Proper Orthogonal Decomposition Snapshot Selection for State ...

AIAA 2006-1400

44th AIAA Aerospace Sciences Meeting and Exhibit 9 - 12 January 2006, Reno, Nevada

Proper Orthogonal Decomposition Snapshot Selection for State Estimation of Feedback Controlled Flows Stefan G. Siegel*, Kelly Cohen†, Jürgen Seidel‡ and Thomas McLaughlin§

Downloaded by UNIVERSITY OF CINCINNATI on November 28, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2006-1400

US Air Force Academy, Colorado Springs, Colorado, 81001

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. If the intended use of the low dimensional POD model is in the area of feedback flow control, the low dimensional state of a flow field needs to be accurately estimated as input for a controller. This poses the problem of snapshot selection: For the state to which the feedback controller drives the flow, usually no snapshots are available beforehand. We investigate POD bases derived from steady state, transient startup and open loop forced data sets for the two dimensional circular cylinder wake at Re = 100. None of these bases by itself is able to represent all features of the feedback controlled flow field. However, a POD basis derived from a composite snapshot set consisting of both transient startup as well as open loop forced data accurately models the features of the feedback controlled flow. For similar numbers of modes, this POD basis, which can be derived a priori, represents the feedback controlled flow as well as a POD model developed from the feedback controlled data a posteriori. These findings have two important implications: Firstly, an accurate POD basis can be developed without iteration from unforced and open loop data. Secondly, it appears that the feedback controlled flow does not leave the subspace spanned by open loop and unforced startup data, which may have important implications for the performance limits of feedback flow control.

I.

Introduction

T

HE usefulness of POD in modeling and, analyzing complex flow fields is well established in literature. Several features of POD prove very advantageous when applied to flow field data: First, the optimality in terms of capturing most of the energy content with the least possible number of modes. This allows reducing large data sets obtained from computational fluid dynamics or particle image velocimetry drastically, while still preserving the most important features of the flow. Second, when applied to time periodic flow fields, the temporal coefficients represent vortex shedding phenomena with good accuracy, and can therefore be used for feedback flow control1. However, strictly speaking, a POD model is only valid for the flow situation from which it is derived. Therefore, any modification of the flow field by means of feedback flow control may bring about a reduction of the validity of the model. To address this problem, modified POD models have been proposed both by Siegel et al.1 and Gerhard et al.2. These ad hoc modifications do capture the change in mean flow reasonably well, but fail to address the changes seen in the fluctuating modes. Therefore, a POD procedure that can be adaptive in real time is advantageous, since it maintains the validity of the model through transient flow situations. The first step in developing such a procedure is to explore the behavior of the POD method as the snapshot ensemble is reduced in size. Siegel et al.3 demonstrated that a short time POD procedure (SPOD) leads to physically correct spatial modes, as long as snapshot ensembles of exactly one cycle of the highest frequency present in the flow are used. “Short” in this context refers to the length of the snapshot ensemble compared to a cycle of the fundamental frequency in the flow field. However, the main question in using a POD basis derived from snapshots obtained without the presence of feedback is how well this basis represents the feedback controlled flow. This turns out to be a “chicken and an egg” type problem: Without applying feedback, a meaningful snapshot basis for the feedback controlled flow may not be *

Assistant Research Associate, Department of Aeronautics, Senior Member. Visiting Researcher, Department of Aeronautics, Senior Member. ‡ Visiting Researcher, Department of Aeronautics, Member. § Director, Aeronautics Research Center, Department of Aeronautics, Associate Fellow. †

1 American Institute of Aeronautics and Astronautics This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

derived successfully. But in order to apply state based feedback, an effective POD basis is needed. The most obvious solution to this conundrum is a spiral approach: The validity of a POD basis gradually declines as the flow is driven away from the states represented in the basis. By recalculating the POD basis including the feedback controlled snapshots from an initial, at least partially successful, feedback simulation, a better POD model can be derived and thus a subsequent simulation can make use of the improved basis and—ideally—better achieve the control goal. This procedure is both inelegant and tedious. In this paper we investigate the ability of different snapshot ensembles obtained from open loop and unforced flow simulations to accurately model a feedback controlled flow field. While no single ensemble is able to successfully model the feedback controlled flow, a composite model produces promising results.


II.

