数理研渦波09.dvi

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1 (Yasuhide Fukumoto) Faculty of Mathematics, Kyushu University (Makoto Hirota) Japan Atomic Energy Agency Helmholtz [9] (vorticity) 1859 Helmholtz [10] Green Kirchhoff Green 20 Lighthill [20] 20 = [15]. [18] 3 Kelvin [27, 5]. [16] Kelvin [7] 1 3 Moore-Saffmann-Tsai-Widnall (MSTW ) [24, 27, 3, 5] 2 Kelvin Hamilton Krein [1, 2]. Lamb [19] p

2 1: U, ũ ũ α u u = U + ũ; ũ = αũ α2 ũ 02 + (1) 0 α αũ 01 Kelvin 1 Kelvin ΔH Kelvin ΔH = 1 u 2 dv 1 U 2 dv = αh α2 H 2 + ; (2) (ũ2 ) H 1 = U ũ 01 dv, H 2 = 01 + U ũ 02 dv. (3) (U 0) α 2 Kelvin ũ 02 α 2 ũ 02 [5] Kelvin (isovortical sheet) Euler 2

3 [1,2,14] 1 2 H 2 [12, 13] [7] H 2 H 1 =0 (H 2 0) U (3) H 2 H 2 0 ũ 02 ( ) ũ 02 : U ũ 02 dv < 0 isovortical Euler Lagrange Lagrange 2 [7]3 [11] Kelvin ( [12, 13] ) Malkus [21] MSTW ( [4, 16] ) Euler Kelvin [29, 26] 2 Kelvin 2 1 ) Lagrange [7] 3 [23] (Hamiltonian normal form) [17] [29, 26] Euler Euler MSTW [7] isovortical 3 3

4 [22] [25] Euler 2 Lagarage 2 3 Euler [11] 4 Kelvin 5 Kelvin 2 [8] 2 Lagrange Arnol d [1, 2] - Euler Lagrange [14] [12, 13, 7] Lagrange 2 2 D R 3 SDiff(D) Lie g <, > R g g u g v g < u,v> g <, > ( ) Lie [, ] g ad ad(u 1 )u 2 =[u 1,u 2 ]=(u 2 )u 1 (u 1 )u 2 for u 1,u 2 g. (4) u i = u i g g F 1 F 2 Lie-Poisson [ δf1 {F 1,F 2 } = δv, δf ] 2,v δv (5) Euler Poisson F/ t = {F, H} g ad ξ g ad u, ad(ξ) v = ad(ξ)u, v. Poisson v g v t = ad ( ) δh v. (6) δv [1] δh/δv u(t) g Euler-Poincaré [14] Euler ad 4

5 i [ad (ξ)v] i =[ ξ ( v)+ f] i (i =1, 2, 3) (7) f D ξ D f v g D D D D (6) v(t) =Ad ( ) ϕ 1 t v(0) ϕt δh/δv SDiff(D) t {Ad (ϕ)v(0) g ϕ SDiff(D)} (isovortical sheet) t 0 u(t 0 )= ( t ϕt ϕ 1 t 0 t0 ) = δh δv (8) t0 SDiff(D) ϕ t g v(t) t =0 v(0) ϕ α,0 SDiff(D) v α (0) = Ad ( ϕα,0) 1 v(0) α R v α (0) v(t) SDiff(D) v α (t) t ϕ α,t SDiff(D) v α (t) v(t) v α (t) =Ad ( ) ( (ϕα,t ) ) ϕ 1 α,t v(t) =Ad 1 ϕ t v(0). (9) α, ϕ α,t t Lie ξ α (t) g ϕ α,t =expξ α (t) ξ α α O(α 2 ) ξ α = αξ α2 ξ 2 +. (10) Ad ( ϕ 1 α,t) = n=0 [ ad (ξ α )] n /n!, (9) v α = v + αv α2 v 2 + ; v 1 = ad (ξ 1 )v, v 2 = ad (ξ 2 )v +ad (ξ 1 )ad (ξ 1 )v. (11) (v = v g ), v 1 = P [ξ 1 ω], v 2 = P [ ξ 1 ( (ξ 1 ω) ) + ξ 2 ω ] (12) 5

