Use of proper orthogonal decomposition to visualize...

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

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Use of proper orthogonal decomposition to visualize coherent structures from

time resolved PIV data

Mario Jensch1, Martin Brede1, Alfred Leder1, Octavian Frederich2, Frank Thiele2

1: Chair of Fluid Mechanics, University of Rostock, Rostock, Germany, [email protected]

2: Institute of Fluid Mechanics and Engineering Acoustics, Berlin Institute of Technology, Berlin, Germany, [email protected]

Abstract The results presented arise from a collaborative study of the flow around a circular cylinder of finite length with an aspect ratio of two at a Reynolds number of ReD = 200000. The study combines both measurements and numerical simulations. The measurements were performed using time resolved stereo particle image velocimetry, whereby large eddy simulation is used for the numerical simulation. Additionally the proper orthogonal decomposition technique (POD) is applied to both datasets and a linear mapping between the spatial POD modes of the measured parallel planes is introduced. The results include a 3D reconstruction of the POD modes in the unsteady wake of the cylinder. 1. Introduction The three dimensional flow around a bluff body is a standard topic in fluid mechanics. Regarding the wake of a cylinder with low aspect ratio the list of previous experiments is very limited and mostly contains steady state results. A detailed review of the research on low aspect ratio cylinders has been given by Pattenden (2005). Results of LDA experiments have been discussed by Leder (2003). With the introduction of the time resolved PIV measurement technique, new measurement capabilities are available. The unsteady flow can be obtained now, covering an entire 2D flow field. From this the problem arises that a large amount of data has to be analyzed. One means to achieve a better understanding of the phenomena in unsteady flows is to reduce the data using the proper orthogonal decomposition technique (Lumley 1970) based on the work of Karhunen (1946) and Loéve (1955). A given flow will be decomposed into orthogonal eigenmodes using the snapshot method of Sirovich (1987), which not only contains information of basic flow structures but also of the energy content. Experimental investigations were performed for the turbulent separated flow around a circular cylinder of finite length with one end fitted to an end plate at a Reynolds number of ReD = 200000. The flow was investigated by means of time-resolved stereo Particle Image Velocimetry (TR-PIV) and numerical simulation using the LES approach. The proper orthogonal decomposition method for experimental data reveals valuable new information about the coherent structures of the unsteady flow. 2. Experimental setup A time-resolved stereo PIV employing two high speed CMOS cameras with Scheimpflug optic and a frequency doubled Neodym-YAG laser is used to perform the measurements. The TR-PIV system allowed a frame rate of 2000 Hz over 2.7 seconds, using a camera resolution of 1024x768 pixels.


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For the experimental investigations a Göttingen type wind tunnel with a 0.6 x 0.6 m² open test section has been used. It is operated at a flow speed of U∞ = 26 m/s. The flow setup in the measurement section is given with geometric quantities in Figures 1 and 2. The Reynolds number based on the diameter of the cylinder is 200000, which is the critical value for transition of the boundary layer flow. Additionally, the transition of the plate’s boundary layer is fixed by a trip wire close to the rounded leading edge. The measurements were performed in vertical planes aligned with the mean flow with a spacing between the planes of 0.1D in the lateral direction. The observation area of each plane covers 220 mm x 124 mm with interrogation areas of 32 x 32 pixels and an overlap of 50 %. 3. Proper orthogonal decomposition The starting point for the proper orthogonal decomposition is the snapshot ensemble

( ) ( ): ,mmt=u x u x of a velocity field ( ){ }

1

Mm

m=u x at M discrete time steps tm. The key is to find an

orthonormal system of modes ( ){ } ( )0

Ni i

N M=

≤u x that describe the field with the Galerkin approximation

[ ] ( ) ( ) ( )0

, :N

Nm i m i

it a t

=

=∑u x u x (1)

in an energetic optimal sense, where 0 1a = and 0 =u u describe the time average flow. The two main elements in the POD snapshot method are the inner product of the velocity variation around the time average

Cylinder geometry diameter D = 120mm

length L = 2D trip wire position x0 = -1.5D

End plate geometry b = 5.833 D

l = 10.833 D d = 0.15 D

Flow parameter ReD = 200000

flow speed U∞ = 26m/s turbulence intensity Tu = 0.5%

Figure 1 (above): Confi-guration of the experimental setup. Figure 2 (left): Experimental setup in the wind tunnel.

U∞


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( )u ,v : u vi i i idΩ

Ω

= Ω∫ (2)

and the correlation matrix

( ) ( )1 j mjmC d

M Ω

= ∫u x u x x , (3)

which comprises the second order structural information of the unsteady flow (Sirovich 1987). 4. Results

Using the POD for the time resolved stereo PIV data, it is possible to investigate separately different coherent structures in the flow. Generally, all POD modes derived from the velocity vector field represent the deviation from the mean flow. The unsteady turbulent wake is recognized by the large amount of POD modes necessary to capture the complete amount of energy concerning the unsteady flow. For example, the first seven modes resolve only 38 % of the fluctuation, but 100 modes are necessary to resolve approximately 80 %. Figure 3 (left) shows the vector field of the first POD mode in the measurement plane y = 0.5D, with incident flow from the left. The vectors show the relative velocity field of the in-plane components u and w, whereas the out-of-plane component v is displayed by colors. It can be seen that a fully three dimensional vortex structure is being shed. The spectral analysis of the first time

coefficient shows a periodicity of the first mode at a Strouhal number DSr fU∞

= of 0.16. From

this the primary vortex street can be identified as the dominant coherent structure in the wake of the finite cylinder, although the Strouhal number is lower than values given in the literature of about Sr = 0.2 for circular cylinders between two plates (Roshko 1954). From the complex three dimensional vortex structures, the problem arises that the velocity vector fields from measurements are only given as the POD modes in planes, so that they can only be investigated separately from each other.

Figure 3: Vector field of the first POD mode in the measured plane y/D = 0.5 (left) and a spectral analysis

of the corresponding Fourier coefficient (right).


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Mode #1 before mapping Mode #1 after mapping

y = -0.5D

y = -0.4D

y =-0.3D

y =-0.2D

y =-0.1D

Figure 4: Contour plot from the out-of-plane velocity component of the first POD mode before (left) and after mapping (middle) in different lateral planes (right).


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The plots tabulated in Figure 4 show the out-of-plane velocity component of the first POD mode at different measurement planes, starting at the tangential plane to the cylinder y = -0.5D down to y = -0.1D. The left column shows the results obtained from POD, containing the maximum energy of the unsteady flow. It can be recognized, that the plane y = -0.5D (top left position) does not contain the same structures as the other planes (remainder of the left column). The reason for this is that every plane is decomposed on its own and the modes are sorted by the energy content of the measured plane and not of the whole velocity field of the wake. A different mode is therefore found with the maximum energy content. As a matter of fact a structure similar to that in the adjacent plane y = -0.4D can be found in mode two in the plane y = -0.5D. Furthermore, the first POD modes calculated in the planes y = -0.2D and y = -0.1D look similar, but have the opposite sign because of their time coefficients. For investigating the modes in space the approach is made that parallel planes that contain the same coherent structures ought to share non orthogonal modes even if they do not have the same mode number or sign. For mapping the modes a linear mapping is introduced between the modes of the vector velocity fields of two neighboring measurement planes.

( ) ( )1 1,2 2i ij jC=u x u x (4)

Where ( )1iu x represents the calculated POD modes of one plane and ( )2

ju x the modes of the

parallel neighboring plane. The operator 1,2ijC gives the degree of linearity between the different

modes of both planes. On the one hand it is now possible to calculate the modes of plane one through convolution of C with the modes of plane two, or by convolving its inverse C− with modes from field one giving the modes of plane two. On the other hand it is a filter which identifies the same structures in different planes even if they have a different mode number or sign. By analyzing the matrix elements of 1,2

ijC , coherent modes and their sign can be identified.

Table 1. Rounded coefficients C form plane y=-0.1D, y=-0.2D -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0

Table 1 shows the rounded matrix elements of C obtained from the first six POD modes, which belong to the planes y = -0.1D and y = -0.2D. A value of 1 denotes that in both planes the same structure exists with the same alignment, whereas a value of -1 identifies the same structures with opposite alignment. This connection is pointed out by the first element in Table 1 and the corresponding contour plots in Figure 4 (left). Furthermore the element 3,5C describes that mode 3 in plane y = -0.1D corresponds to mode 5 in plane y = -0.2D. A value of zero implies no coherence. Starting at a reference plane it is now possible to map the corresponding modes by filtering all planes. The resulting set of modes is not necessarily sorted by the energy content of each measured plane but sorted by the energy content of the reference plane. The plots in the middle of Figure 4 depict the out-of-plane velocity component from the mapped first modes for different measured planes from y = -0.5D down to y = -0.1D with the reference plane chosen at y = -0.1D. It is clear that the same structure is identified in all measured planes with the same alignment. Because of a very complex flow topology the method requires small distances between measured planes. The plots tabulated in Figure 5 show the results from the mapped POD modes from the


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experimental PIV data connected to a 3D volume and the POD modes computed from numerical data. The plots obtained from experimental data show isosurfaces representing a value of

0.3iu U∞= ± whereas the isosurfaces based on the numerical data represents a value of 0.15iu U∞= ± , and red indicates positive and blue indicates negative values. Clearly the rearranged

modes compose a smooth ensemble that allow the calculation of isosurfaces but do not correspond to the results based on numerical simulation. The basic problem is the limited field of view of the dataset from the measurements and also the numerical simulations. The velocity field outside of the measured area cannot be considered in the experimental data but must obviously have an account for the POD modes in the case of the numerical data. Frederich et al. (2007a) show that a reduction of the numerical data to the experimentally measured planar area results in similar modes for both and point out (2007b) that size and position of the decomposed domain is an important issue. The findings have to be analyzed in more detail in the next future.

Figure 5: First (line one) and second (line three) POD mode rearranged from TR-PIV data and computed from LES data (line two and four) with incident flow from the top.


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5. Conclusions and Outlook Proper orthogonal decomposition has been applied to experimental data sets based on time resolved stereo PIV. The measurements were performed in the turbulent separated wake of a circular cylinder of finite length at high Reynolds number. A new method for arranging the POD modes was presented, showing valuable new results for investigating the spatial distribution of coherent structures in the wake. Although the modes differ from the results of the numerical data due to the limited field of view, the method is clearly suitable for the detection of coherent structures. In the ongoing work the correlation method described will be validated by a POD weighted on the reduced flow field as suggested by Frederich et al. (2007b). Acknowledgements Financial support for this work was provided by the German Research Foundation (DFG) in the project 1147 “Imaging based Measuring Methods for Fluid Flow Analysis”. References Frederich, O., Wassen, E., Thiele, F., Jensch, M., Brede, M., Hüttmann, F., Leder, A. (2007a) Numerical simulation of the flow around a finite cylinder with ground plate in comparison to experimental measurements. Notes on Numerical Fluid Mechanics and Multidisciplinary Design Vol. 96, Springer, Berlin, Heidelberg, New York. Frederich, O., Scouten, J., Luchtenburg, M., Thiele, F. (2007b) Database Variation and Structure Identification via POD of the Flow Around a Wall-Mounted Finite Cylinder. Proceeding of the Fifth Conference on Bluff Body Wakes and Vortex-Induced Vibrations. Leder, A. (2003) 3D-flow structures behind truncated circular cylinders. Proceedings of FEDSM’03, Fourth ASME-JSME Joint Fluids Engineering Conference, Honolulu, USA. Lumley, J. (1970) Stochastic tools in turbulence. Academic Press, New York Loéve, M. (1955) Probability Theory, Van Nostrand, Princeton, New Jersey. Karhunen, K. (1946) Zur Spektraltheorie stochastischer Prozesse, Annaleas Academiae Scientiarum Fennicae, Vol. 34. Pattenden, R. J., Turnock, S. R., Zhang, X. (2005) Measurement of the flow over a low-aspect-ratio cylinder mounted on a ground plate. Experiments in Fluids, Vol.39, p. 10-21. Roshko, A. (1954) On the development of turbulent wakes from vortex streets. NACA Report 1191. Sirovich, L. (1987) Turbulence and the dynamic of coherent structures part 1-3. Quarterly of Applied Mathematics, Vol. 45, No. 3, p. 516-590.

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