3.1. Ensemble average of invariants

We first examine the impact of polymers on the ensemble-averaged values of invariants appeared in (3.1) and (3.2), namely $\langle Q_s \rangle$ , $\langle Q_\omega \rangle$ , $\langle R_s \rangle$ and $\langle R_\omega \rangle$ . We use the $R_\lambda = 203$ case here. In figure 1(a), we plot the second-order moments $\langle Q_s \rangle$ and $\langle Q_\omega \rangle$ , and one can see that their magnitudes decrease significantly as the polymer concentration $\phi$ increases, which is consistent with previous studies (De Angelis et al. Reference De Angelis, Casciola, Benzi and Piva2005; Liberzon et al. Reference Liberzon, Guala, Lüthi, Kinzelbach and Tsinober2005, Reference Liberzon, Guala, Kinzelbach and Tsinober2006; Cai et al. Reference Cai, Li and Zhang2010). The well-known suppression of the energy cascade in the inertial range due to polymer additions (Ouellette et al. Reference Ouellette, Xu and Bodenschatz2009; Xi et al. Reference Xi, Bodenschatz and Xu2013; Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021) explains the observed reduction in $-\langle Q_s \rangle$ and $\langle Q_\omega \rangle$ . Furthermore, given the definition of mean energy dissipation rate $\varepsilon = 2\nu \langle S_\textit{ij} S_\textit{ij} \rangle$ , the results shown in figure 1(a) can be interpreted as drag reduction in bulk turbulence, as reported by previous DNS studies (Perlekar et al. Reference Perlekar, Mitra and Pandit2006, Reference Perlekar, Mitra and Pandit2010; Cai et al. Reference Cai, Li and Zhang2010). In figure 1(b), we plot the ensemble-averaged third-order quantities, strain self-amplification $\langle R_s \rangle$ and vortex stretching $\langle R_\omega \rangle$ . Similar to the second-order moments, they also decrease significantly with increasing $\phi$ . This is consistent with our intuition that the depression of third-order moments leads to the depression of the small scales in polymeric turbulence (Liberzon et al. Reference Liberzon, Guala, Lüthi, Kinzelbach and Tsinober2005, Reference Liberzon, Guala, Kinzelbach and Tsinober2006; Cocconi et al. Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017), given that the nonlinear third-order terms contribute to the generation of the turbulent system. The insets of figure 1(a,b) show the ratios $-\langle Q_s \rangle / \langle Q_\omega \rangle$ and $-\langle R_s \rangle / \langle R_\omega \rangle$ , respectively. Remarkably, the data points remain close to 1 within the considered concentration range, which validates the homogeneity condition discussed above (Betchov Reference Betchov1956), i.e. the fluctuation velocity field could be treated as locally homogeneous, allowing spatial averages to replace ensemble averages. We also notice that some DNS studies introduced the concept of polymer stress, and found that while the nonlinear generation terms are suppressed, the polymer stress terms sustain the small scales of turbulence (Cai et al. Reference Cai, Li and Zhang2010; Watanabe & Gotoh Reference Watanabe and Gotoh2014; ur Rehman et al. Reference ur Rehman, Lee and Lee2022), but due to technical limitations, our current experimental set-up cannot precisely quantify the impact of polymers on turbulent dynamics.

Figure 1.

The ensemble-averaged invariants as functions of $\phi$ : (a) second-order moments $\langle Q_s \rangle$ and $\langle Q_\omega \rangle$ ; (b) third-order moments $\langle R_s \rangle$ and $\langle R_\omega \rangle$ . Insets show the ratios between two second- or third-order moments. (c) The skewness of $\boldsymbol{S}$ , $\langle R_s \rangle / (-\langle Q_s \rangle )^{3/2}$ . In (a–c), we use the $R_\lambda =203$ dataset. (d) Plot of $\langle Q_s \rangle$ normalised by the corresponding Newtonian values ( $\phi =0$ ) for three Reynolds numbers, $R_\lambda = 168,203,235$ .

For the rate-of-strain tensor $\boldsymbol{S}$ , we have discussed its second- and third-order moments, $\langle Q_s \rangle$ and $\langle R_s \rangle$ . Both of these moments are significantly suppressed with increasing polymer concentration $\phi$ . In addition, figure 1(c) plots the skewness of $\boldsymbol{S}$ , $\langle R_s \rangle / (-\langle Q_s \rangle )^{3/2}$ , as a function of $\phi$ . Within the considered range of $\phi$ , the skewness consistently maintains a non-zero value, indicating a skewed distribution of $R_s$ (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997; Meneveau Reference Meneveau2011) as we mentioned before. However, as $\phi$ increases, the skewness gradually decreases. Consequently, we conclude that polymers not only reduce the magnitude of the second- and third-order moments of $\boldsymbol{S}$ (or $\boldsymbol{A}$ ), but also weaken its skewed distribution, as will be further demonstrated through the PDFs of invariants in the next subsection.

