The optimal path to turbulence in shear flows

Dan Henningson

Collaborators:

Antonios Monokrousos, Luca Brandt, Alex Bottaro, Andrea Di Vita

Monokrousos et al. PRL 106, 134502, 2011

Outline

• Transition scenarios and threshold amplitudes for subcritical transition- How low amplitude can a disturbance have and still cause

transition to turbulence? Mechanisms?

• Optimal control theory applied to transition optimization- Objective function from thermodynamic considerations

• Results for transition optimization in plane Couette flow- Analysis of non-linear optimal disturbance evolution

• Conclusions

Transition thresholds and basin of attraction • Lundbladh, Kreiss, Henningson JFM 1994

- Transition thresholds in plane Couette flow, incl NL bound• Reddy, Schmid, Bagget, Henningson JFM 1998

- Transition thresholds for streaks and oblique waves in channel flows• Bottin and H. Chaté EPJB 1998

- Statistical analysis of the transition to turbulence in plane Couette flow• Hof, Juel, Mullin PRL 2002

- Scaling of the Turbulence Transition Threshold in a Pipe• Faisst, Eckhardt JFM 2004; Lebovitz NL 2009

- Complex boundary of basin of attraction – varying lifetimes• Viswanath & Cvitanovic JFM 2009

- Low amplitude disturbances evolving into lower branch travelling waves

• Duguet, Brandt, Larsson PRE 2010- Optimal perturbations combination of linear optimal modes

• Pringle, Kerswell PRL 2011- Non-linear optimal disturbance (optimization not including transition)

Shear flow transition scenarios, BL example

Simulations performed byPhilipp Schlatter

Non-modal instability

Subcritical bypass transition

Low disturbance levels

High disturbance levels

Modal instability

Classical supercritical transition

Consider small periodic box as model problem

Bypass transition: 2 main scenarios

Streak breakdown Oblique transition

oblique modeinduced streakfundamental mode

streak/vortexfundamental mode

Streak breakdown in shear flows

Lundbladh, Kreiss & Henningson JFM 1994

Oblique transition in shear flows

streaks are triggered by a pair of oblique waves

Schmid & Henningson PF 1992

Transition thresholds in Poiseuille and Couette flows

Localized oblique transition in channel

• Inital disturbance with energy around pair of oblique waves (1,1)

• Non-linear interaction forces energy around (0,2), (2,2), (2,0)

• Majority of growth in the (0,2) components

• Streaky disturbance in quadratic part

Linear part

Quadratic part

t = 15 Henningson, Lundbladh &

Johansson JFM 1993

Growth mechanisms in oblique transition

• Initial disturbance at (1,1) utilizes some transient growth

• Forced solution largest where sensitivity to forcing largest at (0,2)

Sensitivity to forcing

Transient growth

Phase-space view

Dynamical system

Nonlinear optimal perturbation

Edge state

Turbulence

Basin of attraction boundary

Laminar fixed point

Non-linear optimal disturbances

• Searching for the optimal path to transition- Initial disturbances with minimum energy

• Objective function: time average including turbulent flow- Disturbance kinetic energy- Viscous dissipation

• Flow: Plane Couette

Objective function from Malkus principle

• Malkus 1956- Outline of a theory of turbulent shear flow

Malkus heuristic principle: A viscous turbulent incompressible Channel flow in

statistically steady state maximizes viscous dissipation

• Glansdorff, Prigogine 1964- On a general evolution criterion in macroscopic physics

Di Vita (2010) derived a general criterion for stability in several diverse physical systems far from equilibrium in a statistically steady state, used by to show Malkus principle

Optimization using a Lagrange multiplier technique

– Lagrange Function:

• Find extrema of functional under specific constrains

Constraint

Optimal initial conditionLooking for the initial condition that maximizes the time integral of viscous dissipation

Governing equations and objective function

Lagrange functional

• and : Lagrange multipliers

• : very small initial amplitude as close as possible to the laminar – turbulent boundary

– Variations of the Lagrange function with respect to each variable

– Set each term to zero independently

• Standard non-linear Navier-Stokes

• Adjoint Navier-Stokes (retrieved using integration by parts)

• Normalization condition

Optimal initial condition

– Integration by parts give

• Spatial boundary terms– We choose boundary conditions for the adjoint

system so that all the terms cancel out, implying same periodic and Dirichlet BC as forward problem.

• Temporal boundary terms give the initial conditions for the adjoint and forward problem

Optimal initial condition-Boundary terms

Power iteration algorithm

Choose u*(T)=u(T)

Update u(0) with u*(0) and normalize

u(0) is the answer!

