Nonlinear Evolution of a Vortex Ring

Transcription

1 Nonlinear Evolution of a Vortex Ring Yuji Hattori Kyushu Institute of Technology, JAPAN Yasuhide Fukumoto Kyushu University, JAPAN EUROMECH Colloquium 491 Vortex dynamics from quantum to geophysical scales September 2007, University of Exeter

2 Main Result Captured Curvature Instability of a Vortex Ring by DNS! (1,0) 10-4 (2,1) (3,1) (4,1) (1,1) t

3 Outline of Talk 1. Background: Curvature Instability 2. Previous DNS NO curvature Instability Critical Layer 3. New DNS Vortex Ring in a Torus Results 4. Weakly Nonlinear Analysis 5. Summary

4 Background: Instability of a Vortex Ring Vortex Ring: Curved Vorticity Lines Widnall Instability (= Elliptic Instability) Curvature Instability (Fukumoto & Hattori, 2002) Normal-Mode Analysis (F&H, 2002; F&H 2005) Short-Wavelength Stability Analysis (H&F, 2003) Mechanism: Parametric Resonance Widnall & Tsai, 1977 Naitoh et al., 2002

5 Linear Stability (Theory) Fukumoto & Hattori, 2005; Hattori & Fukumoto, 2003 Assumption: inviscid, incompressible Parametric Resonance Widnall Instability strain resonates (m, m+2) modes Short wave limit: Curvature Instability dipole field resonates (m, m+1) modes Short wave limit: max 2 = Stronger than Widnall instability σ σ max ε log 16 4 ε = ε 256 m: wavenumber in θ

6 Next Question Can we observe curvature instability? We should take account of: Nonlinear effects: Finite Amplitude Viscous effects Vorticity distribution Mode analysis: Kelvin s vortex ring Experiments: close to Gaussian distribution Weakly Nonlinear Analysis & DNS

7 DNS (Previous) 3D Navier-Stokes equations Method 1 Periodic Box (periodic in x, y, z) Pseudo-Spectral Method Method 2 Periodic Cylinder (periodic in z) Pseudo-Spectral + Finite Difference (cylindrical coordinate system) Initial Vorticity: Gaussian core Disturbance: Widnall: 9 waves along the ring Curvature: tried (0,1), (1,2) decayed Reyonlds number: n ε = k z

8 Method 1: Energy Energy of disturbance Energy spectrum (disturbance) t log E Ek 1e-06 1e-08 1e-10 1e-12 1e k t=100 t=110 t=120 t=130 t=140 t=150

9 Method 2: Iso-surface of vorticity magnitude Widnall Instability is observed Curvature Instability is NOT observed

10 Method 2: Mode energy (Widnall instability) E e-005 Growth Rate DNS: Theory: e-010 1e-015 Symmetry Main Rest_1 Rest_ t

11 NO curvature instability: Why? Kelvin waves on vortices Rankine vortex: no critical layer Smooth Vorticity: critical layer exists Linear theory (Le Dizes, 2004 etc.) If there is a critical layer, Kelvin waves decay exponentially Equation for Kelvin wave = dr r Δ dr dr rσ dr Δ dr Σ r d p 1 1 dδ dp 2m dω Ω dδ k Δ m p V Ω= ω m r Critical Layer Σ = + Ω Σ=0 critical layer 0

12 Critical Layer Unstable Waves on Vortex Ring (smooth vorticity) Bending wave (Widnall Instability): no critical layer Curvature Instability: has a critical layer Potentially unstable waves decay exponentially Vortex Ring in a Torus NO critical layer Curvature Instability Can Be Observed

13 Numerical Method 3D Navier-Stokes equations Vortex Ring in a Torus Spectral Method (toroidal coordinate system) Free From Critical Layer Consistent with Weakly Nonlinear Analysis Base Flow: Kelvin s Vortex Ring Poisson Equation: Iteration Method

14 Numerical Method Toroidal Coordinate System Equation: evolution of U 2 2 U U V U εw U V εw + U + + t r r Uθ 1+ εrsinθ s r 1+ εrsinθ 2 P 2 2= VRU[ U] ε cosp θ = + ν U V 2 r tr θ r r( 1+ εrsinθ) 2 ε 2 W U sin θ + V cosθ sinθ + 2 sinθ 2 ( 1+ εr sinθ) s s θ r Equation: incompressibility U =0 U U 1 V ε Usinθ V cosθ W = 0 r r r θ 1+ εrsinθ s

15 Numerical Method Spatial Discretization r: compact scheme -> Chebyshev Collocation θ, s: Fourier Collocation Procedure 1. Solve Poisson equation 2. Advance by U t Ns-fold Symmetry in s is assumed = U 2 P R[ ] = R[ U] P periodic

