Stokes preconditioning for

Channelflow and Openpipeflow

Laurette TuckermanJake LanghamAshley Willis

Computation of steady state in plane Couette flow using Channelflow (Gibson)

Fourier in x,z / Chebyshev in y

Computation of traveling waves in pipe flow using Openpipeflow (Willis)

Fourier in θ, z / finite differences in r

Methods used in hydrodynamics for computing steady states and traveling waves

Newton with full direct Jacobian inversion: from 1980s: Keller, Busse & Clever, Nagata, Generalis, Kerswell, Eckhardt scales like M3, so limited to small resolutions

non-Newton or variants of Newton: Newton-Picard with subspace iteration: Lust & Roose Selective frequency damping: Akervik Recursive projection method: Keller & Shroff

Newton with matrix-free Jacobian inversion via Stokes preconditioning: from 1989: Tuckerman, Henry/Bergeon/Beaume, Mercader, Mellibovsky

Newton with matrix-free Jacobian inversion via integration: from 2004: Sanchez-Umbria, Viswanath, Duguet, Gibson, Willis

Other computations using Stokes preconditioning

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Rayleigh-Bénard convection in a cylinder of aspect ratio with radius / height =2 Pseudospectral computation with (Mr,Mθ,Mz)=(60,130,30), leading to 500 000 degrees of freedom. From Boronska & Tuckerman Phys Rev E (2010).

3D binary fluid convection in a porous medium in cavity of size 6 x 6 x 1. Spectral element computation with 6 elements per quarter each with (23,23,17) points. From LoJacono, Bergeon & Knobloch FDR (2017).

Other computations using Stokes preconditioning

1800 1900 2000 21000

0.02

0.04

0.06

Rayleigh number

Nu −

1

SOC15

2PH15

2PC15

θ

R=2111 (2PH15)

Cθ

R=1955 (2PH15)

Cθ

R=1938 (2PC15)

C

Nuss

elt -

1

Other computations using Stokes preconditioning

Binary fluid convection in a domain with aspect ratio width / height = 14. Spectral computation. From Mercader, Batiste, Alonso & Knobloch JFM (2011).

Other computations using Stokes preconditioning

Steady states of a streamwise-independent reduced model for plane Couette flow.at Re ≈ 1000. Spectral element computation. From Beaume, Chini, Julien & Knobloch PRE (2015).

Ulam= y ex Ulam= (1-y2) exUlam= (1-r2) ez

For a long time, the only regular solutions known were the analytic laminar ones. First new solution found numerically for plane Couette flow by Nagata (1990). Then many many many many other solutions were found. Is turbulence a chaotic attractor with these solutions forming a skeleton?

Gibson, Halcrow, Cvitanovic, J. Fluid Mech. (2008)

Experimental Observation of Nonlinear Traveling Waves in Turbulent Pipe FlowHof, van Doorne, Westerweel, Nieuwstadt, Faisst, Eckhardt, Wedin, Kerswell, WaleffeScience (2004)

Experiment: instantaneous snapshots

Numerics: computed traveling wave solutions

∂U∂t

= LU + N(U)

0 = LU + N(U)

Navier-Stokes

Steady states

Steady states OR T-periodic orbits0 = U(T ) − U(0) ≡ (ΦT − I)U(0)

ΦT ≈ (BΔt)T/Δt

In theory ΦT : U(0) → U(T )

In practice

BΔt ≡ (I − ΔtL)−1(I + ΔtN)

where a single small timestep is, for example:

≈ Δt(L + N)≈ − L−1(L + N) for Δt large

for Δt small{

BΔt − I = (I − ΔtL)−1(I + ΔtN) − I= (I − ΔtL)−1[(I + ΔtN) − (I − ΔtL)]= (I − ΔtL)−1Δt(L + N) has same roots as L+N

Hence Δt interpolates between no preconditioner and preconditioning by L-1

Find roots via Newton’s method Solve linear systems of Newton’s method via BiCGSTAB

GUu = G(U)U ← U − u

G(U), GU G(U), G2U G(U), … Gk

U G(U), … GK−1U G(U)

Solve linear systems of Newton’s method via BiCGSTAB or GMRES or IDR or … Construct estimate of solution u within Krylov space:

Find roots via Newton’s method

Cost of making K-space should be product of: —number K of Krylov vectors constructed = actions of GU —cost of single action of GU

