Flow around an oscillating grid in water and shear-thinning polymer solution at low Reynolds number

The principle of oscillating grid turbulence (OGT) is to produce turbulence in a fluid initially at rest by making a grid oscillate, usually vertically, at a frequency f and with an amplitude or stroke 2S. It is commonly said that the jets and wakes behind the grid’s holes and bars interact to generate turbulence (Herlina 2005; Hopfinger and Toly 1976; Thompson and Turner 1975; Voropayev and Fernando 1996), which then diffuses away from the grid. It is of great interest for turbulence study since it theoretically creates almost no mean flow and, therefore, allows to study the effects of turbulence alone, which is not the case for fixed grid systems. OGT is also supposed to yield a quasi-homogeneous and isotropic turbulence in horizontal planes when measured far enough from the grid’s top position.

The first oscillating grid apparatus was designed by Rouse and Dodu (1955) to study turbulent diffusivity across a density stratification layer located above the grid and consisted in a square mesh grid in a cylindrical tank. Later, Bouvard and Dumas (1967) performed hot-fibre velocity measurements of grid generated turbulence in water, using a perforated plate as an oscillating grid. A full characterization of OGT in prismatic tanks came with the pioneer works of Thompson and Turner (1975) and Hopfinger and Toly (1976). Thompson and Turner (1975) studied several types of grids with different mesh sizes M and different bar shapes, and showed that the best homogeneity and intensity of turbulence was achieved with square section bars and a solidity parameter \(\varXi =\frac{d}{M} \left( 2-\frac{d}{M} \right)\) lower than 0.4, d being the width of grid bars. Hopfinger and Toly (1976) added that the walls of the tank should be planes of symmetry for the grid, to minimize the recirculation effects in the tank. For the same reason, the grid’s average position should be at least \(2.5\times M\) above the bottom of the tank (Xuequan and Hopfinger 1986). Horizontal homogeneity of turbulence is reached when the distance from the grid’s mean position is greater than an empirically determined value \(Z_{h}\) expressed in numbers of mesh parameter, with \(2M<Z_{h}<4M\) depending on the study. Since the ultimate interest is to achieve turbulence homogeneity, many studies of OGT use this \(Z_h\) parameter as a critical minimal distance at which the grid has to be placed from the item one wants to study the interaction with turbulence. Only recently, McCorquodale and Munro (2018b) evidenced the spatial homogenization of turbulence from the grid’s swept zone to the \(Z_h\) region. Once the far-grid, homogeneity region is reached, turbulence properties can be described by various evolution laws, first derived by Thompson and Turner (1975) and Hopfinger and Toly (1976) and later completed by other studies (Mcdougall 1979; Nokes 1988; Shy et al. 1997; Silva and Fernando 1994; Wan Mohtar 2016). These turbulence decay relationships are well described for water, but not in shear-thinning polymer cases. This yet will not be discussed in the present paper, as it is the subject of another work that has been published separately (Lacassagne et al. 2019). It is worth noting that OGT has almost never been simulated numerically. Only the recent work by Zhang et al. (2017) presents a Lattice Boltzmann simulation of OGT and numerically visualizes the interactions between jets and wakes behind the grid bars and holes leading to turbulence generation.

At first, the concept of OGT generating no mean flow but only isotropic and homogeneous turbulence was rather well believed. Yet with the development of PIV techniques allowing further investigations of the flow inside OGT tanks, it became clear that mean flow exist even when matching the requirements detailed above. McKenna and McGillis (2004b) studied the mean flows in an oscillating grid tank using PIV and showed that persistent mean flow cells always existed, with relative high kinetic energy level as compared to the turbulent kinetic energy. Moreover, these mean flows seem to be poorly reproducible and strongly dependent on initial conditions (Herlina 2005). It is, therefore, really hard to predict the mean flow that could occur in the oscillating tank during a specific measurement, and this is one of the main limitations of OGT systems.

A method for reducing mean flows in OGT is to use an inner box placed inside the stirred tank to separate the wall-induced vortices from the rest of the flow (Dickinson and Long 1983; Hopfinger and Toly 1976; McCorquodale and Munro 2018b). The complementary work previously referred to (Lacassagne et al. 2019) also suggests that the mean flow may be enhanced by the presence of polymer in the dilute entanglement regime. Investigations of flow properties in the grid region may allow to explain the origin of this mean flow both in water and in shear-thinning polymer solutions.

