カルマン渦列の発生の物理と数理 (オイラー方程式の数理 : カルマン渦列と非定常渦運動100年)

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$3 (B\ enard)[2] 100 ( ) 2 McKoen[3] McKoen Taneda[4] Taneda ${\rm Re}_{d}=3.2$ ${\rm Re}_{d}=30$ Taneda $B$, ${\rm Re}_{d}=47$ Jackson[51 Jackson ${\rm Re}_{d}=46.184$ Jackson Taneda McKoen Taneda$

1 (Yukio Takemoto) (Syunsuke Ohashi) (Hiroshi Akamine) (Jiro Mizushima) Department of Mechanical Engineering, Doshisha University 1 (Theodore von Ka rma n, l ) [1]. 3 (B\ enard)[2] 100 ( ) 2 McKoen[3] McKoen Taneda[4] Taneda ${\rm Re}_{d}=3.2$ ${\rm Re}_{d}=30$ Taneda $B$, ${\rm Re}_{d}=47$ Jackson[51 Jackson ${\rm Re}_{d}=46.184$ Jackson Taneda McKoen Taneda

$$t\iota$ 29 (Briggs(1964)[6], (1991)[7] ). - ( ) ( ) 1 [8, 9, 10, 11, 12, 13, 14, 15, 16]. ${\rm Re}_{d}\sim 25$ [12], 3. $5d$ [16].$

2 $t\iota$ 29 (Briggs(1964)[6], (1991)[7] ). - ( ) ( ) 1 [8, 9, 10, 11, 12, 13, 14, 15, 16]. ${\rm Re}_{d}\sim 25$ [12], 3. $5d$ [16]. Takemoto and Mizushima[17] Takemoto and Mizushima 100 Taneda[18] ( ) [19, 20, 21, 22]. Durgin and Karlsson[21] $a$ $a/h>0.366$ Sato and Kuriki[22] ( ) 2 2 ( 100 ) Inasawa, Nakano and Asai [26] ) $w$, $d$ $A=w/d$ 1 $A=0.4$ $A$ } 02 2

$$\hat{\psi}$ $\hat{\omega}$ 30 2 Takemoto and Mizushima[17] Takemoto and Mizushima 2.$

3 $\hat{\psi}$ $\hat{\omega}$ 30 2 Takemoto and Mizushima[17] Takemoto and Mizushima 2.1 $U$ $d$ $\prime x$ $\psi(\prime x, y,t)$ $y$ 2 $\psi$ $\omega$ $\omega(\prime x, y, t)$ ${\rm Re}\equiv Ud/\nu$ $\nu$ )j $\overline{\psi}(x, -y)=-\overline{\psi}(x,y)$ $\overline{\omega}(\prime x, -y)=-\overline{\omega}(x, y)$ ${\rm Re}=50$ $-5\leq x\leq 16$ 3. $0d$ $-5\leq y\leq\check{o}$ 1(a) }- $({\rm Re}=50)$ $\hat{\psi}$ $(x, y)=(30,0)$ $(x,y)=(30,0)$ $t=[0,1x10^{-3}]$ $\hat{\psi}=1xlo^{-3}$ $\hat{\psi}=0$ $t=0$ $\psi=\overline{\psi}+\hat{\psi}$ $\omega=\overline{\omega}+\omega$ $\psi=\overline{\psi}+\hat{\psi}$ $(\hat{\psi},\hat{\omega})$ $\omega=\overline{\omega}+\omega$ (1) (2) $(\hat{\psi},\hat{\omega})$ (a) 1: ( ).. ${\rm Re}=50$ $(a)$ ( ). $t=83$.

$$\sigma_{a}$ $\hat$ $\sigma_{p}$ $\sigma_{a}$ 31 22 $(x, y)=(30,0)$ ${\rm Re}=50$ 1 ${\rm Re}_{g}$ Jackson[5] 1 2(a) $\hat{\psi}$ $R=35$ $t=0$ $t=60$ $x=30$ $t=0$ $t=60$ $t=60$ $0\leq x\leq 100$ A$

