On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data

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1 On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data Monika Nitsche Department of Mathematics and Statistics University of New Mexico Collaborators: Darryl Holm Robert Krasny Jack Moffitt Vacktang Putkaradze Mark Taylor Guangdong Zhu This work is supported by the National Science Foundation.

2 Outline I. Background II. Vortex Blob Regularization III. Euler-α Regularization IV. Regularization by fluid viscosity V. Scaling behaviour VI. Summary 1

3 I. BACKGROUND Shear Layer y Vortex Sheet Model y u u Shear layer: thin transition region between fluids of distinct velocity, vorticity ω = v x u y is large in layer, small outside Vortex sheet: inviscid model, asymptotic approximation of outer flow, vorticity = δ function on surface, zero outside (irrotational) 2

4 Governing Equations and Discretization (2D) Γ j x( Γ,t), y( Γ,t) x (t), y (t) j j Incompressible Euler Equations: Circulation Γ = Lagrangian. Velocity given by: dx dt = 1 2π pv y ỹ (x x) 2 + (y ỹ) 2d Γ dy dt = 1 2π pv x x (x x) 2 + (y ỹ) 2d Γ Point vortex approximation (PVA): discretize by x j (t), y j (t), corresponding to circulations Γ j, dx j dt = 1 2π N k=1 k j y i y k (x j x k ) 2 + (y j y k ) 2 Γ k dy j dt = 1 2π N k=1 k j x j x k (x j x k ) 2 + (y j y k ) 2 Γ k 3

5 Kelvin-Helmholtz Instability Linear stability analysis: Consider a small perturbation of the flat sheet x(γ, t) = Γ + p(γ, t), y(γ, t) = q(γ, t) where p = Pe ωt e ikγ, q = Qe ωt e ikγ. Substitute into linearized system dispersion relation w = ±k ω ω Vortex sheet k Shear Layer k a growing and a decaying mode. The higher the wavenumber, the faster the growthrate. illposed in the sense of Hadamard. By consistency, PVA is illposed as well. 4

6 Problem 1: Exponential growth of high wavenumbers introduced by roundoff (Krasny 1986) Krasny filter Problem 2: Singularity forms in finite time, approximation does not converge past this time (Moore 1979, Meiron, Baker, Orszag 1982, Krasny 1986, Shelley 1992, Caflisch, Ercolani, Hou, Landis 1993, Cowley, Baker, Tanveer 1999) Regularization Vortex blob regularization (Rosenhead 1933): Introduce a numerical parameter δ dx dt = 1 y ỹ 2π (x x) 2 + (y ỹ) 2 + δ 2d Γ Discretization is well-posed dy dt = 1 2π x x (x x) 2 + (y ỹ) 2 + δ 2d Γ 5

7 Alternative Regularizations 1. Vortex blob regularization 2. Euler-alpha regularization 3. Fluid viscosity 4. Finite thickness constant vorticity layer 5. Surface tension Questions: Do the regularized solutions approximate the real fluid flow? Does the limiting solution of vanishing regularization solve the Euler Equations? (DiPerna & Majda 1987ab, Delort 1991,1992, Liu & Xin 1995: weak solution) Do the limits of different regularizations agree, for vortex sheet initial data? (Baker & Pham 2006, Beale & Majda 1985) 6

8 II. VORTEX BLOB REGULARIZATION 1. Vortex ring formation (Nitsche & Krasny, JFM 276, 1994) We compared results using vortex sheet model to experiments by Didden (1979) 7

9 8

10 2. Roll-up of initially flat sheets (Krasny & Nitsche, JFM 454, 2002) (a) (b) Initial condition: potential flow past a flat plate (planar) and disk (axisymmetric) 9

11 Evolution, computed with δ = 0.2 Sheet moves in direction of initial impulse as it rolls up at its edges, forming a vortex pair (planar) and vortex ring (axisymmetric) 10

12 11

13 Streamlines and vorticity at quasi-steady state particle position streamlines regularized vorticity Vorticity is maximal at center, streamlines similar to pair of point vortices 12

14 Periodically perturbed flow (Rom-Kedar, Leonard, Wiggins 1992) elliptic point heteroclinic orbit hyperbolic point heteroclinic tangle KAM curve resonance band Rotation number: T(I)f Frequency of perturbation : f T(I): time for particle on streamline to one complete circuit Sufficiently irrational rotation number: closed orbits remain closed under perturbation (KAM Theorem) Rational rotation number: closed orbits break up into sequence of hyperbolic and elliptic points, forming a resonance band (eg, see Arnold & Avez 1968) 13

15 Oscillations of core vorticity contours 14

16 Oscillation frequency 15

17 Poincare maps 16

18 Questions Are chaotic dynamics present with other regularizations? With viscosity? What is the limit as regularization vanishes? 17

19 III. EULER-ALPHA REGULARIZATION (Holm, Nitsche and Putkaradze, JFM 555, 2006) Originally proposed to model turbulence. Obtained by averaging the Euler equation along Lagrangian trajectories. General Framework (Holm, Marsden & Ratiu 1998, Foias, Holm & Titi 2001 ): v: a possibly singular velocity field u: smoothed velocity, u = L(v) = h v, L commutes with and 2 Euler-Poincare equation: t v u ( v) Π = 0 (preserves Hamiltonian structure of Euler Equations, satisfies Circulation Theorem) Take curl: t q + u q = q u where q = v is the singular vorticity (Note: blob method) Note: regularized vorticity ω = u satisfies ω = L(q) = h q = G q Hence h = G is the regularized vorticity associated to a delta source q. The regularization is determined by specifying either L = h, L 1, or G where u = (G q) 18

20 Three-dimensional flow Euler-alpha regularization : The G, h corresponding to L 1 = 1 α 2 are G α (r) = 1 e r/α 4πr Vortex blob regularization :, h α (r) = e r/α 4πα 2 r h α r 35 Given by Rosenhead s kernel G δ (r) = 1 4π r 2 + δ 2, h δ (r) = 3δ 2 4π(r 2 + δ 2 ) 5/2 h δ r Notice that the regularized vorticity h associated to a point source q is unbounded in the Euler alpha case but bounded in the blob. Euler-alpha closer to unregularized case (blob more regularizing). 19

21 Two-dimensional flow Euler-alpha regularization : h α (r) = 1 r ) o( 2πα 2K α Vortex blob regularization : h δ (r) = δ 2 π(r 2 + δ 2 ) 2 Axi-symmetric flow Euler-alpha regularization : ω δ (x, y; x 0, y 0 ) = y 0 4πα 2 2π 0 e ρ/α ρ cos θ dθ Vortex blob regularization : ω δ (x, y; x 0, y 0 ) = 3y oδ 2 4π 2π 0 cos θ (ρ 2 + δ 2 ) 5/2 dθ 20

22 Linear Stability Consider a small perturbation of the flat sheet x(γ, t) = Γ + p(γ, t), y(γ, t) = q(γ, t) where p = Pe ωt e ikγ, q = Qe ωt e ikγ. Substitute into linearized equations, get w 2 α = 1 α 2I 1(αk)I 2 (αk) k 1 w 2 δ = ke kδ (1 e kδ ) 4δ ke kδ 0 where I 1 (k) = 1 π I 2 (k) = I 1 1 π 1 uk 1 (u) (1 cos ku)du u 2 0 K o(u)(1 cos ku)du 21

23 Linear growth rates w(k) Euler-alpha (solid), Blob (dashed), Unregularized (dotted) 22

24 Comparing Blob and Euler-alpha, α = δ = 0.2 Top: vortex blob, Bottom: Euler-alpha 23

25 Closeup at t =

26 Evolution of axisymmetric sheet (Top: vortex blob, Bottom: Euler-alpha) 25

27 Closeup at t = 60 Core vorticity oscillations and chaotic dynamics present in both cases, but different frequencies 26

28 IV. REGULARIZATION BY FLUID VISCOSITY (with M.Taylor and J.Moffitt) Fundamentally different: non-hamiltonian regularization Approach: 1. Regularize initial vortex sheet by vortex blob operator with δ > 0 2. Evolve smooth vorticity by Navier-Stokes Equations with viscosity ω t + u ω = µ ω, Ψ = ω, u = Ψ, ω(x,0) = ω 0 (x) using 4th order finite differences. 3. Investigate the limit δ, ν 0. (Note: For fixed δ, the limit ν 0 converges to the Euler solution.) Questions: Does Navier-Stokes solution approximate the vortex blob solution? Do the limiting solutions as δ, α, ν 0 agree? Are the irregular dynamics present with δ, ν sufficiently small? 27

29 Initial conditions, using δ = 0.4, 0.2, 0.1 Vorticity contours ω = 2 j, j = 7,...,5 28

30 Comparison : vortex blob (top) and viscous (bottom) Position of particles initially on plate, computed with δ = 0.2, ν =

31 Comparison at t = 10: Particle position ν = 10 3 ν = 10 4 ν = 10 5 vortex blob δ = 0.2 δ = 0.1 δ =

32 Closeup at t = 120: vortex blob, δ = 0.2, viscous, δ = 0.2, ν =

33 Oscillations in core vorticity, viscous, δ = ν=10 4 r ν=10 5 r ν=10 6 r ν=10 7 r t As δ 0, ν 0, oscillations persist for increasingly longer times. 32

34 V. SCALING BEHAVIOUR AS δ, α, ν 0 Semi-infinite sheet Consider evolution of a semi-infinite initially flat sheet dx dt = 1 2π dy dt = 1 2π 0 0 y ỹ (x x) 2 + (y ỹ) 2 + δ 2 d Γ, x(γ,0) = 0 x x (x x) 2 + (y ỹ) 2 + δ d Γ, y(γ,0) = Γ2 2 2 Unregularized case δ = 0: self-similar solution (Pullin 1978) equations invariant under rescaling of time and space Regularized case δ > 0: not self-similar, however for any constant T, solution does remain invariant under the change of variables x = x T 2/3, ŷ = y T 2/3, Γ = Γ T 1/3, t = t T and δ = δ T 2/3 That is, solution for any (t, δ) is same as with ( t, δ) up to scale Example: (t = 1, δ =.009) same as ( t = 1/10, δ = (0.009)10 2/3 ) 33

35 Solution at t = 1, with δ =

36 Solution at t = 0.1, with δ = 0.009/10 2/ It follows that for semi-infinite sheet timescales scale as δ 3/2 and frequencies f δ 3/2 35

37 Oscillation frequency for finite sheet δ 3/ blob frequency f 1 computations f δ 3/ ν=10 4 ν=10 5, δ δ Frequency of the elliptical component of the ω = ω max /2 contour. In the viscous case, frequency increases slightly as center is approached. 36

38 Rotation numbers The time T for particles to move around the vortex center once converges as δ 0 Since f as δ 0, it follows that the average radius of resonance bands of given rotation number T f must shrink. time T δ=0.4 δ=0.2 δ=0.1 δ=0.07 rotation number Tf r r 37

39 Resonance bands of given rotation number (with Guangdong Zhu) So what is the limit? 38

40 Emerging Picture Resonance of all rational rotation number must appear, by theory. We see the low p : q first. Each resonance bands average radius 0 as δ 0 Qualitatively, this behaviour should be same for all regularization. Caveat: effect of frequency variations in viscous case? 39

Topics in Fluid Dynamics: Classical physics and recent mathematics

Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog: