POD
 analysis of oscillating grid turbulence in water and shear thinning polymer solution

Present address Tom Lacassagne, Department of Mechanical Engineering, University College London, London, UK Abstract Oscillating grids are frequently used with water and Newtonian fluids to generate controlled turbulence and mixing. Yet, their use with shear thinning fluids still requires experimental characterization. Proper orthogonal decomposition (POD) is applied to PIV measurements of the flow generated by an oscillating grid in water and a shear thinning dilute polymer solution (DPS) of xanthan gum. The aims are to investigate the ability of POD to isolate periodic flow structures, and to use it to describe the effects of the shear thinning property. A dominance of the low order POD modes is evidenced in DPS. The methods applied in blade stirred tanks to identify oscillatory motion fail here. However, a strong mode coupling in the grid swept region is observed, determined by the working fluid and by an underlying chaotic nature of the flow. Possibilities of reconstructing turbulence properties using high order modes are discussed.


| INTRODUCTION AND BACKGROUND
Improving the understanding of the flow dynamics of non-Newtonian fluids is a current goal for many industrial and environmental applications. In particular, turbulence in non-Newtonian liquids occurs in several industrial applications for the food or pharmaceutical industries. 1,3 For example, bacteria fermentation for the production of vaccines requires the stirring of liquid media with a rheology transitioning from a low viscosity Newtonian behavior to a high viscosity, shear thinning and viscoelastic behavior. 4,5 The level of understanding has not reached that of turbulence in Newtonian fluid yet. This is mostly because of flow interactions with suspended matter giving the fluid its non-Newtonian properties are multiple and complex, and strongly depend on the microscopic nature and properties of this suspended matter. Nevertheless, it is of great importance to improve the understanding of turbulence in non-Newtonian media, since it influences other physical phenomena present in such industrial processes such as passive or reactive scalar mixing, cells growth, or multi-phase mass transfers.
The best way to achieve this, is through fundamental approaches, such as laboratory experiments or numerical simulations. In both type of studies, the situation is often simplified, in terms of geometry and mean flow features, compared to real life conditions, so that the focus can be made on turbulence alone, or turbulence interacting with other phenomenon. The three keys to a relevant experimental approach are the use of accurate measurement techniques, good modeling of the real-life fluid's complex rheology, and use of an efficient way to generate controlled turbulence with a less intense as possible mean flow. Optical techniques such as particle image velocimetry (PIV), are good candidates to fulfill the first condition. They further provide instantaneous and multi-point measurement of the liquid phase velocity, and allow to estimate many relevant turbulence properties (fluctuating velocities, length scales, time scales ...). They can however only be used in optically transparent solutions, which restricts the set of fluids that can be studied. Fluids that are not optically transparent need to be modeled, which appeals to the second condition listed above. Model solutions often consist in additives dissolved in Newtonian solvents.
Polymers are widely studied as flow additives because they are at the origin of the non-Newtonian properties of the fluid in many applications, and also because the wide range of existing polymers allows to use them to build model fluids reproducing the behaviors of other non-Newtonian media. 6 A well-known feature of dilute polymer solutions is their elasticity, which comes from the ability of polymer molecules to deform and exchange energy with the base flow, 7-10 and leads to the well-known drag reduction phenomenon, among others. 11,12 But polymer solutions also commonly exhibit shearthinning behaviours 13,14 : the viscosity of the solution decreases with increasing shear rate. This, for example, is known to have an impact on both oscillatory motions and turbulence in stirred tanks. 5,15 The last condition is most probably the hardest to meet: generating controlled turbulence, if possible with a negligible mean flow, has been the challenge of many research teams in the last 50 years, in Newtonian fluids to begin with, but also by extension in non-Newtonian fluids. Several experimental designs have been developed to try to achieve turbulence with moderate mean flow, for example arrays of randomly actuated jets, 16 rotating grids, 17 or "washing machine" experiments. 18 In this paper, we focus on one of the most easy to design and commonly used in water: oscillating grid turbulence (OGT). This method consists in making a rigid grid oscillate in a fluid initially at rest. It is usually said that the jets and wakes behind the grid's holes and bars interact to generate turbulence, [19][20][21][22] which then diffuses away from the grid. In other words, turbulence comes from the periodic shearing of the fluid by the grid.