4 $\sigma_{a}$ $\hat$ $\sigma_{p}$ $\sigma_{a}$ $(x, y)=(30,0)$ ${\rm Re}=50$ 1 ${\rm Re}_{g}$ Jackson[5] 1 2(a) $\hat{\psi}$ $R=35$ $t=0$ $t=60$ $x=30$ $t=0$ $t=60$ $t=60$ $0\leq x\leq 100$ A $B$ B A A ${\rm Re}=35$ ( $<$ Rg) 2 $20\leq t\leq 100$ $t=20$ ${\rm Re}=35$ ${\rm Re}=50$ $($ $2(c))$ $t\leq 60$ $x=25$ 23 $\sigma_{a}$ $\sigma_{a}$ 3(a) $\sigma$ ${\rm Re}_{a}=48.1$ ${\rm Re}_{a}$ ${\rm Re}_{g}$ $arrow$ ${\rm Re}_{a}={\rm Re}_{g}=48.1$ Jackson[5] ${\rm Re}_{g}=46.184$ 4% $\sigma_{a}$ ${\rm Re}=35$ $x=24$ 3(a) ${\rm Re}_{g}\leq{\rm Re}=50$ $x=25$ $x\geqq 25$ $\sigma_{p}$ ${\rm Re}=50$ $x\geqq 25$ ${\rm Re}_{g}\leq{\rm Re}$ ${\rm Re}<{\rm Re}_{g}$ ${\rm Re}={\rm Re}_{g}$

5 32 (a) $-4\cross 10^{-8}arrow$ $-6x10^{-8}---- $ (c) $t=40\underline{\sim-}$ $t=20 $ $ \hat{\psi}$ 2: (a). ${\rm Re}=35$ $$ $(c)$ ${\rm Re}=50$. 2.4 ${\rm Re}=35$. ${\rm Re}=35$ $\mathfrak{e} -$ $x_{p}$ 4(a) ${\rm Re}=50$ $4$ $x=25$ 4(a) 4 $\dagger\grave$ $f^{-}$ ${\rm Re}=50$ $({\rm Re}=35)$ $({\rm Re}=50)$ $t$ } x-t $x_{f}(t),$ $x_{p}(t),$ $x_{t}(t)$ $x_{f}(t)$ 5(a) $\triangleright$ ${\rm Re}$ ${\rm Re}_{g}$ $x_{t}(t)$ $x_{p}(t)$ $x=0$ ${\rm Re}>{\rm Re}_{g}$ 5 $x_{f},$ $x_{p}$ $x_{t}$

6 $\sigma_{a}$. 33 (a) $\sigma_{a}$ $\sigma_{p}$ $1\overline{0}202\overline{v}303\overline{o}$ ${\rm Re}$ (a) $$ 3: $\sigma_{\text{ }}$ (a) $t$ $t$ so 100 4: (a) ${\rm Re}=35,$ ${\rm Re}=50$. 5 $80<t<100$ $x_{f}$ ( 0.9 ) $x_{p}$ Rg $\iota$ $x_{t}$ ${\rm Re}_{g}$ 0 Takemoto and Mizushima 2 3

7 $v_{\ell}$, $v_{\ell}$, - $arrow$ $v_{\ell}$, 34 (a) $t$ $v$ ${\rm Re}$ 5: $v_{p}$, (a) $v_{\ell}$ $v_{p}$, $v_{t}$ : : $v_{p}$, $v_{t}$ $U$ $w$, $A=w/d$ $d$ $\psi(x, y, t)$ $y$ ( 6). 2 $\psi$ $\omega(x, y, t)$ $\omega$ $d$ $U$ $\frac{\partial\omega}{\partial t}=j(\psi,\omega)+\frac{1}{{\rm Re}}\Delta\omega$, (1) $\Delta\psi=-\omega$, (2) $J(f, g)= \frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}$ $\Delta=(\frac{\partial^{2}}{\partial, x^{2}}+\frac{\partial^{2}}{\partial y^{2}})$ ${\rm Re}\equiv Ud/\nu$ $\nu$ $u=\frac{\partial\psi}{\partial y}=0$, $v=- \frac{\partial\psi}{\partial x}=0$ (3) $\psi$ $p$ $C$ $\partial p/\partial s$ $U$ $\oint_{c}\partial p/\partial ds=0$ $\frac{\partial^{2}\psi}{\partial x^{2}}=0$, $\frac{\partial\omega}{\partial x}=0$ (4) $\frac{\partial\psi}{\partial t}+c\frac{\partial\psi}{\partial x}=0$ $\frac{\partial\omega}{\partial, t}+c\frac{\partial\omega}{\partial x}=0$ (5) $\langle u\rangle$ $c$