16 Poisson Equation = U 2 P R[ ] 1 εsinθ 1 ε cosθ ε r r r 1+ εrsinθ r r θ r 1+ εrsinθ θ 1 ε sinθ s = ( ) ( + r ) Iteration Method 2 2 = 0 +ε L 1 1 = r r r r θ s ε L sinθ cosθ 1 = + + ε 1 1+ εrsinθ r r( 1+ εrsinθ) θ ( 1+ εr sinθ) s [ U] P = R ε LP 2 ( n+ 1) ( n) 0

17 Initial Conditions u 01 : Pair of Kelvin waves Case 1: (m,m+1)=(1,2), Ns=7, ε= Case 2: (m,m+1)=(2,3), Ns=6, ε= Number of Points: Dispersion Curves ω k 0

18 Evolution of Energies of Unstable Modes (Cases 1 & 2) m=1: (1,1) m=1: (2,1) m=2: (2,1) m=2: (3,1) theory t Growth Rates: Case (m) Theory DNS Case1 (m=1) Case2 (m=2)

19 Case 2: (m, m+1)=(2, 3) Evolution of Mode energies 10-2 ( ) 2 Enl (, ) = u r; n, l rdr u ( r n l) i( nθ ln s) u= ;, exp + s 10-3 (1,0) 10-4 Enl (, ) } 10-5 l = 2, n = 2 ~ (4,1) (1,1) (2,1) (3,1) t

20 Evolution of Vorticity Fields (Case 2) vorticity fields in s=0 t=0 t=16 t=32 t=48 Total Flow Disturbance

21 Evolution of Disturbance (Case 2) Disturbance fields in s=0 t=0 t=16 t=32 t=48 Pressure Vorticity

22 Structure of Unstble Modes (Case 2) Iso-surface of Disturbance in s=const., t=32 Pressure Vorticity

23 Structure of Unstble Modes (Case 2) vorticity (s-component) fields in s=const., t=32 6s=0 π/4 π/2 3π/4 π Large Vorticity in left semi-circle vortex stretching in s-direction

24 Unstable Parameter Region Curvature Instability: Parametric Resonance Vortex Ring: ε is quantized Wavenumber k has a bandwidth ε = k/ Ns growth rate Unstable 0 pairs exist 0.3 for 0.4 all 0.5 ε ε Theory Theory+DNS Divide by ε

25 Weakly Nonlinear Analysis Expansion u ε εαu + εα u u02 α u03 = U + εu + + αu + α :(core radius)/(ring radius) α : amplitude of disturbance Vortex Ring in a Torus: Boundary Conditions are simplified (slip wall at r=1) Substitute to the Euler equations

26 Important modes U 0 :( 0, 0,0 ) U 1 :( 1, 0, 0) cc.. u :( m, 1, ) ( m+ 1, 1, 1) cc u02 :( 2m, 2, 2) ( 2m+ 1, 2, 2) ( 2m + 2, 2, 2) ( 1, 0, 0) cc.. u 03 Rankine Vortex Dipole Field Linear Modes :( m,,) 11 ( m+ 1,,) 11 ( m+ 2,,) 11 ( 3m, 3, 3) ( 3m + 1, 3, 3 ) c. c. ( θ nks ω t) ( m, n, l) exp i m + l 0 ( m, 1, 1 ),( m+ 1, 1, 1) contribute Nonlinear Effects u 11 :( m,,) 1 1 ( m+ 1,,) 1 1 ( m+ 2,,) 1 1 cc.. Curvature Instability

27 Amplitude Equation ε α 2 da dt db + dt da + dt db dt dc dt dc dt + ( ) = ab + i c A + c B + c A + c B + d C + d C A + ieab B ( ) = aa + ic A + c B + c A + c B + d C + d C B + iebaa ( ) = ab + i c A + c B + c A + c B + d C + d C A + iea B B = aa 2 + ( i c ) 3+ A + c4+ B + c3 A+ + c4 B+ + d2+ C + d2 C+ B + ie2b+ A A+ = A B + A B = AB + AB Symmetry Breaking (Knobloch et al., 1994) O(2) SO(2) O(2)

28 Invariants E F 2 2 +, ± A ± B ± C ± = E + E E = K A + K B + K C ( C C ) = A + A a + ( ) G = B + B b C + C H ± ( e ) ± 1 ± 2 1 ± = e A + e B ae + b C Number of independent invariants: 3 E = K F + K G e F A 2 1 B + e G = H + + H

29 Example Spatial Discretization: Chebyshev collocation Singular Value Decomposition Modes: (m,m+1)=(1,2), σ= ω k 0

30 Evolution of Energy Symmetric IC Asymmetric IC Mode Energy t E A + E B + E A - E B - Mode Energy t E A + E B + E A - E B - Asymmetric Initial Conditions Chaotic Behavior Larger Energy

31 Summary Captured Curvature Instability by DNS New Numerical Method for Toroidal Coordinate System Growth Rate: Agrees with Theory Structure of Unstable Waves is revealed Unstable for all ε