Large Eddy Simulations of Flow Over a Circular Cylinder Using Unstructured Grids

R. King (Ed.): Active Flow Control, NNFM 95, pp. 105–118, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

State Estimation of Transient Flow Fields Using Double Proper Orthogonal Decomposition (DPOD)

Stefan Siegel, Kelly Cohen, Jürgen Seidel, and Thomas McLaughlin

US Air Force Academy, Colorado Springs, Colorado 81001, USA

Abstract

For successful feedback flow control, an accurate estimation of the flow state is necessary. Proper Orthogonal Decomposition (POD) has been used to achieve this. However, if the POD modes are derived from a set of snapshots obtained from one flow condition only, the resulting modes will become less and less valid for a flow field that is for example altered by the effect of feedback flow control. In the past, a shift mode has been added to account for the change in the mean flow. Here, we present a new scheme that allows for the derivation of shift modes for all of the original POD modes. This DPOD mode set thus may span a range of flow conditions that are different in forcing, Reynolds number or other parameters affecting the modes. Artificial Neural Network Estimation (ANNE) allows for real time monitoring of the time coefficients associated with these DPOD modes.

1 Introduction

The usefulness of POD in estimating and analyzing complex flow fields like wake flows is well established in literature. Two features of POD prove very advantageous when applied to flow field data: First, the optimality in terms of capturing most of the energy of the flow 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. And second, when applied to time periodic flow fields, the temporal coefficients represent the physical vortex shedding phenomena with good accuracy, and can therefore be used for feedback flow control (Siegel et al. 2003). However, a POD model is strictly speaking 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 and in some cases absolute nullification of the validity of the model. In order to address this problem, modified POD models have been proposed both by Noack et al. (2003) and Siegel et al. (2003). Both modifications improved the POD model by allowing adjustments of the mean flow using a shift or mean flow mode in addition to the POD derived modes themselves. These ad hoc modifications do capture the change in the mean flow reasonably well, but fail to address the spatial changes seen in the fluctuating modes. Therefore, a POD procedure that can adapt the fluctuating modes as well may be advantageous since it would maintain the validity of the model through transient flow situations. This has been realized by other research

106 S. Siegel et al.

groups as well, and different attempts have been made to interpolate sets of similar POD modes. For example, Luchtenburg et al. (2006) use the shift mode information in order to select a proper set of fluctuating modes from a larger data basis. The advantage of this approach is that the number of POD modes used for flow estimation at any point in time is the same as what was used as the initial POD basis. However, it is impossible to derive a low dimensional model from the associated time coefficients of these modes without further analysis and simplifying assumptions about the spatial changes in these modes. The reason is that the dynamic behavior of the slower mode changing phenomena is not represented by the time coefficients that may be derived for this type of decomposition.

In this work, we seek to develop a low dimensional POD basis that spans transient data sets with the least possible modes, and includes the entire dynamic behavior of the flow in the resulting time coefficients. The first step we take in developing such a procedure is to explore the behavior of the POD method as the snapshot ensemble is reduced in size. This has been done by Siegel et al. (2005) using a short time POD procedure (SPOD) that leads to physically correct spatial modes. “Short” in this context refers to the length of the snapshot ensemble compared to a cycle of the fundamental frequency in the flow field. Siegel et al. (2005) explored the SPOD method by applying it to various transient data sets. The goal of this research was to find a better set of POD modes in order to obtain an accurate global estimate of the flow field. It was demonstrated that SPOD delivers accurate spatial POD modes for snapshot ensembles as short as one shedding cycle. This capability can be used to analyze transient flow situations in great detail by deriving POD bases for each individual shedding cycle. However, this results in as many POD bases as there were shedding cycles in the original data set, which makes the resulting assembly of bases relatively large and in general unusable for both real time estimation and low dimensional model building. The SPOD mode basis is therefore very similar to the “tunable model“ described by Luchtenburg et al. (2006).

