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 mode
induced streak
fundamental mode
streak/vortex
fundamental 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 condition
Looking 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
No
Yes
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
Convergence
Find 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 function
DNS 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