1. Introduction
The addition of polymers to turbulent flows has attracted considerable attention due to their remarkable effects – most notably, the phenomenon of drag reduction in wall-bounded turbulence (Toms Reference Toms1948). Despite ongoing advances in the field (Lumley Reference Lumley1969; Procaccia, L’vov & Benzi Reference Procaccia, L’vov and Benzi2008; White & Mungal Reference White and Mungal2008; Xi Reference Xi2019), the underlying mechanisms remain incompletely understood. A particularly thought-provoking perspective is that polymers significantly alter the energy cascade – a process in turbulence that spans a wide range of spatial and temporal scales. Seminal theories by Lumley (Reference Lumley1969, Reference Lumley1973) and de Gennes (Reference de Gennes1986) propose that turbulence–polymer interactions become prominent below a characteristic scale, where polymers absorb and additionally dissipate kinetic energy, thereby disrupting the classical energy cascade from energy injection to dissipation scales. To analyse the energy transfer in three-dimensional turbulent flows, three main diagnostic tools are commonly employed: structure functions, energy spectra and the filtering method. Among these, structure functions and energy spectra have been extensively applied in the context of polymer-laden turbulence. For example, in Newtonian turbulence, the inertial subrange exhibits an approximately scale-independent flux of kinetic energy towards smaller scales (Kolmogorov Reference Kolmogorov1941); however, this behaviour is significantly altered in the presence of polymers. Experimental and numerical studies have reported an attenuation in the small-scale regime of velocity structure functions (De Angelis et al. Reference De Angelis, Casciola, Benzi and Piva2005; Ouellette, Xu & Bodenschatz Reference Ouellette, Xu and Bodenschatz2009; Xi, Bodenschatz & Xu Reference Xi, Bodenschatz and Xu2013; de Chaumont Quitry & Ouellette Reference de Chaumont Quitry and Ouellette2016), energy spectra (Kalelkar, Govindarajan & Pandit Reference Kalelkar, Govindarajan and Pandit2005; Berti et al. Reference Berti, Bistagnino, Boffetta, Celani and Musacchio2006; Perlekar et al. Reference Perlekar, Mitra and Pandit2006, Reference Perlekar, Mitra and Pandit2010; Cai, Li & Zhang Reference Cai, Li and Zhang2010; Vonlanthen & Monkewitz Reference Vonlanthen and Monkewitz2013; Watanabe & Gotoh Reference Watanabe and Gotoh2013, Reference Watanabe and Gotoh2014; Nguyen et al. Reference Nguyen, Delache, Simoëns, Bos and El Hajem2016; Valente, da Silva & Pinho Reference Valente, da Silva and Pinho2016; Fathali & Khoei Reference Fathali and Khoei2019; ur Rehman, Lee & Lee Reference ur Rehman, Lee and Lee2022; Zhang & Xi Reference Zhang and Xi2022) and flow structures (Cai et al. Reference Cai, Li and Zhang2010; Watanabe & Gotoh Reference Watanabe and Gotoh2013; Zhang et al. Reference Zhang, Wang, Peng and Xi2025), consistent with the suppression of small-scale energy transfer predicted by earlier theories (Lumley Reference Lumley1969, Reference Lumley1973; de Gennes Reference de Gennes1986). This suppression has also manifested as a new scaling behaviour in recent studies (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Rosti, Perlekar & Mitra Reference Rosti, Perlekar and Mitra2023). To further investigate the energy cascade in polymer-laden turbulence, the present study introduces the third approach – the filtering method (Germano Reference Germano1992) – which, to date, has seen limited application in this context.
A low-pass filter at a specified scale effectively removes small-scale fluctuations while preserving coherent large-scale motions, thereby providing a valuable tool for probing the interactions between large and small scales (Meneveau & Katz Reference Meneveau and Katz2000). In particular, it enables a quantitative assessment of scale-to-scale energy transfer in turbulence, expressed as
\begin{equation} \varPi = -\tau _{\textit{ij}}\tilde {S}_{\textit{ij}} , \end{equation}
where
$\tau _{\textit{ij}}$
denotes the subgrid-scale (SGS) stress tensor and
$\tilde {S}_{\textit{ij}} = ({1}/{2}) ( \tilde {A}_{\textit{ij}} + \tilde {A}_{\textit{ji}} )$
is the filtered strain-rate tensor. Here,
$\tilde {\boldsymbol{\cdot }}$
indicates the filtering operation and
$\tilde {A}_{\textit{ij}} = {\partial \tilde {u}_i}/{\partial x_{\!j}}$
is the filtered velocity gradient tensor. The SGS stress tensor
$\tau _{\textit{ij}}$
originates from the nonlinear convective term in the Navier–Stokes equations after filtering, and accounts for the effects of unresolved small-scale motions on the large scales. It is defined as
\begin{equation} \tau _{\textit{ij}} = \widetilde {u_i u_{\!j}} - \tilde {u}_i \tilde {u}_{\!j} . \end{equation}
From (1.1), it is evident that the energy flux
$\varPi$
results from the interaction between a stress tensor and a strain-rate tensor, highlighting the critical importance of the geometric structures of
$\tau _{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
– as well as their relative alignment – in the energy transfer process. Furthermore, considerable effort has been devoted to understanding the statistical geometry of the unfiltered strain-rate tensor and vorticity vector (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Meneveau Reference Meneveau2011), owing to their relevance to the vortex stretching mechanism (Taylor Reference Taylor1938). A well-established observation is that the vorticity tends to align with the intermediate eigendirection of the strain-rate tensor – associated with a weakly extensional eigenvalue – rather than with the most extensional direction (Meneveau Reference Meneveau2011). Interestingly, this phenomenon has also been examined experimentally in polymeric turbulence (Wang et al. Reference Wang, Zhang, Yang and Xi2025c
), where polymers are found to enhance this alignment while reducing the corresponding strain-rate eigenvalue. However, relatively few studies have explored the framework based on the filtered tensors
$\tau _{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
(Tao, Katz & Meneveau Reference Tao, Katz and Meneveau2002; Higgins, Parlange & Meneveau Reference Higgins, Parlange and Meneveau2003; Yang & Wang Reference Yang and Wang2016; Ballouz & Ouellette Reference Ballouz and Ouellette2018), and even fewer have addressed these statistics in the context of polymer-laden turbulence. To advance our understanding of how polymers suppress energy transfer, this study focuses on SGS statistics, with particular attention to the structures and alignment properties of the
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
tensors, as well as the behaviour of the energy flux
$\varPi$
.
The SGS stress tensor
$\tau _{\textit{ij}}$
plays a central role in large-eddy simulation (LES) of turbulent flows. In simulations of viscoelastic fluids, LES enables higher Reynolds numbers than those attainable by direct numerical simulation (DNS), while still preserving the temporal dynamics of the flow. Recent efforts have extended LES to polymeric turbulence. For instance, a temporal LES model for viscoelastic turbulent channel flow demonstrated a significant reduction in computational cost compared with DNS (Thais et al. Reference Thais, Tejada-Martinez, Gatski and Mompean2010), and was later applied to forced homogeneous isotropic turbulence with polymers (Wang, Cai & Li Reference Wang, Cai and Li2014). Ferreira et al. developed a new SGS closure model for viscoelastic fluids in homogeneous isotropic turbulence (Ferreira, Pinho & da Silva Reference Ferreira, Pinho and da Silva2016), which also proved effective in inhomogeneous conditions such as jet flow (Parvar, da Silva & Pinho Reference Parvar, da Silva and Pinho2021). Additionally, using DNS data from polymeric channel flow, Masoudian et al. conducted a priori tests to evaluate the modification of SGS terms in polymer-laden turbulence (Masoudian, da Silva & Pinho Reference Masoudian, da Silva and Pinho2016). These advancements have been supported by effective modelling of SGS stress and other terms arising from non-Newtonian fluid behaviour, as emphasised by Meneveau & Katz (Reference Meneveau and Katz2000) in the context of Newtonian turbulence. The present work provides an experimental investigation of polymeric turbulence, with an emphasis on SGS stress modelling. This enables us to conduct a priori tests using real experimental data for LES of flows with polymers. In practical LES applications, the most widely used SGS model is Smagorinsky’s model (Smagorinsky Reference Smagorinsky1963), which introduces a scalar eddy viscosity. However, this model shows relatively low correlation with the actual stress field (Liu, Meneveau & Katz Reference Liu, Meneveau and Katz1994). In contrast, tensor eddy-viscosity models, such as the nonlinear model, perform significantly better in predicting SGS stress (Borue & Orszag Reference Borue and Orszag1998). In polymeric turbulence, a priori tests have revealed the limitations of the Smagorinsky model (Ferreira et al. Reference Ferreira, Pinho and da Silva2016; Masoudian et al. Reference Masoudian, da Silva and Pinho2016), despite its application in practical LES scenarios (Ferreira et al. Reference Ferreira, Pinho and da Silva2016; Parvar et al. Reference Parvar, da Silva and Pinho2021). Therefore, it is promising to evaluate the performance of both scalar and tensor SGS models using real experimental data. This constitutes a secondary objective of the present work, with the aim of offering new insights into LES for viscoelastic fluids.
The filtering method is recognised as a powerful tool for investigating turbulence–polymer interactions via SGS statistics and energy flux. In this study, we conduct experiments on three-dimensional turbulence with polymer additives in a bulk flow, thereby minimising the interference typically introduced by solid boundaries and allowing for a clearer observation of the energy cascade. The bulk flow is generated in a von Kármán swirling (VKS) flow system, under conditions comparable to those in previous experiments (Ouellette et al. Reference Ouellette, Xu and Bodenschatz2009; Xi et al. Reference Xi, Bodenschatz and Xu2013; de Chaumont Quitry & Ouellette Reference de Chaumont Quitry and Ouellette2016; Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021, Reference Zhang, Wang, Peng and Xi2025), which differ mainly in the employed measurement techniques. In those earlier studies, the velocity field was obtained either from Lagrangian trajectories or from two-dimensional measurements within a spatial plane. However, Lagrangian data are spatially sparse, rendering the filtering method difficult to implement. Two-dimensional data acquired via planar particle image velocimetry (PIV) support spatial filtering, but even when supplemented by simplifying assumptions (Liu et al. Reference Liu, Meneveau and Katz1994), such data are subject to several limitations, including the need for amplitude corrections in energy flux estimates (Natrajan & Christensen Reference Natrajan and Christensen2006). These challenges may explain the limited adoption of the filtering approach in previous experimental investigations. In the present work, we employ tomographic PIV (Tomo-PIV), which provides three-dimensional, three-component velocity fields (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006), to study polymer-laden and Newtonian turbulence under comparable conditions. Building on previous work in Newtonian turbulence using holographic PIV (Tao et al. Reference Tao, Katz and Meneveau2000, Reference Tao, Katz and Meneveau2002), this study seeks to uncover previously unreported SGS properties in polymeric turbulence.
In this paper, we investigate the local energy flux and other SGS statistics in the bulk region of three-dimensional polymer-laden turbulence. Using Tomo-PIV, we measure the three-dimensional, three-component velocity field in the central region of a turbulent VKS flow, and apply spatial filtering to obtain the relevant SGS quantities. Specifically, we focus on an intermediate scale within the inertial range, which has recently been identified as a new scaling regime in polymeric turbulence (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Rosti et al. Reference Rosti, Perlekar and Mitra2023). The structure of the paper is as follows. Section 2 outlines the experimental set-up, Tomo-PIV measurements, general flow characterisation and data processing procedures including spatial filtering. Section 3 presents the main results, with a focus on SGS statistics. Section 4 discusses the implications of these findings for SGS modelling in LES of viscoelastic flows. Finally, § 5 summarises the key findings of this study.
2. Experimental methodology
2.1. Basic flow characterisation
The datasets used in this study are taken from our previous work (Wang et al. Reference Wang, Zhang and Xi2025a ), and here we briefly summarise the experimental set-up. For detailed descriptions of the configuration and measurement methods, the reader is referred to our earlier publications.
Experiments were conducted in a VKS flow system, consistent with previous studies (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Zhang & Xi Reference Zhang and Xi2022; Wang et al. Reference Wang, Zhang and Xi2025b
). In this closed cylindrical tank, two face-to-face baffled disks rotate in opposite directions to generate intense turbulence. The rotation frequency of the disks was set to
$f = 2$
Hz, corresponding to a large-scale Reynolds number of
$\textit{Re} = (2 \pi R_d f) R_d / \nu = 1.5 \times 10^5$
, where
$R_d = 110$
mm is the disk radius and
$\nu = 1 \times 10^{-6} \ \mathrm{m^2}\,\mathrm{s}^{-1}$
is the kinematic viscosity. Measurements were focused on the central region of the tank, sufficiently distant from solid boundaries, where the flow is referred to as bulk turbulence. Three-dimensional, three-component velocity fields were acquired using Tomo-PIV, both with and without polymer additives. The measurement volume was
$31\,{\rm mm} \times 29\,{\rm mm} \times 6 \, \mathrm{mm}$
in the
$x$
,
$y$
and
$z$
directions, resolved by a grid of
$113 \times 102 \times 22$
vectors with a spatial resolution of 0.28 mm.
Reynolds decomposition was applied to extract the fluctuating velocity components
$u_i(\boldsymbol{x}, t)$
, where
$i = x, y, z$
in the laboratory frame. At the high Reynolds numbers considered here, the mean flow is substantially weaker than the turbulent fluctuations (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Wang et al. Reference Wang, Zhang and Xi2025a
), and polymer additives do not modify the overall topology of the mean flow (Wang et al. Reference Wang, Zhang and Xi2025b
). More importantly, our focus is on the inertial-range dynamics of the turbulent fluctuations, and thus the term ‘velocity’ here refers specifically to the fluctuating component. The velocity field within the measurement domain is nearly homogeneous, allowing spatial averages to approximate ensemble averages. The root-mean-square (r.m.s.) velocity at a point is defined as
\begin{align} u^\prime (\boldsymbol{x}) = \left \langle \frac {1}{3} u_i (\boldsymbol{x}, t) u_i (\boldsymbol{x}, t) \right \rangle _t^{1/2}, \end{align}
and the global r.m.s. velocity is obtained as
\begin{align} u^\prime = \left \langle u^\prime (\boldsymbol{x}) \right \rangle _V\! , \end{align}
where
$\langle \boldsymbol{\cdot }\rangle _t$
and
$\langle \boldsymbol{\cdot }\rangle _V$
denote temporal and spatial averaging, respectively. In the Newtonian case, the global r.m.s. velocity was
$u^\prime = 0.22 \ \mathrm{m\,s}^{-1}$
, with component-wise r.m.s. values of
$0.24$
,
$0.16$
and
$0.25 \ \mathrm{m}\,\mathrm{s}^{-1}$
in the
$x$
,
$y$
and
$z$
directions, respectively, indicating anisotropy between the horizontal (
$x$
and
$z$
) and vertical (
$y$
) components.
The mean energy dissipation rate, estimated using Kolmogorov’s second-order structure function scaling, was
$\varepsilon = 0.11 \ \mathrm{m^2}\,\mathrm{s}^{-3}$
(additional technical details on the determination of the dissipation rate can be found in Wang et al. (Reference Wang, Zhang and Xi2025a
)). Based on this value, the Kolmogorov scales are calculated as
\begin{align} \eta = (\nu ^3/\varepsilon )^{1/4}, \quad \tau _\eta = (\nu /\varepsilon )^{1/2}, \quad u_\eta = (\nu \varepsilon )^{1/4}, \end{align}
and the Taylor microscale Reynolds number is
\begin{align} R_\lambda = \left ( \frac {15 u^{\prime 4}}{\nu \varepsilon } \right )^{1/2}\!. \end{align}
The polymer used is polyacrylamide with a weight-average molecular weight of
$M_w = 18 \times 10^6$
atomic mass units and a relaxation time of
$\tau _p = 38 \ \mathrm{ms}$
. Experiments were performed with four concentrations:
$\phi = 5$
,
$10$
,
$20$
and
$30$
parts per million (ppm) by weight. For the polymeric case, we define the Weissenberg number as
$Wi = \tau _p / \tau _\eta$
, which characterises the degree of polymer stretching. In our study,
$Wi = 12.7 \gt 1$
, indicating that polymers are significantly stretched by the turbulent flow (Lumley Reference Lumley1969). All characteristic scales listed above were calculated from the Newtonian case and used consistently for the polymeric cases, including the Kolmogorov time scale used in the definition of
$Wi$
. A summary of all key parameters is provided in table 1.
Flow characteristics in the current experiment. Quantities are computed from the Newtonian case and applied across all polymer concentrations.
2.2. Scale range of interest
In the present experimental dataset, the maximum accessible scale is limited in the
$z$
direction due to the shorter extent of the measurement volume along this axis. As an example, we calculate the second-order longitudinal velocity structure functions, defined as
$D_{\textit{LL}}(r_i) = \left \langle [ u_i (\boldsymbol{x} + \hat {\boldsymbol{r}}_i r, t) - u_i (\boldsymbol{x}, t) ]^2 \right \rangle$
, where
$\hat {\boldsymbol{r}}_i$
is the unit vector in the
$i$
direction (
$i = x,y,z$
), and the angle brackets denote ensemble averages over time and space. Figure 1 presents the structure function
$D_{\textit{LL}}(r)$
obtained by averaging
$D_{\textit{LL}}(r_i)$
over the three coordinate directions. All structure functions are normalised by the Kolmogorov length scale
$\eta$
and velocity scale
$u_\eta$
. In the current three-dimensional dataset, the maximum separation
$r$
(specifically
$r_z$
) extends slightly beyond
$100\eta$
, whereas in a single direction, such as
$x$
, the maximum
$r_x$
can reach up to
$500\eta$
.
(a) Second-order longitudinal velocity structure functions,
$D_{\textit{LL}}(r)$
, averaged over the three directions for different polymer concentrations
$\phi$
. All curves are normalised by the Kolmogorov length scale
$\eta$
and the Kolmogorov velocity scale
$u_\eta$
based on the Newtonian case. The dashed, dash-dotted and dotted lines indicate slopes of
$2/3$
,
$2$
and
$1.45$
, corresponding, respectively, to the inertial, dissipation and elastic scaling ranges reported in Zhang et al. (Reference Zhang, Bodenschatz, Xu and Xi2021). (b) Compensated representation of (a), emphasising the elastic scaling range.
For the Newtonian case (marked with blue circles) shown in figure 1(a),
$D_{\textit{LL}}$
agrees with Kolmogorov’s predictions for both the inertial and dissipation ranges, as evidenced by the slopes
$2/3$
and
$2$
, respectively. When polymers are added, the structure functions exhibit behaviours consistent with previous studies (De Angelis et al. Reference De Angelis, Casciola, Benzi and Piva2005; Ouellette et al. Reference Ouellette, Xu and Bodenschatz2009; Perlekar, Mitra & Pandit Reference Perlekar, Mitra and Pandit2010; Xi et al. Reference Xi, Bodenschatz and Xu2013; Watanabe & Gotoh Reference Watanabe and Gotoh2013; de Chaumont Quitry & Ouellette Reference de Chaumont Quitry and Ouellette2016; Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Rosti et al. Reference Rosti, Perlekar and Mitra2023). As
$\phi$
increases,
$D_{\textit{LL}}$
is significantly suppressed at small scales, and an additional scaling range emerges at the lower end of the inertial interval. This new range, characterised by a power-law exponent of approximately
$1.45$
(dotted line), originates from polymer-induced elasticity and is commonly referred to as the elastic range. To more clearly reveal the elastic scaling, figure 1(b) presents the compensated form of
$D_{\textit{LL}}$
. The compensated curves clearly display increasingly broad plateaus as the polymer concentration rises, highlighting the robustness of the
$\sim r^{1.45}$
scaling. These trends are fully consistent with earlier observations of the elastic range (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Wang et al. Reference Wang, Zhang and Xi2025a
).
In conclusion, despite the relatively limited scale extent, the current three-dimensional dataset is sufficient to observe the typical scaling behaviours, including the inertial and elastic ranges, as shown in figure 1. This supports our objective of conducting SGS analysis in three-dimensional turbulence. In particular, we focus on the scale
$r = 97\eta$
, which lies within the inertial or elastic range depending on the fluid type, and is marked by the magenta vertical line in figure 1, as a representative scale for comparison.
2.3. Filtering procedure
To extract the large-scale features of a volumetric field such as the velocity field
$\boldsymbol{u}(\boldsymbol{x})$
, a three-dimensional low-pass spatial filter with characteristic scale
$\varDelta$
is applied. This operation is mathematically defined as
\begin{equation} \tilde {u}_i (\boldsymbol{x}) = \iiint _{-\infty }^\infty u_i (\boldsymbol{y}) \varphi _\Delta (\boldsymbol{x}-\boldsymbol{y}) \mathop {}\!\mathrm{d}^3 \boldsymbol{y} ,\end{equation}
where
$\tilde {u}_i (\boldsymbol{x})$
denotes the filtered velocity component. Since the flow is statistically steady, the time dependence is omitted for brevity. The filtering kernel
$\varphi _\Delta$
is chosen to be a cube box filter of scale
$\varDelta$
, defined as
\begin{equation} \varphi _\Delta (\boldsymbol{x}) = \left \{ \begin{array}{llll} \frac {1}{\Delta ^3}, & \vert \boldsymbol{x}\vert \leqslant \frac {\varDelta }{2} \\[9pt] 0, & \vert \boldsymbol{x}\vert \gt \frac {\varDelta }{2}. \end{array} \right . \end{equation}
The box filter is chosen primarily for its computational efficiency and its ability to reach larger filtering scales, making it more suitable for the current dataset than smoother filters such as the Gaussian filter, which entail higher computational costs and more limited filtering scales. Previous studies have shown that the choice of filter has minimal impact on SGS analysis, provided the filter is positive in physical space, i.e.
$\varphi _\Delta (\boldsymbol{x}) \gt 0$
(Liu et al. Reference Liu, Meneveau and Katz1994; Borue & Orszag Reference Borue and Orszag1998).
From the filtered velocity field
$\tilde {\boldsymbol{u}}(\boldsymbol{x})$
, we can directly compute the filtered velocity gradient
$\tilde {\boldsymbol{A}} = \boldsymbol{\nabla }\tilde {\boldsymbol{u}}$
and the filtered vorticity
$\tilde {\boldsymbol{\omega }} = \boldsymbol{\nabla }\times \tilde {\boldsymbol{u}}$
using second-order finite differences. This allows us to calculate the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and the SGS stress tensor
$\tau _{\textit{ij}}$
. To ensure consistency with prior experimental studies (Tao et al. Reference Tao, Katz and Meneveau2002), we remove any small dilatational components from the strain-rate tensor, enforcing its tracelessness in all subsequent calculations.
In practice, data points near the boundaries of the measurement domain are discarded due to incomplete filtering windows, and only the central region is retained for statistical analysis. Even at the largest filtering scale
$\varDelta$
, where the number of available points is smallest, the sample size remains of the order of
$10^8$
, which is sufficient to ensure statistical convergence. The range of
$\varDelta$
values used in the present experiment is shown by filled markers in figure 1. These values span parts of both the inertial and elastic ranges and satisfy the criteria necessary for robust analysis. Unless otherwise specified, a filtering scale of
$\varDelta = 97\eta$
, indicated by the magenta vertical line in figure 1, is adopted as a representative value in the results presented in the following sections.
3. Subgrid-scale statistics
This section investigates the SGS statistics in turbulent flows, both with and without polymer additives, focusing on their dependence on polymer concentration
$\phi$
. The analysis is conducted at a filtering scale of
$\varDelta = 97\eta$
, which lies within the inertial range for Newtonian turbulence and the elastic range for polymeric turbulence. By comparing results for different values of
$\phi$
, we aim to identify and quantify the modifications induced by the polymers. Focusing on the local energy flux relation
$\varPi = -\tau _{\textit{ij}} \tilde {S}_{\textit{ij}}$
, § 3.1 examines the statistical structure of the two fundamental tensors: the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and the SGS stress tensor
$\tau _{\textit{ij}}$
. This includes an analysis of their eigenvalues, eigendirections and tensorial invariants. In § 3.2, we explore the ensemble-averaged value and the probability density function (PDF) of the local energy flux
$\varPi$
, while also addressing the anisotropic characteristics of SGS energy and its flux, which reflect the large-scale anisotropy of the VKS flow. Finally, to further understand the interaction between
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
, we adopt the concept of energy transfer efficiency recently introduced by Ballouz et al. (Reference Ballouz and Ouellette2018), which characterises the inner product between the two tensors.
3.1. Tensor structures
3.1.1. Filtered strain rate
We first present the results for the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
. Its three eigenvalues are denoted by
$\lambda _i^s$
with
$i = 1, 2, 3$
, ordered such that
$\lambda _1^s \gt \lambda _2^s \gt \lambda _3^s$
. Owing to incompressibility, their sum is zero. The PDFs of these eigenvalues for different polymer concentrations
$\phi$
are shown in figure 2(a), with magnitudes normalised by the standard deviation of the tensor norm
$\Vert \tilde {S}_{\textit{ij}} \Vert$
, denoted as
$\sigma _s$
. Here, the norm
$\Vert \boldsymbol{\cdot }\Vert$
represents the magnitude of a tensor or vector, defined as
$\Vert T_{\textit{ij}} \Vert = (T_{\textit{ij}} T_{\textit{ji}})^{1/2}$
for a generic tensor
$T_{\textit{ij}}$
, or
$\Vert p_i \Vert = (p_i p_i)^{1/2}$
for a generic vector
$p_i$
.
(a) The PDFs of the eigenvalues
$\lambda _i^s$
of the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
at
$\varDelta = 97\eta$
for different values of polymer concentration
$\phi$
. The magnitudes of the three eigenvalues (dashed lines for
$i=1$
, solid lines for
$i=2$
and dotted lines for
$i=3$
) are normalised by the standard deviation of the norm of
$\tilde {S}_{\textit{ij}}$
, denoted as
$\sigma _s$
. (b) The PDFs of the dimensionless parameter
$\lambda _2^{s,\ast }$
at
$\varDelta = 97\eta$
for different values of
$\phi$
.
Due to the incompressibility condition, the largest eigenvalue
$\lambda _1^s$
is always positive, while the smallest eigenvalue
$\lambda _3^s$
is always negative. For the intermediate eigenvalue, figure 2(a) shows that the most probable value of
$\lambda _2^s$
is positive, and its magnitude is significantly smaller than those of the other two. These results are consistent with previous experimental observations (Tao et al. Reference Tao, Katz and Meneveau2002), and no significant qualitative change is found in the polymeric cases, except for a slight increase in the peak value of the corresponding PDF.
To further examine the effects of polymers on the eigenvalues of
$\tilde {S}_{\textit{ij}}$
, we consider the dimensionless intermediate eigenvalue, defined as (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Lund & Rogers Reference Lund and Rogers1994)
\begin{equation} \lambda _2^{s,\ast } = \frac {\sqrt {6}\lambda _2^s }{ \left [ (\lambda _1^s)^2 + (\lambda _2^s)^2 + (\lambda _3^s)^2 \right ]^{1/2} }. \end{equation}
In incompressible flow, this parameter is bounded by
$\lambda _2^{s,\ast } = \pm 1$
. Using the same dataset as in figure 2(a), the PDFs of
$\lambda _2^{s,\ast }$
for different values of
$\phi$
are plotted in figure 2(b). The distributions are clearly asymmetric, with the most probable value skewed towards positive values, consistent with the trend observed in figure 2(a). This indicates that the local flow structure associated with
$\tilde {S}_{\textit{ij}}$
is predominantly characterised by biaxial extension, in agreement with previous studies (Tao et al. Reference Tao, Katz and Meneveau2002; Higgins et al. Reference Higgins, Parlange and Meneveau2003). As
$\phi$
increases, the PDF of
$\lambda _2^{s,\ast }$
becomes more symmetric, and its peak gradually shifts towards
$\lambda _2^{s,\ast } = 0$
, corresponding to the eigenvalue ratio
$\lambda _1^s : \lambda _2^s : \lambda _3^s = 1:0:-1$
. This trend suggests that polymer additives suppress the biaxial extension state of
$\tilde {S}_{\textit{ij}}$
and tend to promote a more two-dimensional configuration.
Next, we analyse the alignment between the filtered vorticity vector
$\tilde {\omega }_i$
and the eigenvectors
$\boldsymbol{e}_i^s$
of the strain-rate tensor
$\tilde {S}_{\textit{ij}}$
. This alignment plays an important role in the inertial range processes such as vortex stretching (Lozano-Durán et al. Reference Lozano-Durán, Holzner and Jiménez2016; Danish & Meneveau Reference Danish and Meneveau2018), and its unfiltered counterpart is known to be crucial for small-scale turbulent dynamics (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Elsinga & Marusic Reference Elsinga and Marusic2010; Meneveau Reference Meneveau2011). Figure 3 shows the PDFs of the cosine of the angle between
$\tilde {\omega }_i$
and
$\boldsymbol{e}_i^s$
at
$\varDelta = 97\eta$
, for various values of
$\phi$
. Here,
$\vert \cos (\tilde {\omega }_i, \boldsymbol{e}_i^s)\vert = 1$
indicates perfect alignment between the two vectors, while
$\vert \cos (\tilde {\omega }_i, \boldsymbol{e}_i^s)\vert = 0$
indicates orthogonality.
(a–c) The PDFs of the cosine of the angle between the filtered vorticity vector
$\tilde {\omega }_i$
and the eigenvectors
$\boldsymbol{e}_i^s$
of the strain-rate tensor
$\tilde {S}_{\textit{ij}}$
at
$\varDelta = 97\eta$
for different values of
$\phi$
.
In the Newtonian case (blue lines), the filtered vorticity
$\tilde {\omega }_i$
within the inertial range tends to align weakly with
$\boldsymbol{e}_1^s$
and
$\boldsymbol{e}_2^s$
, and to be nearly perpendicular to
$\boldsymbol{e}_3^s$
, consistent with earlier simulations (Danish & Meneveau Reference Danish and Meneveau2018). As polymer concentration
$\phi$
increases, the alignment of
$\tilde {\omega }_i$
with
$\boldsymbol{e}_2^s$
becomes stronger. Meanwhile,
$\tilde {\omega }_i$
tends to be orthogonal to both
$\boldsymbol{e}_1^s$
and
$\boldsymbol{e}_3^s$
, with the probability density at
$\vert \cos (\tilde {\omega }_i, \boldsymbol{e}_1^s)\vert = 0$
being lower than that at
$\vert \cos (\tilde {\omega }_i, \boldsymbol{e}_3^s)\vert = 0$
.
In addition to eigenvalues and eigenvectors, we now examine the invariants of the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
. The second- and third-order invariants are defined as
$Q_s = - ({1}/{2}) (\tilde {S}_{\textit{ij}} \tilde {S}_{\textit{ji}})$
and
$R_s = - ({1}/{3}) (\tilde {S}_{\textit{ij}} \tilde {S}_{\textit{jk}} \tilde {S}_{\textit{ki}})$
, respectively. The joint PDFs of
$Q_s$
and
$R_s$
for
$\phi = 0$
ppm and
$\phi = 30$
ppm are shown in figures 4(a) and 4(b), respectively. These invariants are normalised by the second-order invariant of the filtered rotation-rate tensor
$\tilde {\varOmega }_{\textit{ij}}$
, defined as
$Q_\omega = ({1}/{4}) (\tilde {\omega }_i \tilde {\omega }_i)$
, where
$\tilde {\varOmega }_{\textit{ij}} = ({1}/{2}) ( \tilde {A}_{\textit{ij}} {-} \tilde {A}_{\textit{ji}} )$
. The solid lines in the plots represent the discriminant
$D_s \equiv ({27}/{4}) (R_s^2 + Q_s^3 = 0)$
, derived from the characteristic equation of
$\tilde {S}_{\textit{ij}}$
. Since
$\tilde {S}_{\textit{ij}}$
is real and symmetric, its invariants should lie below these discriminant curves. In the Newtonian case (figure 4
a), the contours align closely with the
$D_s = 0$
line in the region of
$R_s \gt 0$
, similar to the behaviour observed in the unfiltered case (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994; Blackburn, Mansour & Cantwell Reference Blackburn, Mansour and Cantwell1996). This alignment suggests an eigenvalue ratio of
$\lambda _1^s : \lambda _2^s : \lambda _3^s = 1:1:-2$
and a local flow structure characterised by axisymmetric extension. In contrast, for the polymeric case (figure 4
b), the contour tail contracts and shifts towards the vertical line at
$R_s = 0$
, reflecting the tendency towards two-dimensionalisation, as previously seen in figure 2(b), since
$R_s = -\lambda _1^s \lambda _2^s \lambda _3^s$
(Johnson Reference Johnson2021).
Joint PDFs of the invariants of the
$\tilde {S}_{\textit{ij}}$
tensor (a,b) and the
$\tilde {A}_{\textit{ij}}$
tensor (c,d) at
$\varDelta = 97\eta$
for (a,c)
$\phi = 0$
ppm and (b,d)
$\phi = 30$
ppm. All contours are logarithmically spaced and represent the same iso-values for both Newtonian and polymeric cases.
We also analyse the joint PDFs of the second- and third-order invariants of the filtered velocity gradient tensor
$\tilde {A}_{\textit{ij}}$
, defined as
$Q = - ({1}/{2}) (\tilde {A}_{\textit{ij}} \tilde {A}_{\textit{ji}})$
and
$R = - ({1}/{3}) (\tilde {A}_{\textit{ij}} \tilde {A}_{\textit{jk}} \tilde {A}_{\textit{ki}})$
. Since
$\tilde {A}_{\textit{ij}} = \tilde {S}_{\textit{ij}} + \tilde {\varOmega }_{\textit{ij}}$
includes both the filtered strain-rate and rotation-rate contributions, it directly relates to the energy cascade process (Johnson & Wilczek Reference Johnson and Wilczek2024). Therefore, these invariants provide valuable insight into the underlying flow dynamics. In the Newtonian case, shown in figure 4(c), the joint PDF is concentrated along the right branch of the so-called Vieillefosse tail, forming a teardrop-shaped pattern. This structure resembles that of the unfiltered case and reflects the universal characteristics of small-scale turbulence (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994; Elsinga & Marusic Reference Elsinga and Marusic2010; Meneveau Reference Meneveau2011). In the polymeric case with
$\phi = 30$
ppm (figure 4
d), the distribution becomes notably narrower along the Vieillefosse tail. This suppression of
$R$
indicates that polymer additives weaken the turbulent production mechanisms associated with the energy cascade in the inertial range (Johnson & Wilczek Reference Johnson and Wilczek2024).
3.1.2. Subgrid-scale stress
Next, we present the results for the structure of SGS stress
$\tau _{\textit{ij}}$
, following the approach used in the previous section. We introduce its deviatoric part,
$\overline {\tau }_{\textit{ij}} = \tau _{\textit{ij}} - ({1}/{3}) (\tau _{kk} \delta _{\textit{ij}})$
, where
$\delta _{\textit{ij}}$
is the Kronecker delta. The eigenvalues of
$\overline {\tau }_{\textit{ij}}$
, denoted by
$\lambda _i^\tau$
with
$\lambda _1^\tau \gt \lambda _2^\tau \gt \lambda _3^\tau$
, satisfy
$\lambda _1^\tau + \lambda _2^\tau + \lambda _3^\tau = 0$
. The PDFs of these eigenvalues for various values of
$\phi$
are shown in figure 5(a). The most extensive eigenvalue
$\lambda _1^\tau$
is consistently positive, while the most compressive eigenvalue
$\lambda _3^\tau$
is always negative. The intermediate eigenvalue
$\lambda _2^\tau$
is predominantly negative and similar in magnitude to
$\lambda _3^\tau$
. These results for the Newtonian case agree with previous studies (Tao et al. Reference Tao, Katz and Meneveau2002). Upon the addition of polymers, the peak positions of all three eigenvalues shift towards zero, and the heights of the peak values increase.
(a) The PDFs of the eigenvalues
$\lambda _i^\tau$
of the deviatoric SGS stress tensor
$\overline {\tau }_{\textit{ij}}$
at
$\varDelta = 97\eta$
for various values of polymer concentration
$\phi$
. The eigenvalues (dashed lines for
$i=1$
, solid lines for
$i=2$
and dotted lines for
$i=3$
) are normalised by the standard deviation of the norm of the tensor
$\Vert \overline {\tau }_{\textit{ij}} \Vert$
, denoted as
$\sigma _\tau$
. (b) The PDFs of the dimensionless intermediate eigenvalue
$\lambda _2^{\tau ,\ast }$
of
$\overline {\tau }_{\textit{ij}}$
at
$\varDelta = 97\eta$
for different
$\phi$
.
The concept of non-dimensionalised eigenvalues is also applied to the SGS stress, similar to the case of the filtered strain-rate tensor (Tao et al. Reference Tao, Katz and Meneveau2002). The dimensionless intermediate eigenvalue
$\lambda _2^{\tau ,\ast }$
is defined as
\begin{equation} \lambda _2^{\tau ,\ast } = \frac {\sqrt {6}\lambda _2^\tau }{ \left [ (\lambda _1^\tau )^2 + (\lambda _2^\tau )^2 + (\lambda _3^\tau )^2 \right ]^{1/2} }, \end{equation}
which satisfies
$|\lambda _2^{\tau ,\ast }| \leqslant 1$
in incompressible flow. Figure 5(b) shows the PDFs of
$\lambda _2^{\tau ,\ast }$
. The distribution is negatively skewed, with the most probable value being negative, indicating a biaxial compression structure of the eigenvalues of
$\overline {\tau }_{\textit{ij}}$
, consistent with earlier findings (Tao et al. Reference Tao, Katz and Meneveau2002; Higgins et al. Reference Higgins, Parlange and Meneveau2003). As the polymer concentration
$\phi$
increases, the probability associated with negative values of
$\lambda _2^{\tau ,\ast }$
rises. Simultaneously, the eigenvalue ratio approaches
$\lambda _1^\tau : \lambda _2^\tau : \lambda _3^\tau = 2:-1:-1$
, suggesting that polymer additives promote an axisymmetric compressive state in the SGS stress.
We similarly define the eigenvectors of
$\overline {\tau }_{\textit{ij}}$
as
$\boldsymbol{e}_i^\tau$
(
$i=1,2,3$
) and examine the alignment between the filtered vorticity vector
$\tilde {\omega }_i$
and these eigendirections, as illustrated in figure 6. In the Newtonian case, figure 6(a) reveals a strong misalignment between
$\tilde {\omega }_i$
and
$\boldsymbol{e}_1^\tau$
, consistent with previous experimental findings (Tao et al. Reference Tao, Katz and Meneveau2000, Reference Tao, Katz and Meneveau2002). In contrast,
$\tilde {\omega }_i$
tends to align with the other two eigendirections – those corresponding to compressive directions – evidenced by the peaks of the PDFs at
$\vert \cos (\tilde {\omega }_i, \boldsymbol{e}_2^\tau )\vert = 1$
and
$\vert \cos (\tilde {\omega }_i, \boldsymbol{e}_3^\tau )\vert = 1$
in figures 6(b) and 6(c), respectively. These alignments reflect the axisymmetric structure of the SGS stress tensor, in agreement with earlier experimental observations (Higgins et al. Reference Higgins, Parlange and Meneveau2003). In the polymeric case, the misalignment of
$\tilde {\omega }_i$
with
$\boldsymbol{e}_1^\tau$
becomes more pronounced, while the alignment with
$\boldsymbol{e}_2^\tau$
is enhanced. Together with the eigenvalue results from figure 5, these observations indicate that the SGS stress in polymeric turbulence tends towards a more pronounced axisymmetric state, with an approximate eigenvalue ratio of
$\lambda _1^\tau : \lambda _2^\tau : \lambda _3^\tau = 2:-1:-1$
.
(a–c) The PDFs of the cosine of the angles between the filtered vorticity vector
$\tilde {\omega }_i$
and the eigenvectors
$\boldsymbol{e}_i^\tau$
at
$\varDelta = 97\eta$
for various values of polymer concentration
$\phi$
.
Finally, we analyse the invariants of the SGS stress. Following Tom, Carbone & Bragg (Reference Tom, Carbone and Bragg2021), we define the non-dimensional anisotropic tensor of
$\tau _{\textit{ij}}$
as
\begin{equation} b_{\textit{ij}}= \frac {\tau _{\textit{ij}} - \frac {1}{3}\tau _{kk} \delta _{\textit{ij}} }{ (\tau _{\textit{mn}} \tau _{\textit{mn}})^{1/2} } = \frac {\overline {\tau }_{\textit{ij}}}{\Vert \tau _{\textit{mn}} \Vert }, \end{equation}
from which two scalar invariants can be constructed:
\begin{equation} \zeta = -\sqrt {6} b_{\textit{ij}} b_{\textit{jk}} b_{\textit{ki}}, \quad \chi = \big ( b_{\textit{ij}} b_{\textit{ji}} \big )^{3/2}. \end{equation}
These quantities are constrained by
$0 \leqslant \chi \leqslant 1$
and
$|\zeta | \leqslant \chi$
, defining a triangular region in the invariants plane. The origin,
$\zeta = \chi = 0$
, corresponds to the isotropic state, while
$\chi = 1$
represents the maximally anisotropic state. The boundary lines
$|\zeta | = \chi$
signify axisymmetric states; in particular, the segment with
$0 \leqslant \chi \leqslant 1$
and
$\zeta = -\chi$
corresponds to axisymmetric compression. Additionally, the values
$|\zeta | = \chi \approx 0.19$
and
$0.54$
represent the limiting cases of two components and one component, respectively, as identified in Tom et al. (Reference Tom, Carbone and Bragg2021).
The joint PDFs of the invariants
$\chi$
and
$\zeta$
are presented in figure 7. In both the Newtonian and polymeric cases, the PDF contours are predominantly concentrated in the lower-left region of the invariants triangle. The observation that
$\chi$
is much less than
$1$
suggests that
$\tau _{\textit{ij}}$
remains far from a fully anisotropic state, primarily due to the constraint of positive SGS energy, i.e.
$\tau _{kk} \geqslant 0$
. Moreover, the alignment of the contours along the left edge of the invariants triangle – and their tendency to remain nearly parallel to it – is consistent with the eigenvalue structure of
$\overline {\tau }_{\textit{ij}}$
shown in figure 5, where the most probable eigenvalue ratio is
$\lambda _1^\tau : \lambda _2^\tau : \lambda _3^\tau = 2:-1:-1$
. This observation further supports the conclusion that the SGS stress possesses a biaxial compressive structure. In the Newtonian case (figure 7
a), the PDF also displays a slight inclination towards an isotropic or two-component axisymmetric state, as indicated by the green square marker at approximately
$\zeta = -\chi \approx -0.19$
, which corresponds to a configuration where one eigenvalue approaches zero.
Joint PDFs of the invariants of the
$b_{\textit{ij}}$
tensor (or equivalently
$\tau _{\textit{ij}}$
, as referred to in the text) at
$\varDelta = 97\eta$
for (a)
$\phi = 0$
ppm and (b)
$\phi = 30$
ppm. All contours are logarithmically spaced and share the same iso-values for both Newtonian and polymeric cases.
Upon the addition of polymers, the structure of the SGS stress tensor shifts further away from isotropic or two-component axisymmetric configurations towards a one-component axisymmetric state, characterised by a single dominant eigenvalue equal to
$\tau _{kk}$
. As illustrated in figure 7(b), the high-probability contours become more curved and move towards the cyan diamond marker, located at approximately
$\zeta = -\chi \approx -0.54$
. This polymer-induced shift in SGS stress structure introduces additional challenges for modelling, especially in capturing the one-component axisymmetric configuration. Nevertheless, it provides valuable insight into the complex dynamics of SGS stress in polymer-laden turbulent flows.
In summary, this subsection provides a systematic analysis of the statistical structures of the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and SGS stress tensor
$\tau _{\textit{ij}}$
, as well as the effects of polymer additives. We find that the axisymmetric extension state of
$\tilde {S}_{\textit{ij}}$
is slightly suppressed by polymers, leading it to tending towards a two-dimensional eigenvalue structure with a ratio of
$\lambda _1^s : \lambda _2^s : \lambda _3^s = 1:0:-1$
, resembling the unfiltered case, as confirmed by previous experimental studies (Wang et al. Reference Wang, Zhang, Yang and Xi2025c
). Conversely, the axisymmetric compression state of
$\tau _{\textit{ij}}$
becomes more pronounced in polymeric turbulence. These geometric features of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
are further emphasised through their respective invariants. Additional comparisons across different filtering widths show that the influence of
$\varDelta$
modifies the PDFs only in magnitude rather than in their qualitative shape. Furthermore, we observe that in polymeric turbulence, the filtered vorticity
$\tilde {\omega }_i$
tends to align more with
$\boldsymbol{e}_2^s$
of
$\tilde {S}_{\textit{ij}}$
and with
$\boldsymbol{e}_2^\tau$
,
$\boldsymbol{e}_3^\tau$
of
$\overline {\tau }_{\textit{ij}}$
, while remaining more perpendicular to the other eigenvectors. These findings provide a foundation for examining the alignment relation between the two tensors in subsequent sections, contributing to a deeper understanding of the local energy flux
$\varPi = -\tau _{\textit{ij}} \tilde {S}_{\textit{ij}}$
, especially in the inertial range for Newtonian turbulence and in the elastic range for polymeric turbulence.
3.2. Local energy transfer
3.2.1. Statistics of energy flux
We begin by examining the ensemble-averaged local energy flux,
$\left \langle \varPi \right \rangle$
, as a function of polymer concentration
$\phi$
, as shown in figure 8(a). The filtering scale is fixed at
$\varDelta = 97\eta$
, corresponding to the inertial range in Newtonian turbulence and the elastic range in the polymeric case. When normalised by the mean energy dissipation rate
$\varepsilon$
, the directly computed value of
$\left \langle \varPi \right \rangle$
for the Newtonian case shows excellent agreement with
$\varepsilon$
estimated from second-order velocity structure functions, with a deviation of less than
$10\%$
. However, as
$\phi$
increases to 30 ppm,
$\left \langle \varPi \right \rangle$
exhibits a dramatic reduction of approximately
$85\,\%$
. This substantial suppression of the energy flux is consistent with previous findings from structure function analyses in polymer-laden turbulence (Ouellette et al. Reference Ouellette, Xu and Bodenschatz2009; Xi et al. Reference Xi, Bodenschatz and Xu2013; de Chaumont Quitry & Ouellette Reference de Chaumont Quitry and Ouellette2016; Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Wang et al. Reference Wang, Zhang and Xi2025a
).
(a) The ensemble-averaged local energy flux
$\langle \varPi \rangle$
as a function of
$\phi$
at
$\varDelta = 97\eta$
. (b) The PDFs of
$\varPi$
corresponding to the same data as in (a).
In contrast to earlier experimental studies, the present work provides access to instantaneous and pointwise measurements of the local energy flux
$\varPi$
, thus enabling a detailed analysis of its probability distribution – a feature previously unavailable. To further investigate the variations in energy flux, we compute the PDFs of
$\varPi$
for various polymer concentrations
$\phi$
, as shown in figure 8(b), with each PDF normalised by its average value. All distributions exhibit a positively skewed shape, suggesting that forward (direct) cascade events – associated with positive values of
$\varPi$
– are statistically more prevalent than inverse cascade events. With the addition of polymers, the tails of the normalised PDFs become increasingly elevated on both sides, and this effect intensifies with growing
$\phi$
. This broadening of the distribution implies a modest increase in intermittency at the filtering scale
$\varDelta = 97\eta$
, which corresponds to the elastic range in polymeric turbulence. Previous studies (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Rosti et al. Reference Rosti, Perlekar and Mitra2023) evaluated intermittency using higher-order velocity structure functions and, based on spherically averaged scaling exponents, reported that the intermittency corrections are the same for the polymeric case as for the Newtonian case, but in those previous experimental works, with spherical averaged manipulation, this approach could not evaluate the directional dependence of the intermittency. And in the following paragraphs (see figure 11 and the related discussion), we examine in detail how anisotropy influences the energy transfer processes. We show that intermittency in polymeric turbulence is direction-dependent and that polymer additives enhance the anisotropic features of energy transfer. In contrast, earlier studies relied on direction-averaged structure functions, which extract only the isotropic contributions and effectively average out any directional dependence. Our results therefore extend the previous discussions and demonstrate that intermittency in polymeric turbulence exhibits a pronounced directional dependence.
To further elucidate the influence of polymer additives on the energy flux, we compute both the ensemble averages and the PDFs of
$\varPi$
across a range of available filtering scales. Figure 9(a) presents the ensemble-averaged energy flux
$\left \langle \varPi \right \rangle$
as a function of filtering scale
$\varDelta$
for different polymer concentrations
$\phi$
. In the Newtonian case,
$\left \langle \varPi \right \rangle$
increases with
$\varDelta$
and asymptotically approaches the mean energy dissipation rate
$\varepsilon$
. In contrast, for polymeric flows, increasing
$\phi$
leads to a systematic suppression of
$\left \langle \varPi \right \rangle$
across all scales, in agreement with earlier findings based on velocity structure functions (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Wang et al. Reference Wang, Zhang and Xi2025a
). The specific case corresponding to
$\varDelta = 97\eta$
– highlighted by a magenta vertical line – marks the transition from the inertial range in Newtonian turbulence to the elastic range in polymeric turbulence.
(a) The ensemble-averaged local energy flux
$\langle \varPi \rangle$
as a function of
$\varDelta$
for different values of
$\phi$
. The PDFs of
$\varPi$
as a function of
$\varDelta$
for
$\phi = 0$
ppm (b) and
$\phi = 30$
ppm (c).