In figure 1(d), we further explore the Reynolds number effect on the moments of VGT in polymeric turbulence; as an example, we plot $\langle Q_s \rangle$ normalised by its corresponding Newtonian value for various $R_\lambda$ (and also $\textit{Wi}$ ). We observe that the normalised $\langle Q_s \rangle$ decreases differently as a function of polymer concentration $\phi$ . Specifically, both curves exhibit a rapid descent at a particular $\phi$ value, and as $R_\lambda$ increases, this descent occurs at higher polymer concentration. On the other hand, at sufficiently high $\phi$ , the results for all three $R_\lambda$ seem to level off to almost the same extent. Experimental work in VKS with baffled disks (Ouellette et al. Reference Ouellette, Xu and Bodenschatz2009) and smooth disks (Zhang & Xi Reference Zhang and Xi2022) also reported a similar phenomenon. Recall that a reduction in the amplitude of $\langle Q_s \rangle$ or $\langle Q_\omega \rangle$ corresponds to a drag reduction; the results in figure 1(d) as well as previous VKS experiments may represent a maximum drag reduction, suggesting a balanced energy exchange between polymers and turbulent flows.

3.2. The PDFs of invariants

Compared to the ensemble average, PDFs provide more comprehensive information about physical quantities. Figure 2 illustrates the PDFs of $- Q_s$ and $ Q_\omega$ at $R_\lambda =203$ . Consistent with the reduction of mean value shown in figure 1(a), all the PDF curves in figure 2 contract as polymer concentration $\phi$ increases. Intense events with high dissipation or enstrophy are significantly suppressed, which has been shown in extensive experimental (Liberzon et al. Reference Liberzon, Guala, Lüthi, Kinzelbach and Tsinober2005, Reference Liberzon, Guala, Kinzelbach and Tsinober2006; Zhang & Xi Reference Zhang and Xi2022) and numerical (Perlekar et al. Reference Perlekar, Mitra and Pandit2010; Watanabe & Gotoh Reference Watanabe and Gotoh2013; Cocconi et al. Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017; ur Rehman et al. Reference ur Rehman, Lee and Lee2022) research. In addition, we can see that although the mean values of $Q_s$ and $Q_\omega$ balance each other ( $- \langle Q_s \rangle = \langle Q_\omega \rangle$ ), the tails of the PDFs of $Q_\omega$ are always broader than the tails of $Q_s$ for different $\phi$ , which is consistent with the Newtonian cases (Buaria & Pumir Reference Buaria and Pumir2022; Gotoh, Watanabe & Saito Reference Gotoh, Watanabe and Saito2023).

In the insets of figure 2(a,b), we demonstrate that these PDFs exhibit similar functional forms across different polymer concentrations when normalised by their respective standard deviations. This observation is qualitatively consistent with previous experimental and numerical studies (Perlekar et al. Reference Perlekar, Mitra and Pandit2010; Watanabe & Gotoh Reference Watanabe and Gotoh2013; Zhang & Xi Reference Zhang and Xi2022). Upon closer examination, we find that this collapse is particularly good within approximately ten standard deviations, i.e. in the core region of the PDFs. In contrast, the tails exhibit some scatter, which may arise from limited statistical convergence. Alternatively, the observed differences in the tails could reflect a physical effect: the presence of polymers may alter the occurrence and distribution of extreme dissipation events, which are often associated with coherent flow structures. As is shown later, our results indicate that polymer additives cause significant changes in these flow structures, which could lead to the observed modification of the PDF tails for $-Q_s$ and $Q_\omega$ . Slight tail deviations have also been reported in previous studies (Perlekar et al. Reference Perlekar, Mitra and Pandit2010; Watanabe & Gotoh Reference Watanabe and Gotoh2013; Zhang & Xi Reference Zhang and Xi2022), and our findings offer a possible physical explanation for this phenomenon.

Figure 2.

PDFs of (a) $- Q_s$ and (b) $ Q_\omega$ for different concentrations $\phi$ . The insets show the same PDF curves but normalised by their corresponding standard deviations $\sigma$ . Here, we use the $R_\lambda =203$ dataset.

Next, we present the PDFs of third-order invariants $R_s$ and $-R_\omega$ in figure 3. In figure 3(a,c), the PDF curves exhibit a clear trend of shrinking as $\phi$ increases, consistent with the reduction of ensemble average shown in figure 1(b). Similar to the PDFs of $- Q_s$ and $ Q_\omega$ shown in figure 2, the third-order quantities are also significantly depressed by the polymer additives, as reported in previous studies (Liberzon et al. Reference Liberzon, Guala, Lüthi, Kinzelbach and Tsinober2005, Reference Liberzon, Guala, Kinzelbach and Tsinober2006; Cai et al. Reference Cai, Li and Zhang2010; Cocconi et al. Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017; ur Rehman et al. Reference ur Rehman, Lee and Lee2022). Notably, both $R_s$ and $-R_\omega$ are positively skewed, which remains consistent in polymeric turbulence. And this skewness plays a crucial role in the turbulent energy transfer process. In the context of decaying homogeneous isotropic turbulence, Cai et al. (Reference Cai, Li and Zhang2010) analysed the changes in production terms, and concluded that polymer addition reduces the strength of vortex stretching, leading to the drag-reducing phenomenon.

Figure 3.

The PDFs of (a) $R_s$ and (c) $-R_\omega$ for different polymer concentrations $\phi$ . (b,d) The PDFs using the same data as in (a,c), but normalised by their corresponding standard deviations, and the curves of higher $\phi$ have been shifted downwards by $10^{0.6}$ relative to their lower-value neighbours, for clarity. The insets show the original, unshifted data. These results are based on the $R_\lambda =203$ dataset.

