frac{\partial\psi }{\partial x}=0$ (11) (7) (8) (1) (2) (1) (2) $(\psi,\omega)$ $(\overline{\psi},\overline{\omega})$ 3.2 $\psi =\omega =0$ (5) (6) 2 SOR SOR $\frac{\partial^{2}\psi}{\partial x^{2}}=0$ $\frac{\partial\omega }{\partial, x}=0$ (12) $(\hat{\psi},\omega$ $)$ $k-1$ $\psi_{i,j}^{(k-1)}$ (11) $k$ $\psi_{i,j}^{(k)}$ (12) $(\psi,\omega )$ $(\psi,\omega)\text{_{}}$ $10^{-8}$ $(\langle\psi\rangle, (\omega))$ Inasawa and Asai [10] (1) (2) $x$ $y$ ( $\delta x$ $\delta y$ $(\delta x=\delta y)$ $)$ $A=w/d$ 1 (1) (2) 2 SOR (Succsessive Over Relaxafion Method) 3 ${\rm Re}=80$ $A=0.5$ $A=1$ $(i\delta x,j\delta y)$ ( ) $x=-10$ $n\delta t$ $\psi(i\delta x,j\delta y,n\delta t)$ $x=100$ 3(a) $k-1$ $\psi_{i,j}^{n(k-1)}$ $k$ $\psi_{:,j}^{n(k)}$ $(A=1)$ $10^{-0}$ 3(b) $(A=0.5)$ $40d$

$38 (a) $A=1$ $A=0.5$ (b) (a) (c) (b) (d) 3: $($ $, {\rm Re}=80)$. (a) $A=1$. (b) $A=0.5$. (e) $A=0.5$ ${\rm Re}=30$ 140 (f) 4 4(a) ${\rm Re}=30$ ( ) $x$ (g) 4(b) $B\epsilon=40$.$

5 38 (a) $A=1$ $A=0.5$ (b) (a) (c) (b) (d) 3: $($ $, {\rm Re}=80)$. (a) $A=1$. (b) $A=0.5$. (e) $A=0.5$ ${\rm Re}=30$ 140 (f) 4 4(a) ${\rm Re}=30$ ( ) $x$ (g) 4(b) $B\epsilon=40$. $($$4(c))$ ( ) 4(d) (h) $R\epsilon=90$ $($$4(e))$ $40d$ $(x=40)$ 4: ( ) ( 4(0 $)$. $A=0.5$. (a), (c), (e), (g) ( ). (b), (d), (f), (h) ( ). (a), (b). ${\rm Re}=30$ (c), (d). ${\rm Re}=40$ $(e),$ $(f){\rm Re}=90$. $(g)$, (h)re $=120$. ${\rm Re}=40$ 90 ${\rm Re}=120$ ${\rm Re}\sim 40$ $($$4(d))$ 2 $($$4(h))$. $20d\sim 30d$

$39 5: $x_{1}$ 1 2 $x_{2}$ ${\rm Re}_{c}$ $y$ $0$ ${\rm Re}_{c}$ $y$ 90(4(0) 1 $40d\sim 50d$ $x_{2}$ $v_{2}$ $a_{2}$ 1 ${\rm Re}_{c}$ $60d$ ($x$ ) ${\rm Re}>{\rm Re}_{c}$ $a_{2}$ ${\rm Re}=120$ 1$

6 39 5: $x_{1}$ 1 2 $x_{2}$ ${\rm Re}_{c}$ $y$ $0$ ${\rm Re}_{c}$ $y$ 90(4(0) 1 $40d\sim 50d$ $x_{2}$ $v_{2}$ $a_{2}$ 1 ${\rm Re}_{c}$ $60d$ ($x$ ) ${\rm Re}>{\rm Re}_{c}$ $a_{2}$ ${\rm Re}=120$ 1 ${\rm Re}=90$ ${\rm Re}$ ${\rm Re}\sim 90$ 2 1 $a_{2}$ $0$ $x_{2}=100$ Durgin and Karlsson[4] Karasudani and Funakoshi [7] ${\rm Re}\sim 100$ $a_{2}$ ${\rm Re}_{c}=39.7$ a ${\rm Re}_{c}\sim 40$ ${\rm Re}\sim 40$ $x=100$ ${\rm Re}=100$ $B\epsilon\sim 100$ $A=0.5$ $x$ $x_{1}=20$ $x_{2}=100$ $y$ $v_{1}$ $v_{2}$ $a_{1}$ $a_{2}$ $x_{1}$ 1 $x_{2}$ 2 ${\rm Re}$ $a_{1}$ $a_{2}$ 6 $x_{1}=20$ $v_{1}$ $a_{1*}$ $x_{2}=100$ $v_{2}$ $a_{1}\propto({\rm Re}-{\rm Re}_{c})^{1/2},$ $({\rm $a_{2}$ Re}_{c}=39.7)$

$$\omega$ $\lambda$ $\lambda_{r}$ $\omega$ 40 35 36 37 38 39 40 6: $a_{1}$ $a_{2}$ ( ). $A=0.5$. : $a_{1}(x_{1}=20)$. : $a_{2}(x_{2}=100)$. ${\rm Re}$ $\lambda_{r}$ 7: Rek $=39.$

7 $\omega$ $\lambda$ $\lambda_{r}$ $\omega$ : $a_{1}$ $a_{2}$ ( ). $A=0.5$. : $a_{1}(x_{1}=20)$. : $a_{2}(x_{2}=100)$. ${\rm Re}$ $\lambda_{r}$ 7: Rek $=39.7$ 1 $r=1$ ${\rm Re}\sim 100$ 2 $\psi_{r}$ $\psi_{r}$ 8 8(a) 1 2 ${\rm Re}=40$ ( ) ${\rm Re}=90$ 8(b) 2 $(x\sim 60)$ (5) Durgin and Karlsson[4] ( $(\overline{\psi},\varpi)$ (6) $)$ (9) (10) (11) (12) $\lambda_{i}$ $\lambda_{r}$ 1 7 $\lambda_{r}>0$ 2 $\lambda_{r}<0$ 45 $\lambda_{r}=0$ ${\rm Re}_{c}$ 7 $R\epsilon_{c}=38.2$ ${\rm Re}_{c}=39.7$ $x$ Pl $((x,y)=(20,0))$ 3 Matsui and Okude[8]

$41 (a) 0.2 (b) $fi,$. 015 $f_{2}01\ovalbox{\tt\small REJECT}$ $o^{o^{\phi}}$ @. $o^{o^{o\circ O\circ O}}$. 0.05 $0:f_{1}(P_{1})$ $:h(p_{2})$ 8: ( ). $\psi$r( ). (a). $(b)b\epsilon=90$.$

8 41 (a) 0.2 (b) $fi,$. 015 $f_{2}01\ovalbox{\tt\small REJECT}$ $o^{o^{o\circ O\circ O}}$ $0:f_{1}(P_{1})$ $:h(p_{2})$ 8: ( ). $\psi$r( ). (a). $(b)b\epsilon=90$. ${\rm Re}=40$ $0_{0}$ : $P_{1}$ $f_{2}$ $fi$ P $O:f_{1}(P_{1}, x_{1}=20)$. $fi$ 9 2 $P_{1}(x_{1}=20)$ P2 $(x_{2}=100)$ $T_{2}$ 2 9 Pl 5 P2 $(\psi,\omega)$ 100 Pl P2 $(\langle\psi\rangle, \langle\omega))$ ${\rm Re}=100$ 2 ${\rm Re}=100$ ${\rm Re}<100$ P2 $a_{2}$ $P_{1}$ $a_{1}$ (7) (8) (11) (12) Pl $(\psi,\omega )$ P2 $f1$ $f_{2}$ 2 1 $(\psi,\omega)$ $(\langle\psi\rangle, \langle\omega\rangle)$ 1 $(\psi,\omega)-((\psi\rangle, \langle\omega\rangle)$ ${\rm Re}=115$ 10(a) $x=90$ 10(a) 4.5 $\blacksquare:f_{2}(p_{2}, x_{2}=100)$. $x=110$ 10(b) (a) ( )

9 42 [8] Matsui, T., Okude, M., In Structure of Complex Turbulent Shear Flow, IUTAM Symposium, Marseille (Springer, Berlin, 1983) pp [9] Takemoto, Y., Mizushima, J., Phys. Rev. E., Vol. 82, (2010), pp (a) [10] Inasawa, A., Asai, M., Private communication (2010). (b) 10: $(Be=115)$. (a) (b) [1] B\ enard, H., C. R. Acad. Sci. Paris, Vol. 147, (1908), pp [2] Von $K4rm\acute{a}n$, Th., Nachr. Ges. Wlss. G\"oningen, Math.-phys. Kl., (1911), pp , (1912), pp [3] Taneda, S., J. Phys. Soc. Japan., Vol. 14, (1959), pp [4] Durgin, W. W., Karlsson, S. K. F., J. Fluid Mech., Vol. 48, (1971), pp [5] Sato, H., Kuriki, K., J. Fluid Mech., Vol. 11, (1961), pp [6] Cimbala, J. M., Nagib, H. M. Roshko, A., J. Fluid Mech., Vol. 190, (1988,) pp [7] Karasudani, T., Funakoshi, $M$, Fluid Dyn. Res., Vol. 14, (1994), pp

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