九州大学学術情報リポジトリ Kyushu University Institutional Repository 楕円回転流の弱非線形安定性のためのオイラー ラグランジュ混合法 福本, 康秀九州大学大学院数理学研究院 Fukumoto, Yasuhide Faculty of Mathematics
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1 九州大学学術情報リポジトリ Kyushu University Institutional Repository 楕円回転流の弱非線形安定性のためのオイラー ラグランジュ混合法 福本, 康秀九州大学大学院数理学研究院 Fukumoto, Yasuhide Faculty of Mathematics, Kyushu University 出版情報 : 応用力学研究所研究集会報告. 22AO-S8 (19), pp , 九州大学応用力学研究所バージョン : 権利関係 :
2 No.22AO-S8 COE Reports of RIAM Symposium No.22AO-S8 Development in Nonlinear Wave: Phenomena and Modeling Proceedings of a symposium held at Chikushi Campus, Kyushu Universiy, Kasuga, Fukuoka, Japan, October 28-30, 2010 Co-organized by Kyushu University Global COE Program Education and Research Hub for Mathematics - for - Industry Article No. 19 (pp ) FUKUMOTO Yasuhide Received 21 December 2011 Research Institute for Applied Mechanics Kyushu University March, 2011
3 (FUKUMOTO Yasuhide) 2 3 Kelvin Hopf ( ) 1 Moore-Saffman-Tsai-Widnall (MSTW) 3 [18, 21, 3, 5] MSTW Kelvin m 2 2 Kelvin Fukumoto [5] m m + 2 Kelvin (m,m + 2) = ( 1,1) (m,m + 2) = (1,3) (0,2) [4, 14] Malkus [15] 2 ( [4] ) MSTW MSTW Waleffe [23] Sipp [20] Kelvin Mason & Kerswell [16] MSTW 2 Kelvin Rodrigues & Luca [19] [2, 11, 12] Fukumoto & Hirota [7] Kelvin (= ) 3 Kelvin 1
4 (action) MSTW 2, 3 4 Kelvin 5 Kelvin 6, 7 MSTW [5] 3 (isovortical) [2] (isovortical) 2 [11, 12]. 2 [7]. [10] - [2, 13] [11, 12, 7] D( R 3 ) SDiff(D) Lie g Lie g g u g v g < u,v > 1 g <, > L 2 Lie g Lie [, ] g ad(u 1 )u 2 = [u 1,u 2 ] = (u 2 )u 1 (u 1 )u 2 (u 1,u 2 g) (2.1) g 2 F 1, F 2 Lie-Poisson [ δf1 {F 1,F 2 } = δv, δf ] 2,v δv Poisson F/ = {F,H} ad ad u,ad(ξ ) v = ad(ξ )u,v, (ξ g) Poisson v g [1]: ( ) v δh = ad v. (2.3) δv 2 (2.2)
5 δh/δv u(t) g [13] ad [ad (ξ )v] i = [ ξ ( v) + f ] i (i = 1,2,3) (2.4) f D ξ D f v g D D D D (2.3) v(t) = Ad ( ϕt 1 ) v(0) ϕt δh/δv SDiff(D) 1 {Ad (ϕ)v(0) g ϕ SDiff(D)} (isovortical sheet) ϕ t u(t 0 ) = ( ϕt ϕ 1 ) δh t 0 = t0 δv SDiff(D) ϕ t, g ( ) v(t) (t = 0) ϕ α,0 SDiff(D) v(0) v α (0) = Ad ϕα,0 1 v(0) α R v α (0) v α (t) t ϕ α,t SDiff(D) v(t) v α (t) v α (t) = Ad ( ϕα,t 1 ) ( ( ) ) v(t) = Ad 1 ϕα,t ϕ t v(0) (2.6) α, ϕ α,t ξ α (t) g ϕ α,t = expξ α (t) α O(α 2 ) ξ α = αξ 1 + α 2 ξ 2 /2 +. Ad ( ϕ 1 α,t ) = n=0 [ ad (ξ α )] n /n! (2.6) v α = v + αv 1 + α 2 v 2 /2 + ; (2.5) v 1 = ad (ξ 1 )v, v 2 = ad (ξ 2 )v + ad (ξ 1 )ad (ξ 1 )v (2.7) v 1 = P [ξ 1 ω], v 2 = P [ ξ 1 ( (ξ 1 ω) ) + ξ 2 ω ] (2.8) v = v g ω = v v P ϕ α,t ϕ t u α (t 0 ) = ( ϕα,t ϕ t ϕt 1 0 t0 = u(t 0 ) + n=0 ϕ 1 α,t 0 ) 1 (n + 1)! [ad(ξ α)] n ( ξα ad(v)ξ α ) 2 u α = u + αu 1 + α 2 u 2 /2 + (2.9) u 1 = ξ 1 ad(u)ξ 1, u 2 = ξ 2 ad(u)ξ 2 + ad(ξ 1 ) ( ξ1 ad(u)ξ 1 ) (2.10) 3