In figure 3(b,d), we present the same data as in figure 3(a,c), but normalised by their own standard deviations. Note that in the insets, the normalised PDF curves with different $\phi$ exhibit similar forms. However, upon closer examination, we observe that right-branch (positive values) curves, representing a positive generation of enstrophy, decrease more rapidly with increasing $\phi$ than left-branch curves (negative values). In the main plots of figure 3(b,d), these curves for different $\phi$ have been uniformly shifted down $10^{0.6}$ equidistant from each other for clearer comparisons. An interesting finding is that the asymmetric distribution of $R_s$ or $-R_\omega$ is suppressed by polymers, resulting in a decreased value of skewness, which is consistent with figure 1(c). Cocconi et al. (Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017) first noticed that polymer additions have a stronger depressive effect on the positive values of vortex stretching $\omega _i S_\textit{ij} \omega _j$ than on the negative parts. Subsequently, ur Rehman et al. (Reference ur Rehman, Lee and Lee2022) studied the orientation between vorticity $\omega _i$ and $S_\textit{ij} \omega _j$ . They found that the probability of parallel alignment between these vectors decreases, while the probability of their being antiparallel increases, which favours negative values of vortex stretching $\omega _i S_\textit{ij} \omega _j$ . Our current experiments yield results similar to those shown in figure 9(a) of ur Rehman et al. (Reference ur Rehman, Lee and Lee2022) (not shown for simplicity). In summary, polymers suppress intense events of the third-order moments of the VGT, with a more substantial impact on positive values of vortex stretching. Our experimental results at higher Reynolds numbers are fully consistent with previous findings at lower Reynolds numbers.

Figure 4.

Joint PDFs of (a–d) $Q$ and $R$ , (e–h) $Q_s$ and $R_s$ , (i–l) $Q_s$ and $Q_\omega$ , non-dimensionalised by the second-order moment $\langle Q_\omega \rangle$ . The values of the PDF contours are logarithmic, and are $0.01,0.1,1$ in (a–d), $0.005, 0.05,0.5$ in (e–h), and $0.01,0.1,1$ in (i–l). Results are for (a,e,i) the Newtonian case, (b, f, j) $\phi =5\,{\textrm{ppm}}$ , (c,g,k) $\phi =10\,{\textrm{ppm}}$ , (d,h,l) $\phi =25\,{\textrm{ppm}}$ . Black curves correspond to the $R_\lambda =203$ case, and (d,h,l) also provide the results from another two Reynolds numbers with $\phi =25\,{\textrm{ppm}}$ .

3.3. Local flow topology

In addition to basic statistics such as ensemble averages and PDFs, the impact of polymers on small-scale turbulence can be further investigated through the local flow topology, which could be characterised by the invariants introduced earlier in this section. In figure 4, we present joint PDFs of these invariants, which often exhibit universal features across various turbulent flows. The joint PDFs of $Q$ and $R$ , $Q_s$ and $R_s$ , $Q_s$ and $Q_\omega$ are shown in the left, middle and right columns of the figure, respectively. The first row corresponds to the Newtonian case, while the three rows below show the polymeric cases with polymer concentrations $\phi = 5$ , $10$ and $25\,{\textrm{ppm}}$ , respectively, to better illustrate the concentration dependence of the results. The black curves correspond to $R_\lambda =203$ , and in order to clarify the Reynolds number dependence, we also include the $\phi =25\,{\textrm{ppm}}$ case for two additional Reynolds numbers in figure 4(d,h,l).

In figure 4(a–d), we show the joint PDFs between $Q$ and $R$ , non-dimensionalised by $\langle Q_\omega \rangle$ . Contours with values $0.01,0.1,1$ have been selected. The Vieillefosse tails denoted by $D \equiv ({27}/{4}) R^2+Q^3 = 0$ are also shown in figure 4, where $D$ represents the discriminant of the characteristic equation of $\boldsymbol{A}$ . The different regions in this $R$ – $Q$ map characterise the local flow topology: the $D\gt 0, R\lt 0$ and $D\gt 0, R\gt 0$ regions correspond to local vortex stretching and vortex compression, respectively, while the $D\lt 0, R\lt 0$ and $D\lt 0, R\gt 0$ regions correspond to deformation motion subjected to uniaxial extension and biaxial extension, respectively (Vieillefosse Reference Vieillefosse1982, Reference Vieillefosse1984; Cantwell Reference Cantwell1992; Meneveau Reference Meneveau2011). As shown in figure 4(a) for the Newtonian case, one can easily recognise the teardrop shape of the PDF, which is consistent with previous experimental results (Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014; Buxton et al. Reference Buxton, Breda and Chen2017; Fiscaletti et al. Reference Fiscaletti, Ragni, Overmars, Westerweel and Elsinga2022; Warwaruk & Ghaemi Reference Warwaruk and Ghaemi2024). This teardrop shape of the PDF on the $R$ – $Q$ map is believed to be a universal phenomenon for small scales in turbulent flows (Elsinga & Marusic Reference Elsinga and Marusic2010; Meneveau Reference Meneveau2011).