Start with random IC, u(0)

DNS

Adj DNS

NoYes

Store u(t)

Numerical Code– Fully-Spectral numerical code

• Fourier series in the wall-parallel directions• Chebyshev polynomials

– MPI parallelization with capabilities more than 104

processors• Open-MP support for smaller scale simulations

– Capabilities:• Couette, Plane channel, boundary layers with and

without acceleration, sweep, etc.• Suitable for both fully turbulent flows as well as a

accurate stability analysis of laminar flows• DNS & LES

Numerical Simulations– Fully turbulent field converged

– Computational challenges• Storing of the full 3D, time dependant solution of the forward

problem used as a base flow for the adjoint• O(102-103) Direct numerical simulations for one optimal initial

condition (expensive)

Box size:

Resolution:

X Y Z

Re: 1500

ConvergenceFind minimum amplitude with

power iterations – relaxed with previews iterates: “averaged optimal”

Example of convergence

Optimizing for the amplitude

The red star is the optimal!

The blue squares correspond to optimisation around the laminar flow (Pringle & Kerswell)

• Start with high optimization amplitude run until convergence

• Compute transition threshold for optimized disturbance using bisection algorithm lowers amplitude (green circles)

• Reduce amplitude and repeat until flow always re-laminarizes.

• Lowest amplitude where transition occur is optimal initial condition (red star)

• Fastest path to transition is the optimal path for lowest initial amplitude

• Transition thresholds for lower optimization amplitudes are higher than optimal initial condition (blue squares)

Objective function vs Optimization amplitude

– Green circles: Turbulent flow, Blue squares: Laminar flow

– The objective is maximized for each amplitude separately

– For constant optimization time flows with higher initial amplitudes spend longer time in turbulent state since transition is faster, thus larger value of objective function

Optimal initial condition localized

• Total initial energy of disturbance constant during optimization

• Local amplitude can be higher for same total energy if initial condition is localized

• Transition caused by large local non-linear interactions

Optimal path to turbulence: different Reynolds numbers

• Convergence at lower Re more difficult– longer time to transition– timescale larger for reaching statistically steady state

• Convergence at larger Re more difficult– higher resolution required

• Optimal path close to edge trajectories– steady for lower Re– chaotic for higher Re

Optimal path to turbulence

Initial condition Vortex pair

Streak Turbulence

Initial condition -> Vortex pair

Orr mechanism: backward tilting structures lean against shear

Similar to Orr mechanism generating 2D wavepacket

2D optimal disturbance: Initial backward leaning structures amplifies when tilted forward by the shear

Vortex pair-> Streak

Oblique waves non-linearly force streaks which grow due to lift-up effect

Streak-> Turbulence

Secondary instability of streak causes flow to break down to turbulence

Comparison of the threshold values

– Reddy, et al 1998 Monokrousos et all 2011

– The numbers correspond to energy density of the initial disturbance

– Significant reduction O(10) from the values relative to previous studies

– Combination of several mechanisms to gain more energy (Orr, oblique forcing, lift-up, ...)

(Re=1500)

Same growth mechanism in pipe flow

Pringle, Willis, Kerswell (2011) arxiv.org/pdf/1109.2459v1

Orr-mechanism

Localized/oblique

Lift-up

Conclusions

– Non-linear optimization of turbulent flow using adjoints

– Average viscous dissipation better choice than disturbance energy as objective function

– Transition threshold reduced relative to previous studies

– Fully localized optimal initial condition

– Disturbance evolution utilizes combination of several growth mechanisms efficiently triggering turbulence (Orr, oblique, lift-up)

– Scenario general, also present in pipe flow

Thank you!

A few quantities from the DNS

Streak breakdown and oblique transition in channel flows

Threshold for streak breakdown in Couette flow

Nonlinear optimals and transitionLinear optimals and weakly nonlinear approaches:

vortices and streaks

Suboptimal perturbations: oblique scenario (Viswanath & Cvitanovic 2010, Duguet et al. 2010)

Nonlinear optimization: localized disturbances (Pringle & Kerswell, Cherubini et al.,)

Plane Couette flow:different box size and Re

• State-space formulation– Define pressure through Poisson

– Norm:

– Define the adjoint operator:

• Lagrange Function:– Find extrema of functional

Basic Formulation-Technique

Optimal initial conditionLooking for the initial condition that maximizes the time integral of viscous dissipation

• Governing equations and objective function

Lagrange functional

• Lagrange multipliers: and

• Variations of the Lagrange functionDNS of NS

DNS of Adjoint NS

Set the IC amplitude

Optimizing for the amplitude

• Reducing the initial energy until turbulence can not be achived

• The red star is the optimal!

• “Stochastic” objective function & initial condition

Phase-space view

Associated dynamical system

Associated metricsNonlinear optimal

perturbation

Edge state

Turbulence

Non-linear optimals and Transition• Optimization

- Power iterations & Conjugate gradient- Time stepper

• Different approaches- Linear optimals- Weakly non-linear (extension of the linear problem)- Fully non-linear (Turbulence)

• Flow: Plane Couette