Oscillating grid apparatus have been commonly used in experimental fluid mechanics since the seminal works of. 20,21 Numerous applications can be found in the literature, among which the study of interactions between turbulence and solid impermeable boundaries, 23,24 of turbulence and gas-liquid mass transfer at a free surface, 19,[25][26][27][28] in stratified media, 20,21,29,30 or to study the behavior of bubbles, cells, fibers, and flocculation aggregates suspended in a turbulent liquid phase. [31][32][33][34][35] Oscillating grid stirred tanks are thus tools that allow to generate and study controlled turbulence. They can be used as reference cases to help improve the understanding of conventional (bio)chemical reactors.
The most frequent configuration is that of a grid oscillating vertically in a rectangular section tank with a solid flat bottom and a free surface, and for which the properties of turbulence are studied either above or below the grid swept region, far enough from the liquid/gas or liquid/solid boundary condition. Thompson with n = 1, and where u 0 z and u 0 x are respectively the vertical and horizontal turbulent velocity fluctuations, and <. > rms denotes horizontal average (along x, on a −3M < X < 3M span) of ensemble rms values. From an historical perspective, the purpose of such setup has mainly been to generate controlled turbulence away from the grid. As a consequence, until recently, 24,40 no study had focused on oscillatory flows generated by the grid motion in and around its swept region, or on how energy is transferred from the oscillatory forcing to turbulence or mean flow.
The influence of non-Newtonian properties on oscillating grid turbulence has first been studied by Liberzon et al. 41 and Wang et al. 42,43 , who addressed questions on turbulence in viscoelastic solutions. 41 observed the propagation velocity of the boundary between turbulent and nonturbulent regions in the grid-stirred tank, at the first instants after the onset of the grid's oscillations.
They found that the turbulent/nonturbulent interface moved globally faster in dilute polymer solution than in water 42,43 later used a two oscillating grid system to study the viscoelastic effects of surfactants and dilute polymers on coherent structures of OGT. Using proper orthogonal decomposition (POD), they show that the addition of polymer tends to decrease the small scale effects of turbulence, and that this decrease can not only be attributed to the overall viscosity increase, since it is not associated with a decrease of the turbulent kinetic energy (TKE). Hence the non-Newtonian property of the fluid seems to strongly modify the turbulent spectrum. They show promising results for the use of this method to analyze OGT flow properties in complex fluids such as DPS. It is also worth noting that many studies on fixed (passive) grid turbulence in viscoelastic polymer solutions exist in the literature (see for example, 44,45 ), but as the shear forcing is not periodic and thus intrinsically different from the one considered in the present work, they will not be reviewed here in details for the sake of brevity.
The behavior of OGT acting in shear thinning solutions remained unexplored until two of our recent articles. 40,46 It has been showed in 46 that polymer addition tends to promote the mean flow's intensity and structure, and that Equation 1 and 2 remains valid for polymer solutions in the dilute entanglement regime with a concentration dependent n exponent. In Reference 40, the influence of polymer on oscillatory features of the flow around the grid have been described using a phase resolved measurements approach. Despite its suitability for the analysis of periodic motion, this phase averaged method required the collection of important quantity of data and an elaborate data processing. 40,47 It was also only focused on the grid neighborhood only, whereas the nonphase resolved measurements in Reference 46 studied the bulk flow without considering the near grid region.
POD has been recently used in stirred tanks to identify periodic flows induced by blade motions and study their interactions with mean flows and turbulence, 15,48,49 without having to resort to the previous phase averaged measurements. It is also now established as a quick and efficient tool for the study of turbulent flows. 48,[50][51][52] In this work, POD is thus applied to PIV measurements of OGT in water, and in a shear thinning xanthan gum (XG) solution at a 100 ppm concentration. In that way, we shall connect the results of our two existing studies in terms of both oscillatory flow and turbulence, and investigate several aspects related to OGT in water and dilute polymer solution (DPS) through the following key questions: • Can POD be used to identify organized motion (local coherent structures or flow periodicity), oscillating velocity fluctuations, and turbulence in an oscillating grid stirred tank?
where ρ is the fluid density (equivalent to that of water given the very small mass of polymer added) and μ the reference fluid viscosity. The  Viscoelasticity is checked to be negligible for such a XG concentration (elastic modulus G' and storage modulus G" have been measured by oscillatory rheological measurements, and G' is much lower than G" for C XG = 100 ppm), as expected. 55

| PIV measurements
Measurements are performed in the vertical central plane of the grid, in two regions: the close grid region (CG) centered around the grid swept zone (SZ), and the full tank region (FT) between the top of the grid stroke and the bottom of the tank (see Figure 1). PIV acquisition and processing parameters for close grid and full tank studies are indicated in Table 1.