Figures 9(b) and 9(c) display the PDFs of
$\varPi$
as a function of
$\varDelta$
for the Newtonian case and the polymeric case with
$\phi = 30$
ppm, respectively. In both scenarios, the PDFs retain a positively skewed profile across all filtering scales, indicating a predominance of forward cascade events. As
$\varDelta$
decreases, the tails of the PDFs broaden, which is consistent with recent DNS (Johnson Reference Johnson2021). A direct comparison of figures 9(b) and 9(c) reveals that the PDFs for the polymeric case exhibit broader tails than those for Newtonian turbulence at each filtering scale. This observation echoes the results at
$\varDelta = 97\eta$
in figure 8(b). Furthermore, it is noteworthy that the PDF curves for both Newtonian and polymeric flows tend to collapse at relatively large filtering scales. Thus, this collapse suggests that the filtering scale has entered either the inertial range for Newtonian turbulence or the elastic range for polymeric turbulence, where the statistical features of the local energy flux become less sensitive to the values of
$\varDelta$
.
As shown in § 2.1, the VKS flow exhibits axisymmetry and anisotropy, with horizontal velocity fluctuations approximately
$50\,\%$
greater in r.m.s. magnitude than those in the vertical direction. To investigate how this large-scale anisotropy influences the local energy flux, we examine the anisotropic behaviour of SGS statistics at the intermediate filtering scale
$\varDelta = 97\eta$
, focusing on the SGS energy and its flux. The diagonal components of the SGS stress tensor
$\tau _{\textit{ij}}$
represent directional contributions to the SGS energy, with their sum
$\tau _{kk}$
quantifying the total SGS energy. Figure 10(a) shows that as polymer concentration
$\phi$
increases, the SGS energy decreases, which is consistent with the reduction in energy flux observed in figure 8(a). Since the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
is traceless due to incompressibility, these diagonal components of
$\tau _{\textit{ij}}$
do not contribute to
$\varPi$
. Therefore, the concurrent suppression of both SGS energy and energy flux with increasing
$\phi$
highlights the damping effect of polymer additives. Figure 10(b) presents the normalised diagonal components of
$\tau _{\textit{ij}}$
, revealing that their relative proportions change very little with
$\phi$
. Notably,
$\left \langle \tau _{11} \right \rangle$
and
$\left \langle \tau _{33} \right \rangle$
are consistently larger than
$\left \langle \tau _{22} \right \rangle$
, indicating that the SGS energy is preferentially concentrated in the horizontal directions.
(a) The ensemble-averaged SGS energy
$\langle \tau _{kk} \rangle$
and (b) its three components (corresponding to the three directions) as a function of
$\phi$
at
$\varDelta = 97\eta$
. In (b), normalisation by
$\langle \tau _{kk} \rangle$
is applied so that the sum of the three components equals one.
We conclude by analysing the anisotropic features of the energy flux
$\varPi$
, which can be decomposed into
$\varPi = \varPi _x + \varPi _y + \varPi _z$
, where
$\varPi _x = -\tau _{1j} \tilde {A}_{1j}$
,
$\varPi _y = -\tau _{2j} \tilde {A}_{2j}$
and
$\varPi _z = -\tau _{3j} \tilde {A}_{3j}$
, corresponding to energy flux terms in the equations of
$\tilde {u}_x^2$
,
$\tilde {u}_y^2$
and
$\tilde {u}_z^2$
, the energy of the three filtered components. Figure 11(a) shows their ensemble averages normalised by
$\left \langle \varPi \right \rangle$
. In the Newtonian case, the flux is clearly anisotropic, with
$\left \langle \varPi _x \right \rangle$
and
$\left \langle \varPi _z \right \rangle$
being comparable and slightly larger than
$\left \langle \varPi _y \right \rangle$
. As polymer concentration increases,
$\left \langle \varPi _y \right \rangle$
rises while the other two components decline, suggesting that the flux in the vertical direction becomes more pronounced in polymeric turbulence, even though SGS energy remains less concentrated there. This intriguing phenomenon is likely related to the inherent anisotropy of VKS flow, though the precise influence of polymers remains uncertain due to the lack of information about the polymer in the experiments.
(a) The three contributions to
$\langle \varPi \rangle$
,
$\langle \varPi _x \rangle$
,
$\langle \varPi _y \rangle$
and
$\langle \varPi _z \rangle$
, as a function of
$\phi$
at
$\varDelta = 97\eta$
, normalised by the corresponding values of
$\langle \varPi \rangle$
. The (b) PDFs of
$\varPi _x$
and (c) PDFs of
$\varPi _y$
corresponding to the same data as in (a).
We further examine the PDFs of
$\varPi _x$
and
$\varPi _y$
– the energy flux components along two orthogonal directions – for various polymer concentrations, as shown in figures 11(b) and 11(c). Each distribution is normalised by its respective mean values. Both components display intermittent behaviour, but polymers influence them differently. In the Newtonian case, the PDFs of
$\varPi _x$
and
$\varPi _y$
are nearly identical. However, with increasing
$\phi$
, the distribution of
$\varPi _x$
becomes notably more intermittent and less skewed, while
$\varPi _y$
remains largely unchanged, aside from a slight reduction in backscatter events. These findings suggest that polymers modulate the local energy flux anisotropically.
In summary, we have analysed the local energy flux
$\varPi$
and its dependence on polymer concentration. At the intermediate scale
$\varDelta = 97\eta$
, representative of the inertial or elastic range, polymers significantly reduce the amplitude of
$\varPi$
and enhance its intermittency – a trend consistent across scales. Moreover, anisotropy emerges as a key factor: the decomposition of
$\varPi$
reveals that the vertical component
$\left \langle \varPi _y \right \rangle$
becomes increasingly dominant with increasing
$\phi$
, despite the weaker SGS energy
$\left \langle \tau _{22} \right \rangle$
in this direction. We further examined anisotropy across a range of filtering scales, considering both the SGS energy and its associated flux, and found that anisotropy decreases as the scale becomes smaller. At the same time, polymer additives consistently strengthen the overall anisotropic characteristics of the flow. These anisotropic features of energy transfer in the VKS flow may offer broader implications for understanding anisotropy in other complex turbulent systems, such as drag-reducing boundary layers.
3.2.2. Alignment of filtered strain rate and SGS stress
In § 3.1, we analysed the statistical structures of the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and the SGS stress tensor
$\tau _{\textit{ij}}$
. We now focus on their inner product, which plays a central role in characterising the local energy flux (1.1) and in modelling SGS stress (see (4.2)). Following the convention of Tao et al. (Reference Tao, Katz and Meneveau2002), we define the eigenvectors of
$-\overline {\tau }_{\textit{ij}}$
as
$\boldsymbol{e}_i^{-\tau }$
, where
$i = 1, 2, 3$
. In this notation,
$\boldsymbol{e}_3^{-\tau }$
corresponds to the most extensive eigenvalue of
$\overline {\tau }_{\textit{ij}}$
(i.e.
$\boldsymbol{e}_1^\tau$
),
$\boldsymbol{e}_2^{-\tau }$
to the intermediate eigenvalue (
$\boldsymbol{e}_2^\tau$
) and
$\boldsymbol{e}_1^{-\tau }$
to the most compressive eigenvalue (
$\boldsymbol{e}_3^\tau$
). To quantify the alignment between
$-\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
, we compute the cosine of the angle between corresponding eigenvectors
$\boldsymbol{e}_i^{-\tau }$
and
$\boldsymbol{e}_i^s$
. The results, shown in figure 12, are evaluated at a filtering scale of
$\varDelta = 97\eta$
for various polymer concentrations
$\phi$
.
(a–i) The PDFs of the cosine angles between the eigenvectors
$\boldsymbol{e}_i^{-\tau }$
of the (negative) SGS stress and
$\boldsymbol{e}_i^s$
of the filtered strain rate at
$\varDelta = 97\eta$
for different values of
$\phi$
. The subscripts
$i$
range from 1 to 3, resulting in a total of 9 pairs of combinations.
Several key observations can be drawn from figure 12. First, strong alignments are found between
$\boldsymbol{e}_1^{-\tau }$
and
$\boldsymbol{e}_2^s$
, as well as between
$\boldsymbol{e}_2^{-\tau }$
and
$\boldsymbol{e}_2^s$
, while
$\boldsymbol{e}_3^{-\tau }$
shows clear misalignment with
$\boldsymbol{e}_2^s$
. These results are consistent with the earlier findings in figures 3 and 6, where the filtered vorticity tends to align with
$\boldsymbol{e}_2^s$
,
$\boldsymbol{e}_2^{\tau }$
(or
$\boldsymbol{e}_2^{-\tau }$
) and
$\boldsymbol{e}_3^{\tau }$
(or
$\boldsymbol{e}_1^{-\tau }$
), while being nearly orthogonal to
$\boldsymbol{e}_1^{\tau }$
(or
$\boldsymbol{e}_3^{-\tau }$
). The PDFs for the remaining eigenvector pairs show peak positions less than unity, indicating partial misalignment. In particular, figure 12(a,e,i) shows the alignment between corresponding eigenvectors of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
. Previous studies (Tao et al. Reference Tao, Katz and Meneveau2002; Higgins et al. Reference Higgins, Parlange and Meneveau2003) report peak positions around
$0.85$
for directions
$1$
and
$3$
, marked by vertical dashed lines in figure 12(a,i). Our Newtonian results show similar behaviour. However, as
$\phi$
increases, these peaks shift to lower values, indicating a weakening of alignment between the corresponding eigenvectors of
$-\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
. This weakening may contribute to the reduced effectiveness of the Smagorinsky model in polymeric turbulence, as discussed in § 4.1.
Furthermore, the alignment between eigenvectors of
$-\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
directly influences the local energy flux, which can be expressed as (Johnson Reference Johnson2021)
\begin{equation} \varPi = \sum _{i=1}^{3} \sum _{j=1}^{3} \lambda _i^{-\tau } \lambda _{\!j}^s \cos ^2 (\boldsymbol{e}_i^{-\tau }, \boldsymbol{e}_{\!j}^s) , \end{equation}
where
$\lambda _i^{-\tau }$
denotes the eigenvalue of
$-\overline {\tau }_{\textit{ij}}$
corresponding to eigenvector
$\boldsymbol{e}_i^{-\tau }$
, and the relationship between
$\lambda _i^{-\tau }$
and
$\lambda _i^\tau$
is given by:
$\lambda _1^{-\tau } = -\lambda _3^\tau$
,
$\lambda _2^{-\tau } = -\lambda _2^\tau$
and
$\lambda _3^{-\tau } = -\lambda _1^\tau$
. To quantify the contribution of each eigenvector pair to the energy flux, following Tao et al. (Reference Tao, Katz and Meneveau2002), we define the component-wise flux as
$\varPi _{(i,j)} = \lambda _i^{-\tau } \lambda _{\!j}^s \cos ^2 (\boldsymbol{e}_i^{-\tau }, \boldsymbol{e}_{\!j}^s)$
.
Figure 13 shows the ensemble average of
$\varPi _{(i,j)}$
as a function of polymer concentration
$\phi$
, normalised by
$\left \langle \varPi \right \rangle$
, such that the sum of the nine components equals unity, as indicated by the black asterisk. In the Newtonian case, the component
$\left \langle \varPi _{(3,3)} \right \rangle$
dominates, contributing the largest portion of
$\left \langle \varPi \right \rangle$
, consistent with previous experimental findings (Tao et al. Reference Tao, Katz and Meneveau2002). As
$\phi$
increases, the contribution of most components to the total
$\left \langle \varPi \right \rangle$
remains nearly constant, with notable exceptions for
$\left \langle \varPi _{(3,1)} \right \rangle$
and
$\left \langle \varPi _{(3,3)} \right \rangle$
. As seen in figure 12, the alignment changes for these components are more pronounced, resulting in significant increases or decreases in their contributions. Furthermore, these two components exhibit opposite signs, which causes their variations to nearly cancel each other out.
The ensemble-averaged local energy flux components
$\langle \varPi _{(i,j)} \rangle$
calculated by (3.5) and normalised by
$\langle \varPi \rangle$
, as a function of
$\phi$
at
$\varDelta = 97\eta$
. The sum of nine different components should equal one, as indicated by the black asterisk.
