Stefan Siegel

Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition

For the systematic development of feedback flow controllers, a numerical model that captures the dynamic behaviour of the flow field to be controlled is required. This poses a particular challenge for flow fields where the dynamic behaviour is nonlinear, and the governing equations cannot easily be solved in closed form. This has led to many versions of low-dimensional modelling techniques, which we extend in this work to represent better the impact of actuation on the flow. For the benchmark problem of a circular cylinder wake in the laminar regime, we introduce a novel extension to the proper orthogonal decomposition (POD) procedure that facilitates mode construction from transient data sets. We demonstrate the performance of this new decomposition by applying it to a data set from the development of the limit cycle oscillation of a circular cylinder wake simulation as well as an ensemble of transient forced simulation results. The modes obtained from this decomposition, which we refer to as the double POD (DPOD) method, correctly track the changes of the spatial modes both during the evolution of the limit cycle and when forcing is applied by transverse translation of the cylinder. The mode amplitudes, which are obtained by projecting the original data sets onto the truncated DPOD modes, can be used to construct a dynamic mathematical model of the wake that accurately predicts the wake flow dynamics within the lock-in region at low forcing amplitudes. This low-dimensional model, derived using nonlinear artificial neural network based system identification methods, is robust and accurate and can be used to simulate the dynamic behaviour of the wake flow. We demonstrate this ability not just for unforced and open-loop forced data, but also for a feedback-controlled simulation that leads to a 90 % reduction in lift fluctuations. This indicates the possibility of constructing accurate dynamic low-dimensional models for feedback control by using unforced and transient forced data only.

FEEDBACK CONTROL OF A CIRCULAR CYLINDER WAKE IN EXPERIMENT AND SIMULATION (INVITED)

The effect of feedback flow control on the wake of a circular cylinder at a Reynolds number of 100 is investigated in both water tunnel experiment and direct numerical simulation. Our 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. We demonstrate 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. While 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, while the far wake exhibits vortex shedding at a reduced intensity. We achieved a drag reduction of close to 90% of the vortex-induced drag, and lowered the unsteady lift force by the same amount.

Plasma-Based Actuators for Cylinder Wake Vortex Control

A number of methods have been developed over the years to affect the wake of a circular cylinder. These include oscillation of the cylinder normal to the direction of flow, rotational oscillation, and blowing and suction through slots. While each of these methods has been found effective for operation at low Reynolds numbers, their low bandwidth, resonance, or form factor make them generally unsuitable for robust operation. As the field of flow control aims to affect flows of Reynolds numbers of technical interest, there is a need for actuators with higher bandwidth and non-resonant behavior for robust mitigation of Karman Vortex Street-induced drag and oscillatory lift. The dielectric barrier discharge (DBD) plasma electrode has been developed as a flow control actuator over the last few years, showing the ability to affect flow behavior in a range of applications. In this effort, the DBD plasma actuator is applied to a circular cylinder at a Reynolds number of 7400. Hot film measurements show that vortex shedding frequency can be driven to the actuator forcing frequency, exhibiting lock-in behavior similar to that previously shown with the other forcing methods. Preliminary data indicates that plasma actuators are effective in controlling vortex shedding frequency and in achieving spanwise coherent shedding. They also can alter the vortex shedding frequency within forcing amplitude/frequency bands similar to that for the above methods at lower Reynolds number. Flow visualization shows the actuators have significant authority in affecting flow separation and wake behavior.

Feedback control of a cylinder wake low-dimensional model

Nomenclature Ak = time-dependent coefx8e cient of the kth proper orthogonal decompositionmode Aest k = estimation of Ak B = matrix containing the bk coefx8e cients bk = coefx8e cients associatedwith the control input ck i j = coefx8e cients of the nonlinear function gk fa = control input resulting form external forcing of the cylinder gk = nonlinear function of the time-dependent low-dimensionalmodes J = Jacobian of the open-loop low-dimensionalmodel Jc = closed-loop Jacobian K p = proportional gain of the linear feedback controller

Performance characteristics of a compact position-sensitive LSO detector module

The authors assembled a compact detector module comprised of an array of small, individual crystals of lutetium oxyorthosilicate:Ce (LSO) coupled directly to a miniature, metal-can, position-sensitive photomultiplier tube (PSPMT). The authors exposed this module to sources of 511-keV annihilation radiation and beams of 30- and 140-keV photons and measured spatial linearity; spatial variations in module gain, energy resolution, and event positioning; coincidence timing; the accuracy and sensitivity of identifying the crystal-of-first-interaction at 511 keV; and the effects of intercrystal scatter and LSO background radioactivity. The results suggest that this scintillator/phototube combination should be highly effective in the coincidence mode and can be used, with some limitations, to image relatively low-energy single photon emitters. Photons that are completely absorbed on their first interaction at 511 keV are positioned by the module at the center of a crystal. Intercrystal scatter events, even those that lead to total absorption of the incident photon, are placed by the module in a regular connect-the-dot pattern that joins crystal centers. As a result, the accuracy of event positioning can be made to exceed 90%, though at significantly reduced sensitivity, by retaining only events that occur within small regions-of-interest around each crystal center and rejecting events that occur outside these regions in the connect-the-dot pattern.

Numerical Simulations of a Feedback-Controlled Circular Cylinder Wake

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%.

PROPER ORTHOGONAL DECOMPOSITION MODELING OF A CONTROLLLED GINZBURG-LANDAU CYLINDER WAKE MODEL

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 nonlinear one-dimensional GinzburgLandau wake model at 20% above the critical Reynolds number was studied. The numerical model, which is a nonlinear partial differential equation with complex coefficients, was solved using the FEMLAB/MATLAB package and validated by comparison with published literature. Based on computationally generated data obtained from solving the unforced wake, a low-dimensional model of the wake was developed and evaluated. The lowdimensional 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 an 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 for a proportional gain of 0.06. While the controller uses only the estimated temporal amplitude of the first mode of Im(A(x,t)), 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.

Short Time Proper Orthogonal Decomposition for State Estimation of Transient Flow Fields

Proper Orthogonal Decomposition (POD) has been used extensively in the past for estimation and low dimensional modeling of both steady and time periodic flow fields. We demonstrate how POD can be successfully applied to transient flow fields by using snapshots acquired over short time periods. Using snapshot ensembles of exactly one cycle of the dominant flow field frequency in length, it is possible to correctly model and estimate transient flow fields. The onset of vortex shedding in a circular cylinder wake simulation as well as a feedback controlled cylinder wake flow are used to demonstrate the capabilities of this short time proper orthogonal decomposition (SPOD) procedure. In comparison to traditional and recently published modified POD methods, SPOD improves the estimation accuracy while reducing the number of sensors employed from 35 to 8.

Sensor Placement Based on Proper Orthogonal Decomposition Modeling of a Cylinder Wake

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 Karman 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 direct numerical 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 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 2 modes was within 20% of the desired values. This level of error is acceptable for a moderately robust controller.

Effective Sensor Placements for the Estimation of Proper Orthogonal Decomposition Mode Coefficients in von Karman Vortex Street

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.