(Hiroshi Okamoto) (Jiro Mizushima) (Hiroshi Yamaguchi) 1,.,,,,.,,.,.,,,.. $-$,,. -i.,,..,, Fearn, Mullin&Cliffe (1990),,.,,.,, $E

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1 (Hiroshi Okamoto) (Jiro Mizushima) (Hiroshi Yamaguchi) 1 $-$ -i Fearn Mullin&Cliffe (1990) $E=3$ $Re_{C}=4045\pm 015\%$ ( $Re=U_{\max}h/2\nu$ $U_{\max}$ $h$ ) $-t$ Ghaddar Korczak&Mikic (1993) 2) $=_{\wedge}$ $Re_{c}=975$ $\mathrm{t}$-s T-S (Kelvin-Helmholtz) Sobey&Drazin $(1986)^{3)}$ \nearrow $=-$

2 129 Roberts $(1994)^{4)}$ $=$ $Re_{c}=100$ (Kelvin-Helmholtz) $(Re\simeq 160)$ $=$ Fig 1 Channel geometry 2 Fig1 $h$ AB $h$ $\mathrm{g}\mathrm{h}$ DEJK $E=\mathrm{D}\mathrm{K}/\mathrm{c}\mathrm{L}=$ 3 $=3$ DE $=\mathrm{j}\mathrm{k}=l$ $A$ $A=L/\cdot 3h$ Fig1 $\mathrm{d}\mathrm{k}$ $x$ $y$ $h/2$ $U_{\max}$ $\omega(x y t)$ $\psi(x y t)$ $\frac{\partial\omega}{\partial t}+\frac{\partial\psi}{\partial y}\frac{\partial\omega}{\partial x}-\frac{\partial\psi}{\partial x}\frac{\partial\omega}{\partial y}=\frac{1}{re}(\frac{\partial^{2}\omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}})$ (1) $\omega=-(\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}})$ (2)

3 130 $\nu$ $Re\equiv U_{\max}h/2\nu$ $(u v)=(\partial\psi/\partial y -\partial\psi/\partial x)$ AB $\frac{\partial\psi}{\partial x}=0$ $\frac{\partial\omega}{\partial x}=0$ $\psi=\int_{-1}^{y}udy=\int_{-1}^{y}(1-y^{2})dy=y(1-y^{2}/3)+2/3$ (at $x=-l_{1}$ ) (3) $\mathrm{g}\mathrm{h}$ $\frac{\partial^{2}\psi}{\partial x^{2}}=0$ $\frac{\partial^{2}\omega}{\partial x^{2}}=0$ (at $x=l+l_{2}$ ) (4) $\mathrm{g}\mathrm{h}$ $\frac{\partial\psi}{\partial t}+c\frac{\partial\psi}{\partial x}=0$ $\frac{\partial\omega}{\partial t}+c\frac{\partial\omega}{\partial x}=0$ (at $x=l+l_{2}$ ) (5) $c$ $c$ $u$ $\psi=\psi_{1}=0$ (on BCDEFG) $\psi=\psi_{2}=4/3$ (on HIJKLA) $u= \frac{\partial\psi}{\partial x}=0$ $v=- \frac{\partial\psi}{\partial y}=0$ (on BCDEFG and HIJKLA) (6) 3 $=$ SOR = $(x y)=(l/2-1)$ $(L/20)$ $(L/21)$ $(L+L_{2}/20)$ 4 $y$ $v$ SOR SOR $(x=0)$ $\omega(xy)=-\omega(x -y)$ $(x>0)$ $\omega=0$ (on $x=0$) (7) $y<0$ $y>0$

