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Vorticity characterization in a turbulent boundary layer using PIV and POD analysis

Vorticity characterization in a turbulent boundary layer using PIV and POD analysis
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  4th International Symposium on Particle Image VelocimetryG¨ottingen, Germany, September 17-19, 2001 PIV’01 Paper 1184 Vorticity characterization in a turbulent boundary layerusing PIV and POD analysis A. Liberzon, R. Gurka, G. Hetsroni Abstract We investigate coherent structures in turbulent boundary layer flow using Particle ImageVelocimetry(PIV) technique. Flow coherent patterns have been identified and characterizedusing Proper Orthogonal Decomposition technique, applied to the instantaneous vorticityfields. Spatial characteristics, previously reported in the literature, such as non-dimensionallength scale ( x + ≈  1000), and the average expansion angle ( ∼  9 ◦ ), have been found fromlinear combination of three first POD modes. 1 Introduction Wall bounded turbulent flows have been extensively discussed in the literature regarding thecoherent structures in the near-wall region (see for example Panton, 1997). The investigation of the mechanisms that produce and control this phenomenon in a turbulent boundary layer requiresthe identification of the coherent patterns.Vorticity is one of the main parameters that characterize coherent structures and it plays adominant role in the dynamics of turbulence in general and in the energy cascade in particular(Tsinober, 2000). A high spatial resolution demand for the vorticity calculation has been achievedusing Particle Image Velocimetry (PIV) measurement technique (Raffel et al., 1998).It had been recognized for a long time, that raw data does not help to characterize the coherentstructures. Without subjective conditional sampling techniques, usually applied to measurementdata, neither vorticity nor another flow characteristic parameter, would provide a mechanism todistinguish between the coherent and non-coherent parts of the flow.In the present work, the unbiased statistical analysis method, Proper Orthogonal Decompo-sition (POD) (Berkooz et al., 1993) has been applied to the PIV data measured in a flume. Inaddition to the common POD analysis of the velocity maps, we propose to analyze the modesof the instantaneous vorticity field in order to extract the most energetic parts of the vorticityfluctuations. 2 Experimental analysis The investigated flow field is a turbulent boundary layer in a flume. The flume is made fromglass and has dimensions of 4 . 9 × 0 . 32 × 0 . 1  m . A schematic diagram of the configuration is shownin figure 1. The Reynolds number was 20000 based on the water height. The velocity field wasmeasured 2.5 m from the entrance and the field of view was 0 . 1 × 0 . 1  m  at the wall-normal (X-Y)plane. A PIV system was utilized for this work. Hollow glass sphere particles were seeded with anaverage diameter of 10  µm . A choice of 32 × 32 pixels square interrogation areas, a 100  µm/pixel A. Liberzon, R. Gurka, G. Hetsroni, Mechanical Engineering Department, Technion,Haifa, IsraelCorrespondence to:Roi Gurka, Mechanical Engineering Department, Technion, Haifa 32000, Israel.E-mail: Tel: +972-48-29-38611  PIV’01 Paper 1184 Figure 1: Schematic drawing of the experimental setupratio, and repeating the PIV analysis every 8 pixels resulted in 11500 vectors in a given field of view. The calculations were based on 150 image pairs resulting in 150 vector maps. 2.1 Vorticity field calculation The calculation of a vorticity field based on PIV velocity measurements requires the applica-tion of numerical differentiation scheme. In this work, five numerical algorithms of the vorticitycalculation (see e.g. Raffel et al., 1998) were tested using Monte-Carlo numerical simulations basedon two-dimensional Taylor-Green vortex, shown in figure 2. The instantaneous flow field of theTaylor-Green vortex flow is a good choice to evaluate the vorticity and derivatives calculationalgorithms because it includes strong rotation. The simulation results show that the least squaremethod is the most robust and accurate for PIV data. Moreover, this second order differencescheme provides the smallest error in instantaneous vorticity fields. Figure 3 presents the rel-ative error propagation versus additive noise levels (i.e., 2%, 5%, and 7.5% from r.m.s. of theTaylor-Green velocity magnitude) for three number of simulation runs.In order to investigate coherent structures in shear layers one probably needs the vorticityfluctuations that are calculated by removing the average vorticity field. The average vorticity isrelated as a background masking noise and has to be removed from each instantaneous vorticitymap. Therefore, the accuracy of the averaging is critical and it was tested by comparison betweenthe averaged vorticity based on the averaged velocity and by averaging the instantaneous vorticityfields. Figure 5 demonstrates that there is no difference between these two average vorticity fields. 3 Identification analysis 3.1 Proper Orthogonal Decomposition Proper orthogonal decomposition (POD), has been proposed by Lumley (1970) as a methodthat could provide an unbiased identification of coherent structures in turbulent flow. This tech-nique is based on second-order statistical properties which result in a set of optimal eigenfunctions.2  PIV’01 Paper 1184 Figure 2: Taylor-Green vortex flow fieldFigure 3: Relative error of the vorticity value versus different levels of the additive noise for  N   =a) 100, b) 500, and c) 1000 numerical runs.3  PIV’01 Paper 1184 Specially, the method makes use of eigenvalues and eigenvectors of the covariance matrix, indicat-ing the dispersion of the variable distribution,The covariance matrix of random variable X is approximated by: C   = 1 M  M   i =1 ˆ X i  ˆ X T i  (1)where ˆ X are defined as the fluctuation vectors around the average, ¯ X .Due to computational problems, the covariance matrix has been estimated using method of snapshots, as described by Sirovich (1987a). Using this method, the covariance matrix becomes: C  ij  =   ˆ X i ,  ˆ X j  , i,j  = 1 ,...,M   (2)where  · , ·  denotes the usual Euclidean inner product. The matrix is now  M   ×  M   instead of  N   × N   and, assuming  M < N  , we can compute its eigenvalues and eigenvectors more easily. Sincethe covariance matrix is symmetric, we know that its eigenvalues,  λ i , are nonnegative and itseigenvectors, φ i , i  = 1 ,...,M  , form a complete orthogonal set. The orthogonal eigenfunctions of the data are defined as:Ψ [ k ] = M   i =1 φ [ k ] i ˆ X i , k  = 1 ,...,M   (3)where  φ [ k ] i  is the  i -th component of the  k -th eigenvector. Lumley refers to these eigenfunctions ascoherent structures of the data. Whether or not they appear as spatial structures in a laboratoryexperiment is questionable Sirovich (1987b). Nevertheless, there is a common agreement that theywill be present at least indirectly. One of the possibilities is that a coherent pattern will consistof a linear combination of eigenfunctions.The “energy” of the data is defined as being the sum of the eigenvalues of the covariancematrix: E   = M   i =1 λ i  (4)To each eigenfunction we assign an energy percentage based on the eigenfunction’s associatedeigenvalue: E  k  =  λ k E   (5)Using only the first  K   ( K < M  ) most energetic eigenfunctions, we can construct an approximationto the data: X ≈  ¯ X + K   i =1 a i Ψ [ i ] (6) 3.2 POD and vorticity Usually, POD analysis is applied to velocity fields under the assumption that the decomposedvelocity eigenfunctions could be referred as coherent structures of the data. This straightforwardPOD analysis of the PIV data in the present work has not provided additional information aboutthe coherent part of the flow in the flume. Nevertheless, the first modes accurately recover theinstantaneous velocity fields (see figure 4. Assuming that the vorticity in turbulent flows is more4  PIV’01 Paper 1184 Figure 4: a) Original and b) reconstructed velocity magnitude fields of an instantaneous map.meaningful property in characterization of coherent structures then the velocity (e.g., Hunt andVassilicos, 2000) we proceeded to the investigation of the vorticity fluctuations.The combination of POD and vorticity is not straightforward and requires clarification. Similarto the interpretation of the POD modes as the energetic part of the flow for velocity fields, onecan relate the first decomposed vorticity modes as the energetic part of the instantaneous vorticityfield. In the present work we subtracted the average part from each instantaneous vorticity map,following by consequential POD analysis. Both reconstructed vorticity map and linear combinationof the first three modes are shown in the following section. 4 Results The averaged vorticity is shown in figure 5. Beside of the clear evidence of three regions of theflow (outer, entrainment and near wall layers, from top to bottom), it is not possible to extract anyadditional information regarding the coherent part of the flow. Figure 6 presents one instantaneousvorticity field together with a reconstructed vorticity field using ten POD modes. Although, fewsmall features (probably included in higher modes) do not appear in the reconstructed field, theoverall map could be successfully reconstructed by only several first modes. The linear combinationof three first POD modes of the vorticity fluctuations is shown in figure 7. Two large-scale coherentpatterns with positive and negative vorticity values are located side-by-side in the near-wall regionof the flow. Repeated measurements and analysis have shown the consistent spatial characteristicsof those features: the streamwise non-dimensional length ( x +  1000) and the average expansionangle (8 . 5 0 ).5
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