カルマン渦列の消滅と再生成のメカニズム

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$Taneda[3] Taneda 100 ( d) $50d\sim 100d$ $100d$ Taneda Durgin and Karlsson[4] 1 2 1 $h$ 2 $a$ $a/h>0.$

1 (Jiro Mizushima) (Hiroshi Akamine) Department of Mechanical Engineering, Doshisha University 1. [1,2]. Taneda[3] Taneda 100 ( d) $50d\sim 100d$ $100d$ Taneda Durgin and Karlsson[4] $h$ 2 $a$ $a/h>0.366$ Sato and Kuriki[5] ( ) 2 Cimbala, Nagib and Rosho[6] 2 1 Karasudani and Funakoshi[7] $a$ $h$ $a/h$ 2 Okude and Matsui[8, 9] [6]

$$h$ $d$, $A=h/d$ 1 Pl P2 2 1 2 $U$ 2 $d$ $(x^{*},y^{*})=(x/d, y/d)$, $(u^{*},v^{*})=(u/u, v/u)$ $*$ 2$

2 Inasawa and Asai [10] $w$, $A=w/d$ 1 $d$ $A$ $A=0.4$ $A$ : $U$ $O$ $y$ ( 1). $h$ $d$, $A=h/d$ 1 Pl P $U$ 2 $d$ $(x^{*},y^{*})=(x/d, y/d)$, $(u^{*},v^{*})=(u/u, v/u)$ $*$ 2

3 99 $\psi(x, y, t)$ $\omega(x, y, t)$ $\psi$ $\omega$ $(u=\partial\psi/\partial y, v=-\partial\psi/\partial x)$. $\frac{\partial\omega}{\partial t}=j(\psi, \omega)+\frac{1}{re}\triangle\omega$, (1) $\Delta\psi=-\omega$, (2) $J(f,g)= \frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}, \Delta=(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}})$ $Re$ $\nu$ $Re\equiv Ud/\nu$ $W$ 1 $L_{1}+L_{2}$, $ABCD$ $\psi=0, \frac{\partial\psi}{\partial n}=0$ (3) $n$ ( $AD$ ) $U$ $u=1)$ ( $\psi=y, \omega=0$ (4) ($AB,DC$ ) $\psi=\psi_{1}$ (AB), $\psi=\psi_{2}$ (DC), $\omega=0$ (AB, DC) (5) ($BC$ ) $\frac{\partial\psi}{\partial t}+c\frac{\partial\psi}{\partial x}=0, \frac{\partial\omega}{\partial t}+c\frac{\partial\omega}{\partial x}=0$ (6) $(\psi,\omega)$ (1) (2) (3) $-(6)$ 2.2 ( (1) 0 ) (2) $(\overline{\psi},\overline{\omega})$ (3) $-(6)$ $J( \overline{\psi},\overline{\omega})+\frac{1}{{\rm Re}}\Delta\overline{\omega}=0$, (7) $\overline{\omega}=-\delta\overline{\psi}$ (8) ( 1 ) $($ (3)$-(6))$ 1 $100d$ 2 2 1

4 $\lambda_{r}$ $(\overline{\psi},\overline{\omega})$ $\lambda_{i}$ $\omega$ $(\langle\psi\rangle, \langle\omega\rangle)$ 2.3 $\omega $ $\psi,$ $\psi$ $(\overline{\psi},\overline{\omega})$ $\psi=\overline{\psi}+\psi, \omega=\overline{\omega}+\omega $ (9) (1) $\frac{\partial\omega }{\partial t}=\frac{1}{re}\delta\omega +J(\psi,\overline{\omega})+J(\overline{\psi},\omega )+J(\psi,\omega )$ (10) $(\psi =\hat{\psi}(x,y)e^{\lambda t},$ (10) (10) $\omega =$ $(X, y)e^{\lambda t})$ $= \frac{1}{re}\delta\hat{\omega}+j(\hat{\psi},\overline{\omega})+j(\overline{\psi},\hat{\omega})$ $\lambda$. (11) (11) $\hat{\omega}=-\delta\hat{\psi}$ (12) $\lambda$ ( ) $(\hat{\psi},\hat{\omega})$ : $\hat{u}=\frac{\partial\hat{\psi}}{\partial y}=0, \hat{v}=-\frac{\partial\hat{\psi}}{\partial x}=0$ (13) $\hat{\psi}=\hat{\omega}=0$ $\frac{\partial^{2}\hat{\psi}}{\partial x^{2}}=0, \frac{\partial\hat{\omega}}{\partial x}=0$ (14) $(\langle\psi\rangle, \langle\omega\rangle)$ 2.4 (1) (2) $y$ $\Delta x$ $\Delta y$ $(\Delta x=\delta y)$ (1) 1 2 (2) 2 $SOR$ (Succsessive Over Relaxation Method)

