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