However, after adding polymers, one can clearly see that the contours in figure 4(b–d) shrink in the $R$ direction, contrasting with the Newtonian case. Furthermore, as $\phi$ increases, the degree of shrinkage becomes more pronounced. This attenuation of $R$ indicates the suppression of dissipation and enstrophy generation, consistent with previous discussions on the depression of mean values and PDFs of $R_s$ and $R_\omega$ . Interestingly, our findings here are significantly different from previous results reported by Liberzon et al. (Reference Liberzon, Guala, Kinzelbach and Tsinober2006) and Perlekar et al. (Reference Perlekar, Mitra and Pandit2010). They observed that the teardrop shape still exists on the $R$ – $Q$ map for polymeric turbulence, albeit reduced in area. These differences may arise from the relatively small Reynolds numbers and Weissenberg numbers in the literature. A low Reynolds number implies underdeveloped turbulence, while a low Weissenberg number indicates that the flow is not vigorous enough to effectively stretch polymer molecules, leading to limited polymer–turbulence interactions.

This dramatic change of $R$ – $Q$ PDFs might be understood by the polymer dynamics in turbulent flow. It is expected that the coiled polymer molecule can be stretched by the flow into long chain structures (White & Mungal Reference White and Mungal2008), as clearly shown by a number of numerical simulations (Terrapon et al. Reference Terrapon, Dubief, Moin, Shaqfeh and Lele2004; Peters & Schumacher Reference Peters and Schumacher2007; Watanabe & Gotoh Reference Watanabe and Gotoh2010). In Watanabe & Gotoh (Reference Watanabe and Gotoh2010), by plotting the polymer extension conditioned on the different regions of the $R$ – $Q$ map, they reported that the longest polymer or maximum extension is found preferentially in the upper left or lower right regions. These regions correspond to extreme vortex stretching or biaxial extension motion, where polymers experience significant stretching. Analogous to a dumbbell spring, once stretched, the polymer may dampen the flow, especially in these specific regions. We also notice that people have suggested the interaction between polymer and near-wall vortices as an explanation of turbulent drag reduction (Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004), and indeed, people observed depression of the VGT, especially $R$ in polymeric wall turbulence (Mortimer & Fairweather Reference Mortimer and Fairweather2022; Warwaruk & Ghaemi Reference Warwaruk and Ghaemi2024), which might be a relevant phenomenon for our findings here in bulk turbulence.

In figure 4(e–h), we present the joint PDFs of the non-dimensionalised invariants of $\boldsymbol{S}$ , $Q_s$ and $R_s$ , with contour values $0.005, 0.05,0.5$ . Since the rate-of-strain tensor $\boldsymbol{S}$ is real and symmetric, the PDFs should lie below the curve defined by $D_s \equiv ({27}/{4}) R_s^2+Q_s^3$ , i.e. $D_s$ should be less than $0$ , where $D_s$ denotes the discriminant of the characteristic equation of $\boldsymbol{S}$ . In reality, a few events with $D_s\gt 0$ arise due to unavoidable divergence errors in the experiments, which can also be seen in previous experimental results (Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014; Warwaruk & Ghaemi Reference Warwaruk and Ghaemi2024). These events are rare (approximately 1 % of overall single-point statistics) and do not significantly impact our conclusions. In general, this joint PDF of $Q_s$ and $R_s$ characterises the local topology of deformation motion, and given that $R_s = -\lambda _1 \lambda _2 \lambda _3$ in incompressible flows, $R_s$ also marks the sign of $\lambda _2$ . In the Newtonian case shown in figure 4(e), the contours are inclined to the right-hand dashed line with $D_s=0, R_s\gt 0$ , which corresponds to $\lambda _1 : \lambda _2 : \lambda _3 = 1:1:-2$ . This result is consistent with the previous finding in the literature that the ratios $\lambda _1 : \lambda _2 : \lambda _3 = 3:1:-4$ or $2:1:-3$ are most probable (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994; Blackburn et al. Reference Blackburn, Mansour and Cantwell1996), suggesting a geometry of biaxial stretching. However, in the polymeric case shown in figure 4( f–h), the contours gradually approach the vertical dashed line as $\phi$ increases, corresponding to two-dimensional stretching with $\lambda _1 : \lambda _2 : \lambda _3 = 1:0:-1$ . This is consistent with the results shown in figures 3(b) and 4(b–d), since the third-order invariant $R_s$ vanishes in this two-dimensional structure.

Next, in figure 4(i–l), we examine the joint PDFs of $Q_s$ and $Q_\omega$ , which characterise the flow topology associated with energy dissipation. We have chosen contour values $0.01$ , $0.1$ and $1$ . In the Newtonian case shown in figure 4(i), the PDF concentrates towards the horizontal axis. This region represents the event with strong enstrophy but weaker dissipation, corresponding to a vortex tube structure (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994; da Silva & Pereira Reference da Silva and Pereira2008). Surprisingly, when polymers are added, the PDF shape changes dramatically, as shown in figure 4( j–l), where we observe that the PDF increasingly concentrates along the diagonal line as the polymer concentration increases. This region represents the events with comparable strength between enstrophy and dissipation, corresponding to vortex sheet structures (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994).

Combining the results in figure 4, we can conclude that the local flow topology in polymeric turbulence differs significantly from the Newtonian case. The local flow patterns of vortex stretching and biaxial extension, which are abundant in Newtonian fluid turbulence and essential for the stretching of polymer molecules, are greatly suppressed. Furthermore, as the concentration increases, this suppression effect exhibits a nearly continuous trend and becomes progressively stronger. Additionally, figure 4 exhibits statistical characteristics resembling a vortex sheet structure with two-dimensional properties, which has not been reported in other studies. Finally, results with different $R_\lambda$ in figure 4(d,h,l) demonstrate that these findings become more pronounced as $R_\lambda$ increases.