8 $J(\overline{\psi},\overline{\omega})$ $\Delta\overline{\psi}$ 35 6: 312 $(x$ $)$ $\overline{\psi}(x, -y)=-\overline{\psi}(x, y)$ $\overline{\omega}(x, -y)=-\overline{\omega}(x, y)$ (1) $+$ $\frac{1}{{\rm Re}}\Delta\overline{\omega}=0$, (6) $=$ $-\overline{\omega}$ (7) [21]. ( $(\psi\},$ $(\omega\})$ $(\langle\psi\},$ $(\omega\})$ 1 2 Pl $((x, y)=(20,0))$ P2 $((x, y)=(100,0))$ $T_{1}$ 1 2 $T_{2}$ $T_{1}$ $T_{2}$ $T$ $(\langle\psi\rangle, \langle\omega\rangle)$ 5 3J4 $\omega $ $\psi $ $\psi=\overline{\psi}+\psi $ $( \psi,\omega )$ $\omega=\overline{\omega}+\omega $ $(\psi,\omega )$ (1) (2) (6) (7) $\frac{\partial\omega }{\partial t}=j(\psi,\overline{\omega})+j(\overline{\psi},\omega )+\frac{1}{{\rm Re}}\triangle\omega $, (8) $\triangle\psi =-\omega $ (9) $(\{\psi\rangle,$ $\langle\omega\rangle)$

9 $\lambda_{r}$ $\lambda_{i}$ 36 $\lambda$ $\psi =\hat{\psi}(x, y)\exp(\lambda t),$ $\omega =\hat{\omega}(x, y)\exp(\lambda t)$ ( ) (8) (9) $\psi(x, y)$ $\hat{\omega}(x,y)$ $\lambda\hat{\omega}=j(\hat{\psi},\overline{\omega})+j(\overline{\psi}, \omega$ $)+ \frac{1}{{\rm Re}}\Delta\omega$ (10) $\Delta\hat{\psi}=-\hat{\omega}$ (11) (lo) (11) $( \psi,\omega )$ $(\hat{\psi},\hat{\omega})$ : $u= \frac{\partial\psi }{\partial y}=0$, $v=- \frac{\partial\psi }{\partial x}=0$ (12) $\nu\backslash$, $\psi =\omega =0$ $\frac{\partial^{2}\psi }{\partial x^{2}}=0$ $\frac{\partial\omega }{\partial, x}=0$ (13) $(\hat{\psi},\hat{\omega})$ (12) $(\psi, \omega )$ (13) $( \hat{\psi},\omega$ $(\{\psi\rangle,$ $(\omega\rangle)$ $)$ 315 (1) (2) $\delta x$ $\delta y$ $y$ $(\delta x=\delta y)$ (1) 1 2 (2) 2 SOR (Succsessive Over Relaxation Method) $(i\delta x,j\delta y)$ $\psi_{i,j}^{n(k)}$ $n\delta t$ $\psi(i\delta x,j\delta y, n\delta t)$ $\psi_{i,j}^{n(k-1)}$ $k-1$ $k$ $10^{-6}$ $\delta t=$ $\delta x=\delta y=0.1$ $\delta x=\delta y=0.05$ 2% (8) (9) (1) (2) $(\psi,\omega)$ (1) (2) $\psi =0$ $\omega^{l}=0$ $\backslash$, 316 (6) (7) 2 SOR SOR $k-1$ $\psi_{;,j}^{(k-1)}$ $k$ $\psi_{i,j}^{(k)}$ $10^{-8}$