Numerical Method

For the numerical simulations reported in this paper, Cobalt Solutions’ Cobalt solver was used. Cobalt is a commercial finite-volume method developed for solution of the compressible Navier-Stokes equations and described in Strang et al.4 The numerical method is a cell-centered finite volume approach applicable to arbitrary cell topologies (e.g. hexahedra, prisms, tetrahedral). The spatial operator used the exact Riemann Solver of Gottlieb and Groth5, least squares gradient calculations using QR factorization to provide second order accuracy in space, and TVD flux limiters to limit extremes at cell faces. A point implicit method using analytic first-order inviscid and viscous Jacobians is used for advancement of the discretized system. For time-accurate computations, a Newton subiteration scheme is employed, and the method is second order accurate in time. For parallel performance, Cobalt uses the domain decomposition library ParMetis to provide nearly perfect load balancing with a minimal surface interaction between zones. Communication between processors is achieved using Message Passing Interface (MPI), with parallel efficiencies above 95% on as many as 1024 processors. Simulation of rigid-body motion is achieved through an Arbitrary Lagrangian Eulerian (ALE) formulation, where the grid is neither stationary nor follows the fluid motion. The conservation equations are solved in an inertial reference frame, but the spatial operator is modified so that the advection terms are relative to the (non-inertial) grid reference frame. This requires simple modifications of many boundary conditions and of the initial conditions for the Riemann problem. The inviscid and viscous work terms due to the grid velocity must also be removed from the spatial operator. The ALE formulation also forces certain modifications to the time-centered implicit temporal operator. At the beginning of a time step, all geometric quantities are transformed to their values at the end of the given time step, according to the specified motion. This ensures the fluxes, which an implicit scheme computes at the end of the time step, are consistent with the geometry. Such quantities include centroid locations and least-squares weights vectors, but since the motion is rigid, volume and area are invariant under the transformation. A number of Newton sub-iterations are used to reduce errors associated with integrating over the time step with an implicit temporal operator. In order to evaluate the snapshot selection procedure, the two-dimensional circular cylinder wake at a Reynolds number of 100 was used as a prototype flow field. This flow was represented by an unstructured two-dimensional grid with 63700 nodes/31752 elements. The grid extended from -16.9 cylinder diameters to 21.1 cylinder diameters in the x (streamwise) direction, and ±19.4 cylinder diameters in y (flow normal) direction. The boundary conditions at the far field were Riemann invariant; at the cylinder surface an adiabatic no-slip wall boundary condition was implemented. Cobalt allows for the specification of advection and diffusion damping coefficients, which are nondimensional with a range from zero to one. Their purpose is to resolve potential instability problems in the solver at the cost of numerical accuracy. In the limiting case of zero, no artificial damping is added. For the present investigation, an advection coefficient of 0.01 and a diffusion coefficient of 0.0 were used. Other pertinent simulation parameters: • Reynolds Number (Re) = 100 • 32 Iterations for matrix solution scheme • 3 Newton sub-iterations • Non-dimensional time step Δt* = Δt.U/D=0.05 An initial disturbance in the simulation was triggered by skewing the incoming mean flow by α = 0.5 degrees at the first time step. This breaks the symmetry of the flow and accelerates the development of the limit cycle. A grid and time resolution study showed good convergence for the simulation parameters outlined above. For further validation of the unforced cylinder wake CFD model at Re = 100, the resulting value of the mean drag coefficient, cd, was compared to experimental and computational investigations reported in the literature. At Re = 100, experimental data, reported by Oertel6 and Panton7, point to cd values ranging from 1.26 to 1.4. Furthermore, Min 2 American Institute of Aeronautics and Astronautics

and Choi8 report on several numerical studies that obtained drag coefficients of 1.35 and 1.337. The COBALT CFD model used in this effort results in a cd =1.35 at Re = 100, which compares well with the reported literature. Another important benchmark parameter is the non-dimensional vortex shedding frequency, the Strouhal number (St) for the unforced cylinder wake. Experimental results at Re = 100, presented by Williamson9, show Strouhal numbers ranging from 0.163 to 0.166. The Strouhal number obtained from the COBALT CFD model used in this effort is St = 0.163 at Re = 100, which also compares well. The simulations were performed on a Beowulf Linux cluster consisting of 64 Pentium III processors operating at 1 Ghz. When running on 8 processors, typically a time step took on the order of 6 s to compute. Employing larger number of processors yielded disproportionately small improvements in execution time, due to network and disk access overhead for saving the results at the end of each time step.


III.

