Real-time reactive control of stochastic disturbances in forced turbulent jets

In this work we perform reactive control of stochastic disturbances in forced turbulent jets based on destructive interference. The study is motivated by the success of recent studies in applying this type of control on instability waves in transitional boundary layers and free-shear flows. Linear convective mechanisms in the initial region of turbulent jets are explored in order to perform reactive control, wherein the actuation signal is updated in real time based on sensor measurements performed upstream, resulting in an inverse feedforward approach. The control law is based on empirical transfer functions of the jet response to stochastic forcing and actuation, which are measured experimentally. Since turbulent jets have energy content spread in a number of azimuthal wave numbers, we apply axisymmetric forcing at the nozzle lip in order to be able to perform control using a reduced number of sensors and actuators. The external forcing produces axisymmetric wave packets which possess stochastic phases and amplitudes, akin to turbulent fluctuations found in unforced jets. We demonstrate the successful implementation of real-time reactive control of these disturbances, achieving order-of-magnitude attenuations of associated velocity fluctuations. Control is shown to reduce fluctuation levels over an extensive streamwise range.

Figure 1

Schematic of control experiment showing the relative position of inputs and outputs. Forcing (at the nozzle lip) and actuation consist of synthetic jets generated by loudspeakers; the sensors are microphones situated in the near-pressure field of the jet, immediately outside of the shear layer; the objective consists of streamwise velocity measurements performed by a hot wire at the jet centerline. The boundary layer is tripped inside the nozzle by a strip of carborundum particles placed 2.5 diameters upstream of the exit plane, so as to produce a fully turbulent jet.

Figure 2

(a) Mean boundary layer (BL) profiles at the nozzle exit plane. The untripped boundary layer is compared to the Blasius solution for a laminar boundary layer. ${\overline{U}}_x$ is the mean streamwise velocity and $h$ is the normal distance from the wall. (b) Tripped BL diagnostic plot. The present data is compared to experiments from Örlü and Alfredsson [98] and direct numerical simulation (DNS) data from Schlatter and Örlü [99].

Figure 3

Schematic of the inverse feedforward scheme showing the relative positions of the system inputs and outputs. The transfer functions between different elements are indicated, and the arrows indicate the direction where information is traveling. (a) Measurement-based control; (b) disturbance-based control.

Figure 4

Growth rate of the Kelvin-Helmholtz mode as a function of Strouhal number, computed through a locally parallel linear stability analysis carried out with velocity profiles measured at $x/D=0.3$ (black squares) and $x/D=2$ (red circles). Growth rates are shown with inverse sign, so that positive values mean exponential growth along positive $x$.

Figure 5

Comparison of jet response to harmonic forcing at $\text{St}=0.5$ and stochastic forcing in different bandwidths. (a) Power spectral density of velocity fluctuations measured at the jet centerline at $x/D=2$. (b) Velocity fluctuation time signal of a jet forced harmonically; (c) velocity fluctuation time signal of a jet forced stochastically in the band $0.3⩽\text{St}⩽0.85$. (d) Velocity fluctuation time signal of an unforced jet. Velocity fluctuations have been normalized by their maximum values. Stochastic forcing produces stochastic phases and amplitudes in the response, similar to what is observed in the unforced jet.

Figure 6

Control kernels in (a) frequency and (b) time domain for a jet forced in the band $0.3⩽\text{St}⩽0.65$. The abscissa in (b) represent a nondimensional time, $t^{*}=t{U}_j/D$. Kernels are normalized by their maximum value for the sake of comparison.

Figure 7

Jet response to stochastic forcing in the band $0.3⩽\text{St}⩽0.65$ as a function of forcing amplitude. Response at three selected Strouhal numbers are shown. Amplitudes correspond to the voltage applied to the forcing system. For sufficiently low amplitudes, clear linear response regimes can be obtained at different Strouhal numbers within the forcing band. Response curves for other forcing bandwidths (not shown) displayed similar behavior.

Figure 8

Power spectral densities of streamwise velocity fluctuations, $u_x$, measured at the objective position ($x/D=2$ at the centerline) of controlled and uncontrolled jets forced at three different bandwidths. $\mathbf{\text{——}}$: baseline case (forced jet); $\mathbf{\text{——}}$: controlled jet with reduction kernel, $K_{y,d}^r$; $\mathbf{\text{——}}$: controlled jet with amplification kernel, $K_{y,d}^a$; $\mathbf{\text{– – –}}$: unforced jet. (a),(c),(e) Control based on the external disturbances, $d$, as input to the control law; (b),(d),(f) control based on flow measurements, $y$, as input to the control law. Forcing bandwidths, represented by the gray shaded areas, are $0.3⩽\text{St}⩽0.45, 0.3⩽\text{St}⩽0.65$, and $0.3⩽\text{St}⩽0.85$.

Figure 9

Coherences associated to the control results shown in Fig. 8. (a)–(c) Comparison between sensor/objective (solid line), ${\gamma }_{yz}$, and disturbance/objective (dashed line), ${\gamma }_{dz}$, coherences measured with increasing frequency bands of forcing. (d) Actuator/objective coherences, ${\gamma }_{uz}$, measured for different actuation frequency bands. The color code for frequency band is the following: $\mathbf{\text{——}}$: $0.3⩽\text{St}⩽0.45$; $\mathbf{\text{——}}$: $0.3⩽\text{St}⩽0.65$; $\mathbf{\text{——}}$: $0.3⩽\text{St}⩽0.85$.

Figure 10

Effect of control downstream of the objective position. (a),(c),(e) Radial profiles of streamwise rms velocity; (b),(d),(f) power spectral densities of streamwise velocity fluctuation measured at the jet centerline. The uncontrolled case corresponds to the baseline jet, forced in the bandwidth $0.3⩽\text{St}⩽0.45$, and the controlled case was obtained with a reduction-aimed kernel, $K_d^r$.

Figure 11

Amplitudes (a) and phases (b) of the system transfer functions. The black dashed lines delimit the frequency band of forcing and actuation, $0.3⩽\text{St}⩽0.65$. The mean phase speed (averaged in the frequency band of forcing) of the hydrodynamic perturbation is indicated.

Figure 12

Experimental and fitted mean flow profiles. The circles correspond to mean experimental profiles measured at $x/D=0.3$ and the squares correspond to measurements at $x/D=2$. The solid lines are the fitted profiles.