T7M7 4 1 cm 3 cm/sec FreeFEM++ FreeFEM++
Dynamics of rotating column put in fluid T7M7 Makoto NOJI Advisor Tutomu IKEDA Graduate Course of Applied Mathematics and Informatics Graduate School of Science and Technology Ryukoku University Abstract In recent years, breaking pitches in a family of non-rotating balls attract marked attention of people such as a knuckleball of baseball and a non-rotating shot of football. The purpose of the present paper is to clarity why such a non-rotating ball moves unpredictably by numerical computation. Because of the restricted ability of a software utilized for calculation of fluid dynamics and the restriction from computation time, the ball is replaced with a column and a two-dimensional numerical computation is performed. Moreover, a value of 4 was assumed for the Reynolds number, which corresponds to the column 1cm diameter moving in the water at about 3cm/sec. We assume that the column can rotate with an arbitrary speed, and we adopt the FreeFEM++ for calculating the velocity and presure fields of moving fluid. Moreover, the lift and drag acting on the column are computated also by using the FreeFEM++. We note that the lift and drag are decided by corresponding area integration in order to keep the accuracy of numerical calculation. Our numerical computations consist of two parts. In the first part, although the lift and drag act on the column, we fix the center of the column and calculate the lift. We find that the lift oscillates by the effect of the Karman vortex appearing behind the column. Moreover, the period and amplitude of oscillation of the lift is independent of the rotation speed of the column while the average of the absolute value of lift increases with the rotation number. In the second part, the dynamics of the column by the lift are numerically computed in the situation where the center of the column can move only in the direction perpendicular to the stream line. We note that the effects of drag are not taken into account still. In our numeral simulation the column is released after the flow field becomes periodic in time. The result is as follows : 1. the column moves more rapidly as the rotation speed becomes higher. 2. the dynamics depends on the timing of release for the slowly rotating column, while it dose not for the rapidly rotating column.
T7M7 21 1 28