$2$ ${\rm Re}=30$ 120 7 $7(a)$ ${\rm Re}=30$ ${\rm Re}=40$ ${\rm Re}=80$ $($ $7(c))$ $30d$ $(x=30)$ ${\rm Re}=40$ 80 ${\rm Re}=100$ 2 $($ $7(d))$.$

10 $A=0.5$ 1 Inasawa, Nakano and Asai[26] $A=0.2$ 1 2 $A=0.2$ ${\rm Re}=30$ $7(a)$ ${\rm Re}=30$ ${\rm Re}=40$ ${\rm Re}=80$ $($ $7(c))$ $30d$ $(x=30)$ ${\rm Re}=40$ 80 ${\rm Re}=100$ 2 $($ $7(d))$. ${\rm Re}=40$ $7)$ $($ $75d$ $75d$ $($ 80 $7(c))$ $20d\sim 25d$ 30 ( $d$ ) ${\rm Re}=100$ 1 ${\rm Re}=80$ 2 1 Durgin and Karlsson[21] Karasudani and Funakoshi [24] ${\rm Re}_{c}\sim 3\overline{o}$ ${\rm Re}\sim 35$ 1 $x=100$ 2 ${\rm Re}\sim 90$ $x_{1}=20$ $x_{2}=100$ $y$ $v_{1}$ $v_{2}$ $x_{2}$ 1 2 $a_{1}$ $a_{2}$ ${\rm Re}$ $a_{1}$ $a_{2}$ 9 $a_{1}\propto({\rm Re}-{\rm Re}_{c})^{1/2}$ $x_{1}=20$ $v_{1}$ $a_{1}$, $x_{2}=100$ $v_{2}$ $a_{2}$, $({\rm Re}_{c}=35.5)$ $x_{1}$ $x_{2}$ ${\rm Re}_{c}$ ${\rm Re}_{c}$ $y$ $y$ 1 $x_{2}$ $v_{2}$ $a_{2}$ 1 Re ${\rm Re}>{\rm Re}_{c}$ $a_{2}$ ${\rm Re}$ ${\rm Re}\sim 60$

11 38 (a) r. (c) (d) 7: ( ). $A=0.2$. (a). ${\rm Re}=30$ ${\rm Re}=40$. $(c){\rm Re}=80$. $(d){\rm Re}=100$. 8: 1 2 $A=0.2$ $a_{2}$ $x_{2}=100$ ${\rm Re}\sim 90$ $a_{2}$ 2 9 ${\rm Re}_{c}=35.5$ ${\rm Re}=90$ $A=0.2$ ${\rm Re}_{c}=35.6$ 1 ${\rm Re}\sim 90$ 2

12 $\lambda$ $\lambda_{r}$ ${\rm Re}$ $\hat{\psi}_{r}$ 39 9: $a_{1}$ $a_{2}$ ( ). $A=0.2$. : $a_{1}(x_{1}=20)$. : $a_{2}(x_{2}=100)$ [A] (6) (7) (12) (13) (10) (11) $\lambda_{i}$ $\lambda_{r}$ $\lambda_{r}>0$ 10 $\lambda_{r}<0$ $\lambda_{r}=0$ ${\rm Re}_{c}$ 10 ${\rm Re}_{c}=35.6$ 1 ${\rm Re}_{c}=35.5$ $\lambda_{r}$ 10:. $A=0.2$. $\hat{\omega}_{r}=1$ $P_{1}((x, y)=(20,0))$ $\psi_{r}$ 11 11(a) ${\rm Re}=40$ ( ) ${\rm Re}=80$ 11 $(x\sim 50)$ ${\rm Re}=100$

13 40 Durgin and Karlsson[21] ( ) (a) (c) 11: ( ). $\psi$r( ). (a) ${\rm Re}=40$. ${\rm Re}=80$. $(c){\rm Re}=100$ $f1$ 12 Pl $(x_{1}=20)$ P2 $(x_{2}=100)$ Pl P2 90 Pl P2 ${\rm Re}<90$ P2 $a_{2}$ $P_{1}$ $a_{1}$ $f_{1}$ 90 2 Pl P2 $f_{2}$ $T_{1}$ $T_{2}$ 1 2 $T_{3}$ $T_{3}$ $(\psi,\omega)$ 5 $((\psi\rangle,$ $(\omega\rangle)$ ${\rm Re}=90$ 2 ${\rm Re}=90$ (8) (9) (12) (13)