4 $\frac{\partial\psi}{\partial y}\frac{\partial\omega}{\partial x}-\frac{\partial\psi}{\partial x}\frac{\partial\omega}{\partial y}=\frac{1}{re}(\frac{\partial^{2}\omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}})$ $A=438/37/324/3$ 6 $L_{1}$ $L_{2}$ $L_{1}=3h$ $L_{2}=3h$ $\Delta x=\triangle y=01$ SOR $(Re<120)$ 0001 $\Delta t=$ $(Re\geq 120)$ SOR $\triangle t=$ $\epsilon$ $\epsilon=15$ SOR $\psi_{ij}^{n}$ $10^{-5}$ $n\triangle t$ maxi $j \psi_{ij}^{n+1}-^{\psi^{n}1}ij<10^{-10}$ $\psi_{ij}^{n}\equiv\psi$ ( xj\delta y ) $n\delta t$ $n\delta t$ $(xy)=(i\delta x$ j\delta $(x y)=(l/2-1)$ $(L/20)$ $(\dot{l}/21)$ $(L+L_{2}/20)$ 4 $y$ $v$ 4 $v$ 32 SOR (8) $\omega=-(\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}})$ (9) $=$ $\epsilon$ SOR SOR $07\leq\epsilon\leq 10$ SOR $\psi_{ij}$ $\omega_{ij}$ $\max_{ij} \omega_{ij}^{k+1k}-\omega i$ $j<10^{-10}$ $\psi_{ij}^{k}$ $k$ $\text{ }\mathrm{m}ij \psi^{k+}ij-^{\psi_{y)}^{k}}ij kj\text{ }\psi(i\triangle xj\triangle 1<10^{-1}0_{Jy}\text{ }\omega(i\delta x)$

5 132 (a) Fig 2 Flow pattern for $Re=35$ $A=7/3$ $00$ Fig3 Flow pattem for $Re=50$ $A=7/3$ (a) asymmetric flow Fig4 $v_{1}$ Vertical velocity at (b) unstable symmetric flow and $v_{1}^{2}(\mathrm{b})_{\mathrm{v}}\mathrm{s}$ Reynolds number (c) disturbance $A=7/3$ 4 $x=08(\mathrm{a})$ Fig 2 $A=7/3$ $Re=35$ (Moffatt) $\mathrm{a}=7/3$ $Re=50$ Fig 3 $3(\mathrm{a})$ Fig $Re=35$ $Re=50$ $\mathrm{c}\mathrm{l}$ 04 $(x=08)$ $y$ $v_{1}$ $v_{1}$

6 $n\overline{-}\infty$ 133 $4(\mathrm{a})$ Fig $y$ (A-B) $y$ $v_{1}$ $v_{1}$ $4(\mathrm{b})$ $v_{1}^{2}$ (B-C B-E) Fig $v_{1}^{2}$ $Re_{c1}=4770$ $v_{1}^{2}\propto(re-re_{\text{ }}1)$ Fearn(1990) 1) $3(\mathrm{b})$ Fig $A=7/3$ $Re=50$ $3(\mathrm{c})$ Fig $A=\infty$ $A=\infty$ $Re_{c1}=4000$ Fearn (1990) 1) $Re_{\text{ }1}=4045\pm \mathrm{o}17\%$ $A=4$ $Re_{\text{ }1}=4000$ A $=\infty$ $A=4$ $A=204/3$ $Re=50\sim 100$ Fig 5 $A_{\text{ }}\simeq 23$ $\cup$ $A$ Fig5 Critical Reynolds number $\mathrm{v}\mathrm{s}$ aspect ratio $A$

7 $v_{1}^{2}$ 134 $A=7/3$ $Re=70$ Fig 6 $A=7/3$ $Re=55$ $Re=70$ $v_{1}(x=08$ $y=0$ $y$ ) Fig 7 $A=7/3$ $v_{1}$ $7(\mathrm{a})$ Fig (C-D E-D) $v_{1}$ $Re_{\text{ }2}=6524$ $v_{1}=0$ $7(\mathrm{b})$ Fig $Re_{\text{ }2}=6524$ $v_{1}^{2}\propto(re_{\text{ }2}-Re)$ Fig 5 $A=\infty$ Fearn(1990) 1) $A=7/3$ 15 $(x=170)$ $y$ $v_{2}$ Fig 8 $A=7/\cdot 3$ $Re=$ $875$ $v_{2}$ $v_{2}$ $Re=120$ $Re=875$ $v_{2}$ Fig 6 Flow pattemm for $Re=70$ $A=7/3$ $/\circ 1$ $\mathrm{o}\alpha)05$ $v_{1}$ 0\alpha Q5 $00$ $ w$ $3\cup$ $\propto J$ $/\cup$ Fig7 Vertical velocity $v_{1}$ at Reynolds number $A=7/3$ $x=08(\mathrm{a})$ and $v_{1}^{2}(\mathrm{b})\mathrm{v}\mathrm{s}$