In this effort, we build on the success of the SPOD method and develop means to reduce the many SPOD bases in an effective manner. This process we refer to as Double Proper Orthogonal Decomposition or DPOD, since the POD decomposition is applied twice in order to arrive at the DPOD basis. We demonstrate the accuracy of the resulting basis for the estimation of transient flow fields, and discuss features that make it particularly suitable for flow field estimation and low dimensional model building, as well as feedback flow control. While we demonstrate the DPOD procedure on a single transient data set, the method can be applied to an arbitrarily large number of SPOD bases that may include actuation effects, Reynolds number changes and other parameter variations. The use of both linear and nonlinear projection methods to perform sensor based flow state estimation is demonstrated.

For low-dimensional control schemes to be implemented, a real-time estimation of the modes present in the flow is necessary, since it is not possible to measure them directly especially in real-time. After the time histories of the temporal coefficients of the POD model are determined using the DPOD procedure, the estimation of the low-dimensional states is needed for closing the loop. Sensor measurements may take the form of wake velocity measurements, as in this paper, or for an application be based on surface-mounted pressure measurements and/or shear stress sensors. For practical applications it is desirable to reduce the information required for estimation to the

State Estimation of Transient Flow Fields Using DPOD 107

minimum. In this research effort, an 8 sensor configuration is employed together with a non-linear dynamic estimator which is based on a Artificial Neural Networks design.

The remainder of the paper is structured as follows: Section II describes the numerical simulation of the Navier Stokes equations, using a computational fluid dynamics (CFD) solver. This is followed by a short description of the DPOD procedure in section III. Sections IV and V provide an extended discussion of the results for the SPOD and DPOD decompositions, respectively. Section VI demonstrates how the time coefficients of the DPOD model can be estimated from sensor readings using linear and nonlinear methods. Section VII provides conclusions of the current research. An outlook for future work is presented in section VIII.

2 D Shaped Cylinder Simulation

For the scope of this work, a D-Shaped two-dimensional bluff body geometry was chosen as a generic wake flow developing a von Karman vortex street. The geometry is a semi ellipse with an aspect ratio of 35:2. This geometry was chosen to provide room for actuator implementation in future wind tunnel studies. The reference length for Reynolds and Strouhal numbers is the base height, H. The Reynolds number based on H was 300. The wake can be controlled by two blowing and suction slots which are located at the rear corners of the body and are angled at 30 degrees to the free stream direction.

A structured, body fitted “D” grid with about 200,000 points is used for the CFD simulation (see Figure 1). This grid is stretched to cluster the grid points tightly in the shear layers of the near wake, as well as around the blowing and suction slots.

Uniform flow boundary conditions at the far field are imposed using Riemann invariants.

To ensure computational efficiency, the Mach number is set to M=0.1. The time step is ∆t=0.005d/U, where U is the free stream velocity. A previous study using this grid showed good comparison of the resulting Strouhal number with experimental results (Siegel et al. 2005b).

For the purpose of POD model development, the computational results are truncated in space to the bounding box shown in Figure 1. The bounding box extends

Fig. 1. Body Geometry, CFD gridpoints and POD bounding box


from 1H upstream of the base, to 9H downstream. In the vertical direction, the flow is truncated to ±2H. This region of interest encloses the vortex formation region entirely; outside of it, the flow is mostly steady. The bounding box contains about 40 000 grid points.

During the simulation, the flow is impulsively started from rest at time t=0s. As a result of the wake instability, the limit cycle oscillation known as the von Karman vortex street develops. This is shown in Figure 2 where it can be seen that the flow requires about 20 shedding cycles to develop a limit cycle oscillation as witnessed by the lift force.

1 2 3 4 56

78

910

11121314

1516

1718

1920

21

0 0.5 1 1.5 2 2.5 3

0

0.02

0.02

0.02

0.02

0.04

0.06

Simulation Time [s]

Lift

For

ce [N

]

Lift ForceSegmentation

Fig. 2. Unsteady Lift force during development of limit cycle and SPOD bin segmentation

The goal in modeling this flow field is both in the accurate representation of the flow energy, as well as in deriving spatial modes that capture the physical behavior with one or a pair of modes being attributable to a certain aspect of the physical flow behavior, like mean flow, von Karman vortex shedding modes, and higher order vortex shedding modes. This separation of the flow into individual modes representing physical phenomena is a key requirement for the development of feedback controllers that control these particular flow features.

3 Double POD Decomposition

The POD decomposition for a 2D scalar spatial field u evolving over J time steps as described by Holmes, Lumley and Berkooz (1996) is shown in equation (1).

