Closed-Loop Turbulence Control: Progress and Challenges

Turbulence control roadmap. For details, see text and the respective sections.

Applications of closed-loop turbulence control: (a) homogeneous grid turbulence (Reproduced with permission from T. Corke and H. Nagib.); (b) turbulent jet from Bradshaw et al. [38]; (c) Karman vortex street behind a mountain, photo by Bob Cahalan, NASA GSFC; (d) coherent structures in a mixing layer from Brown and Roshko [39]; (e) thunderstorm; (f) automobile in a wind tunnel, photo by Robert G. Bulmahn; (g) high-speed train; (h) cargo ship; (i) passenger jet; (j) Blue Angles fighter jets; (k) automobile engine; (l) turbo jet engine; (m) aircraft engines; (n) wind turbines; (o) heat exchanger flow; (p) rotating mixer; (q) air conditioner; (r) chocolate mixing; and (s) total artificial heart. Images (e) and (g)–(n) are from the website.¹ Images (c), (f), and (q)–(s) are from the website.² Images (o) and (p) were made using the COMSOL Multiphysics^® software and are provided courtesy of COMSOL.

Heuristics of turbulence control. Here, s are the sensor signals and b are the actuation signals.

Model hierarchy for control design based on Wiener[76].

Schematic illustrating popular choices at the various levels of kinematic and dynamic descriptions of the turbulent system P and choices for designing the controller K.

Open-loop control topology

Closed-loop control topology

Linear-quadratic Gaussian controller. The Kalman filter Kf is a dynamical system that takes sensor measurements s and the actuation signal b to estimate the full-state â. The LQR gain Kr is a matrix that multiplies the full-state to produce an actuation signal b=−Krâ that is optimal with respect to the quadratic cost function in Eq. (18).

Feedback control with disturbances and noise

General framework for feedback control. The input to the controller is the system measurements s, and the controller outputs an actuation signal b. The exogenous inputs w may refer to a reference wr, disturbances wd, or sensor noise wn. The cost function J may measure the cost associated with inaccuracy of reference tracking, expense of control, etc.

Two degrees-of-freedom control with reference tracking and disturbance rejection

Schematic of the closed-loop controller for transition delay of a flat-plate boundary layer (Reproduced with permission from Semeraro et al. [225]. Copyright 2013 by Cambridge University Press). Here, ψ corresponds to sensors s and φ corresponds to actuators b.

Phase portrait of oscillatory linear dynamics (49). The dashed trajectory corresponds to the unactuated dynamics while the solid trajectory corresponds to actuated dynamics with Eq. (50). The chosen parameters are σ^u= 0.1, σ^c= −0.1, ω^u= 1, g = 1 implying k = 0.4.

Phase portrait of weakly nonlinear dynamics. The dashed trajectory corresponds to the unactuated dynamics while the solid trajectory corresponds to actuated dynamics. The chosen parameters of Eq. (52) are σ^u= 0.1, ω^u= 1, α^u= 1, β^u= 1, γ^u= 0, and the forced decay rate σ^c= −0.1. The globally stable limit cycle lies on the parabolic inertial manifold.

Flow visualization of the experimental wake behind a D-shaped body without (a) and with symmetric low-frequency actuation (b) (Reproduced with permission from Mark Pastoor.) The D-shaped body is shown with five pressure sensors on the rear face, and the arrows at the corners indicate the employed zero net mass flux actuators. (a) Natural wake with vortex shedding and (b) actuated partially stabilized wake.

Phase portrait of moderately nonlinear dynamics (54). The dashed trajectory corresponds to the unactuated dynamics while the solid trajectory corresponds to actuated dynamics with Eq. (50). The chosen parameters are enumerated in Table1.

Phase portrait of the shift-mode amplitudes a₅ and a₆, i.e., the slow dynamics in Eq. (54). Same transient solutions as in Fig. 16.

Venn diagram for the classification of nonlinearities. Prototypic examples are for (A), the subcritical flow over backward-facing step with noise excitation [251]; for (B), the supercritical onset of vortex shedding [256]; for (C), the suppression of Kelvin–Helmoltz vortices by high-frequency forcing [47]; and for (D), the decay of 2D turbulence [267].

Input/output characteristics of different dynamics. Left: actuation command; right: sensor signal without forcing (dark), and sensor signal under periodic forcing (light). From top to bottom: a stable fixed point with periodic excitation (linear dynamics); a stable limit cycle with locking periodic forcing (weakly nonlinear dynamics); a stable limit cycle with high-frequency forcing (moderately nonlinear dynamics); and broadband turbulence under periodic forcing (strongly nonlinear dynamics).

Overview of model-free control methods discussed in Sec. 6. Model-free control involves the choice of control law structure as well as the optimization of controller parameters.

Schematic illustrating the components of an ESC. A sinusoidal perturbation is added to the best guess of the input b, passing through the plant, and resulting in a sinusoidal output perturbation. The high-pass filter removes the DC gain and results in a zero-mean output perturbation, which is then multiplied (demodulated) by the same input perturbation. This demodulated signal is finally integrated into the best guess b̂ for the optimizing input b.

Schematic illustrating ESC on for a static objective function J(b). The output perturbation (light) is in phase when the input is left of the peak value (i.e., b < b*) and out of phase when the input is to the right of the peak (i.e., b > b*). Thus, integrating the product of input and output sinusoids moves b̂ toward b*.

Acoustic pressure reduction in combustor experiment with modified ESC algorithm. The main peak is reduced by about a factor of 60 when control is applied (Reproduced with permission from Gelbert et al. [288]. Copyright 2012 by Elsevier).

Illustration of the benefits of opposition control (bottom) in contrast to the unforced system (top). Contours of streamwise vorticity are plotted in a cross-flow plane. Negative contours are indicated with dashed lines (Reproduced with permission from Lee et al. [22]. Copyright 1997 by AIP Publishing LLC).

Illustration of a possible binary representation of parameters used in GAs. This example has two parameters, each represented with a 3-bit binary number.

Illustration of function tree representation used in GP

Genetic operations are used to advance generations of individuals in GAs. Operations are elitism (E), replication (R), crossover (C), and mutation (M). For each individual of generation k + 1, after the elitism step, a genetic operation is chosen randomly according to a predetermined probability distribution. The individuals participating in this operation are selected from generation k with probability related to their fitness (e.g., inversely proportional to the cost function).

Genetic operations are used to advance generations of functions in GP

Schematic of closed-loop feedback control using GP for optimization. Various controllers in a population compete to minimize a cost function J, and the best performing individual controllers may advance to the next generation according to the optimization procedure on the right (Reproduced with permission from Fig. 4 of Duriez et al. [265]. Copyright 2014 by T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.P. Bonnet, L. Cordier, B.R. Noack, M. Segond, M. Abel, N. Gautier, J.L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton).

“Pseudovisualizations of the TUCOROM experimental mixing layer demonstrator for three cases: (I) unforced baseline (width W = 100%), (II) the best open-loop benchmark (width W = 155%), and (III) MLC closed-loop control (width W = 167%). The velocity fluctuations recorded by 24 hot-wires probes are shown as contour-plot over the time t (abscissa) and the sensor position y (ordinate). The black stripes above the controlled cases indicate when the actuator is active (taking into account the convective time). The average actuation frequency achieved by the MLC control is comparable to the open-loop benchmark.” The relative mixing cost function of the natural flow, open-loop forcing, and machine-learning control is shown in (b), and the mixing layer is shown in (a) (Reproduced with permission from Parezanovic et al. [263]. Copyright 2015 by Springer). A lower cost function J indicates improved mixing (Reproduced with permission from Duriez et al. [265]. Copyright 2014 by T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.P. Bonnet, L. Cordier, B.R. Noack, M. Segond, M. Abel, N. Gautier, J.L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, S.L. Brunton).

Flow chart illustrating the hierarchy of active control approaches. This diagram is conservative in giving preference to the most established techniques. If the task is optimization or minimization of measurement time, machine-learning control may be an earlier branch. Top panel depicts a turbulent jet from Bradshaw et al. [38].

Schematic illustrating a roadmap for future development. We envision the synthesis of classical control theory with data-driven methods for the development of hybrid controllers. Both top-down and bottom-up approaches will contribute to a better understanding of nonlinearities, which will in turn contribute to the development of more effective controllers.