Second order statistics / Statistical State Dynamics

Statistical State Dynamics (SSD/DSS) is the dynamics that governs the statistics of the flow rather than the dynamics governing single flow realizations. Claim: a second-order closure of the SSD is adequate (S3T/CE2 for lovers of acronyms).

Consider second-order statistics as data for an inverse problem: Can one identify forcing statistics to reproduce available statistics? Can this be done by white in-time stochastic process?

Bassam Bamieh's talk, given at the IPAM Turbulence workshop 2 years ago, is a tutorial that connects with what Jovanović and Gayme have talked about. Also Jovanović's talk at the IPAM workshop may be of interest (it was right after Bassam’s and it builds on what he presented; it demonstrates that transition can be prevented by reducing the tremendous sensitivity of flow dynamics using either active or passive means). Brian Farrell's talk at the IPAM workshop addresses the mechanism of wall-bounded flows using SSD.

Björn Birnir's talk, given at the KITP Turbulence Program 6 years ago, on the SCT (Stochastic Closure Theory) of Navier-Stokes turbulence, addresses the form of the noise in homogeneous turbulence.

Readings

Farrell and Ioannou, Structural stability of turbulent jets, JAS (2003)
This papers builts on the second-order closure starting from the equations of motions and employing the quasi-linear approximation. It shows that in the infinite ensemble limit we recover the time-dependent Lyapunov equation for the evolution of the second-order perturbation statistics and, in addition, we have an evolution equation for the first moment (mean flow) which is forced by the perturbation statistics via the Reynolds stresses (see section 2).

M. R. Jovanović and B. Bamieh, "Componentwise energy amplification in channel flows", J. Fluid Mech., vol. 534, pp. 145-183, 2005.
Input-output analysis of the linearized Navier-Stokes equations used to provide a detailed characterization of different flow structures (streamwise streaks, oblique waves, and Tollmien–Schlichting waves) in the early stages of transition to turbulence.

A. Zare, M. R. Jovanović, and T. T. Georgiou, "Colour of turbulence", J. Fluid Mech., vol. 812, 2017, pp. 636-680; arXiv.org:1602.05105
Application to channel flow turbulence modeling.

Bakas and Ioannou, Emergence of large-scale structure in barotropic beta-plane turbulence, PRL (2013)
This paper employs the SSD framework to predict that as turbulence intensity increases there exists a symmetry breaking bifurcation of homogeneous turbulence to the emergence of large-scale wave structure and/or large-scale jets. The predictions of the infinite ensemble ideal are then shown to be reflected in single flow realizations of the full nonlinear equations of motion.

Björn Birnir's SCT (Stochastic Closure Theory) of Navier-Stokes is developed in paper 1, book, paper 2, and the recent preprint, where SCT is used to fit the data from the variable density turbulent tunnel in Göttingen Germany.

Optimal partitions

The noise that physical systems are affected by limits the resolution that can be attained in partitioning their state space. We determine the `finest attainable' partition and replace the Fokker-Planck evolution operator by a finite matrix:

Statistical State Dynamics (SSD/DSS) is the dynamics that governs the statistics of the flow rather than the dynamics governing single flow realizations. Claim: a second-order closure of the SSD is adequate (S3T/CE2 for lovers of acronyms).

Consider second-order statistics as data for an inverse problem: Can one identify forcing statistics to reproduce available statistics? Can this be done by white in-time stochastic process?

Discussion session -- Feb. 3, 1:30-3:30Archived talksBjörn Birnir's talk, given at the KITP Turbulence Program 6 years ago, on the SCT (Stochastic Closure Theory) of Navier-Stokes turbulence, addresses the form of the noise in homogeneous turbulence.

ReadingsFarrell and Ioannou, Structural stability of turbulent jets, JAS (2003)

This papers builts on the second-order closure starting from the equations of motions and employing the quasi-linear approximation. It shows that in the infinite ensemble limit we recover the time-dependent Lyapunov equation for the evolution of the second-order perturbation statistics and, in addition, we have an evolution equation for the first moment (mean flow) which is forced by the perturbation statistics via the Reynolds stresses (see section 2).

Farrell, Gayme and Ioannou, A statistical state dynamics approach to wall-turbulence, PTRSA (2017)

A review paper concentrating on the SSD approach applied to understand wall-turbulence.

M. R. Jovanović and B. Bamieh, "Componentwise energy amplification in channel flows", J. Fluid Mech., vol. 534, pp. 145-183, 2005.

Input-output analysis of the linearized Navier-Stokes equations used to provide a detailed characterization of different flow structures (streamwise streaks, oblique waves, and Tollmien–Schlichting waves) in the early stages of transition to turbulence.

A. Zare, Y. Chen, M. R. Jovanović, and T. T. Georgiou, "Low-complexity modeling of partially available second-order statistics: theory and an efficient matrix completion algorithm", IEEE Trans. Automat. Control, 2016, doi:10.1109/TAC.2016.2595761; arXiv:1412.3399

Theory and an efficient matrix completion algorithm.

A. Zare, M. R. Jovanović, and T. T. Georgiou, "Colour of turbulence", J. Fluid Mech., vol. 812, 2017, pp. 636-680; arXiv.org:1602.05105

Application to channel flow turbulence modeling.

Bakas and Ioannou, Emergence of large-scale structure in barotropic beta-plane turbulence, PRL (2013)

This paper employs the SSD framework to predict that as turbulence intensity increases there exists a symmetry breaking bifurcation of homogeneous turbulence to the emergence of large-scale wave structure and/or large-scale jets. The predictions of the infinite ensemble ideal are then shown to be reflected in single flow realizations of the full nonlinear equations of motion.

Björn Birnir's SCT (Stochastic Closure Theory) of Navier-Stokes is developed in paper 1, book, paper 2, and the recent preprint, where SCT is used to fit the data from the variable density turbulent tunnel in Göttingen Germany.

Optimal partitions

The noise that physical systems are affected by limits the resolution that can be attained in partitioning their state space. We determine the `finest attainable' partition and replace the Fokker-Planck evolution operator by a finite matrix: