3.1. Direct numerical simulation data set

To investigate the geometrical statistics of SFEF based on different filter functions, the direct numerical simulation (DNS) of a forced homogeneous isotropic turbulence is considered. The flow field results of the DNS are obtained from the Johns Hopkins Turbulence Database (JHTDB), whose simulation details can be found on the JHTDB website (Perlman et al. Reference Perlman, Burns, Li and Meneveau2007; Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, Chen, Szalay and Eyink2008; Minping et al. Reference Minping, Chen, Eyink, Meneveau, Johnson, Perlman, Burns, Li, Szalay and Hamilton2012). Briefly, the incompressible Navier–Stokes equation is solved using the pseudospectral algorithm in a cubic domain of size $L=2{\rm \pi}$ with periodic boundary conditions, discretized by $1024^3$ grid points. The nonlinear terms of the Navier–Stokes equation are dealiased using the combined phase-shift and $2\sqrt {2} /3$ truncation methods. Therefore, the effective spectral resolution is $k_{max}\eta \approx 1.35$, where $k_{max}$ is the highest wavenumber, $\eta$ is the Kolmogorov length scale and $\eta /L\approx 5\times 10^{-5}$. At the supergrid scale, forcing is applied to all modes within the spectral radius $k\eta \lesssim 6\times 10^{-3}$, i.e. shells with $k\leq 2$. This forcing maintains the turbulent flow stationary at the Taylor microscale Reynolds number of ${Re}_{\lambda } \approx 433$.

Figure 2 presents the volume-averaged SFEF as a function of the filter width $k={\rm \pi} /\varDelta$ based on the $\mathcal {S}$- and $\mathcal {G}$-filtering methods, namely, $\langle \varPi _\mathcal {S}\rangle$ and $\langle \varPi _\mathcal {G}\rangle$ (the subscripts $\mathcal {S}$ and $\mathcal {G}$ represent the corresponding filter functions). Consistent with the previously reported findings (Rivera, Aluie & Ecke Reference Rivera, Aluie and Ecke2014; Buzzicotti et al. Reference Buzzicotti, Linkmann, Aluie, Biferale, Brasseur and Meneveau2018; Alexakis & Chibbaro Reference Alexakis and Chibbaro2020), it can be observed that $\langle \varPi _\mathcal {G}\rangle$ shows smoother variation with wavenumber $k={\rm \pi} /\varDelta$ compared with $\langle \varPi _\mathcal {S}\rangle$. This difference stems from the non-trivial contribution of the non-local interactions in the $\mathcal {S}$-filtering as follows.

Figure 2.

The SFEF based on the $\mathcal {S}$-filtering $\langle \varPi _\mathcal {S} \rangle$ and $\mathcal {G}$-filtering $\langle \varPi _\mathcal {G} \rangle$, as a function of the filter width $k={\rm \pi} /\varDelta$, normalized by the total turbulent kinetic energy dissipation rate, $\epsilon$.

Consider the non-local triadic interaction $\boldsymbol {r}+\boldsymbol {p}+\boldsymbol {q}=0$, in which two coupling modes are in the neighbourhood of the spectral shell boundary $k={\rm \pi} /\varDelta$, say $r\ll p\approx {\rm \pi}/\Delta \approx q$ with mode $p(q)$ just below(above) $k$. In $\mathcal {G}$-filtering with the filter threshold $k$, due to the spectral overlap of the adjacent modes $p\approx q$, their corresponding triadic interaction cannot survive, and consequently, their coupling is averaged out. This situation contrasts the $\mathcal {S}$-filtering, in which the coupling modes $p\approx q$ are separated by a sharp boundary, and their interaction can contribute to the SFEF through its corresponding triadic interaction. Besides the contribution of the non-local interactions, some part of the energy flux in $\mathcal {S}$-filtering appears as the spatial energy transport of the subfilter scales in $\mathcal {G}$-filtering approach (see schematic 7 in Piomelli et al. (Reference Piomelli, Cabot, Moin and Lee1991)).

In the following, the impact of different filtering approaches, namely, $\mathcal {S}$- and $\mathcal {G}$-filtering, on the SFEF is further investigated in terms of their statistical geometry. The filter size used throughout the present research corresponds to $k\eta \approx 0.11$, which is located in the plateau region of the energy flux in figure 2. Interestingly, the results do not show sensitive dependence on the filter width, and changing the filter size over $k\eta \in [0.02,0.2]$ does not show any significant effect on the results.