3.2.3. Efficiency of energy transfer
Given that the alignment between
$-\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
is closely related to the energy flux
$\varPi$
(see (1.1) and (3.5)), Ballouz & Ouellette (Reference Ballouz and Ouellette2018) proposed the concept of an optimal state in which the combinations
$\varPi _{(i,j)}$
contribute to achieve the maximum local energy flux. Based on this, they defined the efficiency of energy transfer, which has been further developed in subsequent studies (Ballouz & Ouellette Reference Ballouz and Ouellette2020; Ballouz, Johnson & Ouellette Reference Ballouz, Johnson and Ouellette2020; Johnson Reference Johnson2021). Those studies reveal that the actual efficiency of energy transfer in three-dimensional turbulence is low (about
$30\,\%$
). In this work, we adopt the definition proposed by Johnson (Reference Johnson2021) and calculate the efficiency as
\begin{equation} \varGamma = \frac {\varPi }{\varPi ^{ {max}}} = \frac {-\overline {\tau }_{\textit{ij}} \tilde {S}_{\textit{ij}}}{\Vert \overline {\tau }_{\textit{ij}} \Vert \Vert \tilde {S}_{\textit{ij}} \Vert }, \end{equation}
where
$\Vert \overline {\tau }_{\textit{ij}} \Vert = \Vert \lambda _i^\tau \Vert$
and
$\Vert \tilde {S}_{\textit{ij}} \Vert = \Vert \lambda _i^s \Vert$
represent the norms of the
$\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
tensors, respectively. This expression indicates that the maximum local energy flux
$\varPi ^{ {max}}$
occurs when the eigenvectors of the two tensors are perfectly aligned, and their corresponding eigenvalues are proportional, i.e.
$\lambda _1^\tau / \lambda _1^s = \lambda _2^\tau / \lambda _2^s = \lambda _3^\tau / \lambda _3^s$
(Johnson Reference Johnson2021). Thus,
$\varGamma =1$
and
$\varGamma =-1$
represent the optimal forward and inverse energy transfer, respectively. For further details on efficiency and its implications, see Ballouz & Ouellette (Reference Ballouz and Ouellette2018) and Johnson (Reference Johnson2021) and studies on tensor alignment (Liu et al. Reference Liu, Meneveau and Katz1994; Yang & Wang Reference Yang and Wang2016).
Figure 14(a) shows the ensemble average of
$\varGamma$
as a function of polymer concentration
$\phi$
at
$\varDelta = 97\eta$
. In the Newtonian case,
$\left \langle \varGamma \right \rangle$
is slightly above
$30\,\%$
in the inertial range, consistent with previous numerical studies (Ballouz & Ouellette Reference Ballouz and Ouellette2018; Johnson Reference Johnson2021). As
$\phi$
increases,
$\left \langle \varGamma \right \rangle$
decreases, agreeing with the reduction of
$\left \langle \varPi \right \rangle$
observed in figure 8. The reduced values of
$\left \langle \varGamma \right \rangle$
are further confirmed by the PDFs shown in figure 14(b). In the Newtonian case, the PDF peaks around
$0.5$
, indicating significant asymmetry, which is consistent with previous studies (Ballouz & Ouellette Reference Ballouz and Ouellette2020; Johnson Reference Johnson2021). However, as
$\phi$
increases, the most probable efficiency shifts towards zero, and the PDF becomes more symmetric. While prior work has shown that polymers weaken the local energy flux, we demonstrate for the first time that they also suppress the efficiency of the energy transfer process, offering new insights into polymeric turbulence.
(a) The ensemble-averaged efficiency of energy transfer
$\langle \varGamma \rangle$
as a function of
$\phi$
at
$\varDelta = 97\eta$
. (b) The PDFs of
$\varGamma$
corresponding to the same dataset as in (a).
We also computed
$\varGamma$
at various filtering scales, similar to the analysis of
$\varPi$
presented in figure 9. Figure 15(a) shows that
$\left \langle \varGamma \right \rangle$
decreases with decreasing
$\varDelta$
for all polymer concentrations. In the Newtonian case,
$\left \langle \varGamma \right \rangle$
saturates as
$\varDelta$
approaches approximately
$100\eta$
, consistent with the inertial-range behaviour (Ballouz & Ouellette Reference Ballouz and Ouellette2018; Johnson Reference Johnson2021). Notably,
$\left \langle \varGamma \right \rangle$
is reduced at all scales in the polymeric case, with a more pronounced suppression in the elastic range compared with the dissipation range. The results for
$\varDelta = 97\eta$
shown in figure 14 are marked by the magenta vertical line, representing polymer-induced suppression of energy transfer efficiency across both the inertial and elastic ranges.
(a) The ensemble-averaged efficiency of energy transfer
$\langle \varGamma \rangle$
as a function of
$\varDelta$
for different values of
$\phi$
. The PDFs of
$\varGamma$
as a function of
$\varDelta$
for
$\phi = 0$
ppm (b) and
$\phi = 30$
ppm (c).
Figures 15(b) and 15(c) show the PDFs of
$\varGamma$
as a function of
$\varDelta$
for
$\phi = 0$
and
$30$
ppm, respectively. In the Newtonian case (figure 15
b), the positively skewed PDF curves become more symmetric as
$\varDelta$
decreases, with the peak shifting towards
$\varGamma = 0$
. These trends are in agreement with previous numerical studies (Ballouz & Ouellette Reference Ballouz and Ouellette2020; Johnson Reference Johnson2021). In the polymeric case (figure 15
c), the PDFs exhibit a narrower and more symmetric distribution, reflecting the reduced efficiency with increasing
$\phi$
. For
$\phi = 30$
ppm, the peak of the PDF curve decreases and the curves flatten as
$\varDelta$
decreases. Furthermore, the PDF curves tend to collapse at larger filtering scales in both cases. In the polymeric case, this collapse becomes apparent for
$\varDelta \gt 50\eta$
, suggesting a transition into the elastic range.
In this section, we presented statistical results for the energy flux
$\varPi$
and examined the alignment between the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and the SGS stress tensor
$\tau _{\textit{ij}}$
(or equivalently,
$-\overline {\tau }_{\textit{ij}}$
). Our analysis reveals that the eigenvector pair corresponding to the second eigenvalue of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
is well aligned, whereas the other two pairs exhibit partial misalignment. This misalignment becomes more pronounced in the polymeric case, which may explain the observed reduction in local energy flux, given that
$\varPi$
depends on both the eigenvalues and the alignment of the corresponding eigenvectors. To further quantify the degree of misalignment, we employed the energy transfer efficiency
$\varGamma$
. Our results demonstrate that polymers not only reduce the local energy flux but also suppress the efficiency of the energy transfer process, providing the first evidence of this behaviour in polymeric turbulence.
3.2.4. Flow visualisation
In the previous sections, we presented statistical analyses of the energy flux and the associated energy transfer efficiency. To offer a more intuitive understanding of these quantities, this section provides visualisations based on instantaneous flow fields.
Figure 16 shows an instantaneous snapshot of the local energy flux
$\varPi$
at a fixed time and at the filtering scale
$\varDelta = 97\eta$
. Figures 16(a) and 16(c) correspond to the Newtonian case and the
$\phi = 30$
ppm polymeric case, respectively, with both fields normalised by the Newtonian energy dissipation rate
$\varepsilon$
. In both cases,
$\varPi$
exhibits pronounced spatial variability, taking both positive and negative values. The addition of polymers significantly suppresses the amplitude of
$\varPi$
, reducing its magnitude by roughly one order of magnitude on average at
$\phi =30$
ppm. At the same time, it seems that the spatial distribution of
$\varPi$
has become more intermittent, with some regions having high
$\varPi$
values that are significantly higher than the surrounding values.
Instantaneous slices of the local energy flux
$\varPi$
at a fixed time and at the filtering scale
$\varDelta = 97\eta$
for (a)
$\phi = 0$
ppm and (c)
$\phi = 30$
ppm, both normalised by the Newtonian energy dissipation rate
$\varepsilon$
. (b,d) The corresponding instantaneous slices of the energy transfer efficiency
$\varGamma$
for the cases in (a,c), respectively.
Figures 16(b) and 16(d) present the corresponding fields of the energy transfer efficiency
$\varGamma$
for the cases shown in figures 16(a) and 16(c), respectively. Similar to
$\varPi$
, the quantity
$\varGamma$
exhibits pronounced spatial variability. However, large values of
$\varGamma$
are not confined to regions where
$\varPi$
is large; in other words, the energy transfer efficiency can attain high values even when the local energy transfer rate is relatively small. This behaviour is consistent with previous observations in Newtonian turbulence (Ballouz & Ouellette Reference Ballouz and Ouellette2018). The addition of polymers slightly suppresses the amplitude of
$\varGamma$
, in agreement with the mean-value trends reported in the previous section. Furthermore, figure 16 appears to show that the spatial distribution of energy transfer events becomes more coherent after the addition of the polymer.
4. Assessment of the SGS model
Building on the statistical analysis of the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and the SGS stress
$\tau _{\textit{ij}}$
, we now examine the implications of these findings for LES, with a focus on modelling
$\tau _{\textit{ij}}$
. In this section, we consider two representative SGS models: the Smagorinsky model and the nonlinear model (also known as the gradient model). We first evaluate model performance by comparing the modelled local energy flux with the true energy flux. Subsequently, we perform a correlation analysis to assess the agreement between the modelled and true SGS stress tensors, following approaches developed in previous works (Clark, Ferziger & Reynolds Reference Clark, Ferziger and Reynolds1979; Liu et al. Reference Liu, Meneveau and Katz1994; Borue & Orszag Reference Borue and Orszag1998). Our results show that the nonlinear model provides a more accurate approximation of the SGS stress than the Smagorinsky model, consistent with previous studies (Liu et al. Reference Liu, Meneveau and Katz1994; Borue & Orszag Reference Borue and Orszag1998). Notably, this superior performance persists in polymeric turbulence, suggesting that the nonlinear model is a promising candidate for LES of viscoelastic fluids.
To quantify the agreement between the modelled and true stresses, we adopt the methodology of Liu et al. (Reference Liu, Meneveau and Katz1994). For two scalar variables
$A$
and
$B$
(e.g. corresponding components of the SGS stress), the joint PDF of
$A$
and
$B$
reflects their correlation: tighter contours around the diagonal imply stronger statistical correlation. Additionally, the correlation coefficient between
$A$
and
$B$
is computed as
\begin{equation} \rho (A, B) = \frac {Cov(A, B)}{\sigma _A \sigma _B} , \end{equation}
where
$Cov(A,B)$
denotes the covariance of
$A$
and
$B$
, and
$\sigma _A$
and
$\sigma _B$
are their standard deviations. The correlation coefficient
$\rho (A,B)$
approaches unity when
$A$
and
$B$
are strongly positively correlated.
4.1. Smagorinsky’s model
We first briefly introduce the SGS stress models. The Smagorinsky model assumes the form
\begin{equation} \tau _{\textit{ij}}^{ {sm}} = -2\nu _T \tilde {S}_{\textit{ij}} , \end{equation}
where the eddy viscosity is given by
$\nu _T = (C_s \Delta )^2 ( 2 \tilde {S}_{\textit{ij}} \tilde {S}_{\textit{ij}} )^{1/2}$
. The Smagorinsky coefficient
$C_s$
may depend on scale (Meneveau & Katz Reference Meneveau and Katz2000); here, we adopt
$C_s = 0.16$
, a typical value for Newtonian turbulence (Lilly Reference Lilly1967; Meneveau & Katz Reference Meneveau and Katz2000), which has also been used in recent LES studies of viscoelastic flows (Ferreira et al. Reference Ferreira, Pinho and da Silva2016; Parvar et al. Reference Parvar, da Silva and Pinho2021).
The modelled energy flux is computed as
$\varPi ^{ {sm}} = -\tau _{\textit{ij}}^{ {sm}} \tilde {S}_{\textit{ij}}$
. Figure 17(a) shows the ensemble-averaged
$\varPi ^{ {sm}}$
as a function of polymer concentration
$\phi$
, normalised by its actual counterpart
$\left \langle \varPi \right \rangle$
, at the filtering scale
$\varDelta = 97\eta$
. In the Newtonian case,
$\left \langle \varPi ^{ {sm}} \right \rangle$
overestimates the true flux by approximately
$50\,\%$
, and this deviation becomes even larger at smaller filtering scales, consistent with prior observations that the Smagorinsky model does not accurately capture the magnitude of the real SGS stress (Liu et al. Reference Liu, Meneveau and Katz1994).