The influence of non-Newtonian behavior on grid turbulence has been mainly studied for fixed grid configuration and viscoelastic fluids (Barnard and Sellin 1969; van Doorn et al. 1999; Fabula 1966; Friehe and Schwarz 1970; Greated 1969; McComb et al. 1977). Recently, Vonlanthen and Monkewitz (2013) used PIV measurements to look at turbulent spectra and scales in grid turbulence of dilute viscoelastic polymer solutions and evidence elastic modifications of turbulence. In their experiments, they found that both the shape of the energy spectrum and the elastic scale evolved with time, which they explained by the destruction of polymer chains by the strong shears in the vicinity of the grid. This shows one of the limits of fixed grid devices for the study of turbulence in polymer solutions: reaching high levels of turbulence requires high flow rate which causes important degradation of the polymer throughout the measurements. Moreover, turbulent velocity fluctuations are even smaller when compared to mean flow velocities than for the water case. One may thus prefer using OGT rather than fixed grid.

The first study of OGT in viscoelastic dilute polymer solutions (PEO) was made by Liberzon et al. (2009), who observed the propagation velocity of the boundary between turbulent and non-turbulent regions in the tank, at the first instants after the onset of the grid’s oscillations. They found that the turbulent/non-turbulent interface moved globally faster in dilute polymer solution than in water. Wang et al. (2015) and Wang et al. (2016) later used a two oscillating grid system to study the viscoelastic effects of surfactants and dilute polymers on coherent structures. 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 flow seems to strongly modify the spectrum behavior of the turbulent structures. They show promising results for the use of this method to analyze OGT flow properties in complex fluids such as DPS.

In some cases, a flow may display a periodic behavior due to external forcing or waves. Blade motion in stirred tanks (Escudié and Liné 2003), periodic vortex shedding at rigid boundaries in free surface flows (Mignot et al. 2016), or pulsating flow in blood vessels induced by periodic heartbeats (Holdsworth et al. 1999) are good examples of flows in which an oscillatory/organised/periodic effect is present. Oscillating grid turbulence used in this work is another one. In such flows, the time average of the oscillatory component of the flow is nil. Hence, when using classical Reynolds decomposition to describe them, periodic fluctuations are included considered together with while they are not strictly speaking turbulent fluctuations.

To be able to distinguish turbulence from organised motion, Hussain and Reynolds (1970) proposed a triple decomposition of the velocity field, that will be called triple decomposition hereinafter. The total instantaneous velocity field is this time written as the sum of the average velocity \({\overline{U}}\), the periodic fluctuation \(u^*\) and the turbulent fluctuation \(u'\):Pressure decomposes the same way, with \(\overline{u^*}=0\), \(\overline{u'}=0\), \(\overline{p^*}=0\), \(\overline{p'}=0\), \(\overline{u^*u'}=0\) and \(\overline{p^*p'}=0\). Turbulent, mean and oscillatory kinetic energies can be defined, respectively, as \(k'=\frac{1}{2}\overline{u'_i u'_i}=\), \(K=\frac{1}{2}{\overline{U}}_i{\overline{U}}_i\) and \(k^*=\frac{1}{2}\overline{u^*_iu_i^*}\). Hussain and Reynolds (1970) derived the following governing equations for, respectively, \(k'\), \(k^*\) and K:where (1) is the accumulation term, (2) is advection by the mean flow,(3) (6) and (7) are diffusion terms, term (8) dissipation, and terms (4) and (5) are exchanges with mean and oscillatory components. \(\left\langle a \right\rangle _{\phi }\) is the phase-averaged quantity of a (see Sect. 3.4).where term (1) is the accumulation term, term (2) is advection by the mean flow, terms (3) and (6) are diffusion, term (7) is dissipation, and terms (4) and (5) are exchanges with mean and turbulent components.where term (1) is the accumulation term, term (2) is advection by the mean flow, term (3) is the pressure velocity correlation, terms (5) and (6) are diffusion terms, term (7) is dissipation, and term (4) is the transfer to other components’ kinetic energy. From Eqs. (2), (3) and (4), it is possible to extract the exchange terms (respectively numbered (5), (4) and (5), and (4)) and identify the expression of the kinetic energy transfer between each components of the velocity field (Escudié and Liné 2003):

