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 11-14 September 2007, University of Exeter
Main Result Captured Curvature Instability of a Vortex Ring by DNS! 10-2 10-3 (1,0) 10-4 (2,1) (3,1) 10-5 10-6 10-7 (4,1) (1,1) 10-8 0 10 20 30 40 50 60 t
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
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
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 9 3 8 15 ε log 16 4 ε 32 165 = ε 256 m: wavenumber in θ
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
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: 10 4 1 n ε = k z
Method 1: Energy Energy of disturbance Energy spectrum (disturbance) 100 120 140 160 180 t 0.01-2 0.0001-4 log E -6-8 -10 Ek 1e-06 1e-08 1e-10 1e-12 1e-14 1 10 k t=100 t=110 t=120 t=130 t=140 t=150
Method 2: Iso-surface of vorticity magnitude Widnall Instability is observed Curvature Instability is NOT observed
Method 2: Mode energy (Widnall instability) E 100000 1 1e-005 Growth Rate DNS: 2.76 10-2 Theory: 5.65 10-2 1e-010 1e-015 Symmetry Main Rest_1 Rest_2 100 120 140 160 180 200 220 240 260 280 t
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 2 2 2 d p 1 1 dδ dp 2m dω Ω dδ k Δ m 2 2 2 p V Ω= ω m r Critical Layer Σ = + Ω Σ=0 critical layer 0
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
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
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
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
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 2 2 2 2 2 = + + + + + 2 2 2 2 2 ( ) ( + r ) Iteration Method 2 2 = 0 +ε L 1 1 = + + + r r r r θ s 2 2 2 2 2 0 ε 2 2 2 2 L sinθ cosθ 1 = + + ε 1 1+ εrsinθ r r( 1+ εrsinθ) θ ( 1+ εr sinθ) s 2 2 2 [ U] P = R ε LP 2 ( n+ 1) ( n) 0
Initial Conditions u 01 : Pair of Kelvin waves Case 1: (m,m+1)=(1,2), Ns=7, ε=0.1699 Case 2: (m,m+1)=(2,3), Ns=6, ε=0.2506 Number of Points: 41 32 16 Dispersion Curves ω 0 4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 k 0
Evolution of Energies of Unstable Modes (Cases 1 & 2) 10-3 10-4 m=1: (1,1) m=1: (2,1) m=2: (2,1) m=2: (3,1) theory 10-5 10-6 0 10 20 30 40 50 60 t Growth Rates: Case (m) Theory DNS Case1 (m=1) 0.0667 0.0597 Case2 (m=2) 0.1363 0.1181
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 ~ 6 10-6 10-7 (4,1) (1,1) (2,1) (3,1) 10-8 0 10 20 30 40 50 60 t
Evolution of Vorticity Fields (Case 2) vorticity fields in s=0 t=0 t=16 t=32 t=48 Total Flow Disturbance
Evolution of Disturbance (Case 2) Disturbance fields in s=0 t=0 t=16 t=32 t=48 Pressure Vorticity
Structure of Unstble Modes (Case 2) Iso-surface of Disturbance in s=const., t=32 Pressure Vorticity
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
Unstable Parameter Region Curvature Instability: Parametric Resonance Vortex Ring: ε is quantized Wavenumber k has a bandwidth ε = k/ Ns growth rate 0.12 0.35 0.1 0.3 0.25 0.08 0.2 0.06 0.15 0.04 0.1 0.02 0.05 0 Unstable 0 pairs 0.1 0.2 exist 0.3 for 0.4 all 0.5 ε ε Theory Theory+DNS Divide by ε
Weakly Nonlinear Analysis Expansion u ε 0 1 11 2 + εαu + εα u 12 01 2 3 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
Important modes U 0 :( 0, 0,0 ) U 1 :( 1, 0, 0) cc.. u :( m, 1, ) ( m+ 1, 1, 1) cc.. 01 1 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
Amplitude Equation ε α 2 da dt db + dt da + dt db dt dc dt dc dt + ( 2 2 2 2 ) = ab + i c A + c B + c A + c B + d C + d C A + ieab B 1 + 1+ + 2+ + 1 2 1+ + 1 + 1 + ( 2 2 2 2 ) = aa + ic A + c B + c A + c B + d C + d C B + iebaa 2 + 3+ + 4+ + 3 4 2+ + 2 + 2 + ( 2 2 2 2 ) = ab + i c A + c B + c A + c B + d C + d C A + iea B B 1 1+ 2+ 1 + 2 + 1+ 1 + 1 + + = aa 2 + ( 2 2 2 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)
Invariants E F 2 2 +, ± A ± B ± C ± = E + E E = K A + K B + K C ( C C ) 2 2 + + = A + A a + ( ) 2 2 + + G = B + B b C + C H ± ( e ) 2 2 2 ± 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
Example Spatial Discretization: Chebyshev collocation Singular Value Decomposition Modes: (m,m+1)=(1,2), σ=0.1962 3.5 3 ω 0 2.5 2 1.5 1 0 0.5 1 1.5 2 2.5 3 3.5 k 0
Evolution of Energy Symmetric IC Asymmetric IC Mode Energy 10 0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 0 200 400 600 800 1000 t E A + E B + E A - E B - Mode Energy 10 0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 0 200 400 600 800 1000 t E A + E B + E A - E B - Asymmetric Initial Conditions Chaotic Behavior Larger Energy
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