6 1 H = D v2 α/2dv H Lie u α (t) g u α (t) = δh δv v α (t) (2.8) (2.10) (t) = v α (t) (2.11) α ξ 1 + (U )ξ 1 (ξ 1 )U = v 1, (2.12) ξ 2 + (U )ξ 2 (ξ 2 )U + (v 1 )ξ 1 (ξ 1 )v 1 = v 2 (2.13) v 1 v 2 (2.8) U = v (2.8) ω = U 1 (2.12) 2 (2.13) [7] 3 (isovortical disturbances) α 2 H (v α ) = H(v) + αh 1 + α 2 H 2 /2 +. ( v/ = 0), 1 (2.7) δh δh H 1 = δv,v 1 = δv, ad (ξ 1 )v = ξ 1, v = 0 (3.1) [1, 2] 2 H 2 = ξ 1, v ( 1 ξ = ω 1 ξ D 1 )dv (3.2) (3.2) (3.2) 2 (2.13) 2 ξ 2 (x,t) (3.2) [2] 2 ξ 2 O(α 2 ) [10] η g J =< η,v > Noether H expη J v α J α =< η,v α > J α J α J α =< η,v > +αj 1 + α 2 J 2 /2 + (2.7) 2 J 1 = η,v 1 = η, ad (ξ 1 )v = ξ 1,ad (η)v, J 2 = η,v 2 = ξ 2,ad (η)v + ξ 1,ad (η)v 1 (3.3) 4
7 v(t) ad (η)v = 0, J 1 = 0 J 2 J 2 = ad(η)ξ 1, ad (ξ 1 )v = ω (ξ 1 L η ξ 1 )dv (3.4) L η ξ 1 = ad(η)ξ 1 ξ 1 η Lie 2 J Kelvin (= Rankine ) Kelvin 1 ( 6 ) z (r,θ,z) 2 r θ U 0 V 0 P 0 0 r 1 D U 0 = 0, V 0 = r, P 0 = r 2 /2 1 (4.1) ũu = α u 01 z z u (m) 01 = A m(t)u (m) 01 (r)eimθ e ikz, A m (t) e iω 0t A m t ω 0 m k Kelvin u (m) 01 L m,k u (m) 01 + p(m) 01 = 0, u (m) 01 = 0 (4.3) L m,k = r u (m) 01 = i ω 0 m + 2 i(ω 0 m) i(ω 0 m) i(ω 0 m) { m r J m(η m r) + ω } 0 m ω 0 m 2 η mj m+1 (η m r) (4.2) (4.4) η m η m 2 = [ 4/(ω 0 m) 2 1 ] k 2, J m m 1 Bessel u (m) 01 = 0 (r = 1) J m+1 (η m ) = (ω 0 m 2)m J m (η m ) (4.6) (ω 0 m)η m [22, 17] 1 m = ±1 (m = 1) (m = 1) m = 1 (k,ω 0 ) = (0,1) m = 1 (k,ω 0 ) = (0, 1) [5, 21] (4.5) 5
8 3 2 1 Ω k 1: Kelvin (m = 1) (m = +1) 5 ũu 2 α 2 u 02 Kelvin α u 01 z z (r ) ε O(εα 2 ) (θ) - [20] (z ) Kelvin [17] O(εα 2 ) O(α 2 ) [7] Kelvin (2.12) u 01 = ξ 1 + (U 0 ) ξ 1 (ξ 1 )U 0 (5.1) U 0 = re θ (5.1) i(ω 0 m)ξ 1 Kelvin (4.2) (5.1) [ ξ 1 = Re ia ] m(t) ω 0 m u(m) 01 (r)eimθ e ikz (5.2) U 0 = re θ U 0 = 2e z (3.4) (2.8) θ z 2 6
9 ξ 2 1 ξ 1 u 02 = P [ξ 1 ( (ξ 1 e z ))] = ξ 1 ξ 1 / z = 4ik (ω 0 m) 2 A m 2 (0,u (m) 01 w(m) 01, u(m) 01 v(m) 01 ). (5.3) k 6 [15, 4] x ε + y2 1 ε = 1 (6.1) ε ε U = U 0 + εu 1 +, P = P 0 + εp 1 + ; U 1 = r sin2θ, V 1 = r cos2θ, P 1 = 0 (6.2) ε O(ε) U 1 2 ( ) θ = π/4 ( ) θ = π/4 2 3 ũu 2 ε α O(α 3 ) u = U + ũu = U 0 + εu 1 + α u 01 + εα u 11 + α 2 u 02 + α 3 u (6.3) O(ε m α n ) u mn ε (6.1) r = 1 + ε cos2θ/2 + O(ε 2 ) n 7 Moore-Saffman-Tsai-Widnall u n = 0 at r = 1 + ε cos2θ/2 (6.4) 4 Kelvin εu 1 Kelvin O(α) e imθ e i(m+2)θ Kelvin εu 1 O(εα) e imθ e i(m+2)θ Kelvin [18, 21, 5] 2 Rankine 3 m m + 2 Kelvin (k,ω) O(εα) [5] [22]. 7