1

( , , ) ( ) ( , )J

j jj

u x y t a t x yϕ=

= (1)

Here, aj are the temporal coefficients of the spatial modes ϕj. While the decomposition yields as many modes as there are snapshots in time in the original


data set, it is typically possible to truncate the POD model at a relatively low number of modes while retaining most of the energy of the original flow field. This can be done by either inspecting the energy distribution in the modes, or by inspection of the spatial modes that will not show any discernible structure beyond a certain mode number. If POD is performed on the flow field without subtracting the mean flow, the first mode will typically be the mean flow followed by large scale fluctuating modes, in the case of the cylinder wake the two modes representing the von Karman vortex street. Since in almost all cases the modes obtained by the decomposition described above have a physical interpretation, we will refer to them as the physical modes. The decomposition works particularly well for flow fields with large, time periodic features like the periodic vortex shedding in wake flows.

Siegel et al. (2005) have shown that for time periodic flows modes identical to those obtained from snapshot ensembles containing large number of shedding cycles can be obtained using snapshot ensembles of small integer number of cycles, down to a minimum of one shedding cycle. This allows for decomposition of a time evolving flow field into individual events the size of exactly one cycle of the frequency of interest:

( ) ( ) ( )

1

( , , ) ( ) ( , ).J

i i ij j

j

u x y t a t x yϕ=

= (2)

The result is a collection of I cycles in individual bins. Note also that these bins may contain a different number of samples in time, or span time intervals of different

length as the period of a cycle changes. Processing the spatial eigenfunctions ( )ijϕ of

these bins using POD (see eqn.(1)) yields insight into how the physical modes change as a result of, for example, transient forcing. Siegel et al. (2005) refer to this type of POD decomposition as short time POD or SPOD, due to its similarity to procedures like Short Time Fourier decomposition. While SPOD yields spatial POD modes that are identical to those obtained from many cycles, it is not as low dimensional as one would wish: The result of SPOD is one entire mode set for each period of the flow.

However, taking the concept of the “shift mode” one step further, we can now develop a “shift mode” for all physical modes j by applying the POD procedure to the

I POD mode sets ( )ijϕ ,i.e., the I physical mode sets ( )i

jϕ are now the “snapshots” for

POD. This gives the ansatz for double POD (DPOD)

( ) ( )

1 1

( , , ) ( ) ( , ).J K

i ijk jk

j k

u x y t t x yα= =

= Φ (3)

for all i. The resulting eigenfunctions can be truncated in both J and K in the same way as a regular POD decomposition. After orthonormalization, the decomposition is again optimal in the sense of POD. In the limit of K = 1, the original POD decomposition is recovered. While the different modes distinguished by the index j remain the physical modes described above, the index k identifies the transient changes of these physical modes: For K >1, the energy optimality of the POD decomposition in that direction leads to modes that are the optimum decomposition of a given physical mode as it evolves throughout a transient data set. If K=2 then modes Φ11 and Φ12 are the mean flow and the “shift mode” or “mean flow mode” as


described by Noack and Siegel, respectively. Thus the modes with indices k>1 can be referred to as first, second and higher order “shift” modes that allow the POD mode ensemble to adjust for changes in the spatial modes. We will refer to all of these additional modes obtained by the DPOD decomposition as shift modes, since they modify a given physical mode to match a new flow state due to either a formation length or recirculation zone length change. This may be due to effects of forcing, a different Reynolds number, feedback or open loop control or similar events. Thus, in the truncated DPOD mode ensemble for each physical mode one or more shift modes may be retained based on inspection of energy content or spatial structure of the mode.

We will now demonstrate how this DPOD procedure can be used to create mode ensembles that cover the entire unforced transient startup of the D-shaped cylinder wake. This mode ensemble will thus cover not just the limit cycle, but also the steady flow state before the onset of the vortex shedding. The latter is of particular importance since vortex shedding suppression control targets the steady flow state as the control goal.