(a) The ensemble-averaged local energy flux
$\langle \varPi ^{ {sm}} \rangle$
from the Smagorinsky model divided by the real flux
$\langle \varPi \rangle$
as a function of
$\phi$
. The inset shows the Smagorinsky coefficient
$C_s$
as a function of
$\phi$
, determined from the present data using (4.3). (b) The correlation coefficients between a given element of the real SGS stress
$\overline {\tau }_{\textit{ij}}$
and that from the Smagorinsky representation
$\tau _{\textit{ij}}^{ {sm}}$
as a function of
$\phi$
. Joint PDFs of the diagonal element
$\overline {\tau }_{11}$
from the real SGS stress and
$\tau _{11}^{ {sm}}$
from the Smagorinsky representation for (c)
$\phi = 0$
ppm and (d)
$\phi = 30$
ppm. The filtering scale is
$\varDelta = 97\eta$
.
As
$\phi$
increases, the modelled energy flux
$\left \langle \varPi ^{ {sm}} \right \rangle$
decreases, mirroring the trend observed for the actual flux
$\left \langle \varPi \right \rangle$
(see figure 8
a). However, figure 17(a) shows that the ratio
$\left \langle \varPi ^{ {sm}} \right \rangle / \left \langle \varPi \right \rangle$
increases with
$\phi$
, deviating further from unity. At
$\phi = 30$
ppm,
$\left \langle \varPi ^{ {sm}} \right \rangle$
exceeds its actual counterpart by a factor greater than four, indicating a significant overestimation by the Smagorinsky model, at least in its current form as expressed in (4.2). While recent LES studies report that the Smagorinsky model can capture flow structures and classical statistics in polymeric turbulence with reasonable accuracy (Ferreira et al. Reference Ferreira, Pinho and da Silva2016; Parvar et al. Reference Parvar, da Silva and Pinho2021), our a priori analysis here is not directly comparable, as it does not involve non-Newtonian fluid modelling in LES simulations.
The Smagorinsky model can also be evaluated by estimating the coefficient
$C_s$
, following standard LES procedures (Lilly Reference Lilly1967; Meneveau & Katz Reference Meneveau and Katz2000). Assuming that the modelled flux matches the true flux, i.e.
$\left \langle \varPi ^{ {sm}} \right \rangle = \left \langle \varPi \right \rangle$
, and substituting into the model expression yields
\begin{equation} \left \langle \varPi \right \rangle = 2\sqrt {2} (C_s \Delta )^2 \left \langle \Vert \tilde {S}_{\textit{ij}} \Vert ^2 \right \rangle ^{3/2}. \end{equation}
We adopt the commonly used approximation
$\left \langle \Vert \tilde {S}_{\textit{ij}} \Vert ^3 \right \rangle \approx \left \langle \Vert \tilde {S}_{\textit{ij}} \Vert ^2 \right \rangle ^{3/2}$
(Lilly Reference Lilly1967; Meneveau & Katz Reference Meneveau and Katz2000), which has been applied in a priori tests of polymeric turbulence (Ferreira et al. Reference Ferreira, Pinho and da Silva2016). This formulation enables the estimation of
$C_s$
at a specified filtering scale. As expected from K41 theory, in the inertial range where
$\Vert \tilde {S}_{\textit{ij}} \Vert \sim \Delta ^{-2/3}$
,
$C_s$
remains scale-invariant and recovers the canonical value of
$0.16$
for Newtonian turbulence. The computed
$C_s$
at
$\varDelta = 97\eta$
, representative of the inertial or elastic range, is shown in the inset of figure 17(a).
In the Newtonian case, the Smagorinsky coefficient
$C_s$
obtained from (4.3) is slightly below the canonical value of
$0.16$
, as indicated by the horizontal dashed line. This minor deviation may be attributed to finite Reynolds number effects (Kang & Meneveau Reference Kang and Meneveau2008). As
$\phi$
increases,
$C_s$
decreases by approximately one third at
$\phi = 30$
ppm, consistent with the increasing discrepancy between
$\left \langle \varPi ^{ {sm}} \right \rangle$
and
$\left \langle \varPi \right \rangle$
shown in figure 17(a). Our results agree with previous DNS findings (Ferreira et al. Reference Ferreira, Pinho and da Silva2016; Masoudian et al. Reference Masoudian, da Silva and Pinho2016), suggesting the potential need to adapt the Smagorinsky coefficient
$C_s$
in LES of viscoelastic fluids. While Ferreira et al. (Reference Ferreira, Pinho and da Silva2016) proposed an expression for
$C_s$
in polymeric turbulence based on analogies with Newtonian turbulence (Lilly Reference Lilly1967; Meneveau & Katz Reference Meneveau and Katz2000), its applicability to our measurements is limited due to the absence of a well-defined elastic range in such flows (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021; Rosti et al. Reference Rosti, Perlekar and Mitra2023), highlighting the need for more flexible modelling strategies.
To further evaluate the Smagorinsky model, we conduct a correlation analysis using experimental data. We focus on the
$i = j = 1$
component of both the modelled and measured SGS stresses. Figures 17(c) and 17(d) present the joint PDFs for the Newtonian and polymeric (
$\phi = 30$
ppm) cases, respectively, where all variables are mean-subtracted and normalised by their corresponding standard deviations, e.g. for
$\overline {\tau }_{11}$
we have
$(\overline {\tau }_{11} - \mu _{11})/\sigma _{11}$
, where
$\mu _{11}$
and
$\sigma _{11}$
are the ensemble average and standard deviation of
$\overline {\tau }_{11}$
, respectively. At the filtering scale
$\varDelta = 97\eta$
, the Newtonian case (figure 17
c) exhibits tilted diamond-shaped contours spanning values from
$0.01$
to
$100$
, consistent with previous findings (Borue & Orszag Reference Borue and Orszag1998), indicating a weak correlation between
$\overline {\tau }_{11}$
and
$\tau _{11}^{ {sm}}$
. Other tensor components show similar behaviour. In the polymeric case (figure 17
d), the joint PDF contours exhibit increased scatter but retain an overall similar shape, further illustrating the limitations of the Smagorinsky model in reproducing the SGS stress, especially in viscoelastic turbulence.
Further quantification of the correlations shown in figures 17(c) and 17(d) is provided by evaluating the correlation coefficient between the modelled and real
$\overline {\tau }_{\textit{ij}}$
. In addition to the
$i = j = 1$
component, we compute correlation coefficients between
$\overline {\tau }_{\textit{ij}}$
and
$\tau _{\textit{ij}}^{ {sm}}$
for all tensor components using (4.1), and the results are shown in figure 17(b). Since
$\tau _{\textit{ij}}^{ {sm}}$
is traceless, we omit the overline notation when referring to it in comparisons. In Newtonian turbulence, the correlation
$\rho (\overline {\tau }_{11}, \tau _{11}^{ {sm}})$
reaches 0.30, which is higher than previously reported values (Liu et al. Reference Liu, Meneveau and Katz1994; Borue & Orszag Reference Borue and Orszag1998), possibly due to differences in the filtering methods employed. As the polymer concentration increases, this correlation gradually declines, reaching approximately 0.05 at
$\phi = 30$
ppm. Similar trends are observed for the other stress components, suggesting that polymer additives tend to significantly weaken the correlation between the modelled SGS stress
$\tau _{\textit{ij}}^{ {sm}}$
and the actual one. This reduction in correlation can be attributed to the diminished alignment between
$-\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
, as shown in figure 12. Since the Smagorinsky model prescribes an exact alignment between these two tensors (see (4.2)), its scalar eddy-viscosity formulation becomes increasingly inaccurate in the presence of polymers. Overall, our experimental results indicate that the Smagorinsky model performs poorly in polymeric turbulence and highlight the need for improved SGS models tailored to viscoelastic flows.
4.2. Nonlinear model
The gradient model is formulated as (Leonard Reference Leonard1975; Meneveau & Katz Reference Meneveau and Katz2000)
\begin{equation} \tau _{\textit{ij}}^{\textit{nl}} = C_n \Delta ^2 \tilde {A}_{ik} \tilde {A}_{\textit{jk}}. \end{equation}
Typically interpreted as a tensorial eddy-viscosity model, this approach is closely related to the scale-similarity method (Bardina, Ferziger & Reynolds Reference Bardina, Ferziger and Reynolds1980; Liu et al. Reference Liu, Meneveau and Katz1994). Additionally, the nonlinear model can also be viewed as the leading-order term in a series expansion of
$\tau _{\textit{ij}}$
(Bedford & Yeo Reference Bedford and Yeo1993; Vreman, Geurts & Kuerten Reference Vreman, Geurts and Kuerten1996), with
$C_n = 1/12$
. The nonlinear model’s predicted local energy flux can be calculated as
$\varPi ^{\textit{nl}} = -\tau _{\textit{ij}}^{\textit{nl}} \tilde {S}_{\textit{ij}}$
. In figure 18(a), we present the ensemble average of the model’s prediction for energy flux,
$\varPi ^{\textit{nl}} $
, divided by the real value
$\left \langle \varPi \right \rangle$
as a function of
$\phi$
, at the filtering scale
$\varDelta = 97\eta$
.
(a) The ensemble-averaged local energy flux
$\langle \varPi ^{\textit{nl}} \rangle$
predicted by the nonlinear model, divided by the real value
$\langle \varPi \rangle$
as a function of
$\phi$
. (b) The correlation coefficients between components of SGS stress predicted by the nonlinear model
$\overline {\tau }_{\textit{ij}}^{\textit{nl}} $
and the real value, as a function of
$\phi$
. Joint PDFs between the components
$\overline {\tau }_{11}$
and the model prediction
$\overline {\tau }_{11}^{\textit{nl}} $
for (c)
$\phi = 0$
ppm and (d)
$\phi = 30$
ppm. The filtering scale is
$\varDelta = 97\eta$
.
In the Newtonian case, the ensemble-averaged energy flux predicted by the nonlinear model,
$\left \langle \varPi ^{\textit{nl}} \right \rangle$
, also deviates from the measured value
$\left \langle \varPi \right \rangle$
, and similar levels of discrepancy appear at other filtering scales. However, as the polymer concentration
$\phi$
increases, the ratio
$\left \langle \varPi ^{\textit{nl}} \right \rangle / \left \langle \varPi \right \rangle$
gradually approaches unity, suggesting that the nonlinear model provides a more accurate representation of the energy flux in polymeric turbulence – an improvement that contrasts with the growing discrepancy observed with the Smagorinsky model. Across other filtering scales, the nonlinear model shows consistently good performance, particularly at higher polymer concentrations.
Next, we assess the performance of the nonlinear model
$\tau _{\textit{ij}}^{\textit{nl}} $
via correlation analysis, as shown in figure 18(c,d). We also introduce its deviatoric part,
$\overline {\tau }_{\textit{ij}}^{\textit{nl}} = \tau _{\textit{ij}}^{\textit{nl}} - (1/3) \tau _{kk}^{\textit{nl}} \delta _{\textit{ij}}$
. In the Newtonian case (figure 18
c), the joint PDF contours form an ellipsoidal shape aligned with the diagonal, reflecting a strong correlation between
$\overline {\tau }_{11}$
and
$\overline {\tau }_{11}^{\textit{nl}} $
. This correlation becomes even stronger in the polymeric case (figure 18
d), where the contours narrow and elongate further. Similar trends are observed for the other components of
$\overline {\tau }_{\textit{ij}}$
(such as
$\overline {\tau }_{22}$
and
$\overline {\tau }_{33}$
, not shown), demonstrating robust correlations in both Newtonian and polymeric flows. These strong correlations, particularly evident in the polymeric case, directly contribute to the improved accuracy of the nonlinear model in predicting local energy flux, as illustrated in figure 18(a). Figure 18(b) presents the correlation coefficients
$\rho (\overline {\tau }_{\textit{ij}}, \overline {\tau }_{\textit{ij}}^{\textit{nl}} )$
for all components, together with the trace terms
$\tau _{kk}$
and
$\tau _{kk}^{\textit{nl}} $
(shown by black dotted lines with pentagram markers). In the Newtonian case, high correlation coefficients are found, consistent with previous studies (Liu et al. Reference Liu, Meneveau and Katz1994; Borue & Orszag Reference Borue and Orszag1998). As the polymer concentration
$\phi$
increases, these coefficients further improve, saturating at values above
$0.95$
for
$\phi \sim \mathcal{O}(10)$
ppm. These results suggest that incorporating the nonlinear model in LES can significantly enhance the accuracy of SGS stress predictions in polymeric turbulence.