Spurious vectors are removed from PIV fields by applying a threshold of 1.2 on the peak ratio, and replaced using median filtering.
It is checked that the proportion of replaced vector is always less than 10% of the total vector number.
The data from CG measurements has been used in Reference 40 for a triple decomposition analysis, and the data from FT measurements has been used in Reference 46 to study the mean flow and turbulence properties between the grid and the free surface. Note that in the CG region, the grid is visible on particle images. This made it possible in our previous work 40 Table 1 are similar to the microscales. This does not allow to evidence energy variations at large wave numbers characteristic of viscoelastic turbulence, and thus limits the analysis to the inertial effects and large scales.

| POD analysis
POD is a linear procedure that decomposes data, here a set of vector fields, into a modal base. It has been used only once to study oscillating grid turbulence 43 in a different setup than the present study (two grids). No conclusion on oscillatory flows in OGT has been drawn from this previous work.
The method applied here for obtaining velocity field decomposition is the snapshot method developed by Sirovich 57 and explained in detail in several articles. 5,49,50 Its principle is briefly recalled here.
The velocity field, is decomposed into a sum of temporal amplitudes θ (in m/s) and spatial modes ϕ (dimensionless): Where N is the number of modes needed to properly reconstruct instantaneous velocity fields, typically of the order of 400-1,000. 5 Instantaneous velocity fields obtained by PIV then have to be reorganized in the form of a sequence of snapshots. To do so, the (2 × R × C) × N snapshot matrix S M is written as a column-wise assembly of instantaneous velocity components, where R and C are the number of rows and columns of the two-components, two dimensions The POD algorithm then searches for the set of modes that gives the optimal approximation of each instantaneous velocity field in a least square sense, while satisfying the condition that each mode has to be orthogonal to all others. According to Reference 57 , this is equivalent to an eigenvalue problem and can be written as Where R = 1 N S M S M ð Þ T is the two point cross correlation matrix between velocity components, and λ (in m 2 /s 2 ) is the eigenvalue associated to mode ϕ. λ quantifies the energy content of its associated mode. It is used to sort out the modes from the most to the least energetic.
Once it is done, instantaneous velocity fields can be estimated by reconstruction up to a given number of modes using Equation (5) with an order of summation lower than N. This allows to suppress the small scale of turbulence contribution (high order modes), measurement noise (higher order modes), or even oscillatory motion contributions (coupled and/or oscillatory modes, see the next paragraph). Combination of mode can be used to isolate a relevant set of velocity scales' contribution, assuming that one can give a physical meaning to this collection of modes.
POD has been widely used since the end of the 2000's as an alternative to phase resolved measurements for the identification of trailing vortices and oscillatory flows in stirred tanks. 5,15,[48][49][50]58 Its main advantage is that it requires only a small number of velocity field sampled (N in Equation (5)) and no knowledge of the blade position for each recorded field, while phase resolved measurements require complex timing set-ups and the accumulation of much more instantaneous fields in order to achieve statistical convergence for each blade position. By analogy to this blade stirred tank case, applying POD to the oscillating grid flow should provide information on turbulence and coherent structures in a much quicker and less data-consuming fashion than the phase averaged measurements detailed in, 40 and identify modes that correspond to the oscillatory component of the flow. 5,15,49,59 This is one of the main interest of applying POD here. In this work, mean velocity is removed from instantaneous fields prior to POD decomposition, hence the mean flow correspond to mode number 0 (i.e., it is not present in the decomposition). A description of the mean flow in similar conditions is available in 46 and its interactions with periodic and turbulence fluctuations is addressed in. 40 Comparing mean flow velocities reported in the two previous works with and POD decomposition in the GN region, the kinetic energy associated to this mean flow is estimated to be an order of magnitude smaller than the kinetic energy attached to the first POD mode for water, and of the same order of magnitude for the polymer case (for which a mean flow enhancement occurs). Note: Spatial resolution denotes the distance between two PIV computed vectors (same along X and Z).