GStokesT ≡ BT − I

finds roots of

L + N =∂U∂t

GintT ≡ (BΔt)T/Δt − I

ΦT − I = U(T) − U(0)

finds roots of

many timesteps (T/Δt) per action of G

single timestep per action of G

(Gibson) in plane Couette flow Computation of steady state using Channelflow

NCBW: Nagata Clever Busse Waleffe solution

GStokes Gint

factor of 11

Computation of steady state using Channelflow

Total number of timesteps is product of K and (T/Δt) Integration method has lower K and higher (T/Δt) Stokes method has higher K and (T/Δt) = 1

Computation of steady state using Channelflow

Computation of traveling waves in pipe flow using Openpipeflow (Willis)

Fourier in θ, z / finite differences in r

∂U∂t

= − C∂U∂x

0 = LU + N(U) + C∂U∂x

Traveling waves

Additional unknown C Additional phase condition

factor of 35

Computation of traveling waves using Openpipeflow

As Re increases, the performance ratio measured by timesteps remains nearly constant. But the performance ratio measured by CPU time decreases drastically.

Why?

GMRES:

G(U), GU G(U), G2U G(U), … Gk

U G(U), … GK−1U G(U)

Construct estimate of solution u within Krylov space:

Each Krylov vector must be orthogonalized to every other. For K small, this cost is negligible. But not for K large.

where M is the number of gridpoints and 1 < α < 2

As Re increases, K increases. Why?

CPU ≈ cactionTΔt

KM1+α + corthK2M

Operator spectra for Re = 500

λmodelk = − k2/Re ± ikU

λStokesk ≈

λmodelk

k2/Re= − 1 ± iURe/k

( 1Re

∇2 + U∂x) u

Quantitative predictions for the spectra1) Most negative real value of λmodel is proportional to 1/Re

λmodelmost negative(Re = 500)

λmodelmost negative(Re = 1750)

=−360−105

= 3.5 ≈1750500

λmodelk = − k2/Re ± ikU

λStokesk ≈

λmodelk

k2/Re= − 1 ± iURe/k

2) Largest imaginary value of λStokes is proportional to Re

λStokeslargest imaginary(Re = 500)

λmodellargest imaginary(Re = 1750)

=35100

=1

2.9≈

5001750

Operator spectra for Re = 1750

3) Radius r of circle surrounding (-1,0) containing 95% of eigenvalues of Gint

r = e−Tk2* /Re ⟹ ln(1/r) = Tk2

* /Re

ln(1/r1)ln(1/r2)

=7.081.877

= 3.77 ≈1750500

=Re2

Re1

Conditioning of Gint depends on T/Re: can compensate for increasing Re by increasing T

Seek to reduce large imaginary eigenvalues of GStokes

k > k*These eigenvalues correspond to

NUu = − [U ⋅ ∇)u + (u ⋅ ∇)U − Ulam ∂xu + 2vU′�lamex]

U = Ulam(y) ex + U

Lu = ( 1Re

∇2 − Ulam∂x) u

∇ ⋅ (Lu) =1

Re∇2(∇ ⋅ u) − Ulam(y) ∂x(∇ ⋅ u) − U′�lam(y) ∂xv

Lu =1

Re∇2u − Ulam ∂xu + vU′�lamex

Seek to reduce large imaginary eigenvalues of GStokes

λStokesk ≈

λmodelk

k2/Re= − 1 ± iURe/k

Lu =1

Re∇2uimplicit NUu = − [U ⋅ ∇)u + (u ⋅ ∇)U]explicit

NUu = − [Ulam(y)∂xu + vU′�lam(y) ex + (U ⋅ ∇)u + (u ⋅ ∇)U]probably most responsible for bad conditioning AND easiest to make implicit!

BUT

SO ∇ ⋅ u = 0⟹∇ ⋅ Lu = 0

redefine

Bad for pressure projection step

redefine

so

WILL IT WORK????? WE HOPE SO!

Summary

Stokes preconditioning method (Tuckerman, 1989)

Computes steady states or traveling waves

Finds roots of L+N via preconditioned L-1(L+N)

Very easily implemented in time-stepping code with implicit viscous step

Can be up to 50 times faster than integration method

But conditioning worsens with increasing Reynolds number (trying to fix with implicit advective)