3.4. Statistics related to eigenvalues and eigenvectors of the rate-of-strain tensor

The joint PDFs of $Q_s$ and $R_s$ shown in figure 4(h) suggest that the two-dimensional strain structure, indicated by the ratios $\lambda _1 : \lambda _2 : \lambda _3 = 1 : 0 : -1$ , dominates the statistics of the rate-of-strain tensor $\boldsymbol{S}$ in polymeric turbulence. In this subsection, we further investigate the statistics related to the eigenframe of $\boldsymbol{S}$ and its dependence on concentration $\phi$ . Notice that the PDFs of $\lambda _1$ , $\lambda _2$ and $\lambda _3$ all shrink significantly with increasing $\phi$ , similar to the trend of invariants shown in § 3.2, which are not shown here for simplicity. We introduce the relative magnitudes of $\lambda _1$ and $\lambda _2$ , defined as $\lambda _1^\ast \equiv \sqrt {6}\,\lambda _1 / \sqrt {\lambda _1^2 + \lambda _2^2 + \lambda _3^2}$ and $\lambda _2^\ast \equiv \sqrt {6}\,\lambda _2 / \sqrt {\lambda _1^2 + \lambda _2^2 + \lambda _3^2}$ , and present the PDFs of $\lambda _1^\ast$ and $\lambda _2^\ast$ for different values of $\phi$ in figure 5(a,b), respectively. For the Newtonian case represented by the blue line marked with a cross symbol, we observe that the PDF of $\lambda _1^\ast$ peaks at approximately 1.5 and is nearly symmetric, while the distribution of $\lambda _2^\ast$ is asymmetric with a peak value at approximately $\lambda _2^\ast = 0.5$ , consistent with previous studies (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Lund & Rogers Reference Lund and Rogers1994; Ganapathisubramani et al. Reference Ganapathisubramani, Lakshminarasimhan and Clemens2008; Buaria, Bodenschatz & Pumir Reference Buaria, Bodenschatz and Pumir2020). It is worth noting that our measured VGTs do not fully satisfy the divergence-free condition, which affects the ranges of values in figure 5. Ideally, incompressible flows should yield values within the ranges $\lambda _1^\ast \in [1,2]$ and $\lambda _2^\ast \in [-1,1]$ .

On the other hand, when polymers are added to the flow, the PDFs of $\lambda _1^\ast$ and $\lambda _2^\ast$ both change. Figure 5(a) shows that as $\phi$ increases, the peak becomes more pronounced and shifts towards $\sqrt {3}$ , indicated by the vertical dashed line. Meanwhile, in figure 5(b), the PDF of $\lambda _2^\ast$ gradually becomes symmetric with respect to $\lambda _2^\ast = 0$ , which is consistent with previous numerical studies on bulk turbulence with polymers (Perlekar et al. Reference Perlekar, Mitra and Pandit2010; Cocconi et al. Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017). With increasing concentrations, we clearly see that the PDFs in figure 5 become contracted, with their peaks increasing and converging toward a certain value. Actually, the changes of the PDFs of $\lambda _1^\ast$ and $\lambda _2^\ast$ are mutually consistent, corresponding to the dominance of the eigenvalue distribution $\lambda _1 : \lambda _2 : \lambda _3 = 1 : 0 : -1$ , due to $\lambda _1 + \lambda _2 + \lambda _3 = 0$ , as we already observed in figure 4(h). Additionally, Liberzon et al. (Reference Liberzon, Guala, Lüthi, Kinzelbach and Tsinober2005) and Cocconi et al. (Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017) listed the ensemble averages of these three eigenvalues in bulk turbulence with and without polymers, which is also consistent with our findings here.

Figure 5.

The PDFs of (a) the normalised extensive eigenvalue $\lambda _1^\ast \equiv \sqrt {6}\,\lambda _1 / \sqrt {\lambda _1^2 + \lambda _2^2 + \lambda _3^2}$ and (b) the normalised intermediate eigenvalue $\lambda _2^\ast =\sqrt {6}\,\lambda _2/\sqrt {\lambda _1^2+\lambda _2^2+\lambda _3^2}$ of $\boldsymbol{S}$ for different values of $\phi$ . Notice that our measured VGTs do not fully satisfy the divergence-free condition. Thus the values of $\lambda _1^\ast$ and $\lambda _2^\ast$ do not completely fall in the ranges $[1,2]$ and $[-1,1]$ , respectively, satisfied by incompressible flows. Dataset $R_\lambda =203$ is used here.

Apart from the eigenvalues $\lambda _i$ of $\boldsymbol{S}$ , the statistics of their corresponding eigenvectors $\boldsymbol{e}_i$ are also of interest, especially their alignments with the vorticity vector $\boldsymbol{\omega }$ (Elsinga & Marusic Reference Elsinga and Marusic2010; Meneveau Reference Meneveau2011). It is well known that in Newtonian turbulence, the vorticity vector has a preferential alignment with the intermediate eigenvector $\boldsymbol{e}_2$ (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Meneveau Reference Meneveau2011), which is also observed in our experiments. In figure 6, we plot the PDFs of $\vert {\cos (\boldsymbol{\omega }, \boldsymbol{e}_i)}\vert$ for different concentrations $\phi$ , and in figure 6(b), the Newtonian result indicated by the blue lines marked with cross symbols clearly shows a pronounced peak at $\vert {\cos (\boldsymbol{\omega }, \boldsymbol{e}_2)}\vert =1$ . Additionally, the tendency for vorticity to be randomly aligned with the extensive eigenvector (resulting in a flat PDF for $\vert {\cos (\boldsymbol{\omega }, \boldsymbol{e}_1)}\vert$ ) and perpendicular to the compressive one (with peaks at $\vert {\cos (\boldsymbol{\omega }, \boldsymbol{e}_3)}\vert =0$ ) can also be observed in figure 6(a,c), respectively.