Mean to turbulent energy transfer:Mean to oscillatory energy transfer:Oscillatory to turbulent energy transfer:Each of these transfer terms is a product of a gradient of mean or oscillatory motion and a stress term of organized and turbulent motion. The direct transfer from mean flow to turbulent motion \({{\mathscr {T}}}_{{\text {mt}}}\) is equivalent to the turbulence production by mean flow in Reynolds classical decomposition. However, in the present case, the mean flow also transfers part of its energy to the periodic one (\({{\mathscr {T}}}_{{\text {mo}}}\)) and part of the turbulent kinetic energy production comes from the organized motion (\({{\mathscr {T}}}_{{\text {ot}}}\)). To fully understand turbulence in oscillatory flows, the knowledge of these transfer terms is required. It is worth noting that all of the above equations reduce to the ones obtained using the classical Reynolds decomposition when \(u^*<<u'\), i.e., in case of negligible oscillatory flow influence.

Shear-thinning properties are conferred to the liquid by addition of a minute amount of a polymer, xanthan gum (XG), to distilled water. Here XG produced by Kelco under the commercial name Keltrol CG-T is used. Its average molar mass is \(M_w=3.4\times 10^6\hbox { g mol}^{-1}\) and its polydispersity equal to 1.12 (Rodd et al. 2000). XG is chosen for its high resistance to strong shears and extreme temperature and pH conditions (Garcia-Ochoa et al. 2000). Such features are useful when using it nearby a rigid oscillating grid. XG yields optically clear non-Newtonian solutions once dissolved in aqueous media. This allows the use of optical methods for liquid phase velocity measurements. This polymer is commonly utilized as a flow additive for non-Newtonian fluid mechanics experiments, but also in the food, process, and oil and gas industries (Katzbauer 1998). Its rheology and its properties have been widely studied in the literature (Cuvelier and Launay 1986; Wyatt and Liberatore 2009; Wyatt et al. 2011). Depending on the polymer concentration \(C_{XG}\), one may distinguish between three entanglement regimes: dilute, semi-dilute, and concentrated (Cuvelier and Launay 1986; Wyatt and Liberatore 2009). In the last two cases, the flow behavior is defined by fluid–polymer but most importantly polymer–polymer mechanical or electrical interactions. In this work the focus is made on the dilute regime, for which polymer molecules only interact with the flow and not between each other. It concerns the \(C_{XG}\le 100\) ppm concentration range (Wyatt and Liberatore 2009). Here the working concentration is \(C_{XG}=100\hbox { ppm}\) (Fig. 1). This last concentration is chosen so that the solution is still in the dilute regime for entanglement, with the strongest possible concentration effects within that regime. One then talks about a dilute polymer solution (DPS). The shear-thinning behavior of aqueous XG DPS can be modelled by a Carreau–Yasuda (CY) equationWith \(\mu _0=32.8\hbox { mPa s}\) and \(\mu _{\inf }=1.1\hbox { mPa s}\) the zero shear rate and infinite shear rate Newtonian plateau, \(t_{{\text {CY}}}=1.58\hbox { s}\) a characteristic timescale of the polymer, \((n-1)=-0.5\) the shear-thinning power law exponent, and \(a=\)2 a parameter for the transition between power law and Newtonian behaviors (all values given for \(C_{XG}=100\hbox { ppm)}\). The evolution of viscosity with the nominal shear rate is shown on Fig. 2. Vertical lines indicates estimate of (1) the typical shear rates caused by the grid (full line, based on its frequency), and (2) the maximum shear rates caused by mean, oscillatory and turbulent motions (based on data shown in Fig. 7). Viscoelasticity is checked to be negligible for such a XG concentration. Steady shear rheological measurements performed before and after experiments allow to check that polymer chains are not destroyed by grid oscillations, or at least that destruction does not affect the shear-thinning property (see Lacassagne (2018), Fig. 3.10).