1 1 2 1 2.1......................... 2 3 2 3.1............................ 2 3.2............................. 4 3.3......................... 6 4 7 4.1..................... 8 4.2..................... 8 5 1 6 1 7 12 8 22 8.1.......................... 22 8.2............................ 24 8.3.............. 25 8.4................ 27 9 34 1 39 11 42 12 43
1,.,,.. ([9]).. ([7]),. ([7]).,. ([7]).. ([5]).. 2,. 1 2.,. 2. 2,. 3..,. 4.,. ([4]). ([3]),,. ([3]),. ([3]) 1
2.1.. ([3]),.,..,.,.,... 3 3.1,. G. Ω. 1, 25. G. G Ω. Ω. Ω = G γ i γ o γ w (1) γ i, γ o, γ w. 1. ([3]). ([9])(3).. ([9]),. ([9])(2).,,. ([9]), u = (2) u t + (u )u + 1 ρ p 1 u = f (3) ν 2
1:., u,, p, ν, f.,. ([9]),. ([9])(3). ũ t + (ũ )ũ + p 1 Re ũ = f (4) (3)..,. ([9]). ([9]),.. ([9]) x 1, x 2, U, σ = [σ ij ], i, j = 1, 2, n = (n 1, n 2 ) G G.. u = (U, )(x γ i ) (5) 3
σ(u, p)n = (x γ o ) (6) t T σ(u, p)n =, u n(x γ w ) (7) t γ w. σ. D ij σ ij (u, p) = pδ ij + 2 Re D ij(u) (8) D ij (u) = 1 2 ( u i x j + u j x i ) (9). G,., ε,. u = ( (2πε)x 2, (2πε)x 1 ) (1) 3.2. ([3])(4) v, Ω. ( p + p ( ) v1 ) dx = pv 1 dx 2 p v 1 dx p v 2 dx(11) Ω x 1 x 2 v 2 Ω x 1 Ω x 2 (11) γ o pv 1 dx 2 p v 1 dx p v 2 dx Ω x 1 Ω x 2 = 2 u1 v 1 dx 2 Re x 1 Ω pdiv(v)dx (12). (4), v, Ω. (4) a(u, v) = 2 2 D i,j (u)d i,j (v)dx (13) Re Ω i,j=1 4
(4). 1 Re {2 u 1 v 1 dx + ( u 1 + u 2 )( v 1 v 2 ) + 2 Ω x 1 x 1 Ω x 2 x 1 x 2 x 1 Ω = 1 Re {( 2 u 1 2 u 1 2 u 2 v 1 dx 2 v x 2 x 1 Ω x 2 1 dx) 2 v 1 Ω x 2 2 dx 2 + ( u 1 v 1 + u 1 2 u 1 u 2 v 1 )dx 1 v x 1 x 2 x 2 Ω x 2 1 dx + v 2 dx 2 2 x 2 x 1 2 u 2 u 1 2 u 1 v Ω x 2 2 dx + v 2 dx 2 v 2 dx 1 x 2 x 2 Ω x 1 x 2 + ( u 2 v 1 + u 2 v 1 )dx 1 x 1 x 1 x 1 Ω Ω u 2 x 2 v 2 x 2 dx} 2 u 2 x 1 x 2 v 1 dx} (14) (14),,,,,,. u 1 x 2 = u 2 x 1 1 Re {2 u 1 v 1 dx + ( u 1 + u 2 )( v 1 v 2 u 2 v 2 ) + 2 dx} Ω x 1 x 1 Ω x 2 x 1 x 2 x 1 Ω x 2 x 2 = 1 Re {2 u 1 2 u 1 2 u 2 v 1 dx 2 2 v x 2 x 1 Ω x 2 1 dx 2 v 1 Ω x 2 2 dx 2 2 u 1 2 u 2 v Ω x 2 1 dx v 2 Ω x 2 2 dx 1 2 u 1 2 u 2 v 2 dx v 1 dx} (15) x 1 x 2 x 1 x 2 Ω. (15), x 1 ( u 1 x 1 + u 2 x 2 )v 1 =,, x 2 ( u 1 x 1 + u 2 x 2 )v 2 =, 2 Re 2 Ω i,j=1 D i,j (u)d i,j (v)dx = 2 Re x 2 u 1 x 1 v 1 dx 2 1 Re. (12). V,Q. ([3]) Ω u vdx (16) V (g, ε) = {v (H 1 (Ω)) 2 ; v = ( (2πε)x 2, (2πε)x 1 )(x G), v = g(x γ i ) v n = (x γ)}v = V (, ), Q = L 2 (Ω)(17) V,Q (H 1 (Ω)) 2, L 2 (Ω). (u, p) : (, T ) V (g, ε) Q ( u t, v) + a 1(u, u, v) + a(u, v) + b(v, p) = (f, v)( v V ) b(u, q) = ( q Q) (18) 5
