Information flow and causality of -roll interactions in wall-bounded turbulence Adrián Lozano-Durán Center for Turbulence Research, Stanford University January 9, 2017
Introduction and Motivation Collaborators: - Javier Jiménez (Universidad Politécnica de Madrid) - Gilles Tissot (Institut de Mathématiques de Toulouse) - Laurent Cordier (Institut PPRIME, Poitiers) - X. Sang Liang (Nanjing Institute of Meteorology) Funded by: - First Multiflow Summer School, Madrid 2014 - CTR Summer Program, Stanford 2016
Self-sustaining process in log-layer wall linear lift-up s log-layer streamwise rolls nonlinear interactions
Self-sustaining process in log-layer wall log-layer linear lift-up streamwise rolls s Butler & Farrell (1993) del Álamo & Jiménez (2006) Pujals et al. (2009) Hwang et al. (2010)... nonlinear interactions
Self-sustaining process in log-layer Wale e (1997) U(y, z) wall log-layer linear lift-up streamwise rolls s nonlinear interactions traveling wave modal inflectional instability in z U(y, z)
Self-sustaining process in log-layer Schoppa & Hussain (2002) U(y, z) wall log-layer linear lift-up vortices/rolls s nonlinear interactions wavy linear transient growth U(y, z)
Self-sustaining process in log-layer U(y, z) wall log-layer linear lift-up streamwise rolls Farrell & Ioannou (2012) Farrell et al. (2016) s nonlinear interactions (torque) Reynolds stress linear parametric instability U(y, z, t)
Self-sustaining process in log-layer wall linear lift-up Jiménez (2013, 2015) Kawahara et al. (2003) s Instability corrugated vortex sheet U(y, z) cross-shear velocity residual disturbances log-layer Orr mechanism mean shear
Self-sustaining process in log-layer wall linear lift-up s log-layer streamwise rolls nonlinear interactions
How to study self-sustaining process?
How to study self-sustaining process? Tools in self-sustaining turbulence: - Exact coherent structures - Energy budget - Correlations - Linear stability analysis - Reduced-order models - DNS data interrogation -...
How to study self-sustaining process? Tools in self-sustaining turbulence: - Exact coherent structures For a complete understanding: cause! e ect - Energy budget - Correlations - Linear stability analysis - Reduced-order models - DNS data interrogation -...
Introduction and Motivation Tools in self-sustaining turbulence: - Exact coherent structures For a complete understanding: cause! e ect - Energy budget - Correlations - Linear stability analysis No clear causal inference - Reduced-order models - DNS data interrogation -...
Introduction and Motivation Tools in self-sustaining turbulence: - Exact coherent structures For a complete understanding: cause! e ect - Energy budget - Correlations - Linear stability analysis No clear causal inference - Reduced-order models - DNS data interrogation -... Goal: Present a tool to quantify causality and application to interaction between s and rolls in the log-layer of a channel flow
Outline - Introduction and motivation - Causality as information transfer - Numerical experiment - Identification of s and rolls - Causality flow between s and rolls - Conclusions
Outline - Introduction and motivation - Causality as information transfer - Numerical experiment - Identification of s and rolls - Causality flow between s and rolls - Conclusions
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) How to measure the causality from v to u?
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) How to measure the causality from v to u? Methods to infer causality: - Time-correlation - Granger causality - Information flow Granger (1969) Information Theory Jiménez (2013) Tissot et al. (2014) Liang and Lozano-Durán (2017)
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) How to measure the causality from v to u? Methods to infer causality: - Time-correlation - Granger causality - Information flow Granger (1969) Information Theory Jiménez (2013) Tissot et al. (2014) Liang and Lozano-Durán (2017)
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) I(u) = log(p(u))
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) I(u) = log(p(u)) Information in u Probability density function of u
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) I(u) = Information in u log(p(u)) Probability density function of u Example: coin flipping I(heads) = p(heads) = 0.5 p(tails) = 0.5 ) log 2 (0.5) = 1 bit
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) I(u) = Information in u log(p(u)) Probability density function of u Example: coin flipping n times p(heads) = 0.5 p(tails) = 0.5 I(head, tails,...)= ) log 2 (0.5 n )=n bits
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du Information in u Probability density function of u
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) I(u) = log(p(u)) average hi(u)i = Z p(u) log(p(u))du Information in u Probability density function of u Information entropy
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u (Shannon, 1948) average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du dhi(u)i dt =?