Turbulence is generated in a transparent prismatic tank of \(L^2\) inner cross section with \(L=277\hbox { mm}\). The fluid height is set at H = 450 mm and the distance between the surface and the average grid position is 250 mm. The vertical axis, oriented upwards, is noted Z, and X and Y are the axis defined by the grid bar (see Fig. 1). The origin of the reference frame is placed at the grid average position (\(Z=0\)) at the crossing between the two central bars. In this study, only polymer concentration is varied and all oscillation parameters are kept constant. The grid bars are \(d=7\hbox { mm}\) thick and the mesh parameter is \(M=35\hbox { mm}\). The frequency is fixed at 1 Hz and the amplitude at \(2S=5\hbox { cm}\), This allows to define a grid-based Reynolds number using the definition of Janzen et al. (2010), but based on the zero shear rate viscosity \(\mu _0\):The density \(\rho\) of the fluid is assumed equal to that of water, because of the very small mass of polymer added. Here one gets \(Re_g=2500\) for water and \(Re_g=76\) for the polymer case. To quantify the ratio between the polymer relaxation timescale and the grid forcing timescale the grid-based Deborah number De is defined aswith T the period of oscillations. Here \(De=1.58\).

Liquid phase velocity measurements in the tank are achieved by particle image velocimetry (PIV). The experimental setup is sketched in Fig. 1. The region of interest (ROI) is a vertical rectangle, in a plane normal to the central bar of the oscillating grid. Its width is close to that of the tank, and it is vertically centred around the grid swept zone. The cameras used is a double frame LaVision SCMOS sensors of 2560 by 2160 pixels, equipped with a 50 mm focal macro lens. A pulsed Quantel Nd:YAG laser emitting at \(\lambda =532\hbox { nm}\) is used to illuminate \(50\,\upmu \hbox {m}\) diameter polyamide particles. Double-frame PIV at an acquisition frequency \(f_{\text {acq}}\) of 10 Hz is performed. The time interval between laser pulses is \(\varDelta t=4\hbox { ms}\). Vector fields are computed with DaVis 8 software using a multipass processing: a first pass with 48 by 48 pixels and 2 following passes with 24 by 24 pixels round Gaussian weighted interrogation windows, at a maximum of \(50\%\) overlapping. Spurious vectors are removed from PIV fields by applying a threshold of 1.2 on the peak ratio, and replaced using median filtering.

The final spatial resolution achieved is 3.4 mm. The smallest Kolmogorov length scale and Taylor micro-scales of turbulence are supposed to be found in the water case, for which the viscosity is always the lowest. The Kolmogorov and Taylor length scales are evaluated to be, respectively, of about 0.47 mm and 3.17 mm for water. For 100 ppm XG solutions, using a constant viscosity equal to \(\mu _{\infty }\), they increase to 0.23 mm and 3.25 mm, and up to 6.45 mm and 18.17 mm when using \(\mu _0\) as the scale viscosity. The 3.4 mm spatial resolution is thus quite coarse and would unfortunately not allow to evidence energy variations at large wave numbers characteristic of viscoelastic turbulence. We shall thus discuss only the inertial and large scale effects.

Measurements performed here are planar (2D PIV) and give access to only two components of the velocity field. As a consequence, it is not possible to compute all the terms of the governing equations introduced in Sect. 2.3, but only a part of the terms involving the components and their gradients along X (i and/or j equal to x) and Z (i and/or j equal to z). In what follows, analysis of the velocity fields is carried out as if the flow was two dimensional, assuming that velocities and gradients along the y direction are nil. This is a strong and reductive hypothesis. To perform a complete analysis of the production, dissipation, and transfer terms of turbulence of the governing equations in the whole volume of interest, full three-dimensional three-component velocity data would be required.

Another way to describe the periodicity of turbulence property is to compute rms values of velocity fluctuations not on the full set of data, but for each specific grid position. Such so-called phase rms \(\left\langle u'_z\right\rangle _{\phi ,{\text {rms}}}^{k}\) are defined at Eq.  15. An example of phase rms field for water for a given grid position is given in Fig. 14a, the log-scaled color bar representing the magnitude of \(\left\langle u'_z\right\rangle _{\phi ,{\text {rms}}}^{k=p=1}\) in m/s. One can clearly evidence the wake of the grid by high turbulent intensity patches. We should stress that the following analysis is not performed in the global Eulerian frame of the tank, for which oscillatory flow has already been considered through non-phased rms of oscillatory fluctuations, or oscillatory stress fields. This time, we are looking at oscillatory flow in a grid-attached Eulerian frame: the oscillatory aspect of the flow is intrinsically accounted for by defining turbulence based on the grid position.