6 ω = v P ϕ α,t ϕ t u α (t 0 ) = ( ) t ϕα,t ϕ t ϕ 1 t 0 ϕ 1 α,t 0 t0 ( ) 1 = u(t 0 )+ (n +1)! [ad(ξ α)] n ξα t ad(v)ξ α (13) n=0 u α = u + αu α2 u 2 + ; u 1 = ξ 1 t ad(u)ξ 1, u 2 = ξ 2 t ad(u)ξ 2 +ad(ξ 1 ) ( ) ξ1 t ad(u)ξ 1 (14) H D u α g v α g u α (t) = δh δv (t) =v α (t). (15) α 1 H = D v2 α /2dV v α (t) u α (t) (12) (14) ξ 1, ξ 2 ξ 1 t +(U )ξ 1 (ξ 1 )U = v 1, (16) ξ 2 t +(U )ξ 2 (ξ 2 )U +(v 1 )ξ 1 (ξ 1 )v 1 = v 2 (17) v 1, v 2 (12) U = v (12) ω = U 1 (16) 2 (17) [7] [7], (17) 6

7 3 α H (v α )=H(v)+αH α2 H 2 + (18) ( v/ t = 0), (11) 1 δh δh H 1 = δv,v 1 = δv, ad (ξ 1 )v = ξ 1, v =0. (19) t Euler [1, 2] 2 H 2 = δh δ 2 δv,v H 2 + δv v 1,v 2 1 = ξ 1, v 1 = t = ξ 1, ad ( ξ1 t ξ 2, v t = ω ) v D ξ 1, v 1 t ( ξ1 t ξ 1 ) dv (20) (20) (20) 2 2 Lagrange ξ 2 (x,t) 1 ξ 1 (x,t) (20) ( [2]), 2 ξ 2 O(α 2 ) [11]. η g, J =< η,v> Noether exp η J v α J α =<η,v α > J α J α J α =<η,v>+αj α2 J 2 + (21) (11) 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 (22) 7

8 v(t) ad (η)v =0, 1 J 1 =0 2 J 2 Euler J 2 = ad(η)ξ 1, ad (ξ 1 )v = ω (ξ 1 L η ξ 1 )dv (23) L η ξ 1 = ad(η)ξ 1 ξ 1 η Lie 2 J 2 1 Lagrange ξ 1 D 4 Kelvin Kelvin O(α) r =1 z- (r, θ, z) 2 r θ U 0 V 0 P 0 0 [26, 23] U 0 =0, V 0 = r, P 0 = r 2 /2 1 (24) ũ = αu 01 3 u (m) 01 = A m (t)u (m) 01 (r)e imθ e ikz, A m (t) e iω 0t A m (t) t ω 0 m( Z) k( R) Kelvin u (m) 01 (r) Euler L m,k = (25) L m,k u (m) 01 + p (m) 01 = 0, u (m) 01 =0. (26) i(ω 0 m) i(ω 0 m) i(ω 0 m) (27) m- 1 Bessel J m { u (m) i 01 = m ω 0 m +2 r J m(η m r)+ ω } 0 m ω 0 m 2 η mj m+1 (η m r), { } v (m) 1 m 01 = ω 0 m +2 r J 2η m m(η m r)+ ω 0 m 2 J m+1(η m r), w (m) k 01 = ω 0 m J m(η m r), p (m) 01 = J m (η m r) (28) 8

9 3 2 1 Ω k 2: Kelvin (m = 1) (m =+1) η m η 2 m =[4/(ω 0 m) 2 1] k 2 (28) u (m) 01 =0atr =1 J m+1 (η m )= (ω 0 m 2)m (ω 0 m)η m J m (η m ) (29) [28, 23] 2 (m = ±1) (m = 1) (m =+1) (m =1) (k, ω 0 )=(0, 1) (m = 1) (k, ω 0 )=(0, 1) 20 [27, 5] (isolated modes) 5 Kelvin Kelvin αu 01 ũ O(α 2 ) α 2 u 02 (θ z ) z- z- θ z Euler [26] O(α 2 ) [7] ( 1) Lagrange 2 Lagrange [7, 8] 9

10 α, (16) u 01 = ξ 1 t +(U 0 ) ξ 1 (ξ 1 ) U 0 (30) (24) U 0 = re θ, (30) i(ω 0 m)ξ 1 Kelvin (25) (30) [ ] iam (t) ξ 1 =Re ω 0 m u(m) 01 (r)e imθ e ikz (31) ξ 1 =0. U 0 = re θ U 0 =2e z (23) (12) (20) 2 Lagrange ξ 2 u 02 = P [ξ 1 ( (ξ 1 e z ))] = ξ 1 ξ 1 / z = 4ik (ω 0 m) A m 2 (0,u (m) 2 01 w(m) 01, u(m) 01 v(m) 01 ) (32) Lagrangian k ω 0 (k, ω 0 ) 3 k 0 2 (32) θ z (20) (32) O(α 2 ) H 2 E 2, O(α 2 ) J 2 = u 02 da (33) (25) (action) μ 0 = E 2 /ω 0 J 2θ = mμ 0, J 2z = k 0 μ 0 μ 0 J 2 (pseudo-momentum) 6 z Rankine Kelvin 10