$101 $( x,j\delta y)$ $n\delta t$ $\psi(i\delta x,j\delta y, n\delta t)$ $10^{-6}$ $k-1$ $\psi_{i}^{n_{j}(k-1)}$ $k$ $\psi_{i}^{n_{j}(k)}$ 0.001 $\Delta t=$ $\Delta x=\delta y=0.$

5 101 $( x,j\delta y)$ $n\delta t$ $\psi(i\delta x,j\delta y, n\delta t)$ $10^{-6}$ $k-1$ $\psi_{i}^{n_{j}(k-1)}$ $k$ $\psi_{i}^{n_{j}(k)}$ $\Delta t=$ $\Delta x=\delta y=0.1$ $\Delta x=\delta y=0.05$ 2% $0$ $\omega $ (10) (1) (2) (1) (2) $(\overline{\psi}, \overline{\omega})$ $(\psi, \omega)$ $\psi =0$ $\omega =0$ (11) (12) 2 $SOR$ $SOR$ $k-1$ $\psi_{i,j}^{(k)}$ $10^{-8}$ $\psi_{i,j}^{(k-1)}$ $k$ $A$ $(A=5.0,4.0,3.0,0.5,0.2)$ $A=0.2$ $A=0.2$ $Re=30$ $($ $2(a), Re=30)$. $Re=40$ $($ $2(b))$. $Re=80$ $x=30d$ $($ $2(c))$. 2(c) 2 $Re=100$ 2 $y$ 2(d) $x=80d$ 1/ $Re\sim 40$ 1 2 $Re\sim 90$ Taneda Taneda

6 102 (a) (b) (c) (d) : ( $Re=80$, ), $A=0.2.$ $(a)$ $Re=30,$ $(b)$ (b) $Re=100.$ $Re=40,$ $(c)$ Pl $((x,y)=(20,0)$ P2 $((x,y)=(100,0)$ $y$ $a_{2}$ $P_{1}$ $v_{1}$ $v_{2}$ $a_{1}$ P2 2 1 $a_{1}$ $a_{2}$ $Re$ 4 4 Pl $v_{1}$ $a_{1}$, $v_{2}$ P2 $a_{2}$ $a_{1}\propto(re-re_{c})^{1/2},$ $(Re_{c}=35.5)$ $Re_{c}=35.5$ $P_{1}$ $Re_{c}$ P2 $y$ $0$ $Re_{c}$ $y$ $x_{2}$ 1 $Re_{c}$ $v_{2}$ $a_{2}$ 1 $Re>Re_{c}$ $a_{2}$ $Re$ $Re\sim 60$ $a_{2}$ $0$ $60<(Re<90$ 0 P2 $Re\sim 90$ $a_{2}$ 2 4 $Re_{c}=35.5$

7 103 3: 1 2 $A=0.2.$ $Re=90$ ( ) 2 4: $a_{2}$ $0 50$Re$ 100$ $( a_{1}, a_{2})$. $A=0.2$. $(P2, (x, y)=(100,0))$ : $a_{1}(p_{1}, (x, y)=(20,0))$ : $Re=100$ $P_{1}$ $(Pl=(20,0), P_{2}=(100,0))$ P2 $y$ $v$

8 104 (a) $t$ (b) $t$ 5: $P_{1}$ P2 $y$ $v_{1}$ $v_{2}.$ $Re=100.$ Pl P $=5.92$ $(fi=0.169),$ $T_{2}=12.5(f_{2}=0.080)$ Pl P2 $fi$ $f_{2}$ $(\blacksquare)$ 6 $P_{1}$ $fi$ $(+)$ $P_{1}$ $f_{2}$ 2 $Re=90$ 2 $fi$ 1 $f_{2}$ $f_{2}$ $*$ $*\cdot$ 0.12 $ ^{ }$ $fi, f_{2} ++++_{+}$ 0.06 $f_{1}$ (DNs) $f2(dns)$ $(b_{0} $ $Re$ 6: ( ). $P_{1}$ $fi$ P2 $f_{2}.$ $(20,0)),$ $+:f_{2}(p_{2}, (x,y)=(100,0))$. $\blacksquare:fi(p_{1}, (x, y)=$

9 $\lambda$ 105 (a) (b) $Re Re$ 7: 4. $\lambda_{r}.$ (a) $(b)$ $\lambda_{i}/2\pi.$ 4.1 $(\overline{\psi}, \overline{\omega})$ (11) (12) $(\hat{\psi},\hat{\omega})$ $\lambda$ $\lambda_{r}$ $\lambda_{j}$ $2\pi$ $\lambda_{i}/2\pi$ $f$ 7(a) 7(b) 7(a) $Re_{c}=35.6$ $Re_{c}=35.5$ ( 4) $Re_{c}=46.7[11]$ 7(b) $Re_{c}=35.6$ $f_{c}=0.1054$ $Re_{c}=35.6$ $f_{c}=0.109$ $Re=40,80,100$ $8(a)-8(f)$ 40 $Re=80$ 100 $2(b)-2(d)$ 1 1 $\Delta x=5\sim 6$ 2 2 (11) (12) 1 ( 8) Verma and Mittal[12] ). (

$2 2 1 1 $T_{2}$ 1 2 $T_{3}$ $T_{3}$ $(\psi, \omega)$ 5 $(\langle\psi\rangle, \langle\omega\rangle)$ 90 2 $Re=95$ 9(a) $Re=95$ 9(b) $(\psi,\omega )$ $(\psi,\omega)$ $(\langle\psi\rangle,$

10 106 (a) (b) (c) $ $ (d) (e) $ $ (f) : ( ). $A=0.2.$ $(a),$ $(b)re=40$. (c), (d) $Re=80.$ (e), (f) $Re=100.$ $(a),$ $(c),$ $(e)$ (b), (d), (f) $T_{2}$ 1 2 $T_{3}$ $T_{3}$ $(\psi, \omega)$ 5 $(\langle\psi\rangle, \langle\omega\rangle)$ 90 2 $Re=95$ 9(a) $Re=95$ 9(b) $(\psi,\omega )$ $(\psi,\omega)$ $(\langle\psi\rangle, \langle\omega\rangle)$ $(\psi, \omega)-(\langle\psi\rangle, \langle\omega\rangle)$

11 107 (a) (b) (c) $ $ $ $ 9: ( $ $ ). $Re=95$. (a) 2 (b) (c) 9(c) 9 $x=85$ $90$ 2 1 ( ) 5. $100d$ 2 1 [1] B\ ENARD, $H$ Formation des centres de giration \ a l arri\ ere d un obstacle en mouvement. C. R. Acad. Sci. Paris. 147,

12 108 [2] VON K\ ARM\ AN, $T$ \"Uber den Mechanismus des Widerstandes, den einen bewegter K\"orper in einer FlSsigkeit erzeugt. Nachr. $Ges$. Wiss. G\"ottingen, Math.-phys. Kl, $S$ [3] TANEDA, Downstream development of the wakes behind cylinders. J. Phys. Soc. Japan 14, [4] DURGIN, W. $W$., KARLSSON, S. K. $F$ On the phenomenon of vortex street breakdown. J. Fluid Mech. 48, [5] SATO, $H$., KURIKI, $K$ The mechanism of transision in the wake of a thin flat plate placed parallel to a uniform flow. J. Fluid Mech. 11, [6] CIMBALA, J. $M$., NAGIB, H. $M$., ROSHKO, $A$ Large structure in the far wakes of two-dimensional bluff bodies. J. Fluid Mech. 190, [7] KARASUDANI, $T$., FUNAKOSHI, $M$ Evolution of a vortex street in the far wake of a cylinder. Fluid $Dyn$. Res. 14, [8] OKUDE, $M$ Rearrangement of Karman s vortex street. nans. Japan Soc. Aero. Space Sci 24, [9] OKUDE, $M$., MATSUI, $T$ Vorticity distribution of vortex street in the wake of a circular cylinder. $\pi ans$. Japan Soc. Aero. Space Sci 33, [10] INASAWA, $A$., NAKANO, $T$., ASAI, $M$ Development of wake vortices and the associated sound radiation in the flow past a rectangular cylinder of various aspect ratios. Proc. Seventh Int. Symposium on $h$rbulence and Shear Flow Phenomena, (Ottawa, 2011) in press. [11] JACKSON,C. $P$ $A$ finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, $A$ $S$ [12] VERMA,. & MITTAL, new unstable mode in the wake of a circular cylinder. $A$ Phys. Fluids 23,

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)

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) 1776 2012 28-42 28 (Yukio Takemoto) (Syunsuke Ohashi) (Hiroshi Akamine) (Jiro Mizushima) Department of Mechanical Engineering, Doshisha University 1 (Theodore von Ka rma n, l881-1963) 1911 100 [1]. 3 (B\