3.2. Relative attitude

In this section, the impact of different filtering approaches on the relative eigenframe alignment between FSR and SFS tensors is investigated. The main statistical parameters of the eigenframe alignment are presented in table 1. Interestingly, it can be observed that for both $\mathcal {S}$-filter and $\mathcal {G}$-filter, $\langle \chi \rangle$, $\langle \theta \rangle$ and $\langle \phi \rangle$ show similar values. Therefore, the filter type does not considerably influence the average alignment between the FSR and SFS eigenframes. However, it will be seen that despite the similar mean behaviour, the statistical distribution of the eigenframe alignment shows a sensitive dependence on the filter kernel. In the following, the statistical distributions of the Euler angle, $\chi$, and Euler axis orientation, $(\theta, \phi )$, are investigated.

Table 1.

Statistics of the Euler angle, $\chi$, and Euler axis orientation, $(\theta, \phi )$, for $\mathcal {G}$- and $\mathcal {S}$-filters.

3.2.1. Euler angle (rotation angle)

Figure 3 demonstrates the p.d.f. of the Euler angle, $P(\chi )$, based on the $\mathcal {G}$-filter and $\mathcal {S}$-filter. Although $\chi \in [0^{\circ }, 180^{\circ }]$, it can be seen that for both filter functions, the probability for $\chi =0^\circ$ and $\chi \geq 120^\circ$ are zero. The first zero probability, namely, $P(\chi =0^\circ )=0$, indicates that the SFS eigenframe is rarely in alignment with the FSR eigenframe.

Figure 3.

Density function (p.d.f.) of the Euler angle, $P(\chi )$, based on the $\mathcal {G}$-filtering and $\mathcal {S}$-filtering.

Before comparing the p.d.f. of the Euler angle for the $\mathcal {S}$- and $\mathcal {G}$-filtering methods, it is worth providing some physical intuition about the specific values of the Euler angle. Considering an archetype flow configuration, consisting of a compression–elongation flow accompanied by a shear motion, it can be shown that the Euler angles $\chi =45^{\circ }$, $90^{\circ }$ and $120^{\circ }$ can be related to the strong and weak shear motions. Under dominant shear conditions, the alignment between SFS and the FSR eigenframes is through a $\chi =45^{\circ }$ rotation angle. On the other hand, in weak shear conditions, the Euler angle can hold either $90^{\circ }$ or $120^{\circ }$ depending on the ratio of compression–elongation of the FSR tensor. If this ratio is within $[1/2 , 2]$ range, which is an almost isotropic compression, the Euler angle is $\chi = 90^{\circ }$. If this ratio exhibits more anisotropic distribution, i.e. the ratio increases as $[2 , \infty ]$, the Euler angle jumps to $\chi = 120^{\circ }$ (see Yang & Wang (Reference Yang and Wang2016) for more details).

Considering the fact that for both filters in figure 3, the Euler angle $\chi = 90^{\circ }$ is the most probable state and $\chi \geq 120^{\circ }$ has zero probability, it indicates that isotropic compression is the most probable state in both filtering formulations, while anisotropic compression rarely happens. Despite these similarities, $P(\chi )$ for the two filtering approaches shows non-trivial differences as follows.

In $\mathcal {G}$-filtering, $P(\chi )$ exhibits a bimodal behaviour with a primary mode at $\chi =90^\circ$ and a secondary mode at $\chi =45^\circ$. This bimodal pattern has also been reported in a previous study (Yang & Wang Reference Yang and Wang2016) based on the box filter, which is similar to the $\mathcal {G}$-filter, a non-projector filter. On the other hand, $P(\chi )$ for the $\mathcal {S}$-filtering does not have a local maximum at $\chi =45^\circ$, while it shows a higher probability for $75\,^\circ \leq \chi \leq 90^\circ$ rotation angle range compared with the $\mathcal {G}$-filtering.

Therefore, in contrast to the $\mathcal {G}$-filtering, in which both weak and strong shear motions show high probability, the $\mathcal {S}$-filtering effectively annihilates the strong shear motion while having an augmentation effect on the normal straining motion. However, despite this different statistical distribution, the average value of the Euler angle for both filters is identical, $\langle \chi \rangle = 64^{\circ }$, as presented in table 1.

3.2.2. Euler axis (rotation axis)

The orientation of the Euler axis $\boldsymbol {q}$ is specified by the colatitude angle $\theta \in [0^\circ, 90^\circ ]$ and the longitude angle $\phi \in [-90^\circ, 90^\circ ]$. Here, the Euler axis statistics are shown in terms of the premultiplied j.p.d.f.s of these angles, i.e. $P_m(\theta,\phi )\equiv P(\theta,\phi )\sin \theta$, in which $\sin \theta$ ensures an unbiased probability distribution for the randomly oriented $\boldsymbol {q}$ with uniform probability distribution (Higgins et al. Reference Higgins, Parlange and Meneveau2003; Yang & Wang Reference Yang and Wang2016).

Figure 4 presents $P_m(\theta,\phi )$ based on the $\mathcal {G}$- and $\mathcal {S}$-filtering. These j.p.d.f.s are obtained by dividing the whole range of $\theta$ and $\phi$ into $50\times 50$ identical bins. Figure 4(a) shows that the most probable orientation of the Euler axis for $\mathcal {G}$-filtering is $(\theta,\phi )\approx (65^\circ,17^\circ )$, which is consistent with the previously reported result $(\theta,\phi )\approx (62^\circ,15^\circ )$ based on the box filtering (Yang & Wang Reference Yang and Wang2016). On the other hand, figure 4(b) shows that $P_m(\theta,\phi )$ based on the $\mathcal {S}$-filtering has a pronounced bimodal distribution, which is qualitatively different from the $\mathcal {G}$-filtering. This bimodal shape of $P_m(\theta,\phi )$ shows an approximately symmetric distribution of the most probable states at $(\theta,\phi )\approx (90^\circ,\pm 45^\circ )$ with equal probabilities. It will be seen that this symmetrical distribution of the rotation axis in the $\mathcal {S}$-filtering approach has a non-trivial effect on the forward–backward energy flux cancellation.

Figure 4.

Statistics of the Euler axis orientation based on the premultiplied j.p.d.f. of the colatitude and longitude angles, $P_m(\theta,\phi )$, for (a) a $\mathcal {G}$-filter, and (b) an $\mathcal {S}$-filter.

Based on these statistics, the most probable relative attitudes between the FSR and SFS eigenframes are visualized in figure 5, while more quantitative information is compiled in Appendix A, figure 12. Interestingly, in contrast to table 1 which shows that the mean alignment, i.e. $\langle \chi \rangle$, $\langle \theta \rangle$ and $\langle \phi \rangle$, are almost identical for both $\mathcal {S}$- and $\mathcal {G}$-filters, figure 5 shows that the most probable alignments between ${\tilde {\boldsymbol{\mathsf{s}}}}$ and $\boldsymbol {\tau }$ are quite different.

Figure 5.

Schematic of the most probable eigenframes of the FSR, denoted by black $\boldsymbol {v}^{\boldsymbol {s}}_\alpha$–$\boldsymbol {v}^{\boldsymbol {s}}_\beta$–$\boldsymbol {v}^{\boldsymbol {s}}_\gamma$ axes, and the SFS eigenframe, visualized on the edges of a cube with red axes $\boldsymbol {v}^{\boldsymbol {\tau }}_\alpha$–$\boldsymbol {v}^{\boldsymbol {\tau }}_\beta$–$\boldsymbol {v}^{\boldsymbol {\tau }}_\gamma$. The most probable state of $(\theta, \phi,\chi )$ for $\mathcal {G}$-filter at (a) $(65^\circ,17^\circ,90^\circ )$ (G1), and (b) $(65^\circ,17^\circ,45^\circ )$ (G2). The most probable state of $(\theta, \phi,\chi )$ for $\mathcal {S}$-filter at (c) $(80^\circ, -45^\circ,90^\circ )$ (S1), and (d $(80^\circ, 45^\circ,90^\circ )$ (S2).

Figures 5(a) and 5(b) show the primary and the secondary configurations based on the $\mathcal {G}$-filtering, i.e. $(\theta, \phi,\chi ) \approx (65^\circ,17^\circ,90^\circ )$ and $(65^\circ,17^\circ,45^\circ )$, respectively. By comparing these two configurations, it can be seen that in the primary mode, the eigenvectors $\boldsymbol {v}^{\boldsymbol {s}}_{\beta }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma }$ show a pronounced alignment, while this strong alignment cannot be observed in the secondary mode. However, in both primary and secondary configurations, the alignment between $(\boldsymbol {v}^{\boldsymbol {s}}_{\alpha },\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$ eigenvectors is more pronounced compared with $(\boldsymbol {v}^{\boldsymbol {s}}_{\beta },\boldsymbol {v}^{\boldsymbol {\tau }}_{\beta })$, and $(\boldsymbol {v}^{\boldsymbol {s}}_{\gamma },\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma })$ (this issue can also be observed quantitatively in figure 12a,e,i by considering the columns G1 and G2). As will be discussed shortly, the alignment between ($\boldsymbol {v}^{\boldsymbol {s}}_{\alpha }$, $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$) eigenvectors has an augmentation effect on the forward energy transfer of SFEF based on the $\mathcal {G}$-filtering.

Figures 5(c) and 5(d) show the most probable configurations in $\mathcal {S}$-filtering, i.e. $(\theta, \phi,\chi ) \approx (80^\circ,\pm 45^\circ,90^\circ )$. It can be seen that in these cases, the eigenvectors corresponding to the smallest eigenvalues are perpendicular, i.e. $\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma } \perp \boldsymbol {v}^{\boldsymbol {s}}_{\gamma }$ (see columns S1 and S2 in figure 12i). However, concerning the interscale energy transfer, the most important point in configurations S1 and S2 is the relative alignment of $\boldsymbol {v}^{\boldsymbol {s}}_{\alpha }$ with $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma }$. It can be observed that in both S1 and S2 configurations, the eigenvectors $\boldsymbol {v}^{\boldsymbol {s}}_{\alpha }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$ are in the same direction, i.e. $(\boldsymbol {v}^{\boldsymbol {s}}_{\alpha }\boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })>0$ (see also figure 12a, columns S1 and S2). On the other hand, the directional tendency between $\boldsymbol {v}^{\boldsymbol {s}}_{\alpha }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma }$ shows different behaviours: $(\boldsymbol {v}^{\boldsymbol {s}}_{\alpha }\boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma })$ is positive in S1, while it is negative in S2 (see also figure 12c, columns S1 and S2). It will be shortly seen that this behaviour has a hampering effect on the forward energy transfer based on the $\mathcal {S}$-filtering.

3.3. Geometrical structure

In this section, the impact of different filtering approaches on the eigenstructures of the FSR and SFS is analysed in terms of a non-dimensional state parameter, $s^*$. Considering the FSR tensor, this non-dimensional parameter, known as the strain-state parameter, is defined as (Lund & Rogers Reference Lund and Rogers1994)

(3.1)\begin{equation} s^* = \frac{-3\sqrt{6}\alpha_s \beta_s \gamma_s}{(\alpha^2_s + \beta^2_s + \gamma^2_s)^{3/2}}, \end{equation}

where $\alpha _s$, $\beta _s$ and $\gamma _s$ are eigenvalues of the FSR tensor. Considering the fact that the incompressible strain rate is a deviatoric tensor, this parameter lies in the range $[-1,1]$ and a random velocity field displays a uniform p.d.f. for $s^*$.

To provide some physical intuition for $s^*$, this parameter is mainly interpreted as a measure of the geometrical character of the local deformations caused by the strain rate. Accordingly, the plane shear corresponds to $s^*=0$, while in axisymmetric extension and contraction, $s^*=1$ and $s^*=-1$, respectively. A schematic of the deformation topology for an initially spherical fluid particle is depicted in the upper part of figure 6(c). It is also found that for an isotropic turbulence, the most probable value of strain-state is axisymmetric extension and this state is well correlated with regions of high dissipation rate (Lund & Rogers Reference Lund and Rogers1994).

Figure 6.