Causality as information transfer 5 Information equation 0 Consider two signals: d u dt v = u(t) v(t) apple Fu (u, v) F v (u, v) + 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Liouville equation for P (u, v) (Liang, 2016) Causality from v to u := Information flow from v to u (Shannon, 1948) average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du dhi(u)i dt =?
Causality as information transfer Information equation d u dt v = apple Fu (u, v) F v (u, v) + Liouville equation for P (u, v) (Liang, 2016) ) dhi(u)i dt = Z P (u, v) @F u(u, v) @u dudv Z P (v u) @(p(u)f u(u, v)) @u dudv dhi(u)i dt =?
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du dhi(u)i dt = Z P (u, v) @F u(u, v) @u dudv Z P (v u) @(p(u)f u(u, v)) @u dudv
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du dhi(u)i dt = dhi self i dt + v!u +
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du dhi(u)i dt = dhi self i dt + v!u + Change of information in u Self-induced Information flow from v Flow from remaining variables
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Causality from v to u := Information flow from v to u average I(u) = log(p(u)) hi(u)i = Z p(u) log(p(u))du dhi(u)i dt = dhi self i dt + v!u + Change of information in u Self-induced Information flow from v Flow from remaining variables
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Normalized Causality from v to u = dhi self i dt v!u + v!u + 2 [0, 1] 0ifv is fully non-causal to u 1ifv is fully causal to u (Liang, 2016)
Causality as information transfer 5 Consider two signals: u(t) v(t) 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t(time) Normalized Causality from v to u Example: = d dt u v dhi self i dt = v!u 2 [0, 1] + v!u + apple Fu (u, v) F v (u, v) x = 0 v!u (Liang, 2016)
Causality as information transfer Normalized Causality from v to u = dhi self i dt v!u + v!u + Analytic expression for all terms but... extremely expensive to compute
Causality as information transfer Normalized Causality from v to u = dhi self i dt v!u + v!u + Analytic expression for all terms but... extremely expensive to compute Estimator: - Only u(t), v(t),... are known - Ergodicity - Linear dynamic model - Gaussian distribution of the variables
Causality as information transfer Normalized Causality from v to u = dhi self i dt v!u + v!u + Analytic expression for all terms but... extremely expensive to compute Estimator: - Only u(t), v(t),... are known - Ergodicity - Linear dynamic model - Gaussian distribution of the variables v!u = C uuc uv C v u C 2 uuc vv C 2 uvc u u C uu C 2 uv C ij! correlation i and j Liang (2014)
Outline - Introduction and motivation - Causality as information transfer - Numerical experiment - Identification of s and rolls - Causality flow between s and rolls - Conclusions
Numerical experiment DNS channel flow Re = 1000 2h h/4 h/2 Minimal channel for the log-layer (Flores and Jiménez, 2010)
Numerical experiment Numerics: DNS channel flow Re = 1000-2 nd order finite di erences - Staggered mesh - 3 rd order RK, Fractional step 2h - MPI h/4 h/2 Minimal channel for the log-layer (Flores and Jiménez, 2010)
Numerical experiment Numerics: DNS channel flow Re = 1000-2 nd order finite di erences - Staggered mesh - 3 rd order RK, Fractional step 2h - MPI Spatial resolution: - x + =6.5, z + =3.2, y + min =0.3 h/4 h/2 Minimal channel for the log-layer (Flores and Jiménez, 2010)
Numerical experiment Numerics: DNS channel flow Re = 1000-2 nd order finite di erences - Staggered mesh - 3 rd order RK, Fractional step 2h - MPI Spatial resolution: h/4 h/2 Minimal channel for the log-layer (Flores and Jiménez, 2010) - x + =6.5, z + =3.2, y + min =0.3 Time-resolved dataset - Total time simulated: 140 eddy turnovers - Time between snapshots: 25 wall units
Numerical experiment Numerics: DNS channel flow Re = 1000-2 nd order finite di erences - Staggered mesh - 3 rd order RK, Fractional step - MPI u Spatial resolution: h/4 - x + =6.5, z + =3.2, y + min =0.3 h/4 0.25h h/2 Time-resolved dataset - Total time simulated: 140 eddy turnovers Minimal channel for the log-layer (Flores and Jiménez, 2010) - Time between snapshots: 25 wall units
Numerical experiment Numerics: DNS channel flow Re = 1000-2 nd order finite di erences - Staggered mesh - 3 rd order RK, Fractional step v - MPI Spatial resolution: h/4 - x + =6.5, z + =3.2, y + min =0.3 h/4 0.25h h/2 Time-resolved dataset - Total time simulated: 140 eddy turnovers Minimal channel for the log-layer (Flores and Jiménez, 2010) - Time between snapshots: 25 wall units