. a 1 (u, u, v) = Ω i=1 2 (u grad)u i v i dx b(v, p) = pdiv vdx Ω., f. 3.3 Ω. ([3]), h.. ([3]) Ω. V h (g, ε) V h, v h (P ) = ( (2πε)x 2, (2πε)x 1 )(P G), v h (P ) = g(p )(P γ i ), (v h n)(p ) = (P γ w ) (19). P. V h = V h (, ). Q h. V h V, Q h Q, N u = dimv h, N p = dimq h. ϕ, i = 1,..., N u V h, ψ, i = 1,..., N p Q h. N u V h. V h ϕ 1, ϕ 2,..., ϕ Nu, ϕ 1, ϕ 2,..., ϕ Nu V h. v h (x 1, x 2 ) = N u i=1 C i ϕ i (x 1, x 2 ), q h (x 1, x 2 ) = N u i=1 D i ψ i (x 1, x 2 ) (2). C i, D i. (18) (u h, p h ) : (, T ) V h (g, ε) Q h ( u h t, v h) + a 1 (u h, u h, v h ) + a(u h, v h ) + b(v h, p h ) = (f, v h )( v h V h ) b(u h, q h ) = ( q h Q h ) (21) 6
.,. ([3]),,. ([3]) u h = N u j=1 u j ϕ j + N u+m u j=n u+1 u(p j )ϕ j, p h = N u j=1 p j ψ j (22). M u. (2), (22) (21), d dt. N u j=1 u j (ϕ j, ϕ i ) + N u = (f, ϕ i ) d dt j,k=1 N u +M u j=n u+1 a 1 (ϕ k, ϕ j, ϕ i )u k u j + N u (ϕ j, ϕ i )u(p j ) N u+m u j,k=n u +1 j=1 a(ϕ j, ϕ i )u j + N p j=1 a 1 (ϕ k, ϕ j, ϕ i )u(p k )u(p j ) b(ϕ i, ψ j )p j Nu+Mu a(ϕ j, ψ i )u(p j )(i = 1,..., N u ) (23) j=n u +1 4,., D = 1 2 C DρU 2 A (24) L = 1 2 C LρU 2 A (25). ([3]), ρ, U, A G. C D, C L,,. ([3]),,. ([3]). 7
4.1 G., 2 D = σ 1,j n j ds (26) L = G j=1 G j=1 2 σ 2,j n j ds (27). ([3])(24), (25), C D = 2 2 σ ρu 2 1,j (u, p)n j ds (28) A C L = 2 ρu 2 A G j=1 G j=1 2 σ 2,j (u, p)n j ds (29). ([3]),. G,. ([3]),. 4.2 L 2 (Ω), H m (Ω). L 2 (Ω) = {v : Ω R; v < + } v = { v 2 dx} 1/2 Ω H m (Ω) = {v L 2 (Ω); v m < + } v m = { D α v 2 } 1/2 α m., α = (α 1, α 2 ), 8
α = 2 α i, i=1 D α = Π 2 i=1( x i ) α i. (u, p) (H 1 (Ω)) 2 H 1 (Ω), v (H 1 (Ω)) 2. ([3]), ( u t, v) + a 1(u, u, v) + a(u, v) + b(v, p) = { u Ω t + (u grad)u 1 Re u + gradp 1 Re grad(divu)}vdx + [σ(u, p)]n vds (3) Ω. ([3])., (u, p) (18). w Ω, { w = 1(x G) (31) w = (x Ω \ G). v D = (w, ), v L = (, w). (3) 1 4 (3), 5 (2),. (3) v v D v L,. ([3]) [σ(u, p)]n v D ds = ( u t, vd ) + a 1 (u, u, v D ) + a(u, v D ) + b(v D, p) (32) Ω Ω [σ(u, p)]n v L ds = ( u t, vl ) + a 1 (u, u, v L ) + a(u, v L ) + b(v L, p) (33) 2 (28), (29) C D = 2 ρu 2 A ( u t, vd ) + a 1 (u, u, v D ) + a(u, v D ) + b(v D, p) (34) 9
C L = 2 ρu 2 A ( u t, vl ) + a 1 (u, u, v L ) + a(u, v L ) + b(v L, p) (35). (28), (29),, w. ([3]). C D, C L (26), (27), D = ( u t, vd ) + a 1 (u, u, v D ) + a(u, v D ) + b(v D, p) (36) L = ( u t, vl ) + a 1 (u, u, v L ) + a(u, v L ) + b(v L, p) (37). ([3]) 2. 5 Ω 2. γ i 45, γ o 3, γ w 15, G 45, Free FEM++. 1486, 7595.,,.,,. ([2]) 6, t. min t = κ u 2 1 + u 2 2 (38) κ.,. (38), min t, min t min t t. t =.1.,. Re, Free FEM++.,. ([2]) 1
2: 11
.25 Navier- Stokes Free FEM++. Re = 4, Re = 5 1, Re = 2.,. Re = 8,, t,,.,,., [3],, Re 25., Re 4. Re 4, Re = 4. Re = UL ν (39) (39),,. ([9])U, L, ν. U, L. Re 4,. L =.1[m], ν =.893 1 6 [m 2 /s] (25 ), U =.36[m/s]. ([6]) 7. 3 4. 3 Re = 4.,., =.3 1, =.5 1.8, =.7 2.5, =.9 3.2. t =.,,.,.,.. 12
2 4 6 8 1 1-1 -2-3 -4 3: (Re=4),. 4 Re = 4.,., =.3, =.5, =.7, =.9., 3.. (a). 5 Re = 4.,., =.3, =.5, =.7.. =.3.. 6 Re = 3.,., =.3, =.5, =.7, =.9. Re = 4,, Re = 4. 13
2 4 6 8 1 4 3 2 1-1 4: (Re=4) 2 4 6 8 1.1.2.3.4.5.6.7.8 5: (Re=4) 14
2 4 6 8 1 1-1 -2-3 -4 6: (Re=3) 7 Re = 3.,., =.3, =.5, =.7, =.9. Re = 4, 6. 8. t = 5.,,,.. ([9]). ([9]).. 9 =.5. t = 5..,.. 15
2 4 6 8 1 4 3 2 1-1 7: (Re=3) 1 = 1.. t = 5.,..,.,.. 16
8: (Re=4) 17
9: =.5 (Re=4) 18
1: = 1. (Re=4) 19
1.72242 -.536827.535415 -.1172.1 1.884 -.965341.115463 -.42833.2 1.7758-1.381227 -.33647 -.848186.3 1.22736-1.752542 -.72986-1.247294.4.91691-2.7147-1.16356-1.619747.5.73366-2.34959-1.67299-1.976936.6.376478-2.46296-2.86482-2.276547.7.13612-2.591496-2.577885-2.584648.8.18-2.95611-2.95593-2.956.9-3.2753-3.2753-3.2753 1. -3.5231-3.5231-3.5231 -.1 1.8492 -.118742.96616.426268 -.2 1.79486.299981 1.379467.84545 -.3 1.2253.725464 1.747966 1.242787 -.4.982 1.156187 2.64189 1.614452 -.5.724632 1.6575 2.33382 1.9715 -.6.355431 2.9846 2.446277 2.27252 -.7.13221 2.58354 2.593575 2.586924 -.8.15 2.95977 2.95992 2.95983 -.9 3.27453 3.27454 3.274531-1. 3.52564 3.52564 3.52564 (a) (Re=4) (a) Re = 4. = 1., = 1.. t = 4 t = 1.... t = 4 1. t = 4 1. =.1, =.2, =.3 =.8. 2
.,..., = 1.,, = 1..,..955132 -.476942.47819.419.1.96788 -.887396.73393 -.496.2.954617-1.288587 -.33397 -.81692.3.914953-1.666716 -.751763-1.215126.4.836795-2.16565-1.17977-1.62726.5.7532-2.331642-1.626322-1.981856.6.42267-2.527773-2.1513-2.318191.7.2278-2.65767-2.63489-2.645197.8.68-3.4477-3.449-3.44357.9-3.38928-3.38928-3.38928 1. -3.6728-3.6728-3.6728 -.1.96139 -.73384.8886.49226 -.2.955999.332346 1.288345.814895 -.3.914171.749582 1.663753 1.21246 -.4.83287 1.178571 2.1658 1.599277 -.5.697987 1.626949 2.324936 1.978916 -.6.417896 2.16598 2.524495 2.317582 -.7.2499 2.63542 2.6641 2.646562 -.8 3.4574 3.4654 3.4648 -.9 3.3977 3.3978 3.39771 (b) (Re=3) (b) Re = 3. (a), t = 4 t = 1. Re = 4, =.5.. 21
,, Re = 4. 11: 11. 35. 1., =.1, =.2, =.3..,.. 8 8.1., 22
..., (4). m d2 y dy + k dt2 dt dy dt = L (4) (4). m, k, L. m, k. m = m ρl 3 (41) k = k ρl 2 (42) (4).. ([1]). ([1]). ([1]).. ([1]). k = 1 CρS (43) 2 C, C. ([1]) C =.5. S. ρ 25 ρ =.997 1 6 k = 2.5 1 3 [g/m]. (42) k = 25.. (4),. y(), y() y( t), ẏ( t). ẏ = ẏ(). m d2 y dt dy() + k 2 dt dy dt = L (44) (44) (4). ẏ(t) = Z(t). Z(t) = Ae k m ẏ() t + 23 L k ẏ() (45)
y(t) = m k k ẏ() Ae m ẏ() t +. A, B. L k ẏ() t + B (46) A = ẏ() L k ẏ() (47) B = y() + m k ẏ() (ẏ(),. L k ẏ() ) (48) 12: 8.2... 12. 12. 24
, =.1, =.2, =.3..,.,...,.. 8.3.,..,.,,. 13.,.,,.,,.,. x y,..,,.,,.,.,.., 25
13: 26
.. 14: 8.4 14. 35., =.1, =.2, =.3. 1cm. 13. =, =.1, =.2, =.3. 1.. 99.7g.,.,. =.1 1 2cm. 27
15: 15 14....,. 16.,..,.,. 17 =.3.,..,.,.,.. 18. 28
16: 17: 29
5 1 15 2 25 3.6.4.2 -.2 -.4 -.6 18: 85..,.. 19. 4.......,... 2. 4.,,.. 3
2 4 6 8 1 12 14.5 -.5-1 -1.5-2 -2.5-3 -3.5-4 19: 2 4 6 8 1 12 14.8.6.4.2 -.2 -.4 -.6 -.8 -.1 2: 31
2 4 6 8 1 5-5 -1-15 -2-25 -3-35 -4 21: = 1. 21 = 1.., =.2, =.4, =.6, =.8, = 1..., = 1.. 22 = 1.., =.2, =.4, =.6, =.8, = 1...,. =.6... 23 = 1.., =.2, =.4, =.6, =.8, = 1...,. =.6 32
2 4 6 8 1 1-1 -2-3 -4 22: = 1. 2 4 6 8 1.5 -.5 -.1 -.15 -.2 -.25 -.3 -.35 -.4 23: = 1. 33
.. 9 m = 1.. 5 1 15 2.3.25.2.15.1.5 -.5 24: 24. m = 1 99.7g, m = 5 49.85g, m = 1 9.97g, m = 1.997g. m = 1, m = 5 m = 5. m = 1 m = 1. m = 1... 34
25 =.1. 2 4 6 8 1-2 -4-6 -8-1 -12-14 25: =.1 m = 1, m = 5, m = 1, m = 1. m = 1. m = 1 t = 1 m = 1.,. 26 =.3. m = 1, m = 5, m = 1, m = 1.. m = 1 m = 1. 27. m = 1, m = 5, m = 1, m = 1. m = 1,,. m = 1,., 35
2 4 6 8 1-25 -2-15 -1-5 26: =.3 5 1 15 2.8.6.4.2 -.2 -.4 -.6 -.8 27: 36
. 5 1 15 2 25 3 35 4.4.2 -.2 -.4 -.6 -.8-1 -1.2 28: =.1 28 =.1. m = 1, m = 5, m = 1, m = 1. m = 1. m = 1.. 29 =.3. m = 1, m = 5, m = 1, m = 1., =.1 m = 1, m = 1. m = 1.. 3. m = 1, m = 5, m = 1, m = 1.,. 31 =.1. 37
2 4 6 8 1 -.6 -.8-1 -1.2-1.4-1.6-1.8-2 29: =.3 5 1 15 2.15.1.5 -.5 -.1 -.15 3: 38
2 4 6 8 1.1.5 -.5 -.1 -.15 -.2 31: =.1 m = 1, m = 5, m = 1, m = 1. m = 1 m = 5, m = 1, m = 1. m = 1. 32 =.1. m = 1, m = 5, m = 1, m = 1. m = 1 m = 5, m = 1, m = 1. 1., x 1 u 1 = 1, u 2 =.. 33 39
2 4 6 8 1 -.3 -.25 -.2 -.15 -.1 -.5 32: =.3 2 4 6 8 1.5 -.5-1 -1.5 33: 4
., =.1, =.2, =.3..,,.,,...,,. 2 4 6 8 1 5-5 -1-15 -2 34: 34., =.1, =.2, =.3..,. 35 41
2 4 6 8 1.5 -.5 -.1 -.15 -.2 -.25 35:., =.1, =.2, =.3.... 11, LATEX,, Linux. 42
12 [1],,,,, [2], F reef EM + + 27, 21 COE, MARCH 27, pp. 71-78 [3],, No. 417, MARCH 1998, pp. 13-19 [4],,,, Vol.48, pp.22-36, 1996. [5]Masahisa TABATA and Daisuke TAGAMI, Error Estimates for Finite Element Approximations of Drag and Lift in Nonstationary Navier-Stokes Flows, Reprinted from the JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS Vol. 17. No. 3, pp.371-389, October 2 [6], http : //www.gaia.h.kyoto u.ac.jp/public h tml/theses/pdf/higuchi24.pdf [7],, 75 [8],,, [9],, 25,