Results

A. Simulation Results The flow geometry under consideration is a two-dimensional circular cylinder wake at a Reynolds number of 100. Three different cases are used to provide transient flow field data: The startup of the computational fluid dynamics (CFD) simulation from uniform velocity, a case where cylinder displacement normal to the free stream is used to force the flow sinusoidally and out of phase to the vortex shedding, and a feedback controlled simulation where the target is suppression of the vortex street. The startup of the CFD simulation leads to a transient flow situation where the Kármán vortex street develops in the originally steady but unstable flow field. The above transient behavior captures the dynamic behavior of the instability of the flow, which makes it a very useful data set to develop low dimensional models. During the startup, the vortex formation location, the strength of the vortices, and the frequency of the vortex shedding vary. Lift and Drag of this flow field are shown in Figure 1(a). Initially, a steady flow field that exhibits a minimum of drag at about 0.8s into the simulation develops. However, due to the instability of this flow field, subsequently vortex shedding with initially exponential growth of the shedding intensity develops, as is evident by the oscillating lift force. Nonlinear saturation limits the growth eventually, and a time periodic flow field, the von Kàrmàn vortex street, develops after t=2s. We also refer to this periodic flow state as “steady state” in the further discussion. Continuing the simulation of the unforced flow field described above, we applied time periodic forcing in the form of translation of the cylinder normal to the incoming flow. Lift, drag and cylinder displacement of one of these out of phase forced simulations are shown in Figure 1(b). Forcing is at the natural vortex shedding frequency with a half-peak amplitude of 20 percent of the cylinder diameter. Forcing is activated at 3.2s and maintained for 15 shedding/forcing cycles. The displacement is a shifted cosine, which minimizes simulation problems due to abrupt acceleration of the cylinder as would be encountered using a sinusoidal forcing function. This type of forcing initially results in a transient reduction of the strength of the vortex shedding, subsequently the vortex shedding resumes in phase with the forcing. The length of the transient state depends on the forcing amplitude and frequency. In addition to the transient dynamics of the flow, this type of simulation captures the interaction and effect of the forcing with the wake. We conducted open loop forcing with a variety of forcing parameters; four of these simulations were used for POD mode development described in the following section. The forcing parameters employed are shown in Figure 1(d) with their relationship to the boundaries where lock-in of shedding with the forcing can be achieved (Koopman 10). The feedback controlled simulation shown in Figure 1(c) was reported by Siegel et al.1, 11. Feedback control is the final target application of this research, and within the scope of this paper we attempt to find a POD basis that correctly represents a feedback controlled flow field of this type. In this investigation, we intend to develop POD bases from the transient startup snapshots, as well as the open loop snapshot data described above. For reference, we compare the performance of these POD bases to the POD basis developed from steady state unforced data, as well as steady state unforced data plus a shift mode as outlined above. We then use these different POD bases to model the feedback controlled flow field from a closed loop simulation that was able to reduce the strength of the von Karman vortex street. Comparison between the original flow field data and the flow field reconstructed from the POD basis is made by subtracting both flow fields from each other and developing a histogram of all data points in terms of the difference in flow velocity. Without loss of generality, within the scope of this investigation, we will only use the streamwise (U) velocity. The same procedure may be applied to a pressure field or velocity vector field, however.

3 American Institute of Aeronautics and Astronautics

(a)

(b) 0.4

Lift Drag

0.05

Displacement [y/D] Drag [N] Lift [N]

0.35 0.04

0.3 0.25

Force [N]

0.03

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0.15 0.1

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5 5.5 Time [s]

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1.2

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Lift / D 0 Drag / D0

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0 −0.01

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Figure 1. (a) Lift and Drag of unforced startup simulation (b) Lift, Drag and cylinder displacement of open loop forced data, f/fn =1, A/D = 0.2. (c) Lift and drag of feedback controlled simulation, controller active 3.1s2s and Startup t=0.5s

Startup

Open Loop

SU Entire Startup

OL f/f0 = 0.9 A/D=0.3 f/f0 = 1.1 A/D=0.3 f/f0 = 1.0 A/D=0.3 f/f0 = 1.0 A/D=0.1 1(b), 1(d)

See Figure 1(a) 1(a) 1(a) Table 1 Summary of Snapshots used for different POD bases


Open Loop + Startup OLSU Open Loop and Startup

1(a) – (d)


C. Estimation Quality of Different POD Models With the above described POD bases developed, we investigate the capability of each basis to accurately represent different flow fields. In a mathematically strict sense, a given POD basis is only valid for the data set that was used to derive the modes. For that data set, the complete POD basis will exactly recover the original set of snapshots used for its derivation. However, one of the important features of POD is its optimality with respect to energy content, which typically leads to a steep drop-off in energy for higher order modes. Exploiting this feature, one can often truncate the mode set at a relatively small number of modes without significant loss in energy. For the steady state unforced flow field, for example, more than 98% of the flow energy can be represented using only four modes. As opposed to steady state conditions, for transient flow fields typically a larger number of modes are needed to capture similar amounts of energy. If one expects the POD basis to span a larger number of flow features, this will also increase the required number of modes. Figure 3 shows a histogram of the unforced startup data set modeled by different POD bases. It can be seen that for the two models that were derived solely from the startup data, there is little improvement of estimation quality for increasing the number of modes from 6 to 10. However, the combined model that was derived from both startup and open loop forced data has, for 6 modes, a much larger estimation error than the model derived from the startup data. This can be attributed to the fact that the combined model for startup and open loop data has to span a much larger subspace of flow phenomena. For the same reason, as the number of retained modes is increased for the combined model, the error approaches the estimation error of the models derived from startup data only. Using 15 modes, the error of the combined OLSU model is comparable to the error of the 6 and 10 mode SU model.

