Probing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods

Transcription

1 Probing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods Bartosz Protas and Diego Ayala Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL: Funded by Early Researcher Award (ERA) and NSERC Computational Time Provided by SHARCNET Topological Fluid Dynamics, Cambridge July 27, 2012

2 Collaborators Charles Doering (University of Michigan) Dmitry Pelinovsky (McMaster University)

3 Agenda Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Optimization Formulation & Numerical Solution Results Summary & Outlook

4 Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem Navier Stokes equation (Ω = [0, L] d, d = 2, 3) v + (v )v + p ν v = 0, in Ω (0, T ] t v = 0, in Ω (0, T ] v = v 0 in Ω at t = 0 Boundary Condition on Γ (0, T ] 1D Case (Burgers) & 2D Case Existence Theory Complete smooth and unique solutions exist for arbitrary times and arbitrarily large data 3D Case Weak solutions (possibly nonsmooth) exist for arbitrary times Classical (smooth) solutions (possibly nonsmooth) exist for finite times only Possibility of blow up ( singularity formation) One of the Clay Institute Millennium Problems ($ 1M!) Equations

5 Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem What is known? Available Estimates A Key Quantity Enstrophy E(t) v 2 dω (= v 2 2) Ω Smoothness of Solutions = Bounded Enstrophy E(t) (Foias & Temam, 1989) NaN? max E(t) <??? t [0,T ] E(0) t 0 t

6 Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem Can estimate using the momentum equation, Sobolev s embeddings, Young and Cauchy Schwartz inequalities,... 27C 2 128ν 3 E(t)3 Remark: incompressibility not used in these estimates... upper bound on E(t) blows up in finite time E(t) E(0) 1 4 CE(0)2 t ν 3 singularity in finite time cannot be ruled out!

7 Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem Burgers equation (Ω = [0, 1], u : R + Ω R ) u t + u u x ν 2 u x 2 = 0 in Ω u(x) = φ(x) at t = 0 Periodic B.C. Enstrophy : E(t) = u x(x, t) 2 dx Questions of sharpness of enstrophy estimates still relevant 3 ( ) 1 1/3 2 π 2 E(t) 5/3 ν Best available finite-time estimate (Doering, 2010) [ max E(t) t [0,T ] E 1/3 0 + ( ) L 2 ( ) ] 1 4/3 3 4 π 2 E 0 ν C 2E0 3 E 0

8 Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem 2D vorticity equation in a periodic box (ω = e z ω) ω + J(ω, ψ) = ν ω t where ψ = ω J(f, g) = f xg y f y g x Enstrophy uninteresting in 2D flows (w/o boundaries) 1 d ω 2 dω = ν ( ω) 2 dω < 0 2 Ω Evolution equation for the vorticity gradient ω ω t Palinstrophy P(t) Ω + (u ) ω = ν ω + ω u }{{} stretching term Ω ( ω(t, x)) 2 dω = Ω ( ψ(t, x)) 2 dω

9 Regularity of Navier Stokes Flows & Growth of Enstrophy Bounds on Enstrophy Growth in 1D Burgers Problem Bounds on Palinstrophy Growth in 2D Navier Stokes Problem Estimates for the Rate of Growth of Palinstrophy dp(t) = J( ψ, ψ) 2 ψ dω ν ( 2 ψ) 2 dω R ν (ψ) dp(t) dp(t) Ω C 1 E P (Doering & Lunasin, 2011) ν C 2 ν K1/2 P 3/2 (Ayala, 2012) Using Poincaré s inequality (may not be sharp) dp(t) Bound on growth in finite time Ω C ν P2, max P(t) P(0) + C 1 L 4 t>0 2ν 2 16π 4 P(0)2 (Doering & Lunasin, 2011)