Figure 6.

Alignment of the vorticity vector $\boldsymbol{\omega }$ with the (a) first ( $\boldsymbol{e}_1$ ), (b) second ( $\boldsymbol{e}_2$ ) and (c) third ( $\boldsymbol{e}_3$ ) eigenvectors of $\boldsymbol{S}$ , for different values of $\phi$ . Dataset $R_\lambda =203$ is used here.

Now we move on to the discussions for polymeric turbulence, and we can see from figure 6 that as the concentration $\phi$ increases, the preferential alignment of $\boldsymbol{\omega }$ with $\boldsymbol{e}_2$ becomes more pronounced, and the peak value of the PDF curve also increases. At the same time, the vorticity vector tends to be orthogonal to $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ , with a more significant change observed for $\boldsymbol{e}_1$ . At the highest concentration $\phi =25\,{\textrm{ppm}}$ considered here, the PDFs of $\vert {\cos (\boldsymbol{\omega }, \boldsymbol{e}_1)}\vert$ and $\vert {\cos (\boldsymbol{\omega }, \boldsymbol{e}_3)}\vert$ are nearly identical, indicating an axisymmetric configuration of vorticity in the eigenframe of strain. Our findings here differ from previous DNS studies, where it was reported that adding polymers has little impact on the alignment between the vorticity and the eigenvectors of the strain tensor $\boldsymbol{S}$ (Watanabe & Gotoh Reference Watanabe and Gotoh2010; Cocconi et al. Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017; ur Rehman et al. Reference ur Rehman, Lee and Lee2022). It is possible that the inconsistency between our experimental results and previous DNS studies is related to the large-scale structure. In the current experiments conducted in VKS, the configuration satisfies only the axisymmetry condition rather than isotropy. We will discuss the effect of large-scale anisotropy in detail in the next subsection.

3.5. Enhanced small-scale anisotropy

In the preceding subsections, we explored the statistics of invariants related to the VGT and the eigenframe of $\boldsymbol{S}$ . Now, in this subsection, we move to the discussions on the impact of large-scale effects on VGT statistics, with a particular focus on the influence of large-scale anisotropy. In real fluid systems, such as wall flows and free-shear flows, large-scale motions are inherently anisotropic. Adding polymers to the flow further amplifies this anisotropy. For instance, in boundary layers, the streamwise velocity fluctuation increases with polymer addition, while the transverse fluctuation decreases (Warholic, Massah & Hanratty Reference Warholic, Massah and Hanratty1999; White, Somandepalli & Mungal Reference White, Somandepalli and Mungal2004). Therefore, the anisotropy becomes more pronounced compared to the Newtonian counterpart. Similar observations have been made in other flow systems as well (Tong et al. Reference Tong, Goldburg, Huang and Witten1990; van Doorn, White & Sreenivasan Reference van Doorn, White and Sreenivasan1999; Boffetta et al. Reference Boffetta, Mazzino, Musacchio and Vozella2010; Lacassagne et al. Reference Lacassagne, Simoëns, El Hajem, Lyon and Champagne2019; Lin, Owolabi & Lin Reference Lin, Owolabi and Lin2022; Peng, Zhang & Xi Reference Peng, Zhang and Xi2023; Wang et al. Reference Wang, Zhang and Xi2025a ; Xu et al. Reference Xu, Liu, Li and Xia2025). In systems with finite Reynolds numbers, the anisotropy of large-scale motions has a substantial influence on small-scale quantities like the VGT, resulting in directional dependence in its statistical properties. The VKS system employed in our study is also anisotropic, satisfying only the axisymmetric condition. As a result, statistical differences arise between the axial and horizontal directions (La Porta et al. Reference La Porta, Voth, Crawford, Alexander and Bodenschatz2001; Voth et al. Reference Voth, La Porta, Crawford, Alexander and Bodenschatz2002; Ouellette et al. Reference Ouellette, Xu, Bourgoin and Bodenschatz2006). An intriguing and crucial question then emerges: how does the presence of polymers affect the anisotropy of the VKS flow, particularly the properties of the VGT that we investigate in this work?