Once mean flow and oscillatory motions are identified, higher order modes remaining should correspond to either turbulence or to some peculiar combinations of interactions of modes or waves. The highest order ones then correspond to noise.
The code used here is the one implemented in DaVis 8 software. 60 The average velocity field over 1,000 fields (which are enough for convergence, as demonstrated in 40,46 ) is computed and subtracted to instantaneous velocity fields. The POD decomposition is then performed over 1,000 modes using the double precision option.
Lower order reconstruction of velocity fields can then be built using a POD is applied, as described above, to three regions of interest (ROI) of the flow in the plane of measurement: • the swept zone (SZ) which is defined as the area covered by the grid motion, and is available in CG measurements DPS. Mode 60 is supposed to be associated with turbulence, since it is a high order mode. Indeed, the structures observed are much smaller than for modes 1 and 2, and their magnitude decays when moving away from the grid. Moreover, in all the three regions, the typical size of a high magnitude patch seems smaller in water than in DPS (see for example mode 60, region SZ in water and DPS). Note since the intensity is carried by the temporal mode coefficients, the above color maps should not be used to compare amplitudes from one mode to another, but only between spatial structures of the modes and spatial distribution of ϕ, as done here.
The above description seems consistent with the general observation that polymer tends to promote larger organized structures and damp the small scales of turbulence and grid turbulence. 43 Turbulence in dilute polymer solutions sees its energy distribution between scales modified. This should be evidenced in the POD decomposition not only by a modification of the high mode structures, but also by an adaptation of the mode by mode energy distribution. Figure 3 shows the eigenvalue spectrum of POD modes, that is to say the magnitude E λi of each modes divided by the sum of all eigenvalues P i E λi . This ratio quantifies the portion of total kinetic energy included in each mode. It should however be kept in mind that this concerns the measured kinetic energy, which might not be exactly the total kinetic energy actually present in the flow due to experimental limitations (e.g., planar and two-dimensional measurements).

| Energy distribution
The first modes are representative of large coherent structures (organized motions) and so contain most of the energy. This is especially the case close to the grid where the grid's motion forces the large structures: the first modes have a higher energy share in the SZ region than in the CG region, and in the CG region than in the FT region. The higher order modes are then said to represent turbulence or noise. According to Reference 15,61, the eigenvalue value spectrum of a POD decomposition of a turbulent flow presents a −11/9 slope at high modes when plotted on a log-log scale (also plotted on the figure for illustration). The range for which the spectrum follows this slope is here limited. This can be due to two reasons: the relatively low grid Reynolds number which may lead to poor scale separation in all fluids, and the limitations in the spatial resolution of PIV measurements which does not allow to describe the smallest turbulent scales (as mentioned previously). However, it seems that this −11/9 slope is reached for lower modes in the FT region, suggesting that such lower modes may there already be related to fine, turbulent scales, as will be confirmed later in the following sections. Note that advanced maximum entropy methods also exist and allow to reconstruct energy density spectra (as a function of wave-number) from POD decomposition and verify their agreement to the conventional −5/3 slope in the inertial turbulence range, as described in Reference 62.
In the DPS case, this energetic dominance of the first modes seems to be enhanced: the first eigenvalues share is always higher in the non-Newtonian case than in water, and the eigenvalue spectrum  (Figure 3c)), one shows that for the first modes, the energetic enhancement is more significant for the FT region, which does not include the swept zone of the grid. Indeed, the negative slope of E DPS λ i =E w λ i is decreased for FT as compared to the slopes of SZ. In other words, the steeper decreasing slope with increasing mode number suggests an even stronger low mode (large scales) enhancement in the FT case regardless of the overall value of the dissipation. This observation leads to the conclusion that the enhancement of energy share of the first mode can not only be attributed to organized motions generated by the grid (which are dominant in SZ region), but is also a feature of turbulence in DPS. It can be that small scales of turbulence are damped by the polymer, as previously observed in previous viscoelastic turbulence studies 8,9 and also using POD analysis. 43,61 This translates into a decrease of the high POD modes energy, that are indicators of these small scales. For regions where the grid forcing is not directly felt (FT), the eigenvalue spectrum is intrinsically flatter (see Figure 3) since mean and organized motion caused by the grid are weaker.
In other words, even in Newtonian flows, the energy share is quite balanced between modes in the FT region whereas in the SZ region, energy is more contained in low order modes because of the grid. A damping of small structures is thus more visible in the FT region for which these structures are significant in the flow's energy, than in the SZ region where they are less influential.
All of the above thus to confirm that the shear-thinning behavior and the proximity to the grid promote large scale flow structures (organized or turbulent).
However, one should keep in mind that the DPS viscosity being always higher than that of water, total kinetic energy dissipation is likely higher. In a Newtonian fluid with viscosity analog to that of DPS, and equivalent total dissipation, smaller scales would be larger than in water. This implies that in such a fluid, a given mode number is already associated to larger flow structures (smaller wave numbers) with intrinsically higher kinetic energy content, owing only to differences in viscosity. When normalizing spectra "by mode number", as done in Figure 3c, the effect would thus also be to promote low order modes. It is very difficult to estimate true local dissipation values in shearthinning fluids, which makes re-scaling of the spectra and normalization "by wave number" challenging. It is thus possible that the slope observed in Figure 3c derives from both differences in "Newtonian" viscous dissipation, and small-scale damping due to the polymer. The latter alone would have potentially led to still decreasing but flatter slopes. • the temporal coefficient of each mode i, θ i (t), shows a sinusoidal trend when plotted versus time, and its PDF is shaped like that of a sinusoidal function (while that of a turbulent mode approaches a Gaussian shape (as discussed later in Figure 6).

| Identification of organized motion
• the scatter plot of two coupled modes arranges in a circle in the  principal, periodic mode. Modes 2 and 3 also seem to have a periodic behavior, but with a higher frequency (lower p, p = 5). In the GN region for water, the sinusoidal behavior of mode 1 disappears and no periodic behavior is observed for the following modes. In the GN region for DPS however, an oscillatory behavior can be observed for mode 1 and 3. Yet, due to the quite low time resolution of measurements, higher flow frequencies cannot be captured by the plots of To a first approach, all these observations are consistent with the previous remarks based on phase averaged measurements: oscillatory motion can logically be found in the SZ region, but is strongly damped when moving away from the grid, and as a consequence almost invisible in the GN region (for water at least). Polymer has an organizing effect on the flow which tends to promote this oscillatory motion and keep it significant in the vicinity of the grid. However, the distinction between periodic and nonperiodic modes is sometimes arduous due to the poor temporal resolution of our measurements, and also to the periodic behavior of modes that can be less pronounced than in stirred tanks. A conclusion is that evolution and PDF of θ coefficient alone may not be sufficient to identify clearly periodic modes. A complementary step is thus to check the degree of coupling between modes.
No circular arrangement similar to the one found for oscillatory motion in stirred tank can be observed, for any region, fluid, or pair of successive coefficients (θ i ,θ i + 1 ). This is to some extent not surprising since in the stirred tank case, circular organization of the temporal coefficients is evidenced for a pair of successive modes that exhibit an equivalent magnitude in the eigenvalue spectrum, and no such pair of modes is observed here (Figure 3). In the SZ region for both water and DPS however, the scatter plots of θ 1 versus θ 2 and θ 2 versus θ 3 F I G U R E 2 Example of spatial POD modes: modes 1,2 and 60 for regions "Swept Zone" (SZ), "Grid Neighborhood" (GN) and "Full Tank" (FT), in water (top half) and DPS at 100 ppm concentration (bottom half). The color map represents the local non dimensional mode magnitude ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ i . Note that these are not reconstructed velocity magnitudes. The velocity intensity is carried by the temporal mode coefficients and modulated by the present maps. The above color maps should not be used to compare one mode amplitude to another, but only between spatial structures of the modes [Color figure can be viewed at wileyonlinelibrary.com] values are not randomly distributed as one would expect for uncoupled modes, but arrange in a more complex characteristic pattern, which can also be seen in three dimension by plotting the 3D scatter plot of θ 1 versus θ 2 versus θ 3 values (see Figure 7). The dispersion around those defined patterns is quite small. The (θ 1 ,θ 2 ) pattern has a similar infinity symbol shape for water and DPS, with a variation of the two lobes' sizes and of the crossing point location. (θ 2 , θ 3 ) patterns also include two lobes, but are yet quite different in the two fluids. In the GN region, this settlement disappears.