4 SPOD Modes of Transient Startup

Figure 2 indicates the boundaries of the bins used to derive SPOD modes. 21 SPOD mode sets were derived, the individual bins are numbered in Figure 2. As may be expected, the POD modes derived from the bins closer to the start of the simulation are quite different than those derived from snapshot sets from the end of the simulation. Figure 3 compares the second POD mode from bins 4 and 18. The second POD mode is the first mode of a mode pair representing the von Karman vortex street, and contains about 45% of the fluctuating energy. The peak amplitude of this mode can be found about 6d downstream of the base for the mode derived from bin 4, and about 2 base heights downstream of the base for the mode derived from bin 18. This indicates that the vortices are forming further downstream at the beginning of the simulation, and move closer to the base as the limit cycle develops.

0 2 4 6 8 10 12

2

1

0

1

2

SPOD Mode 2 Bin 4

x/H

y/H

0 2 4 6 8 10 12

2

1

0

1

2

SPOD Mode 2 Bin 18

x/H

y/H

Fig. 3. SPOD Mode 2 for Bin 4 (left) and Bin 18 (right)

While Figure 3 demonstrates the change of spatial modes during this transient simulation for one mode only, similar effects can be seen for all other SPOD modes (not shown here). This illustrates the need to derive shift type modes not just for the mean flow, but also the fluctuating modes.


5 DPOD Model of the D-Shaped Cylinder

The DPOD procedure introduced in a previous section achieves just that: Derivation of shift type modes for all physical modes that are deemed of importance in a transient flow field. Figure 4 shows a DPOD mode set for the transient simulation introduced in Figure 2, retaining 5 physical modes and 3 shift modes. Modes (1,n) represent the mean flow, Modes (2,n) and (3,n) the von Karman vortex street, while Modes (4,n) and (5,n) represent a higher order vortex shedding mode with twice the spatial frequency of the von Karman vortex street. Inspecting the energy contents of these modes presented in Figure 5, it can be seen that most energy is contained in the mean flow, followed by the von Karman vortex street modes which appear as pairs of approximately equal energy. This behavior is identical to a conventional POD procedure performed on a time periodic flow field. The energy content of the entire mode basis is now a two dimensional energy plane, with steep energy dropoff both towards higher order physical and higher order shift modes. While the energy content dropoff of the shift modes of the mean flow mode (Modes (1,2) and (1,3)) is fairly steep, it can be seen that the dropoff for the shift modes of the von Karman modes (Modes (2,2), (2,3) and (3,2), (3,3)) is far less steep, demonstrating the importance of including these modes in a low order model. Since the dropoff in energy is not uniform for all shift modes, one might retain different number of shift modes for each physical mode based on energy considerations. The spatial distribution of the shift modes demonstrates how they manage to adjust the distribution of each physical mode to the changes in the vortex shedding pattern seen during the transient startup simultation: The first shift mode of the mean flow mode, for example, will extend the length of the recirculation zone which is what can be observed during the development of the vortex shedding limit cycle. This behavior is almost identical the artificially created shift modes suggested by Noack et al. (2003) or the so called mean flow mode introduced by Siegel (2003). A consequence of the much longer recirculation zone at the startup of the limit cycle is also that the vortices are forming further downstream of the body, which can be seen by inspecting the SPOD Modes shown in Figure 3. This effect is modeled mostly by the first shift mode of the two von Karman vortex shedding modes, i.e. Modes (2,2) and (3,2). These have a similar appearance than the von Karman vortex shedding modes themselves, but with their maximum mode amplitude shifted further downstream. The same effect can be seen for the higher order vortex shedding modes, where the maximum of modal activity of the shift modes (Modes (4,2) and (5,2)) is also further downstream than for the modes themselves (Modes (4,1) and (5,1)).

Since all of the spatial modes have been normalized to unity in magnitude, the temporal coefficients provide insight into the energy content as time evolves during the simulation. Moreover, the energy exchange between modes and their shift modes is also captured in these time coefficients. Thus it is of interest to inspect the time coefficients in order to understand the dynamics of energy exchange between the modes during the transient simulation. For example, the time coefficients of the two mean flow mode shift modes, Mode (1,2) and (1,3), reveal that these only contribute


0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (1,1)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (1,2)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (1,3)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (2,1)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (2,2)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (2,3)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (3,1)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (3,2)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (3,3)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (4,1)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (4,2)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (4,3)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (5,1)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (5,2)

x/H

y/H

0 2 4 6 8 10 12

−2

−1

0

1

2

DPOD Mode (5,3)

x/H

y/H

Fig. 4. DPOD Model of transient startup data using 5 physical POD modes (first index) and 3 Shift modes (second index)