It is interesting to notice that the higher turbulent intensity regions are found to correspond to grid holes. Previously, it was observed that normal Reynolds stresses (Fig. 10) and non-phased rms fields (Fig. 9) where maximal above grid bars. Tangential Reynolds stresses were themselves maximal above grid holes. Here we understand that for a given grid position, turbulence is high where the wakes of grid bars meet, i.e., in the jets behind the holes. In this region, both vertical and horizontal fluctuations are found since structures are the result of a lateral collision of structures, advected by the oscillatory trailing vortices shown in Fig. 8. This thus explains the mapping of tangential Reynolds stresses. Vertical Reynolds stresses, as for rms of \(u'_z\), are maximal in the region where turbulence production is the result of vertical sweeping by bars only.

In the rest of this paragraph, five different sampling grid positions are considered (already used in Fig. 8 for the visualization of a typical vorticity phase). They are represented on a grid period in time and space in Fig. 14b). Position 1 correspond to a grid accelerating downwards, position 2 to a grid still going downwards, but decelerating. Position 3 is the extreme low position at \(Z=-S\), where the velocity is nil and the acceleration maximal and oriented upwards. Positions 4 and 5 are for the grid going upwards and respectively accelerating and decelerating. For each grid sampling position \(k=p\), phase rms fields are averaged over the horizontal dimension X. The width-averaged profiles of \(\left\langle u'_z\right\rangle _{\phi ,{\text {rms}}}^{k=p}\) are plotted versus the instantaneous distance from the grid \(Z-Z_g\) normalized by S in Fig. 14c (water) and d (DPS).

For almost all grid positions and for the two fluids considered (except \(p=2\) and \(p=3\) in DPS as discussed later), rms profiles roughly follow power law decay trends such that \(\left\langle u'_z\right\rangle _{\phi ,{\text {rms}}}^{p} \sim (Z-Z_g)^{n}\). All curves are included inside an envelope defined by \(n=-2\) and \(n=-1\) trends. The \(n=-1\) value happens to be the usual exponents of the decay law for oscillating grid turbulence evidenced by Hopfinger and Toly (1976) when plotting non phased rms profiles of \(\left\langle u'_z\right\rangle _{{\text {rms}}}\) as a function of the dimensional Z. We see that here, once phased to the grid motion, the decay profiles of each grid positions has a n value lower than − 1, but higher than − 2.

In water, one could probably expect that when phasing turbulence with the grid position, the decay exponents characteristic of fixed grid turbulence could be retrieved for well chosen grid positions. Indeed, one can consider an Eulerian description on the flow in a reference frame attached to the grid and no longer to the tank. When the grid is going down (respectively up), the configuration for each grid position that of a fixed grid with flow passing through in the \(+Z\) (respectively, \(-Z\) direction), with a relative fluid velocity equal to the grid’s velocity. For fixed grids, Comte-Bellot and Corrsin (1966) and many more recent works (listed in Hearst (2015) evidenced decay exponents for the turbulent kinetic energy in the [− 1.4, − 1] range. For the streamwise velocity fluctuations considered here, this should yield decay exponents in the [− 1.2, − 1] range. Here curves for position \(p=1\) and \(p=2\) in Fig. 14c correspond to the situation analog to that of a fluid flowing upwards through a fixed grid, and should be comparable to fixed grid scaling laws. They are fitted by expressions of type \(\left\langle u'_z\right\rangle _{{\text {rms}}}=a\times\left( Z-Z_g\right) ^{n}\). Values of n obtained for \(p=1\) and \(p=2\) are, respectively, equal to − 1.2 and − 1, with a better fitting for \(p=1\).

This phase analysis suggests that original mechanisms of turbulence generation by the oscillating grid are in fact quite comparable to those behind a fixed grid in a mean flow, at least as a first approximation: a grid moving in a fluid quite logically generated turbulence in a similar fashion that when a fluid moves across a fixed grid. Yet the fundamental difference between the two setups is that turbulent structures generated at previous grid positions are still present in the flow at following grid positions and may interact with the newly generated structures. This is one of the possible reason why the fitting for the \(p=2\) position is less efficient, and why the decay exponent varies between − 1.2 and − 1. Indeed, for \(p=1\) most of the \(Z-Z_g>0\) region lays outside of the grid swept zone, while for \(p=2\), low \(Z-Z_g\) values are found inside this grid swept zone, when pre-existing turbulent structures interact with the \(p=2\) jets and wakes.