The anisotropy of the VGT can be characterised by the orientation of the vorticity vector $\boldsymbol{\omega }$ and the eigenvectors $\boldsymbol{e}_i$ ( $i=1,2,3$ ) of $\boldsymbol{S}$ in the laboratory frame. We denoted the unit vectors of laboratory coordinates as $\boldsymbol{e}_x$ , $\boldsymbol{e}_y$ and $\boldsymbol{e}_z$ . In figures 7 and 8, we plot the statistics of alignments between vorticity and eigenvectors of strain with respect to the laboratory coordinates for different polymer concentrations. Let us begin by examining the Newtonian case, represented by the blue lines marked with cross symbols. In an ideal isotropic system, the vectors $\boldsymbol{\omega }$ and $\boldsymbol{e}_i$ ( $i=1,2,3$ ) should not exhibit any preferential alignment with the laboratory axes $\boldsymbol{e}_a$ ( $a=x,y,z$ ). But from figure 7, we can see that the vorticity vector $\boldsymbol{\omega }$ and the intermediate eigenvector $\boldsymbol{e}_2$ tend to align weakly with the vertical direction $\boldsymbol{e}_y$ , while $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ show a weak tendency to be perpendicular to $\boldsymbol{e}_y$ . We notice that all the results shown in figures 7 and 8 exhibit symmetry between $\boldsymbol{e}_x$ and $\boldsymbol{e}_z$ , consistent with the axisymmetric nature of the VKS flows. Our findings here agree with the previous experimental measurements (Zimmermann et al. Reference Zimmermann, Xu, Gasteuil, Bourgoin, Volk, Pinton and Bodenschatz2010), revealing subtle yet observable anisotropy in the VGT in Newtonian von Kármán turbulence.

Figure 7.

Alignments of the vorticity vector $\boldsymbol{\omega }$ with the laboratory coordinates $\boldsymbol{e}_a$ ( $a=x,y,z$ ) for different concentrations $\phi$ . Dataset $R_\lambda =203$ is used here.

Figure 8.

Alignments of the three eigenvectors $\boldsymbol{e}_i$ ( $i=1,2,3$ ) of $\boldsymbol{S}$ with the laboratory coordinates $\boldsymbol{e}_a$ ( $a=x,y,z$ ) for different $\phi$ . Dataset $R_\lambda =203$ is used here.

When polymers are introduced, figures 7 and 8 show that the trend of alignment gradually strengthens with increasing polymer concentration. Specifically, at the highest concentration $\phi =25\,{\textrm{ppm}}$ , $\boldsymbol{\omega }$ and $\boldsymbol{e}_2$ strongly aligns with $\boldsymbol{e}_y$ , while $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ prefer to be perpendicular to $\boldsymbol{e}_y$ and lie in the $\boldsymbol{e}_x O\boldsymbol{e}_z$ plane. Still, the statistics at all values of $\phi$ exhibit symmetry between $\boldsymbol{e}_x$ and $\boldsymbol{e}_z$ . We can clearly see a pronounced enhancement of small-scale anisotropy due to polymer additions. Furthermore, the turbulent energy cascade is known for erasing anisotropy from large scales through inter-scale energy transfer within the inertial range (Frisch Reference Frisch1995), and as the Reynolds number increases, small-scale statistics tend towards isotropy. However, in our study, despite increasing the Reynolds number from $168$ to $235$ (not shown in the figures), we do not observe this trend when polymers are added. This suggests that the enhancement of small-scale anisotropy may be linked to the suppression of energy cascades caused by polymer additions.

In such an anisotropic flow system, several open questions arise. The first is that, in addition to small-scale local shear, large-scale shear may also act to stretch the polymer molecules. In the centre region of the VKS flow, the turbulent fluctuation is much stronger than the mean velocity field, suggesting that polymers are primarily elongated by the turbulent flow. In our previous work (Wang et al. Reference Wang, Zhang and Xi2025b ), we observed that polymers do not substantially alter the mean flow pattern – i.e. the pumping and shearing motions characteristic of the VKS flow remain largely unchanged. However, polymers do reduce the intensity of large-scale flows while increasing their characteristic length and time scales, effectively weakening large-scale shear. Within the limited measurement volume considered in the present study (approximately 3 cm near the centre of the VKS set-up), these results suggest that large-scale flows may have a limited ability to stretch polymer molecules. That said, we cannot fully exclude the potential influence of large-scale shear in polymeric turbulence. This important question will be addressed in future work, which will entail larger measurement volumes, broader parameter ranges, and alternative experimental configurations.

Another natural question is: does the orientation of polymer molecules have a preferred direction? This issue has been extensively discussed in previous studies (Den Toonder et al. Reference Den Toonder, Hulsen, Kuiken and Nieuwstadt1997; van Doorn et al. Reference van Doorn, White and Sreenivasan1999; Lacassagne et al. Reference Lacassagne, Simoëns, El Hajem, Lyon and Champagne2019; Peng et al. Reference Peng, Zhang and Xi2023) and confirmed through DNS studies (Guimarães et al. Reference Guimarães, Pimentel, Pinho and da Silva2020; Mortimer & Fairweather Reference Mortimer and Fairweather2022). Even in the case of homogeneous isotropic turbulence without a mean flow, observations indicate that polymers tend to be perpendicular to the eigenvector $\boldsymbol{e}_3$ , and prefer to align with $\boldsymbol{e}_2$ at high $\textit{Wi}$ (Watanabe & Gotoh Reference Watanabe and Gotoh2010; Valente, Da Silva & Pinho Reference Valente, Da Silva and Pinho2014; ur Rehman et al. Reference ur Rehman, Lee and Lee2022). Given these findings, it seems plausible that polymers also exhibit a preferred orientation in VKS flows, namely, along the axial direction. This preferential orientation, in turn, modifies the local flow topology and further enhances the anisotropy. Similar observations of polymer-induced small-scale anisotropy have been reported in previous studies (Crawford et al. Reference Crawford, Mordant, Xu and Bodenschatz2008; Boffetta et al. Reference Boffetta, Mazzino, Musacchio and Vozella2010; Vonlanthen & Monkewitz Reference Vonlanthen and Monkewitz2013; Wang et al. Reference Wang, Zhang and Xi2025b ). Moreover, several works have suggested that the alignment of polymer molecules is influenced by local flow topology (Watanabe & Gotoh Reference Watanabe and Gotoh2010; Valente et al. Reference Valente, Da Silva and Pinho2014; ur Rehman et al. Reference ur Rehman, Lee and Lee2022), and is often oriented along the mean flow direction (Guimarães et al. Reference Guimarães, Pimentel, Pinho and da Silva2020; Mortimer & Fairweather Reference Mortimer and Fairweather2022). However, the mechanism by which oriented polymers enhance existing anisotropy remains a topic for further theoretical exploration, which would inevitably improve our understanding of drag-reducing for wall flows when polymers are present.

