Low-complexity stochastic modeling of turbulent flows
Low-complexity stochastic modeling of turbulent flows A RMIN Z ARE , M IHAILO R. J OVANOVI C´ , AND T RYPHON T. G EORGIOU M OTIVATION Control of turbulent flows prevent/suppress turbulence reduce turbulent drag
M ODEL - BASED CONTROL
C OMPLETION OF TURBULENT FLOW STATISTICS �
• view second-order statistics as data for an inverse problem • identify forcing statistics to account for partially known turbulent statistics
Economic impact
Turbulent channel flow Spanwise wall oscillations W (y = ±1, t) = 2α sin( 2π t) T
sustained drag reduction:
efluids photo by: Miguel Visbal
•
Linearized evolution model
•
Challenges
ψt v
- large number of degrees of freedom
= =
Aψ + f Cψ
A=
- complex flow dynamics • •
�
Aos Acp
“optimal” period of oscillations:
0 Asq
�
ψ=
�
Covariance matrix completion problem
Objectives - control-oriented modeling of turbulent flows
v η
�
u v= v w
up to ≈ 45%
T + ≈ 100
drag reduction ≈ α2 fDR (T + )
Perturbation analysis: X(κ) = X0 (κ) + α2 X2 (κ) + O(α4 ) uv2 (y, κ) = diag{Cu X2 (κ) Cv∗ (κ)}
uv(y, κ) ≈ uv0 (y, κ) + α2 uv2 (y, κ)
available correlations:
Ongoing research
⇓
- model-based flow control design
passive
simulations and experiments:
U
active
W
≈ ≈
U0 +
α2
T+
U2
W0 + α2 W2
Symbols: numerical simulations (Quadrio & Ricco, JFM ’04)
R EMARKS • Control-oriented modeling
y
riblets
•
Convex optimization problem minimize X, Z
hot-film sensors and wall-deformation actuators superhydrophobic surface
– stochastically forced linearized NS equations
subject to
(Yoshino et al. 2008)
– colored-in-time forcing accounts for partially observed statistics
�Z�∗
• Sensor-free control
AX + XA∗ + Z = 0 trace (Ti X) = gi , i = 1, . . . , N X � 0
– optimal period of oscillations captured by perturbation analysis – simulation-free approach to predicting full-scale results
• Acknowledgments
– NSF Award CMMI 1363266; UMII Transdisciplinary Fellowship; 2014 CTR Summer Program
• white-in-time excitation −→ too restrictive!
A PPROACH Stochastically forced Navier-Stokes equations
•
Filter design
P UBLICATIONS [1] 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.
• embed observed statistical features of turbulence in controloriented models
kinetic energy
linear stochastic simulations:
λ+ x
t
[2] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Completion of partially known turbulent flow statistics”, in Proceedings of the 2014 American Control Conference, 2014, pp. 1680-1685.
λ+ z
y+
y+
[3] Y. Chen, M. R. Jovanovi´c, and T. T. Georgiou, “State covariances and the matrix completion problem”, in Proceedings of the 52nd IEEE Conference on Decision and Control, 2013, pp. 1702-1707.
M ODEL - BASED CONTROL
C OMPLETION OF TURBULENT FLOW STATISTICS �
• view second-order statistics as data for an inverse problem • identify forcing statistics to account for partially known turbulent statistics
Economic impact
Turbulent channel flow Spanwise wall oscillations W (y = ±1, t) = 2α sin( 2π t) T
sustained drag reduction:
efluids photo by: Miguel Visbal
•
Linearized evolution model
•
Challenges
ψt v
- large number of degrees of freedom
= =
Aψ + f Cψ
A=
- complex flow dynamics • •
�
Aos Acp
“optimal” period of oscillations:
0 Asq
�
ψ=
�
Covariance matrix completion problem
Objectives - control-oriented modeling of turbulent flows
v η
�
u v= v w
up to ≈ 45%
T + ≈ 100
drag reduction ≈ α2 fDR (T + )
Perturbation analysis: X(κ) = X0 (κ) + α2 X2 (κ) + O(α4 ) uv2 (y, κ) = diag{Cu X2 (κ) Cv∗ (κ)}
uv(y, κ) ≈ uv0 (y, κ) + α2 uv2 (y, κ)
available correlations:
Ongoing research
⇓
- model-based flow control design
passive
simulations and experiments:
U
active
W
≈ ≈
U0 +
α2
T+
U2
W0 + α2 W2
Symbols: numerical simulations (Quadrio & Ricco, JFM ’04)
R EMARKS • Control-oriented modeling
y
riblets
•
Convex optimization problem minimize X, Z
hot-film sensors and wall-deformation actuators superhydrophobic surface
– stochastically forced linearized NS equations
subject to
(Yoshino et al. 2008)
– colored-in-time forcing accounts for partially observed statistics
�Z�∗
• Sensor-free control
AX + XA∗ + Z = 0 trace (Ti X) = gi , i = 1, . . . , N X � 0
– optimal period of oscillations captured by perturbation analysis – simulation-free approach to predicting full-scale results
• Acknowledgments
– NSF Award CMMI 1363266; UMII Transdisciplinary Fellowship; 2014 CTR Summer Program
• white-in-time excitation −→ too restrictive!
A PPROACH Stochastically forced Navier-Stokes equations
•
Filter design
P UBLICATIONS [1] 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.
• embed observed statistical features of turbulence in controloriented models
kinetic energy
linear stochastic simulations:
λ+ x
t
[2] A. Zare, M. R. Jovanovi´c, and T. T. Georgiou, “Completion of partially known turbulent flow statistics”, in Proceedings of the 2014 American Control Conference, 2014, pp. 1680-1685.
λ+ z
y+
y+
[3] Y. Chen, M. R. Jovanovi´c, and T. T. Georgiou, “State covariances and the matrix completion problem”, in Proceedings of the 52nd IEEE Conference on Decision and Control, 2013, pp. 1702-1707.