8 Vd $\iota \mathrm{w}$ $\mathrm{z}\iota\mathrm{v}$ $4\angle\iota\wedge J$ $\angle\angle\supset\cup$ $\bullet$ $\bullet$ $\bullet$ 135 $t$ Fig 8 Vertical velocity $v_{2}$ at $t$ time for $Re=875$ $A=7/3$ $x=170\mathrm{v}\mathrm{s}$ (a) $a$ $001\infty$ (b) $a^{2}$ 0075 $Re_{c3}=843$ 0050 $\oint$ $- 005$ $- 010$ $Re_{c3}=843$ $0\alpha)25$ $- 015$ $8\infty$ 850 $9\infty$ $a(\mathrm{a})$ $a^{2}(\mathrm{b})\mathrm{v}\mathrm{s}$ Fig 9 Amplitude and Reynolds number A $=7/3$ Fig $9(\mathrm{a})$ $a$ $a_{2}^{2}$ $9(\mathrm{b})$ Fig $Re_{c3}=843$ $v_{2}$ $a$ $a$ $v_{2}$ $9(\mathrm{b})$ Fig $a^{2}\propto(re-re_{\text{ }}2)$ Fig10(a) $v_{2}$ $Re=875$ $10(\mathrm{b})$ Fig $10(\mathrm{c})$ Fig $v_{2}$ $Re=875$ $T=$ 1350

9 136 Fig10 Flow pattern for $Re=875$ $A=7/3$ (a) instantaneous flow field (b) time average flow field (c) disturbance Table 1 The comparison of plane Poiseuliie flow and present work for $A=7/3$ $Re=875$ $Re_{\text{ }}=5772$ T-S $A=7/3$ $Re_{\text{ }2}=843$ $L_{2}=15h$ $L_{2}$ T-S Table 1 T-S Gaster Gaster $\alpha_{i}=\frac{\alpha c_{i}}{c_{r}+\alpha\partial C\Gamma/\partial\alpha}$ (10) \alpha $\alpha$ $C_{\Gamma}$ \alpha

$137 T-S T-S 5 Discussion $y$ $A=4/3$ 2 $A=\infty$ $Re=40$ $x_{r}=12$ $A=2$ $A=2$ $Re=40$ $\overline{4}=2$ $Re=50$ $A$ $A_{\text{ }1}$ $A_{c1}\simeq 23$$

10 137 T-S T-S 5 Discussion $y$ $A=4/3$ 2 $A=\infty$ $Re=40$ $x_{r}=12$ $A=2$ $A=2$ $Re=40$ $\overline{4}=2$ $Re=50$ $A$ $A_{\text{ }1}$ $A_{c1}\simeq 23$ $A\leq A_{\text{ }1}$ $A_{\text{ }1}$ $(A_{\text{ }1}<a\leq A_{\text{ }2} A_{\text{ }2}\simeq O(10))$ $(A>A_{\text{ }}2 A_{c2}\simeq o(10))$ T-S T-S $E$

11 138 $h$ $h$ 10 5 ( $\simeq 200$ ) 1) R M Fearn T Mullin&K A Cliffe: Nonlinear flow phenomena in a symmetric sudden expansion J Fluid Mech211(1990) $59\mathrm{k}608$ 2) N K Ghaddar K Z Korczak&B B Mikic: Numerical investigation of incompressible flow in grooved channels Part 1 Stability and self-sustained oscillationsphys Fluids A 5 (10) ) I J Sobey&P G Drazin: Bifurcations of two-dimensional channel flows J Fluid Mech171 (1986) ) E P L Roberts: A numerical and experimental study of transition processes in an obstructed channel flow J Fluid Mech260 (1994)

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