Low-dimensional models have proven essential for feedback control and estimation of flow fields. While feedback control based on global flow estimation can be very efficient, it is often difficult to estimate the flow state if structures of very different length scales are present in the flow. The conventional snapshot-based proper orthogonal decomposition (POD), a popular method for low-order modeling, does not separate the structures according to size, since it optimizes modes based on energy. Two methods are developed in this study to separate the structures in the flow based on size. One of them is Hybrid Filtered POD method and the second one is 3D FFT-based Filtered POD approach performed using a fast Fourier transform (FFT)-based spatial filtering. In both the methods, a spatial low-pass filter is employed to precondition snapshot sets before deriving POD modes. Three-dimensional flow data from the simulation of turbulent flow over a circular cylinder wake at Re=20000 is used to evaluate the performance of the two methods. Results show that both the FFT-based 3D Filtered POD and Hybrid Filtered POD are able to capture the large-scale features of the flow, such as the von Karman vortex street, while not being contaminated by small-scale turbulent structures present in the flow.
A short computational program was undertaken to evaluate the effectiveness of a closed-loop control strategy for the stabilization of an unstable bluff-body flow. In this effort, the non-linear one-dimensional Ginzburg–Landau wake model at 20% above the critical Reynolds number was studied. The numerical model, which is a non-linear partial differential equation with complex coefficients, was solved using the FEMLAB®/MATLAB® software packages and validated by comparison with published literature. At first, a model independent approach was attempted for wake suppression using feedback control. The closed-loop system was controlled using a conventional proportional-integral-derivative (PID) controller as well as a non-linear fuzzy controller. A single sensor is used for feedback, and the actuator is represented by altering the boundary conditions of the cylinder. Simulation results indicate that for a single sensor scheme, the increase in the sophistication of the control results in significantly shorter settling times. However, there is only a marginal improvement concerning the suppression of the wake at higher Reynolds numbers. The feedback control design was then augmented by switching over to a model-dependent controller. Based on computationally generated data obtained from solving the unforced wake, a low-dimensional model of the wake was developed and evaluated. The low-dimensional model of the unforced Ginzburg–Landau equation captures more than 99.8% of the kinetic energy using just two modes. Two sensors, placed in the absolutely unstable region of the wake, are used for real-time estimation of the first two modes. The estimator was developed using the linear stochastic estimation scheme. Finally, the loop is closed using a PID controller that provides the command input to the variable boundary conditions of the model. This method is relatively simple and easy to implement in a real-time scenario. The control approach, applied to the 300 node FEMLAB® model at 20% above the unforced critical Reynolds number stabilizes the entire wake. Compared to the model-independent controllers, the controller based on the low-dimensional model is far more effective in the suppression of the wake at higher Reynolds numbers. Furthermore, while the latter approach employs only the estimated temporal amplitude of the first mode of the imaginary part of the amplitude, all higher modes are stabilized. This suggests that the higher order modes are caused by a secondary instability that is suppressed once the primary instability is controlled.
Feedback flow control of the wake of a circular cylinder at a Reynolds number of 100 is an interesting and challenging benchmark for controlling absolute instabilities associated with bluff body wakes. A two dimensional computational fluid dynamics simulation is used to develop low-dimensional models for estimator design. Actuation is implemented as displacement of the cylinder normal to the flow. The estimation approach uses a low dimensional model based on a truncated 6 mode Double Proper Orthogonal Decomposition (DPOD) applied to the streamwise velocity component of the flow field. Sensor placement is based on the intensity of the resulting spatial modes. A non-linear Artificial Neural Network Estimator (ANNE) was employed to map the velocity data to the mode amplitudes of the DPOD model. For a given four sensor configuration, developed using a previously validated strategy, ANNE performed better than two state-of-the-art approaches, namely, a Quadratic Stochastic Estimator (QSE) and a Linear Stochastic Estimator with time delays (DSE).
For feedback control using low-dimensional proper orthogonal decomposition (POD) models, the mode amplitudes of the POD mode coefficients need to be estimated based on sensor readings. This paper is aimed at suppressing the von Kairman vortex street in the wake of a circular cylinder using a low-dimensional approach based on POD. We compare sensor placement methods based on the spatial distribution of the POD modes to arbitrary ad hoc methods. Flow field data were obtained from Navier-Stokes simulation as well as particle image velocimetry (PIV) measurements. A low-dimensional POD was applied to the snapshot ensembles from the experiment and simulation. Linear stochastic estimation was used to map the sensor readings of the velocity field on the POD mode coefficients. We studied 53 sensor placement configurations, 32 of which were based on POD eigenfunctions and the others using ad hoc methods. The effectiveness of the sensor configurations was investigated at Re = 100 for the computational fluid dynamic data, and for a Reynolds number range of 82-99 for the water tunnel PIV data. Results show that a five-sensor configuration can keep the root mean square estimation error, for the amplitudes of the first two modes to within 4% for simulation data and within 10% for the PIV data. This level of error is acceptable for a moderately robust controller The POD-based design was found to be simpler. more effective, and robust compared to the ad hoc methods examined.
The effectiveness of a sensor configuration for feedback flow control on the wake of a circular cylinder is investigated in both direct numerical simulation as well as in a water tunnel experiment. The research program is aimed at suppressing the von Kármán vortex street in the wake of a cylinder at a Reynolds number of 100. The design of sensor number and placement was based on data from a laminar two-dimensional simulation of the Navier–Stokes equations for the unforced condition. A low-dimensional proper orthogonal decomposition (POD) was applied to the vorticity calculated from the flow field and sensor placement was based on the intensity of the resulting spatial eigenfunctions. The numerically generated data was comprised of 70 snapshots taken over three cycles from the steady state regime. A linear stochastic estimator (LSE) was employed to map the velocity data to the temporal coefficients of the reduced order model. The capability of the sensor configuration to provide accurate estimates of the four low-dimensional states was validated experimentally in a water tunnel at a Reynolds number of 108. For the experimental wake, a sample of 200 particle image velocimetry (PIV) measurements was used. Results show that for experimental data, the root mean square estimation error of the estimates of the first two modes was within 6% of the desired values and for the next two modes was within 20% of the desired values. This level of error is acceptable for a moderately robust controller.
The effect of feedback flow control on the wake of a circular cylinder at a Reynolds number of 100 is investigated in direct numerical simulation. The control approach uses a low-dimensional model based on proper orthogonal decomposition (POD). The controller applies linear proportional and differential feedback to the estimate of the first POD mode. The range of validity of the POD model is explored in detail. Actuation is implemented as displacement of the cylinder normal to the flow. It is demonstrated that the threshold peak amplitude below which the control actuation ceases to be effective is in the order of 5% of the cylinder diameter. The closed-loop feedback simulations explore the effect of both fixed-phase and variable-phase feedback on the wake. Whereas fixed-phase feedback is effective in reducing drag and unsteady lift, it fails to stabilize this state once the low drag state has been reached. Variable-phase feedback, however, achieves the same drag and unsteady lift reductions while being able to stabilize the flow in the low drag state. In the low drag state, the near wake is entirely steady, whereas the far wake exhibits vortex shedding at a reduced intensity. A drag reduction of 15% of the drag was achieved, and the unsteady lift force was lowered by 90%.