10 1 k (ω 0 = 0) (ω 0 = 0) (ω 0 0) [21, 3, 5] ω 0 = 0 (m,m + 2) = ( 1,+1) ω 0 = 0 η = 3k u 01 m = ±1 Kelvin u 01 = A u ( ) 01 e iθ e ikz + A + u (+) 01 eiθ e ikz + c.c. (7.1) A ±1 A ± U 1 O(εα) u 11 = { B u ( ) 11 e iθ + B + u (+) 11 eiθ + B 3 u ( 3) 11 e 3iθ + B 3 u (3) 11 e3iθ } e ikz +c.c. (7.2) u (m) 11 (r), u(m+2) 11 (r) O(εα) (6.4) u ( ) du01 2 dr u 01 cos2θ + v 01 sin2θ = 0 at r = 1 (7.3) (7.3) B ± Kelvin 1 A = 1 A + = i 3(3k2 + 1) A + 10 A 10 8(2k 2 = ia. (7.4) + 1) t 10 = εt k (4.6) J 1 (η) = ηj 0 (η) k (ω 0 = 0) a = 3(3k 2 +1)/[8(2k 2 + 1)] [22] A /A + = i. k 2 ω 0 = 0 (k,a) (1.578, ), (3.286, ),. Kelvin [5] Kelvin (3.2) [7] O(α 3 ) e ±iθ e ikz (m = ±1) u (m) 03 L m,k u (m) 03 = N u (m) 01 / 02 2 t 02 = α 2 t N O(α 2 ) O(α 3 ) A ± 8 O(α 3 ) [17] ( 4ik 0, ( A 2 + A + 2) u (+) 01 w(+) 01,( A 2 A + 2) ) u (+) 01 v(+) 01 8 (8.1)
11 (m,m+2) (5.3) Kelvin A = A + O(α 3 ) (= ) da ± dt = i [ εaa + α 2 A ± ( b A± 2 + c A 2)]. (8.2) a (7.4) [ b = 2k (2k 2 + 1) J 0 (η) 2 rj 0 (ηr) 2 J 1 (ηr) 2 dr (11k k 2 + 5)J 0 (η) ], 2 0 [ ] k 2 64k 2 1 c = 12(2k 2 + 1) J 0 (η) 2 rj 0 (ηr) 2 J 1 (ηr) 2 dr + (20k k k 2 27)J 0 (η) 2 0 k (ω 0 = 0) 2 (k;a,b,c) (1.579;0.5312, ,5.222), (3.286;0.5542, 8.286,53.39) (8.2) [9] (5.3) A ± 2 A ±, A 2 A ± [20] A and A + t (8.2) 4 (a,b,c) a > 0, b < 0, c > 0 (8.2) 2 A = A = A + α 2 = ε (8.2) 2 (8.3) da dt = iε ( aa + β A 2 A ) (8.4) β = b + c 2 (Re[A],Im[A]) A = A e iφ A φ d A dt = εa A sin2φ, dφ dt = εacos2φ + εβ A 2 (8.5) A ( 1) A = 0 (xy) φ φ = π/4,3π/4 φ = π/4,3π/4 MSTW instability A φ (8.5) φ = π/4,3π/4 9
12 Im[A] s u s Re[A] : Trajectories in (Re[A],Im[A]) k = (s:, u: ) 9 Kelvin ( 5) [15, 4] MSTW ( ) ( 3 ) [8] ( ) [1] Arnol d, V. I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l hydrodynamique des fluids parfaits, Ann. Inst. Fourier Grenoble 16 (1966), [2] Arnol d, V. I.: Sur un principle variationnel pour les écoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non linéaire, J. Méc. 5 (1966), [3] Eloy, C. and Le Dizès, S.: Stability of the Rankine vortex in a multipolar strain field, Phys. Fluids 13 (2001),
13 [4] Eloy, C, Le Gal, P. and Le Dizès, S.: Experimental Study of the Multipolar Vortex Instability, Phys. Rev. Lett. 85 (2000), [5] Fukumoto, Y.: The three-dimensional instability of a strained vortex tube revisited, J. Fluid Mech. 493 (2003), [6] Fukumoto, Y., Hattori, Y. and Fujimura, K.: Proc. of the 3rd International Conference on Vortex Flows and Vortex Models (The Japan Society of Mechanical Engineers, 2005) pp [7] Fukumoto, Y. and Hirota, M.: Elliptical