14 $\circ 0\circ$ $\cdot$ $\iota\rangle_{o}^{\circ}$ $(\langle \psi\rangle, \langle\omega\rangle)$ 41 $Re$ 12: ( ). ( $P_{1}$ $fi$ $P_{2}$ $\blacksquare$ $f_{2}$ ). : $fi$ $(P_{1}, (x, y)=(20,0))$, $+:f_{2}(p_{2}, (x, y)=(100,0))$. $(\psi, \omega )$ $( \psi),\omega)$ $(\overline{\psi},\overline{\omega})-(\langle\psi\rangle,$ $\{\omega\rangle)$ ${\rm Re}=90$ 13(a) $x=110$ $13$ $x\sim 100$ (a) 2 1 ( ) (a) 13: $({\rm Re}=90)$. (a) [1] Von K\ arm\ an, Th., Nachr. Ges. Wiss. G\"ottingen, Math.-phys. Kl., (1911), pp , (1912), pp [2] B\ enard, H., C. R. Acad. Sci. Paris, Vol. 147, (1908), pp. 839-S42.

15 42 [3] McKoen, C. H., Aeronautical Research Council, Courrent Paper, No. 303, (1956). [4] Taneda, S., J. Phys. Soc. Japan., Vol. 18, (1963), pp. 28S-296. [5] Jackson, C. P., J. Fluid Mech., Vol. 182, (1987), pp [6] R. J. Briggs, Cambridge, MIT Press,(1964). [7] - lil ( 3 ), ( 1991, ), pp [8] Triantafyllou, G. S., Triantafyllou, M. S., Chryssostomidis, C., J. Fluid Mech., Vol. 170, (1986), pp [9] Triantafyllou, G. S., Kupfer, K., Bers, A., Phys. Rev. Lett., Vol. 59, (1987), pp [10] Kupfer, K., Bers, A., Ram. A. K., Phys. Fluids, Vol. 30, (1987), pp [11] Monkewitz, P. A., Nguyen, L. N., J. Fluids Struct., Vol. 1, (1987), pp [12] Monkewitz, P. A., Phys. Fluids, Vol. 31, (1988), pp [13] Hannemann, K., Oertel, H. Jr., J. Fluid Mech., Vol. 199, (1989), pp. 55-S8. [14] Oertel, H. Jr., Annu. Rev. Fluid Mech., Vol. 22, (1990), pp [15] Chomaz, J. M., Huerre, P., Redekopp, L. G., Phys. Rev. Lett., Vol. 60, (1988), pp [16] Huerre, P., Monkewitz, P. A., Annu. Rev. Fluid Mech., Vol. 22, (1990), pp [17] Takemoto, Y., Mizushima, J., Phys. Rev. E., Vol. 82, (2010), pp [18] Taneda, S., J. Phys. Soc. Japan., Vol. 14, (1959), pp [19] Okude, M., Trans. Japan Soc. Aero. Space Sci., Vol. 24, (1981), pp [20] Okude, M., Matsui, T., Trans. Japan Soc. Aero. Space Sci., Vol. 33, (1990), pp [21] Durgin, W. W., Karlsson, S. K. F., J. Fluid Mech., Vol. 48, (1971), pp [22] Sato, H., Kuriki, K., J. Fluid Mech., Vol. 11, (1961), pp [23] Cimbala, J. M., Nagib, H. M. Roshko, A., J. Fluid Mech., Vol. 190, (1988,) pp [24] Karasudani, T., Funakoshi, $M$, Fluid Dyn. Res., Vol. 14, (1994), pp [25] Matsui, T., Okude, M., In Structure of Complex Turbulent Shear Flow, IUTAM Symposium, Marseille (Springer, Berlin, 1983) pp [26] Inasawa, A., Nakano, T., Asai, M., Proc. Seventh Int. Symposium on Turbulence and Shear Flow Phenomena, (Ottawa, 2011), in press.

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)

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) 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] $a$ $a/h>0.366$ Kirm$4n[2]$