11 [24, 27, 28, 3, 5] Kelvin α O(α 3 ) O(α 2 ) Euler ɛ O(α 2 ɛ) [26, 29] isovortical Kelvin O(α 2 ) O(α 3 ) [23, 8] [17] Kelvin O(α 3 ) Kelvin [21, 4] Kelvin 2 Kelvin [22, 6] 2 Euler O(α 2 ) Lagrange [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 linéaires, J. Méc. 5 (1966) [3] Eloy, C. and Le Dizés, S.: Stability of the Rankine vortex in a multipolar strain field, Phys. Fluids 13 (2001) [4] Eloy, C., Le Gal, P. and Le Dizés, S.: Experimental Study of the Multipolar Vortex Instability, Phys. Rev. Lett. 85 (2000)

12 [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.: Weakly nonlinear evolution of an ellipticalflow,inproc. of the 3rd International Conference on Vortex Flows and Vortex Models (ed. K. Kamemoto, the Japan Society of Mechanical Engineers (JSME), 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 an elliptical flow, submitted to Phys. Scr. (2010). [9] Helmholtz, H. von: Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen, Crelle s J. (J. Reine Angew. Math.) 55 (1858) [English translation] by P. Tait: On integrals of the hydrodynamical equations, which express vortex-motion, Phil. Mag. (4) 33 (1867) pp [10] Helmholtz, H. von: Theorie der luftschwingungen in röhren mit offnen enden, Crelle s J. 57 (1860) [11] Hirota, M: Action-angle representation of waves in fluids and plasmas: applications to stability analysis and wave-mean field interactions, In COE Lecture Note Vol. 20 Math-for-Industry Tutorial: Spectral theories of non-hermitian operators and their application (Faculty of Mathematics, Kyushu University, 2009) pp [12] Hirota, M. and Fukumoto, Y.: Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra, J. Math. Phys. 49 (2008) [13] Hirota, M. and Fukumoto, Y.: Action-angle variables for the continuous spectrum of ideal magnetohydrodynamics, Phys. Plasmas 15 (2008) [14] Holm, D. D., Schmah, T. and Stoica, C.: Geometric Mechanics and Symmetry (Oxford University Press, 2009). [15] Howe, M. S.: Vorticity and the theory of aerodynamic sound, J. Eng. Math. 41 (2001) [16] Kerswell, R. R.: Elliptical instability, Annu. Rev. Fluid Mech. 34 (2002) [17] Knobloch, E., Mahalov, A. and Marsden, J. E.: Normal forms for three-dimensional parametric instabilities in ideal hydrodynamics, Physica D 73 (1994)

13 [18] Kobayashi, T., Takami, T,, Miyamoto, M., Takahashi, K., Nishida, A. and Aoyagi, M.: Calculation with Compressible LES for Sound Vibration of Ocarina, In Proc. of Open Source CFD International Conference [19] Lamb, H.: Hydrodynamics, 6th ed. (Cambridge University Press, 1932). [20] Lighthill, J.; On sound generated aerodynamically. I. General theory, Proc.Roy.Soc. London A 211 (1952) [21] 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) [22] Mason, D. M. and Kerswell, R. R.: Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown, J. Fluid Mech. 396 (1999) [23] 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) [24] Moore, D. W. and Saffman, P. G.: Instability of a straiht vortex filament in a strain field, Proc.R.Soc.Lond.A346 (1975) [25] Rodrigues, S. B. and Luca, J. D.: Weakly nonlinear analysis of short-wave elliptical instability, Phys. Fluids 21 (2009) [26] Sipp, D.: Weakly nonlinear saturation of short-wave instabilities in a strained Lamb- Oseen vortex, Phys. Fluids 12 (2000) [27] 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. 66 (1976) [28] Vladimirov, V. A., Tarasov, V. F. and Rybak, L. Ya.: Stability of elliptically deformed rotation of an ideal incompressible fluid in a Coriolis force field, Izv. Atmos. Ocean. Phys. 19 (1983) [29] Waleffe, F. A.: The 3D instability of a strained vortex and its relation to turblence, PhD thesis. MIT. 13

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

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 https://doi.org/10.15017/23403