Infinity shapes, translating into eight-shapes when projected on a 2 modes space, could possibly be related to the frequency of one mode being twice that of the second one (e.g modes 1 and 2 for water, Figures 5 and 7). Not all mode pairs fit this scenario. For example modes 2 and 3 in DPS display a two lobes structure (and not an 8 shaped one), while the frequency of mode 3 is twice that of mode 2 in Figure 5.
In Reference 50, similar patterns of mode coupling are observed in stirred tanks, and used them to evidence a triadic interaction involved in energy transfer between modes. The observation of Figure 7 and the work previously cited still allows to formulate two hypothesis: • These shapes are a consequence of the strong oscillatory motion observed in the SZ region but not in the GN region, and this motion can not be described by a simple two-mode coupling, but a triadic interaction is likely involved As one can see, organized motion is difficult to identify using the common criterion for stirred tanks. In other words, POD fails to identify unequivocally local coherent structures, but still provides an information on the global periodicity of the flow through a set of modes.
One of the interest of POD is that it makes it possible to reconstruct velocity field using a specific set of modes. With the objective of identifying purely turbulent motion free of any oscillatory component, it is thus easy to remove the oscillatory component from non phase resolved measurements, provided that the modes responsible for oscillatory motion can be identified.
From the above, and as an extension of the existing criterion in blade stirred tanks, we suggest that closed patterns in temporal modes scatter plots could be used as an indicator to unambiguously identify modes involved in oscillatory motion description. Further investigation on multi-modes coupling patterns, in three or more mode dimensions, and how these patterns project in two mode dimension space (Figure 7) is yet still needed.

| POD reconstruction of turbulence properties (FT)
What can still be done in a first approximation is define a threshold mode number for turbulence properties reconstruction. Indeed it is now admitted that oscillatory motion is described by coupling of low order modes, especially in the SZ region. In the GN region, even low order modes are difficult to identify as oscillatory and/or coupled using their temporal amplitude θ (see Figures 5-7). By setting a limit number of modes considered to be part of the oscillatory motion, one

| Turbulence spectra
In the FT region, 1D power spectra can be computed from the Fourier transform of the cross correlation coefficient of the velocity field. For that purpose, two-point auto-correlation coefficients R j i M, r j À Á are F I G U R E 5 θ i coefficients as a function of the instantaneous field index. Modes 1, 2 and 3 are represented as modes for which a periodic behavior can be expected. Mode 10 (chosen arbitrarily among highest mode numbers) is used as a reference for nonperiodic modes. Superimposed light gray lines are best sinusoidal fittings using the full series of snapshots for modes exhibiting a periodic behavior. The period P of each sinusoidal fitting, expressed in field numbers, is indicated in each plots [Color figure can be viewed at wileyonlinelibrary.com] computed for each velocity component i, in each direction j, with r j the distance between M and a second point, and i = x or y and j = x or y. First evaluated at every location M of the ROI, correlation coefficients are then spatially averaged on M over x and y, and 1D power spectra P j i computed by Fourier transform of individual correlation coefficient curves R j i . This yields power spectrum for the tangential and longitudinal fluctuating velocity components respectively P l = 0:5 P x x + P y y and P t = 0:5 P y x + P x y . The cumulative power spectrum P is defined as the sum of the last two. 64 It is plotted in Figure 9 for water and DPS using various reconstructions: one including the full set of modes (black dots), some using the first few modes only indicates that at these wave numbers, turbulence can be described by the higher order modes only.
For low wave numbers, fully reconstructed spectra are better matched by the partial reconstructions using high order modes. Yet in the previous paragraph, it was shown that these low order modes are not likely oscillatory or periodic components of the flow. They should thus already correspond to large, random, turbulent structures. This last assumption can be checked by looking at the slopes of the power spectra at low wave numbers in Figure 9. The energy input by the grid occurs at large spatial scales, described by low wave numbers of magni-

| CONCLUSIONS
In water and DPS, most of the kinetic energy of the flow is contained in the first POD modes. This is especially true for regions of the flow close to or containing the grid's swept zone since in this region, the flow is governed by large coherent structures created by the grid's motion which are described by the first modes of the decomposition.