Numerical experiment Numerics: DNS channel flow Re = 1000-2 nd order finite di erences - Staggered mesh - 3 rd order RK, Fractional step - MPI w Spatial resolution: h/4 - x + =6.5, z + =3.2, y + min =0.3 h/4 0.25h h/2 Time-resolved dataset - Total time simulated: 140 eddy turnovers Minimal channel for the log-layer (Flores and Jiménez, 2010) - Time between snapshots: 25 wall units
Outline - Introduction and motivation - Causality as information transfer - Numerical experiment - Identification of s and rolls - Causality flow between s and rolls - Conclusions
Definition of rolls and s in the log-layer u Streaks 22u z x 16u
Definition of rolls and s in the log-layer u Streaks 22u z Proper Orthogonal Decomposition x 16u Straight Mode 1 Mode 2 Meandering
Definition of rolls and s in the log-layer z u Streaks v Rolls w x 16u 22u 2u 2u 2u 2u
Definition of rolls and s in the log-layer z Streaks Rolls u v w x 16u 22u 2u 2u P.O.D. straight 2u 2u meandering
Definition of rolls and s in the log-layer z u Streaks v Rolls w x 16u 22u 2u 2u P.O.D. P.O.D. 2u 2u straight v long meandering v short
Definition of rolls and s in the log-layer z u Streaks v Rolls w x 16u 22u 2u 2u 2u 2u P.O.D. P.O.D. P.O.D. straight v long w long meandering v short w short
Outline - Introduction and motivation - Causality as information transfer - Numerical experiment - Identification of s and rolls - Causality flow between s and rolls - Conclusions
Causality flow between rolls and s straight v long v short meandering w long w short strong causality weak causality
Causality flow between rolls and s straight v long v short meandering w long w short strong causality weak causality Causality s! v meandering straight v long v short
Causality flow between rolls and s straight v long v short meandering w long w short strong causality weak causality Causality s! v meandering from straight to v long v short
Causality flow between rolls and s straight v long v short meandering w long w short strong causality weak causality Causality s! v meandering from straight to v long v short
Causality flow between rolls and s straight v long v short meandering w long w short strong causality weak causality Causality s! v meandering straight v long v short
Causality flow between rolls and s straight v long v short meandering w long w short strong causality weak causality Causality s! v Causality s! w meandering straight v long v short w long w short
Causality flow between rolls and s straight v long w long v short w short meandering strong causality weak causality Causality v! s Causality w! s v short w long v long w short straight meandering straight meandering
Causality flow between rolls and s straight v long w long v short w short meandering strong causality weak causality Causality v! s Causality w! s v short w long v long w short straight meandering straight meandering
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 w long meandering
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 w long meandering unimportant spanwise rolls
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 v long straight
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 v long straight lift-up
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 straight v long
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 straight v long Nonlinear by pressure
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07
Causality flow between rolls and s 0.30 0.35 v short straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 0.07 meandering meandering w long strong causality weak causality w short
Causality flow between rolls and s 0.30 0.35 v short straight v long 0.50 0.25 v short w long w short nonlinear effect/lift up 0.10 0.01 0.02 0.07 meandering meandering w long strong causality weak causality w short low destabilizing effect of w
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 meandering v short
Causality flow between rolls and s 0.30 0.35 straight v long 0.50 0.25 v short w long w short 0.10 0.01 0.02 meandering strong causality weak causality 1! fully casual 0! fully non-casual 0.07 meandering v short Streak breakdown
Conclusions - Causal inference is not straight forward for most common tools used in turbulence research - Proposed causality analysis based on information flow as a new tool for investigating turbulence dynamics - Data-driven method with low computational cost and applicable to a wide variety of scenarios - Application to self-sustaining process in the log-layer of wall-bounded turbulence - Rolls and straight/meandering s identified by POD - Lift-up, breakdown and nonlinear e ects quantified