AIM Seminar U of Michigan

Sep 7 2007

Turbulence? a stroll through 61,506 dimensions Predrag Cvitanovic’ John F Gibson School of Physics Georgia Tech, Atlanta GA, USA

AIM Seminar - U. Michigan Sept 2007

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ChaosBook.org

New experiments: Unstable Coherent Structures Stereoscopic Particle Image Velocimetry ! 3-d velocity field over the entire pipe1

Observed structures resemble numerically computed traveling waves

What lies beyond? 1

Casimir W.H. van Doorne (PhD thesis, Delft 2004); Hof et al., Science (Sep 10, 2004)

Turbulence: A walk through a repertoire of unstable recurrent patterns? As a turbulent flow evolves, every so often we catch a glimpse of a familiar pattern:

=)

other swirls

=)

For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns. The long term dynamics = a walk through the space of such unstable patterns.

Plane Couette flow: large aspect ratio

Navier-Stokes: BCs:

1 2 ∂u + u · ∇u = −∇p + ∇ u, ∂t Re no-slip at walls y = ±1,

Re = 400 Hamilton, Kim, Waleffe (1995) J. Fluid Mech., 287 Waleffe (1997) Phys. Fluids, 9

∇·u=0

periodic in x and z.

.

The devil is in the details

Computational methods for equilibrium solutions • Spectral expansion of u incorporates BCs and ∇ · u = 0  ˆ jk T (y) ei(jαx+kγz) u u(x) = wavelengths Lx , Lz = 2π/α, 2π/γ j,k,

• 48 × 49 × 48 grid =⇒ 105 dimensions • Solve finite-time equilibrium equation f t (u) − u = 0 • Use trust-region Newton algorithm with Krylov-subspace solver (GMRES) • Evaluate f with CFD and linearize with finite-differencing Df t (u)Δu ≈ f t (u + Δu) − f t (u) • Initial guesses from numerical simulation data • Arnoldi iteration for eigenvalues, eigenfunctions

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Turbulent flows cannot be modeled by a few modes Attractor is "low dimensional," but has to be tracked in the full 103 to 105 dimensions

CFD: Geometry and spectral convergence Re Lx Ly Lz ‘‘Minimal’’ PCF ‰ 400 ‰ 2ı 2 ‰ ı Hamilton, Kim, Waleffe (HWK) 400 7ı=4 2 6ı=5 sustained turbulence fourier spectrum 15

chebyshev spectrum

0

10

10

kx

5

−2

10

0 −4

10

−5 −10

−6

10 0

5

10

15

kz −6

−4

−8

10 −2

0

0

10

20

30

ky

Adequate resolution: 32 ˆ 33 ˆ 32 to 48 ˆ 49 ˆ 48 grids

www.channelflow.org

Example channelflow C++ code Integrate initial condition, project onto orthogonal basis set FlowField u("u0"); // read velocity field from disk array e("ef"); // read eigenfunctions from disk for (int m=0; m