instability of a vortex tube and drift current induced by it, Phys. Scr. T132 (2008), [8] Fukumoto, Y., Hirota, M. and Mie, Y.: Lagrangian approach to weakly nonlinear stability of elliptical flow, Phys. Scr. T142 (2010), [9] Guckenheimer, J. and Mahalov, A.: Instability induced by symmetry reduction, Phys. Rev. Lett. 68 (1992), [10] Hirota, M.: Math-for-Industry Tutorial: Spectral theories of non-hermitian operators and their application, COE Lecture Note Vol. 20 (Kyushu University, 2009), pp [11] Hirota, M. and Fukumoto, Y.: Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra, J. Math. Phys. 49 (2008), [12] Hirota, M. and Fukumoto, Y.: Action-angle variables for the continuous spectrum of ideal magnetohydrodynamics, Phys. Plasmas 15 (2008), [13] Holm, D. D., Schmah, T. and Stoica, C.: Geometric Mechanics and Symmetry (Oxford University Press, 2009). [14] Kerswell, R. R.: Elliptical instability, Annu. Rev. Fluid Mech. 34 (2002), [15] Malkus, W. V. R.: An experimental study of the grobal instabilities due to the tidal (elliptical) distortion of a rotating, Geophys. Astrophys. Fluid Dyn. 48 (1989), [16] Mason, D. M. and Kerswell, R. R.: Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown, J. Fluid Mech. 396 (1999), [17] Mie, Y. and Fukumoto, Y.: Weakly nonlinear saturation of stationary resonance of a rotating flow in an elliptic cylinder, J. Math-for-Industry 2 (2010), [18] Moore, D. W. and Saffman, P. G.: Instability of a straiht vortex filament in a strain field, Proc. R. Soc. Lond. A 346 (1975), [19] Rodrigues, S. B. and Luca, J. D.: Weakly nonlinear analysis of short-wave elliptical instability, Phys. Fluids 21 (2009),
14 [20] Sipp, D.: Weakly nonlinear saturation of short-wave instabilities in a strained Lamb-Oseen vortex, Phys. Fluids 12 (2000), [21] Tsai, C.-Y. and Widnall, S. E.: The stability of short waves on a straight vortex filament in a weak externally imposed strain field, J. Fluid Mech. 73 (1976), [22] Vladimirov, V. A., Tarasov, V. F. and Rybak, L. Y.: Stability of elliptically deformed rotation of an ideal incompressible fluid in a Coriolis force field, Izv. Atmos. Ocean. Phys. 19 (1983), [23] Waleffe. F. A.: The 3D instability of a strained vortex and its relation to turblence, PhD thesis. MIT (1989). 12
数理研渦波09.dvi
(Yasuhide Fukumoto) Faculty of Mathematics, Kyushu University (Makoto Hirota) Japan Atomic Energy Agency 1 1858 Helmholtz [9] (vorticity) 1859 Helmholtz [10] Green Kirchhoff Green 20 Lighthill [20] 20
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