3.6. Vortex sheet structures

We have already seen that in a turbulent VKS flow, polymers have strong effects on the statistics of the VGT. Recall the results in figures 4–8; some of them are consistent with previous studies, and some are quite different. To provide a plausible explanation for our findings, let us summarise the key observations for polymeric VKS turbulence.

1. (i) Attenuation of the third-order statistics (such as $R$ or $R_s$ ; see figure 4 b,d).

2. (ii) The ratios between strain eigenvalues tend to be $ \lambda _1 : \lambda _2 : \lambda _3 = 1:0:-1$ (see figures 4 d and 5).

3. (iii) The coexistence of dissipation $Q_s$ and enstrophy $Q_\omega$ with similar strength (see figure 4 f).

4. (iv) The vorticity vector $\boldsymbol{\omega }$ aligns preferentially with $\boldsymbol{e}_2$ and is perpendicular to both $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ (see figure 6).

5. (v) The preferential alignment of $\boldsymbol{\omega }$ and $\boldsymbol{e}_2$ with the aixal direction $\boldsymbol{e}_y$ , and the symmetry between $\boldsymbol{e}_x$ and $\boldsymbol{e}_z$ in all statistics (see figures 7 and 8).

Surprisingly, we find that a vortex sheet model, like the Burgers vortex layer solution for the Navier–Stokes equations, can perfectly explain results (i)–(iv) listed above. We sketch the Burgers vortex sheet solution (see e.g. equation 5 in Andreotti Reference Andreotti1997) in figure 9, where it becomes evident that the Burgers layer corresponds to a simple shear structure. In a simple local shear flow, it is well established that the magnitudes of the second-order invariants $Q_s$ and $Q_\omega$ are equal to each other, and the third-order invariants $R_s$ and $R_\omega$ vanish. Furthermore, the intermediate eigenvector of strain, $\boldsymbol{e}_2$ , aligns with the vorticity vector, and the ratios between strain eigenvalues are precisely given by $ \lambda _1 : \lambda _2 : \lambda _3 = 1:0:-1$ . The observation (v) can be attributed to the vortex layer being extended along the $\boldsymbol{e}_y$ axis and randomly orientated in the local horizontal plane, as depicted in figure 9. In this case, $\boldsymbol{\omega }$ and $\boldsymbol{e}_2$ point towards $\boldsymbol{e}_y$ , while $\boldsymbol{e}_1$ and $\boldsymbol{e}_3$ randomly orientate in the $\boldsymbol{e}_x O \boldsymbol{e}_z$ plane.

Figure 9.

Sketch illustrating the vortex sheet structure, where $\boldsymbol{u}$ denotes the velocity field, $\boldsymbol{\omega }$ is the vorticity, and $\boldsymbol{e}_{i}$ ( $i=1,2,3$ ) are the eigenvectors of $\boldsymbol{S}$ . The laboratory coordinates are denoted by orthogonal axes $\boldsymbol{e}_{a}$ ( $a=x,y,z$ ). Notice that the orientation of $\boldsymbol{e}_{1}$ (and thus $ \boldsymbol{e}_{3}$ ) is random within the $\boldsymbol{e}_{x} O \boldsymbol{e}_{z}$ plane, as depicted by the dashed circle. This figure is inspired by figure 1(b) of Andreotti (Reference Andreotti1997).

In summary, we have discovered that the single-point statistics of the VGT in polymeric turbulence can be well described by the local vortex sheet structure, which has not been reported previously. Our experimental results exhibit differences compared with the existing literature, which finds that with polymer additions, the shape of $R$ – $Q$ PDFs is qualitatively unchanged (Liberzon et al. Reference Liberzon, Guala, Kinzelbach and Tsinober2006; Perlekar et al. Reference Perlekar, Mitra and Pandit2010), and there are no significant differences for vorticity alignment with eigendirections of $\boldsymbol{S}$ from Newtonian fluids (Watanabe & Gotoh Reference Watanabe and Gotoh2010; Cocconi et al. Reference Cocconi, De Angelis, Frohnapfel, Baevsky and Liberzon2017; ur Rehman et al. Reference ur Rehman, Lee and Lee2022). This discrepancy may be attributed to the intrinsic anisotropy present in VKS flow, which is absent in the numerical simulations. In this section, we focus solely on single-point statistics. In the subsequent two sections, by investigating the instantaneous visualisation and conditional statistics on vorticity, we will try to identify the flow structures and explore their relation with our proposed model in figure 9.