Completion of Partially Known Turbulent Flow Statistics
Completion of partially known turbulent flow statistics A RMIN Z ARE , M IHAILO R. J OVANOVI C´ , AND T RYPHON T. G EORGIOU M OTIVATION Modeling and control of turbulent flows reduce turbulent drag
)
• linear stochastic simulations
Turbulent channel flow
kinetic energy
prevent/suppress turbulence
Economic impact
Linearized evolution model
• ˙
= A v = C efluids photo by: Miguel Visbal
•
C OVARIANCE COMPLETION
C OMPLETION OF TURBULENT FLOW STATISTICS
+ Bd
A=
Aos Acp
0 Asq
Lyapunov equation: A X + X A⇤ =
Challenges
=
i
B ⌦B ⇤
2
3
u v=4 v 5 w
v ⌘
+ x
+ z
y+
t
y+
• Recovered off-diagonals - two-point correlations nonlinear simulations
covariance completion
(A XDNS + XDNS A⇤ ) y uu
white-in-time excitation too restrictive!
- large number of degrees of freedom - complex flow dynamics
• •
y
Objective
uv
- control-oriented modeling of turbulent flows
Ongoing research
i
Structured covariance completion problem available correlations:
- model-based flow control design
passive
y
y
MSI R ESOURCES
active
Mesabi compute cluster
riblets y
hot-film sensors and wall-deformation actuators
•
superhydrophobic surface
Convex optimization problem minimize subject to
Stochastically forced Navier-Stokes equations stochastic forcing
linearized dynamics
velocity fluctuations
• embed observed statistics of turbulence in physics-based models • identify forcing statistics to account for available velocity statistics
•
AFOSR Award FA9550-16-1-0009; NSF Award CMMI 1363266 UMII Transdisciplinary Fellowship; 2014 CTR Summer Program
AX + XA⇤ + Z = 0 (CXC ⇤ )ij =
A PPROACH
Doctoral Dissertation Fellowship
kZk?
log det (X) +
X, Z
(Yoshino et al. 2008)
A CKNOWLEDGMENTS
ij
P UBLICATIONS
(i, j) 2 I
Dynamics of colored-in-time forcing
[1] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Color of turbulence”, in J. Fluid Mech., 2016. Note: Submitted; also arXiv:1602.05105.
˙ = A
[2] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Alternating direction optimization algorithms for covariance completion problems”, in Proceedings of the 2015 American Control Conference, 2015, pp. 515-520.
+ Bd
white noise
low-rank modification ˙ = (A + B Cf )
+ Bw
filter
colored noise
white noise
linearized dynamics
modified dynamics
velocity fluctuations
velocity fluctuations
[3] A. Zare, Y. Chen, M. R. Jovanovi´c, and T. T. Georgiou, “Low-complexity modeling of partially available second-order statistics via matrix completion”, in IEEE Trans. Automat. Control, 2014. Note: Submitted; also arXiv:1412.3399v1. [4] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Completion of partially known turbulent flow statistics via convex optimization”, in Proceedings of the 2014 Summer Program, Center for Turbulence Research, Stanford University/NASA.
)
• linear stochastic simulations
Turbulent channel flow
kinetic energy
prevent/suppress turbulence
Economic impact
Linearized evolution model
• ˙
= A v = C efluids photo by: Miguel Visbal
•
C OVARIANCE COMPLETION
C OMPLETION OF TURBULENT FLOW STATISTICS
+ Bd
A=
Aos Acp
0 Asq
Lyapunov equation: A X + X A⇤ =
Challenges
=
i
B ⌦B ⇤
2
3
u v=4 v 5 w
v ⌘
+ x
+ z
y+
t
y+
• Recovered off-diagonals - two-point correlations nonlinear simulations
covariance completion
(A XDNS + XDNS A⇤ ) y uu
white-in-time excitation too restrictive!
- large number of degrees of freedom - complex flow dynamics
• •
y
Objective
uv
- control-oriented modeling of turbulent flows
Ongoing research
i
Structured covariance completion problem available correlations:
- model-based flow control design
passive
y
y
MSI R ESOURCES
active
Mesabi compute cluster
riblets y
hot-film sensors and wall-deformation actuators
•
superhydrophobic surface
Convex optimization problem minimize subject to
Stochastically forced Navier-Stokes equations stochastic forcing
linearized dynamics
velocity fluctuations
• embed observed statistics of turbulence in physics-based models • identify forcing statistics to account for available velocity statistics
•
AFOSR Award FA9550-16-1-0009; NSF Award CMMI 1363266 UMII Transdisciplinary Fellowship; 2014 CTR Summer Program
AX + XA⇤ + Z = 0 (CXC ⇤ )ij =
A PPROACH
Doctoral Dissertation Fellowship
kZk?
log det (X) +
X, Z
(Yoshino et al. 2008)
A CKNOWLEDGMENTS
ij
P UBLICATIONS
(i, j) 2 I
Dynamics of colored-in-time forcing
[1] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Color of turbulence”, in J. Fluid Mech., 2016. Note: Submitted; also arXiv:1602.05105.
˙ = A
[2] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Alternating direction optimization algorithms for covariance completion problems”, in Proceedings of the 2015 American Control Conference, 2015, pp. 515-520.
+ Bd
white noise
low-rank modification ˙ = (A + B Cf )
+ Bw
filter
colored noise
white noise
linearized dynamics
modified dynamics
velocity fluctuations
velocity fluctuations
[3] A. Zare, Y. Chen, M. R. Jovanovi´c, and T. T. Georgiou, “Low-complexity modeling of partially available second-order statistics via matrix completion”, in IEEE Trans. Automat. Control, 2014. Note: Submitted; also arXiv:1412.3399v1. [4] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Completion of partially known turbulent flow statistics via convex optimization”, in Proceedings of the 2014 Summer Program, Center for Turbulence Research, Stanford University/NASA.