10 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Relevant Estimates 1D Burgers 1D Burgers 2D Navier Stokes 2D Navier Stokes 3D Navier Stokes 3D Navier Stokes Best Estimate ( 3 1 ) 1/3 E(t) 5/3 2 π 2 ν [ max t [0,T ] E(t) E 1/3 0 + ( ) L 2 ( 1 ) ] 4/3 3 E 4 π 2 ν 0 dp(t) C ν P2 max t>0 P(t) P(0) + C 1 2ν 2 L 4 16π 4 P(0) 2 27C 2 128ν 3 E(t) 3 E(t) E(0) 1 4 CE(0)2 ν 3 t Sharp? May not be sharp due to Poincaré s inequality

11 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Question #1 ( small ) Sharpness of estimates (at some fixed t) max u max u Question #2 ( big ) de dp (1D, 3D) (2D) Sharpness of estimates (at some time window [0, T ], T > 0) [ ] max max E(t) (1D, 3D) u 0 t [0,T ] [ ] max max P(t) (2D) u 0 t [0,T ]

12 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Relevant Estimates 1D Burgers 1D Burgers 2D Navier Stokes 2D Navier Stokes 3D Navier Stokes 3D Navier Stokes Best Estimate Sharp? ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (2008) [ max t [0,T ] E(t) E 1/3 0 + ( ) L 2 ( 1 ) ] 4/3 3 E 4 π 2 ν 0 dp(t) C ν P2 max t>0 P(t) P(0) + C 1 2ν 2 L 4 16π 4 P(0) 2 27C 2 128ν 3 E(t) 3 E(t) E(0) 1 4 CE(0)2 ν 3 t May not be sharp due to Poincaré s inequality

13 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Finite Time Optimization Problem in 1D (I) Ayala & Protas (2011) Statement max E(T ) φ H 1 (Ω) subject to E(t) = E 0 T, E 0 parameters; φ initial data Optimality Condition φ H 1 1 J λ(φ; φ 2 u t=t ) = u 1 2 φ t=0 0 x 2 t=t dx λ u 0 x 2 t=0 dx

14 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Finite Time Optimization Problem in 1D (II) Gradient Descent φ (n+1) = φ (n) τ (n) J (φ (n) ), n = 1,..., φ (0) = φ 0, where J determined from adjoint system via H 1 Sobolev preconditioning u t u u x ν 2 u x 2 = 0 in Ω u (x) = 2 u (x) at t = T x 2 Periodic B.C. Step size τ (n) found via arc minimization n d n n+1 = { x 2 = E 0}

15 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Two parameters: T, E 0 (ν = 10 3 ) φ * (x) T E max x Fixed E 0 = 10 3, different T E 0 max t [0,T ] E(t) versus E 0 max E(t) CE1.5 0 t [0,T ]

16 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Relevant Estimates 1D Burgers 1D Burgers 2D Navier Stokes 2D Navier Stokes 3D Navier Stokes 3D Navier Stokes Best Estimate Sharp? ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (2008) [ max t [0,T ] E(t) E 1/3 0 + ( ) L 2 ( 1 ) ] 4/3 3 No E 4 π 2 ν 0 Ayala & P. (2011) dp(t) C ν P2 max t>0 P(t) P(0) + C 1 2ν 2 L 4 16π 4 P(0) 2 27C 2 128ν 3 E(t) 3 E(t) E(0) 1 4 CE(0)2 ν 3 t May not be sharp due to Poincaré s inequality

17 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Lu & Doering (2008) [ ] = 8.97 max 10 4 E vorticity field (top branch)

18 Instantaneous Enstrophy Growth in 1D Burgers Problem Finite Time Enstrophy Growth in 1D Burgers Problem Instantaneous Enstrophy Growth in 3D Navier Stokes Problem Relevant Estimates 1D Burgers 1D Burgers 2D Navier Stokes 2D Navier Stokes 3D Navier Stokes 3D Navier Stokes Best Estimate Sharp? ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (2008) [ max t [0,T ] E(t) E 1/3 0 + ( ) L 2 ( 1 ) ] 4/3 3 No E 4 π 2 ν 0 Ayala & P. (2011) dp(t) C ν P2 max t>0 P(t) P(0) + C 1 2ν 2 L 4 16π 4 P(0) 2 27C 2 128ν 3 E(t) 3 E(t) E(0) 1 4 CE(0)2 ν 3 t Yes Lu & Doering (2008)

19 Optimization Formulation & Numerical Solution Results Summary & Outlook dp(t) Original Estimate: C 1 ν E P Palinstrophy and Enstrophy constraints hard to satisfy exactly require projection on intersection of two manifolds in H 4 (Ω) P 0 constraint + Poincaré s inequality = Upper bound on E 0 ( ψ) 2 dω = P 0 Ω = ( ψ) 2 dω C P 0 φ 2 dω C ( φ) 2 dω Ω Ω Ω Simpler maximization problem (one constraint) max R ν (ψ) ψ H 4 (Ω) subject to: Ω ( ψ) 2 dω = P 0

20 Optimization Formulation & Numerical Solution Results Summary & Outlook Discretization of Gradient Flow dψ dτ = H4 R ν (ψ), ψ(0) = ψ 0 ψ (n+1) = ψ (n) τ (n) H4 R ν (ψ (n) ), ψ (0) = ψ 0 Gradient in H 4 (Ω) (via variational techniques) [ Id L 8 4] H4 R ν = L 2 R ν (Periodic BCs) L 2 R ν (ψ) = 2 J( ψ, ψ) + J(ψ, 2 ψ) + J( 2 ψ, ψ) 2ν 4 ψ Constraint satisfaction via arc minimization

21 Results: small P Optimization Formulation & Numerical Solution Results Summary & Outlook 1.0 max dp Upper Branch P 0 Lower Branch

22 Results: large P 0 Analytical estimate: dp(t) C ν P2 max dp Optimization Formulation & Numerical Solution Results Summary & Outlook Upper Branch 10 3 max dp C P0 1.57± P0 Lower Branch max dp C P0 1.5±0.02

23 y Optimization Formulation & Numerical Solution Results Summary & Outlook Upper Branch Lower Branch x ω P0 (x) P α 0 ω (x/p β 0 ) + ω b(x) ω P0 (x) P α 0 ω b (x) Euler Lagrange equation [inviscid limit: ν = 0 (P 0 ) ] 2 J( ψ, ψ) + J(ψ, 2 ψ) + J( 2 ψ, ψ) 2ν 4 ψ = 0

24 Optimization Formulation & Numerical Solution Results Summary & Outlook Relevant Estimates 1D Burgers 1D Burgers 2D Navier Stokes 2D Navier Stokes 3D Navier Stokes 3D Navier Stokes Best Estimate Sharp? ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (2008) [ max t [0,T ] E(t) E 1/3 0 + ( ) L 2 ( 1 ) ] 4/3 3 No E 4 π 2 ν 0 Ayala & P. (2011) dp(t) C ν P2 No (inconclusive) P. & Ayala (2012) max t>0 P(t) P(0) + C 1 2ν 2 L 4 16π 4 P(0) 2? 27C 2 128ν 3 E(t) 3 E(t) E(0) 1 4 CE(0)2 ν 3 t Yes Lu & Doering (2008)??? May not be sharp due to Poincaré s inequality

25 Optimization Formulation & Numerical Solution Results Summary & Outlook References L. Lu and C. R. Doering, Limits on Enstrophy Growth for Solutions of the Three dimensional Navier Stokes Equations Indiana University Mathematics Journal 57, , (2008). D. Ayala and B. Protas, On Maximum Enstrophy Growth in a Hydrodynamic System, Physica D 240, , (2011). B. Protas and D. Ayala, Maximum Palinstrophy Growth in 2D Incompressible Flows: Part I Instantaneous Case, in preparation, (2012).