Given the strong correlation between
$\tau _{\textit{ij}}^{\textit{nl}} $
and
$\tau _{\textit{ij}}$
, it is natural to further assess how well the nonlinear model reproduces their full statistical distributions. We therefore quantify the differences between the real and modelled quantities by comparing their PDFs in figure 19, including both the model-predicted energy flux and the SGS stress. The results correspond to the representative scale
$\varDelta = 97\eta$
, and for comparison, the predictions of the Smagorinsky model are also shown. For the energy transfer rate, the Smagorinsky model yields only positive values, whereas the nonlinear model is able to reproduce both positive and negative events. As illustrated in figure 19(a), both models show noticeable deviations from the true PDF in the Newtonian case, although the nonlinear model performs slightly better. In the polymeric case shown in figure 19(b) for
$\phi = 30$
ppm, the nonlinear model exhibits a substantially closer agreement with the real distribution, whereas the Smagorinsky model shows a substantial deviation from it. These analyses of the PDFs in figure 19(a,b) are consistent with the corresponding mean-value trends reported earlier. For the SGS stress (taking the component
$\overline {\tau }_{11}$
as an example), the Newtonian case in figure 19(c) does not reveal significant differences between the two models. In contrast, for the polymeric case at
$\phi = 30$
ppm (figure 19
d), the PDF predicted by the Smagorinsky model departs noticeably from the true distribution, particularly in the negative-value range. The nonlinear model, by comparison, remains much closer to the real distribution across the entire range.
(a,b) The PDFs of the real local energy flux
$\varPi$
and the reconstructed values obtained from the Smagorinsky model,
$\varPi ^{ {sm}}$
, and the nonlinear model,
$\varPi ^{\textit{nl}} $
. (c,d) The PDFs of the real SGS stress component
$\overline {\tau }_{11}$
and the corresponding reconstructed components,
$\tau ^{ {sm}}_{11}$
and
$\tau ^{\textit{nl}} _{11}$
. In (a–d), the filtering scale is fixed at
$\varDelta = 97\eta$
. (a,c) The
$\phi = 0$
ppm case and (b,d) the
$\phi = 30$
ppm case.
To further quantify the differences between the real SGS stress and the nonlinear model prediction, let
$\lambda _i^\tau$
and
$\boldsymbol{e}_i^\tau$
denote the eigenvalues and eigenvectors of the real SGS stress tensor
$\overline {\tau }_{\textit{ij}}$
, and
$\lambda _i^{\textit{nl}} $
and
$\boldsymbol{e}_i^{\textit{nl}} $
those of the nonlinear model prediction
$\overline {\tau }_{\textit{ij}}^{\textit{nl}} $
. We analyse the PDFs of the eigenvalue ratios
$\lambda _i^\tau / \lambda _i^{\textit{nl}} $
(
$i=1,2,3$
), shown in figure 20(a–c). In the Newtonian case (blue curves), the PDFs peak slightly above
$\log _{10}(\vert \lambda _i^\tau / \lambda _i^{\textit{nl}} \vert ) = 0$
, suggesting that
$\lambda _i^\tau$
is generally larger than its modelled counterpart. With increasing
$\phi$
, the peak positions shift closer to zero, indicating a better agreement between the eigenvalues. A similar trend is observed for the ratio of the traces of
$\tau _{\textit{ij}}$
and
$\tau _{\textit{ij}}^{\textit{nl}} $
(not shown). We further examine the alignment of eigenvectors by computing the PDFs of
$\vert \cos (\boldsymbol{e}_i^\tau , \boldsymbol{e}_i^{\textit{nl}} ) \vert$
in figure 20(d–f), where
$\cos (\boldsymbol{e}_i^\tau , \boldsymbol{e}_i^{\textit{nl}} )$
represents the cosine of the angle between the eigenvectors of the real and modelled tensors. In the Newtonian case, the PDFs peak at
$\vert \cos (\boldsymbol{e}_i^\tau , \boldsymbol{e}_i^{\textit{nl}} ) \vert = 1$
, consistent with prior experimental findings (Tao et al. Reference Tao, Katz and Meneveau2002), indicating strong alignment. As
$\phi$
increases, the peak values rise further, suggesting that the alignment of eigenvectors improves with polymer concentration in polymeric turbulence. Moreover, we confirm that
$\boldsymbol{e}_i^\tau$
and
$\boldsymbol{e}_{\!j}^{\textit{nl}} $
are nearly orthogonal for
$i \ne j$
and nearly parallel for
$i = j$
. These results demonstrate that the nonlinear model
$\tau _{\textit{ij}}^{\textit{nl}} $
not only correlates well with the components of
$\tau _{\textit{ij}}$
, but also accurately captures the magnitudes of eigenvalues and orientations of eigenvectors.
(a–c) The PDFs of the ratios between the eigenvalues
$\lambda _i^\tau$
of the real SGS stress and
$\lambda _i^{\textit{nl}} $
of the nonlinear model predictions. (d–f) The PDFs of the cosine angles between the eigenvectors
$\boldsymbol{e}_i^\tau$
of the real SGS stress and
$\boldsymbol{e}_i^{\textit{nl}} $
of the nonlinear model predictions. The filtering scale is
$\varDelta = 97\eta$
.
In this section, we have evaluated two SGS models – the Smagorinsky model and the nonlinear (or gradient) model – based on our experimental data. At an intermediate filtering scale
$\varDelta = 97\eta$
, both models exhibit notable discrepancies from the actual local energy flux in the Newtonian case. With the addition of polymers, the behaviour of the models diverges: the Smagorinsky model shows increasing deviation with polymer concentration
$\phi$
, whereas the nonlinear model exhibits decreasing deviation. The increasing overestimation of energy flux by the Smagorinsky model suggests that the
$C_s$
coefficient may need adjustment in polymeric turbulence. In contrast, the nonlinear model offers a more accurate representation of the real SGS stress and presents itself as a viable alternative for modelling viscoelastic flows. Correlation analyses reveal that the Smagorinsky model’s performance deteriorates with increasing
$\phi$
, as evidenced by the weak correlation with the true SGS stress, while the nonlinear model retains strong correlation, improving further at higher concentrations. Additional analyses demonstrate that at sufficiently high polymer concentrations, the nonlinear model can nearly reproduce the true SGS stress.
5. Conclusion
Polymers are well known to suppress the energy cascade in three-dimensional turbulence, a phenomenon extensively investigated through velocity structure functions and energy spectra. In this study, we examine polymer-modified energy transfer from the perspective of SGS dynamics, employing a filtering approach. Experiments are conducted in a turbulent VKS flow, with a Taylor-scale Reynolds number of
$R_\lambda = 581$
in the Newtonian case. The three-dimensional, three-component velocity field is measured using Tomo-PIV. A cubic box filter with scale
$\varDelta$
is applied to extract the relevant SGS quantities, with particular focus on the filtered strain-rate tensor
$\tilde {S}_{\textit{ij}}$
and the SGS stress tensor
$\tau _{\textit{ij}}$
. At a representative filtering scale of
$\varDelta = 97\eta$
, we investigate the statistical structure of these two tensors and their mutual alignment, focusing on the eigenvector relationships and their impact on SGS stress modelling under varying polymer concentrations.
We start by analysing the statistical structures of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
. In the Newtonian case, we find that the intermediate eigenvalue of
$\tilde {S}_{\textit{ij}}$
tends to be positive, whereas that of
$\overline {\tau }_{\textit{ij}}$
tends to be negative. In polymeric turbulence, however, the distributions of eigenvalues shift significantly:
$\tilde {S}_{\textit{ij}}$
exhibits fewer positive intermediate eigenvalues, and
$\overline {\tau }_{\textit{ij}}$
shows an increased occurrence of strongly negative intermediate values. These trends approach the eigenvalue ratios
$\lambda _1^s : \lambda _2^s : \lambda _3^s \approx 1 : 0 : -1$
for
$\tilde {S}_{\textit{ij}}$
, indicating a two-dimensional configuration, and
$\lambda _1^\tau : \lambda _2^\tau : \lambda _3^\tau \approx 2 : -1 : -1$
for
$\overline {\tau }_{\textit{ij}}$
, suggesting a one-component axisymmetric configuration in polymeric turbulence. These modifications in the eigenframe are further supported by the joint PDFs of the invariants of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
. To further explore the alignment characteristics, we analyse the relationship between the filtered vorticity vector
$\tilde {\omega }_i$
and the eigenvectors of
$\tilde {S}_{\textit{ij}}$
and
$\overline {\tau }_{\textit{ij}}$
. The results show a preferential alignment of
$\tilde {\omega }_i$
with the intermediate eigenvector
$\boldsymbol{e}_2^s$
of
$\tilde {S}_{\textit{ij}}$
and a notable misalignment with the most extensional direction
$\boldsymbol{e}_1^\tau$
of
$\overline {\tau }_{\textit{ij}}$
.
Next, we examine the local energy flux
$\varPi$
, defined as the inner product of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
. At a filtering scale of
$\varDelta = 97\eta$
, the ensemble-averaged energy flux
$\langle \varPi \rangle$
is significantly reduced with the addition of polymers, consistent with prior findings based on velocity structure functions (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021). Furthermore, the PDF of
$\varPi$
exhibits increasingly heavy tails as the polymer concentration
$\phi$
increases. The scale dependence of the ensemble average and the PDF of
$\varPi$
show qualitatively similar results to those observed in our previous study (Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021). Additionally, we decompose
$\varPi$
into three components corresponding to the energy flux terms in the energy equation for each direction. Along the axial direction of the VKS flow, we observe that although the SGS energy
$\langle \tau _{22} \rangle$
accounts for a small portion of the total energy, its associated energy flux
$\langle \varPi _y \rangle$
constitutes an increasing fraction of the total flux as the polymer concentration
$\phi$
increases.
We further investigate the alignment between the eigendirections of the
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
tensors, focusing on the eigenvector pairs with identical subscripts, i.e. the alignments between
$\boldsymbol{e}_a^s$
and
$\boldsymbol{e}_a^\tau$
(
$a = 1, 2, 3$
). At a filtering scale of
$\varDelta = 97\eta$
, we observe a general misalignment between the eigendirections for
$a = 1$
and
$3$
, and this misalignment becomes more pronounced as polymer concentration increases, which inhibits the energy flux
$\varPi$
. To quantify these effects, we employ the concept of energy transfer efficiency,
$\varGamma$
, which shows a reduction in efficiency and suppression of energy flux in polymeric turbulence.
Finally, we evaluate two SGS stress models – the Smagorinsky model and the nonlinear (gradient) model – using our experimental data. Direct calculations of
$\tau _{\textit{ij}}^{ {sm}}$
and
$\tau _{\textit{ij}}^{\textit{nl}} $
allow us to compare their ability to predict
$\tau _{\textit{ij}}$
. We first assess the models based on their predictions of the local energy flux. At a filtering scale of
$\varDelta = 97\eta$
, both models show a deviation of approximately
$50\,\%$
from the true value of
$\left \langle \varPi \right \rangle$
in the Newtonian case. The deviation for the Smagorinsky model increases with
$\phi$
, likely due to the weakening alignment between the eigenvectors of
$-\overline {\tau }_{\textit{ij}}$
and
$\tilde {S}_{\textit{ij}}$
, suggesting the need for a reduced Smagorinsky coefficient,
$C_s$
, in polymeric turbulence, as indicated by previous DNS studies (Ferreira et al. Reference Ferreira, Pinho and da Silva2016; Masoudian et al. Reference Masoudian, da Silva and Pinho2016). In contrast, the nonlinear model provides a more accurate approximation of the real SGS stress. Further analysis of the component-to-component correlation between the model stresses and the real stress reveals that
$\tau _{\textit{ij}}^{ {sm}}$
shows low correlation with real
$\tau _{\textit{ij}}$
, while
$\tau _{\textit{ij}}^{\textit{nl}} $
exhibits a strong correlation, which increases further in the polymeric case. As
$\phi$
increases, the nonlinear model closely replicates the SGS stress. Based on these findings, we recommend the nonlinear model for estimating SGS stress in LES of polymeric turbulence.
Overall, while the results for the Newtonian case are consistent with previous studies, this work provides novel insights into SGS statistics in polymeric turbulence. We anticipate that the investigation of the statistics of
$\tilde {S}_{\textit{ij}}$
and
$\tau _{\textit{ij}}$
tensors will enhance the understanding of polymer-induced suppression of interscale energy transfer and contribute to the improvement of SGS stress modelling in LES of viscoelastic fluid flows.
Funding
We are grateful to the National Natural Science Foundation of China (NSFC) through grant nos. 12388101, 12572254, 12202452, 12125204 and 12402298, and the 111 Project of China through grant no. B17037. P.-F.Y. also acknowledges financial support from the Fundamental Research Funds for Central University through grant no. D5000240110.
Declaration of interests
The authors report no conflict of interest.