Figure 3. Histogram of time averaged modeling error for startup data set. For legend nomenclature of POD models used refer to Table 1. Figure 4 compares two of the models shown in Figure 3; the 6 mode SU model and 15 mode OLSU model, to selected other models. A 15 mode model derived from open loop data performs relatively poorly, indicating that the startup data dynamics are not captured by the open loop forced simulations. This may be due to the fact that the open loop forced simulations never reach a periodic state as the startup simulation does. Supporting this hypothesis is the fact that the SH model does relatively well in comparison to the SU and OLSU models. On the other hand, the same model without the shift mode performs as poorly as the open loop derived model. It appears that both the open loop and steady state models do not cover the steady flow field at the beginning of the startup simulation, which may explain their poor performance. Figure 4 does, however, clearly demonstrate the value of adding the shift mode. 6 American Institute of Aeronautics and Astronautics


Figure 4. Histogram of time averaged modeling error for startup data set. For legend nomenclature of POD models used refer to Table 1. Figure 5 shows the evolution of the spatially averaged estimation error over time. While the SU models show a balanced error level for all time instances that decreases with model order, the OLSU models of lower order show a large increase in estimation error for the initial portion of the simulation. This indicates that the truncation at these lower levels eliminates modes that are covering the steady flow field. In addition, the 6 and 9 mode combined models are lacking the ability to represent a higher harmonic to the von Karman vortex street, which manifests itself in the oscillation of the error at twice the von Karman shedding frequency. This harmonic is modeled by including mode 10. At order 15, the error of the OLSU model approaches the values of the SU models.

Figure 5. Spatially averaged modeling error for startup data set. For legend nomenclature of POD models used refer to Table 1.



In summary, the only models that are able to represent the startup data well, other than for the SU models themselves, are the OLSU model of order 15, as well as the SH model (albeit at a slight increase in estimation error).

Figure 6. Histogram of time averaged modeling error for the open loop forced data set, f/f0 = 1, A/D = .2. For legend nomenclature of POD models used refer to Table 1. Applying the same models to an open loop forced data set, Figure 6 shows that the only well performing models are the OL model, as well as the OLSU model. All the models that are based on startup or unforced data only do not perform well, including the SH model that did perform well in modeling the startup data as discussed above and shown in Figure 5. The most likely reason lies in the fact that the interaction between the forcing, i.e. the cylinder motion, and the flow can only be captured in models that include data involving forcing. Thus, all models that use forcing data for their derivation perform well, all others do not.

Figure 7. Histogram of time averaged modeling error for feedback controlled data set. For legend nomenclature of POD models used refer to Table 1. In order to address the question if it is possible to estimate a feedback controlled flow field using models derived from unforced and/or open loop forced data, we projected a feedback controlled set of snapshots onto the various 8 American Institute of Aeronautics and Astronautics


models introduced so far. Figure 7 shows that the 15 mode OLSU model, which was able to accurately model both the OL and SU flow fields, also does quite well in modeling the feedback controlled flow field. Individual error bins that show larger than ± 20% of the free stream velocity errors contain less than 1% of the data points each. However, the maximum error reaches up to 80% of the free stream velocity with still significant numbers of data points. The respective temporal evolution of the estimation error is shown in Figure 8.

Figure 8 Spatially averaged modeling error for feedback controlled data set. For legend nomenclature of POD models used refer to Table 1. While the spatially averaged error of the combined model is smaller by a factor of two than for the SS or SH models, we were interested in the spatial distribution of the estimation error in order to identify areas in the flow where the OLSU model shows large differences compared to the original flow. By inspecting the original and reconstructed flow fields at one particular instant of time, as shown in Figure 9(a) and 9(b), small differences are evident. Subtracting these data sets from each other, as shown in Figure 9(c), does reveal large differences on the front of the cylinder surface. These differences may be attributed to the effect of cylinder motion onto the flow field. Since the combined mode set does include the effects of open loop forcing with various combinations of amplitudes and frequencies, it is not immediately obvious why the effect of forcing is not successfully modeled in the feedback controlled case. The answer to the estimation error may be found in the phase between forcing and vortex shedding: In the open loop forcing cases, the locked-in flow field establishes a quite different phase between cylinder motion and the Kármán vortex street than in the feedback controlled flow field. However, the open loop forced model has the phase between forcing and vortex shedding hard-wired in the spatial modes, i.e. only phases that were present in the open loop forced flow fields will be included in the POD spatial modes. To test this hypothesis, we spatially truncated the feedback controlled data set in the stream wise direction in order to exclude the cylinder, i.e. 1