The energetic dominance of the first modes is enhanced by the presence of polymer. The enhanced dominance of the first modes in DPS is a feature that cannot be only attributed to the forcing of the flow, but that is also significant in purely turbulent regions. It endorses the idea that the small structures of turbulence (here represented by the high mode numbers) are significantly damped in dilute polymer solutions as observed previously. 43 Whether this effect is here due to viscosity or to non-Newtonian properties is still open for discussion.
The effect of polymer seems to be felt differently depending on the region: close to the grid it acts on the shape of organized motion, far from the grid on the structure of turbulence.
Criterion for the identification of organized motion in stirred tanks based on eigenvalue spectrum and temporal coefficient mostly fail to reveal oscillatory motion in OGT. This is not surprising in the FT region, since it is assumed that the intensity of oscillatory motion quickly decreases when moving away from the grid. Without identifying precisely the modes associated to oscillatory motion, defining an arbitrary cut-off mode number is obviously not fully satisfactory: removing the first POD modes leads to artificially remove kinetic energy from the reconstructed flow, and to a modification of the power spectra. Modes in the GN region shows only weak if any sign of oscillatory behavior. This is consistent with the fact that the periodic motion evidenced in phase resolved measurements was quite weak energetically compared to periodic motion in the SZ region. 40 In this SZ region however, oscillatory motion is known to be strong. Even if two-mode coupling is not as clear as in stirred tanks, a different type of coupling involving at least three modes seems to appear. The exact reasons to why the link between mode coupling F I G U R E 7 Scatter plots of (θ 1 ,θ 2 ,θ 3  and oscillatory motion seems more complex in the OGT situations than in stirred tank are yet to be understood. A first guess can be advanced: in POD studies applied to stirred tanks, the motion of the blade is always in the same direction. The oscillatory motion, that is to say vortex shedding by the blade, is thus always the same (same vortex propagation direction and rotation), and appears periodically. In the OGT case however, oscillatory eddies rotation is reversed whether the grid goes up or down. In the stirred tank case, turbulent F I G U R E 9 POD reconstruction of turbulence total power spectrum P = P l + P t using all the modes (black dots), modes 0 to 4 (empty blue squares), modes 5 and above (full blue squares), modes 0 to 9 (empty red diamonds), and modes 10 and above (full red diamonds). The grid energy input occurs at wave numbers between 1/S and 1/M (gray region). The expected inertial sub-range power law slope is also represented ( structures once generated by the blade evolve and dissipate freely in its wake. In the grid-stirred case on the other hand, eddies generated by one grid pass almost immediately re-interact with the grid half a period later. All the above strongly could make the model description more complex, likely to be associated with a (chaotic) near-grid mixing process efficient at low Reynolds numbers.
A visual summary of the previous conclusions is made in Figure 10.
It is well known that POD is useful to identify coherent fluid structures in various type of flows. In this work, we showed that the criterion for periodic flow identification may not always succeed. However POD still allows to isolate and remove flow features that are not purely intrinsic to turbulence from subsequent analysis, and avoid the pitfall of misinterpreting coherent fluctuations for turbulence. This is a first and crucial step towards proper modeling of turbulence properties in such flows.
Further investigations on the POD study of OGT should focus on this issue of oscillatory motion identification using modes. The structures observed on 3 modes temporal coefficient scatter plots suggest that a more complex mode coupling may exist, and so that criterion could be developed to identify periodic flows from POD decomposition in grid stirred tanks. This development of more universal criterion is of great interest for in the investigations of non-Newtonian effects on flow organization and oscillatory motion in grid stirred tanks, but also in more complex situations involving periodic forcing. Polymer concentration effect could then be investigated quite easily, even if a first simple approach using the tools of eigenvalue spectrum and Newtoniannormalized eigenvalue spectrum can also be considered (Figure 3). The use of a dynamic modes decomposition method (DMD, 49 ) could also be considered as an interesting alternative to access additional information on temporal variation, despite the low temporal resolution of the data.
Finally, the similarities between the pattern observed in Figure 7 for the SZ region and the shape of chaotic attractors and associated trajectories raises an interesting question: the possible existence of an underlying chaotic process in oscillating grid flows that may be present in laminar cases and persist in turbulent cases, in the grid swept zone. Further works could be done in this sense.