All of the above is valid for water. When considering DPS (Fig. 14d), the first observation is that profiles no longer follow single exponent power law trends, at least in the vicinity of the grid. For \(p=1\) and \(p=5\), a two-slope trend is visible with a change of slope at \((Z-Z_g)/S\simeq 0.5\). For \(p=2\) and \(p=3\), a peak of \(\left\langle u'_z\right\rangle _{{\text {rms}}}\) is observed at the same scaled distance. This shift in decay rate concerns top grid positions, and evidences a boundary between high turbulent intensity, near-grid region and the rest of the flow above the swept zone that for grid positions. The second behavior, namely the rms peak, is characteristic of a turbulent patch shedding by the grid: at bottom grid positions (\(p=2,3\)) and with a grid going down, previously generated turbulent structures are advected away from the grid by the hole jets. Since the grid is slowing down (\(p=2\)) or even stopping (\(p=3\)), turbulence generation is less efficient and up-going turbulent patches break away from the near-grid region. This is no longer observed for (\(p=4\)) since the grid is going up and catches back on the shed turbulent patches. We thus see again an enhanced importance of flow periodicity in term of turbulence generation for the DPS case.

It is worth mentioning that this phase rms study can be another way of describing the periodicity of the flow induced by the grid. It is here addressed not in terms of periodic flow structures produced, but as a periodic behavior of turbulent properties. Figure 14 shows that if this periodicity of turbulence is significant in the swept zone (see sub-figure a as an illustration), it reduces when moving away from the grid, as evidenced by the asymptotic collapse of all curves for \((Z-Z_g)/S>3\). This seems to indicate that outside of the grid swept zone, turbulent properties are also not periodic. However, clearer evidence could be brought by performing time resolved measurement of turbulence generated by the grid and studying the temporal periodicity of correlation functions of turbulent correlations to themselves (or turbulent timescales)in the Eulerian tank-attached frame.

A fast approach for studying polymer influence on the origins of OGT is an estimation of the local viscosity field and its statistics. The main idea is to derive local shear rate fields from instantaneous, phase-averaged or time-averaged velocity fields. Local shear rate values are then used to estimate an apparent viscosity through the constitutive equation of the polymer, here a CY law for the rheological behavior derived in Sect. 3.1 (Eq.  8, Fig. 2).

This method is limited because the velocity data is only 2D and lacks the third velocity component. Shear rate values can thus only roughly approach the real local shear rate, and no precise quantitative study can be performed. However, the method allows to qualitatively estimate the viscosity variations in a plane of measurement.

In practice, the apparent instantaneous shear rate is computed asand the instantaneous apparent viscosity aswith \(f_{{\text {CY}}}\) the Carreau–Yasuda function defined in Eq. (8), and using fitting parameters obtained in Sect. 3.1. Statistics for viscosity are then obtained by ensemble averaging over the whole set of field for \({\overline{\mu }}_{{\text {app}}}\), or grid position by grid position for \(\mu _{{\text {app}}}^*\) exactly like what is done for the velocity field.

Figure 15 shows the fields of ensemble average viscosity \({\overline{\mu }}_{{\text {app}}}\), ensemble rms of turbulent viscosity fluctuations \(\left\langle \mu '_{{\text {app}}} \right\rangle _{{\text {rms}}}\) and fields of maximum and minimum viscosity values over time \(\mu _{{\text {app,max}}}\) and \(\mu _{{\text {app,min}}}\). All values are scaled by the \(\mu _0\) maximum viscosity.

Estimating the uncertainty on instantaneous viscosity field is quite arduous since one would need to know the uncertainty on instantaneous velocity fields and instantaneous local shear rates. However, another approach can be used to evaluate the error associated to this viscosity estimation process. From statistical convergence analysis (see Fig. 5), we estimate the uncertainty on velocity statistics convergence to be at worst of 5\(\%\). Uncertainty propagation on velocity gradient and shear rates can be performed to quantify the resulting relative uncertainty on apparent shear rate, which may be up to \(30\%\). From this value, the CY law (Eq. 8, Fig. 2) can finally be used to assess uncertainty on calculated apparent viscosity at each shear rate value. This last uncertainty is maximal when the slope of the curve in Fig. 2 is the steepest, i.e., at the lowest effective flow index (as defined in Cagney and Balabani (2019)). This is here the case for shear rates around 4 \(s^{-1}\). At this worst case shear rate, the relative uncertainty of apparent viscosity is found to be 14\(\%\).