1 2 3 4 5 6 7 8

10−6

10−4

10−2

100

Shift Mode Number

Mod

e E

nerg

y

Mode 1Mode 2Mode 3Mode 4Mode 5

Fig. 5. Energy Content of DPOD modes shown in Figure 4


0 1 2 3−7000

−6000

−5000Mode 1,1

0 1 2 3−2000

0

2000Mode 1,2

0 1 2 3−1000

0

1000Mode 1,3

0 1 2 3−1000

0

1000Mode 2,1

0 1 2 3−500

0

500Mode 2,2

0 1 2 3−200

0

200Mode 2,3

0 1 2 3−1000

0

1000Mode 3,1

0 1 2 3−500

0

500Mode 3,2

0 1 2 3−200

0

200Mode 3,3

0 1 2 3−500

0

500Mode 4,1

0 1 2 3−200

0

200Mode 4,2

0 1 2 3−200

0

200Mode 4,3

0 1 2 3−500

0

500Mode 5,1

0 1 2 3−200

0

200Mode 5,2

0 1 2 3−200

0

200Mode 5,3

Fig. 6. DPOD Time Coefficients of transient startup. Horizontal axis shows simulation time [s] corresponding to Figure 2, vertical axis non-dimensional mode amplitude correlating with mode energy, since all spatial modes are normalized to unity magnitude.

at the beginning of the limit cycle development, before about 1.5 s into the simulation. For the time periodic portion of the simulation, their contribution approaches zero. The von Karman shift modes show a different behavior: Their activity has a peak around 0.9 seconds, and a minimum at about 1.5 seconds. It can be shown that their phase with respect to their physical mode (Mode (n,1)) changes at that time by about 180 degrees and as a result they add to the main mode’s activity for one portion of the simulation, while they decrease the activity for the other portion of the simulation. In combination with their different spatial distributions, the observed shift in the shedding location is thus modeled correctly by the vortex shedding and their shift modes. The physical mode together with its shift modes thus offer a possibility to develop for example a multi input single output controller that adjusts its feedback


parameters in accordance with the vortex shedding location shift: By applying different gains to the mode and shift modes, the feedback can be tuned to the required phase at any time.

6 Sensor Based State Estimation

The time histories of the temporal coefficients of the POD model are determined by introducing the spatial Eigenfunctions into the flow field data using the least squares technique. For closed-loop control, we are interested in a system that maps, in real-time, velocity measurements provided by the sensors onto the estimates of the fifteen temporal modes. The estimation scheme predicts the temporal amplitudes of the first fifteen POD modes from a finite set of velocity measurements obtained from the CFD solution of the “D” shaped cylinder wake. For each sensor configuration, 341 velocity measurements were used equally spaced at 0.01seconds apart. Of the 341 snapshots, the first 170 were used for training of the estimator, whereas, the final 171 snapshots were used for validation purposes. Only data concerning velocity components in the direction of the flow were used for the sensor placement and number studies reported in this effort. The main approach in this effort is the incorporation of a non-linear dynamic estimator. The decision was to look into universal approximators, such as artificial neural networks (ANN), for their inherent robustness and capability to approximate any non-linear function to any arbitrary degree of accuracy. The ANN employed in this effort, in conjunction with the ARX model is the mechanism with which the dynamic model is developed using the POD time-coefficients extracted from the high resolution CFD simulation. Non-linear optimization techniques, based on the back propagation method, are used to minimize the difference between the extracted POD time coefficients and the ANN while adjusting the weights of the model (Nørgaard et al., 2003). The main hypothesis is that the non-linearity and scaling characteristics of the temporal coefficients lead to numerical stability issues which undermine the development and analysis of effective estimation/control laws. In order to assure model stability, the ARX dynamic model structure is incorporated. This structure is widely used in the system identification community. A salient feature of the ARX predictor is that it is inherently stable even if the dynamic system to be modeled is unstable. This characteristic of ARX models often lends itself to successful modeling of unstable processes as described by Nelles (2001). The artificial neural network (ANN) has the following features:

• Input Layer: Eight sensor signals, n (n = 8), namely, U-Velocity at the given sensor locations. In addition to these readings, in order to obtain a strong representation of the dynamics of the system, the input layer includes 8 past inputs and 15 time delays. No past outputs were used. The eight sensors are placed on the maxima/minima of the first eight modes as described by the heuristic procedure developed by Cohen et al. (2006). The total number of inputs to the net is as follows:

# inputs to ANN = # time delays *[ # past outputs + (# past inputs per sensor)] + bias


By inserting the values chosen after a brief sensitivity study, we obtain:

# inputs to ANN = 15*[0 + 8] + 1 = 121

From the above relation, we can see that the number of inputs is 121 for all the cases irrespective of the number of sensors.

• Hidden Layer: One hidden layer consisting of 15 neurons. The activation function in the hidden layer is based on the non-linear tanh function. A single bias input has been added to the output from the hidden layer.

• Output Layer: Fifteen outputs, namely, the 15 states representing the temporal coefficients of the 15 mode POD reduced order model developed in the previous Section. The output layer has a linear activation function.

• Weighting Matrices: The weighting matrices between the input layer and the hidden layer (W1) and between the hidden layer and the output layer (W2)depend on the number of sensors. In this effort, W1 is of the order of [121*15] and W2 is of the order of [15*16]. These weighting matrices are initialized randomly.

• Training the ANN: Back propagation, based on the Levenberg-Marquardt algorithm, was used to train the ANN using Nørgaard et al’s (2003) toolbox. The training procedure converged near 350 iterations. Generally speaking, the training data fits well and this should not be very surprising. Results for the training data are enclosed in Fig. 7.

• Validating the ANN: The validation data is as described in Table 1. Results for the validation data are enclosed in Fig. 8.

Table 1. RMS Errors [%] for ANNE Estimates

Mode RMS of Training Error [%]

RMS of Validation Error [%]

1,1 0.12 0.12 2,1 0.84 0.93 3,1 0.47 0.63 4,1 3.0 2.4 5,1 2.4 3.0 1,2 1.8 2.5 2,2 2.3 6.3 3,2 4.1 5.6 4,2 16.2 18.5 5,2 11.2 10.5 1,3 2.5 5.7 2,3 11.4 8.7 3,3 9.7 12.6 4,3 27.0 80.3 5,3 67.2 94.5


0.5 1 1.5 2 2.5−5600

−5400

−5200

Mode 1,1

0.5 1 1.5 2 2.5−1000

0

1000Mode 1,2

0.5 1 1.5 2 2.5−1000

0

1000Mode 1,3

0.5 1 1.5 2 2.5

−2000

200

Mode 2,1

0.5 1 1.5 2 2.5

−2000

200

Mode 2,2

0.5 1 1.5 2 2.5−500

0

500Mode 2,3

0.5 1 1.5 2 2.5−500

0

500Mode 3,1

0.5 1 1.5 2 2.5

−2000

200

Mode 3,2

0.5 1 1.5 2 2.5

−100

0

100

Mode 3,3

0.5 1 1.5 2 2.5−200

0

200Mode 4,1

0.5 1 1.5 2 2.5−200

0

200

Mode 4,2

0.5 1 1.5 2 2.5−200

0

200Mode 4,3

0.5 1 1.5 2 2.5−200

0

200Mode 5,1

0.5 1 1.5 2 2.5−50

0

50Mode 5,2

0.5 1 1.5 2 2.5−50

0

50Mode 5,3

Fig. 7. ANNE Estimates for Training Data Set. Horizontal axis shows simulation time [s] corresponding to Figure 2, vertical axis non-dimensional mode amplitude correlating with mode energy, since all spatial modes are normalized to unity magnitude.

2.5 3 3.5 4−5550

−5500

−5450Mode 1,1

2.5 3 3.5 4−1000

0

1000

Mode 1,2

2.5 3 3.5 4

−1000

0

1000

Mode 1,3

2.5 3 3.5 4−500

0

500Mode 2,1

2.5 3 3.5 4−500

0

500Mode 2,2

2.5 3 3.5 4200

300

400Mode 2,3

2.5 3 3.5 4−200

0

200Mode 3,1

2.5 3 3.5 4−200

0

200Mode 3,2

2.5 3 3.5 4−100

0

100Mode 3,3

2.5 3 3.5 4−100

0

100Mode 4,1

2.5 3 3.5 4−100

−50

0Mode 4,2

2.5 3 3.5 4−100

0

100Mode 4,3

2.5 3 3.5 4−100

0

100Mode 5,1

2.5 3 3.5 4

−20

0

20

Mode 5,2

2.5 3 3.5 4

−10

0

10

Mode 5,3

Fig. 8. ANNE Estimates for Validation Data Set. Horizontal axis shows simulation time [s] corresponding to Figure 2, vertical axis non-dimensional mode amplitude correlating with mode energy, since all spatial modes are normalized to unity magnitude.


7 Discussion

We introduce Double Proper Orthogonal Decomposition (DPOD) as a means to derive POD spatial modes that span different flow conditions. We demonstrate its ability by applying it to a transient simulation of the development of the limit cycle of a D shaped cylinder wake.

8 Outlook

The derivation of valid spatial modes and temporal coefficients constitutes the first step towards developing a set of low dimensional equations capturing the dynamic behavior of a flow field. Both linear and nonlinear techniques, such as system identification and Galerkin projection may be used to do so, and we will explore their efficiency for this task in the future.

While we demonstrate the capabilities of DPOD by applying it to a transient flow field which in essence consitutes one parameter change, DPOD may also be used to model the effects of several different parameters on a flow field, like the impact of changes in actuation frequency and amplitude, or Reynolds number. Thus it is a generic technique that can be applied to problems of any degree of complexity, not just the relatively simple sample flow field that was used in this work.

Acknowledgments

The authors would like to acknowledge funding by the Air Force Office of Scientific Research, LtCol Sharon Heise, Program Manager. We would also like to thank Dr. Jim Forsythe of Cobalt Solutions, LLC for their support. The authors appreciate the fruitful discussions with Dr. Young Sug-Shin on system identification using Artificial Neural Networks and for his very helpful insight.

References

Noack, B. R.; Afanasiev, K.; Morzynski, M.; Tadmor, G., Thiele, F., 'A hierarchy of low-dimensional models for the transient and post-transient cylinder wake', J. Fluid Mechanics, 497, 2003

Luchtenburg, M., G. Tadmor, O. Lehmann, B.R. Noack, R. King, M. Morzyinski, 'Tuned POD Galerkin models for transient feedback regulation of the cylinder wake', 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA-2006-1407, 2006

M. Bergmann, L. Cordier, and J.-P. Brancher, 'Optimal rotary control of the cylinder wake using POD reduced order model', 2nd AIAA Flow Control Conference, AIAA 2004-2323, 2004

Siegel, S.; Cohen, K.; McLaughlin, T., 'Feedback Control Of A Circular Cylinder Wake In Experiment And Simulation (invited)', 33rd AIAA Fluid Dynamics Conference, Orlando, AIAA 2003-3569, 2003


Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Short Time Proper Orthogonal Decomposition for State Estimation of Transient Flow Fields', 43rd AIAA Aerospace Sciences Meeting, Reno, AIAA2005-0296, 2005

Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Two Dimensional Simulations Of A Feedback Controlled D-Cylinder Wake', AIAA Fluid Dynamics Conf Toronto, ON, CA, AIAA 2005-5019, 2005

Holmes P., Lumley, J. L., Berkooz, G., 1996, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press

Cohen, K., Siegel S., and McLaughlin T.,"A Heuristic Approach to Effective Sensor Placement for Modeling of a Cylinder Wake”, Computers and Fluids, Volume 35, Issue 1, January 2006, pp. 103-120.

Nelles, O., Nonlinear System Identification, Springer-Verlag, Berlin, Germany, 2001, Chap. 11. Nørgaard, M., Ravn., O., Poulsen, N.K., Hansen, L.K., Neural Networks for Modeling and

Control of Dynamic Systems, 3rd printing, Springer-Verlag, London, U.K., 2003, Chap. 2.

Large Eddy Simulations of Flow Over a Circular Cylinder Using Unstructured Grids

Documents

Transcript of Large Eddy Simulations of Flow Over a Circular Cylinder Using Unstructured Grids