The first sub-figure of Fig. 15 shows that average viscosity is minimal close to \(Z=0\) inside the swept grid region. It then increases when moving away from \(Z=0\) until it reaches its zero shear rate plateau value at \(Z/S>3\). The maximum of rms velocity fluctuations is found just above the grid’s top and bottom positions, around \(X=0\). Here the viscosity varies from a value close to its zero shear rate plateau when the grid is on the other side of its stroke, to a much lower value when the grid departs from this position and releases high shearing eddies. Vertical trails of high viscosity rms are also observed in regions associated to grid holes. Since this plot represents the rms of turbulent viscosity fluctuations, it means that these trails are not due to oscillatory motion, but rather to strong fluctuations of the shear rate fields because of turbulent structures in those regions. The central high viscosity fluctuation path in the \(X=0\) mm, \(Z/S=1.5\) region confirms the existence of preferential paths for turbulence. Turbulent fluctuations are more likely to be found at the centre of the ROI above the swept zone. The minimum viscosity fields shows that viscosity is always higher than the infinite shear rate viscosity, which means that during the experiments, no shear rate (mean, oscillatory or turbulent) is high enough for the fluid to behave like a low viscosity Newtonian fluid, even locally. Finally, the maximum viscosity field shows that almost everywhere in the fluid, the maximum viscosity value may be reached at a time, i.e., that almost everywhere in the fluid, at some time, the shear rate is almost nil. The only locations where this is not the case are just above the grid swept region. Here lower maximum viscosity patches are found which correspond to the shape and location of the mean flow vortices.

The grid periodic motions create oscillatory vortices behind the grid bars. At the walls (on both sides of the ROI), these periodic, non symmetric vortices feed the large mean flow eddies. Oscillatory component is strong within the grid swept region, but quickly vanishes when moving away from the grid’s action. This suggests that a classical Reynolds decomposition can equally be used to describe turbulence properties in the bulk flow. Turbulence is created mostly by oscillatory flows in the swept region. It is highly anisotropic, with a much higher intensity along the vertical direction, as imposed by the grid motion. When turbulent intensity is computed in phase with the grid position, its decay law with the distance to the grid can be compared to fixed-grid correlations for well chosen grid positions. The global mechanisms of turbulence and mean flow production by the oscillating grid is the following: the grid introduces turbulence in the oscillatory fields, which transfers energy to the mean flow and to turbulence.

Polymer addition has several effects on the flow structure and energy exchanges around the grid. The first one is an enhancement of the oscillatory flow on the sides of the ROI near the walls. This leads to a periodic vortex shedding that feeds the mean recirculation and increase its intensity, thus possibly explaining the mean flow enhancement. The second effects is a modification of the spatial distribution of turbulence. Reynolds stress and turbulence intensity are increased in the core of the swept region close to \(Z=0\), and decrease quicker with increasing Z magnitude. Turbulence is enhanced at the centre of the ROI (far from the walls), both in the swept region and far from the grid. Finally, the transfer from oscillatory to turbulent fluctuations seems more complex in DPS than in water, and suggests that oscillatory motion may locally both give and take energy from turbulence, which was not the case for water. Another visualization of the same phenomenon was made possible by an estimation of viscosity from PIV measurements and its statistical analysis.

It is worth reminding that a limitation of the previous analysis is the fact measurements are planar and do not account for the highly three-dimensional behavior of turbulence. Results may be different when considering for example a plane of measurement not coincident with a grid bar, but with a set of grid holes, (i.e., translated of M / 2 in the Y direction, see Fig. 1). With similar 2D PIV measurements in additional planes, and under a specific set of symmetry hypothesis, it would be possible to estimate kinetic energy budgets along Z directions at specific X and Y values, and go further in the modelling of transfer terms and turbulence production. However, three-dimensional three-component measurements of the velocity fields would ultimately be required to measure the total energy transfer terms, or the actual local viscosity mappings. Being now confirmed that the spatial heterogeneity of oscillatory flow is a possible cause of mean flow generation, and that this heterogeneity is induced by spatial confinements effects, further study of the influence of this confinement (e.g., through the M/L ratio) on the flow would be of great interest.