Impact of the filter kernel on the eigenstructure of the FSR and SFS tensors. The p.d.f. of (a) the strain-state, $P(s^*)$, and (b) the stress-state, $P(\tau ^*)$. (c) Functional relation between the normalized eigenvalues, $\alpha ^*$ (Red), $\beta ^*$ (Green), $\gamma ^*$ (Blue) and corresponding state parameter, i.e. relations (3.3). The upper part of (c) provides axisymmetric contraction $s^*=-1$, plane shear $s^*=0$, and axisymmetric extension $s^*=1$ for an initially spherical fluid particle.

Similarly, expression (3.1) can also be extended for geometrically characterizing the SFS tensor by defining the stress-state parameter, $\tau ^*$ (Tao et al. Reference Tao, Katz and Meneveau2002; Higgins et al. Reference Higgins, Parlange and Meneveau2003) as

(3.2)\begin{equation} \tau^* = \frac{-3\sqrt{6}\alpha_\tau \beta_\tau \gamma_\tau}{(\alpha^2_\tau + \beta^2_\tau + \gamma^2_\tau)^{3/2}}, \end{equation}

where $\alpha _\tau$, $\beta _\tau$ and $\gamma _\tau$ are eigenvalues of the deviatoric part of the SFS tensor $\tau ^d_{ij} = \tau _{ij} - 1/3 \tau _{kk}\delta _{ij}$. Similar to $s^*$, the value of $\tau ^*$ is also in the interval $[-1,1]$, and it yields a uniform p.d.f. for a random velocity field. A unique dependence between the normalized eigenvalues and the state parameter exists as (Lund & Rogers Reference Lund and Rogers1994)

(3.3)\begin{equation} \left.\begin{gathered} \alpha^* ={-}\cos \varPsi + \sqrt{3} |\sin \varPsi |\\ \beta^* = 2\cos \varPsi \\ \gamma^* ={-}\cos \varPsi - \sqrt{3} |\sin \varPsi |,\end{gathered}\right\} \end{equation}

where $\alpha ^*$, $\beta ^*$ and $\gamma ^*$ are the eigenvalues normalized by $(\alpha ^2+\beta ^2+\gamma ^2)^{1/2}/\sqrt {6}$, e.g. $\beta ^* = \sqrt {6}\beta /(\alpha ^2+\beta ^2+\gamma ^2)^{1/2}$. Moreover, the angle parameter is defined as $\varPsi = 5/3{\rm \pi} - 1/3 \cos ^{-1}(\bullet )$, in which $(\bullet )$ stands for the corresponding state parameter $s^*$ or $\tau ^*$.

The main statistics of $s^*$ and $\tau ^*$ are gathered in table 2, while figure 6 provides their statistical distributions as well as their eigenstructures. Figure 6(a) presents the p.d.f. of the strain-state parameter, $s^*$, based on the $\mathcal {S}$-filter and $\mathcal {G}$-filter. It can be observed that for both filter kernels, $P(s^*)$ shows almost similar distribution with the most likely state at $s^* \to 1$, corresponding to the local axisymmetric extension. This similar statistical distribution of $s^*$ for the two filters can also be quantitatively recovered from their approximately equal mean values and variances, presented in table 2. Therefore, it can be asserted that different filtering approaches do not significantly affect the eigenstructure of the FSR tensor.

Table 2.

Statistics of the strain-state parameter $s^*$ and the stress-state parameter $\tau ^*$ (for the $\mathcal {S}$-filter, the mean value $\langle \tau ^* \rangle =-0.004$ is presented by $-0.00$).

The eigenstructure of the FSR tensor can be further analysed from relations (3.3), which is visualized in figure 6(c). Considering the most probable strain-state $s^* \to 1$, figure 6(c) shows that the most probable eigenstructure of FSR tends to be $\alpha ^*_s : \beta ^*_s : \gamma ^*_s = 1 : 1 : -2$. Therefore, for both filtering formulations, the descending order based on the absolute values is $|\gamma _s|\geq |\alpha _s| \geq |\beta _s|$, which corresponds to the contractive–extensive–intermediate eigenvectors (note that $\alpha >0$, $\beta$ and $\gamma <0$, correspond to the extensive, intermediate and contractive eigendirections, respectively).

Figure 6(b) presents the p.d.f. for the stress-state parameter $\tau ^*$. It can be observed that in contrast to the FSR tensor, different filtering approaches have a non-trivial impact on the eigenstructure of the SFS tensor. Here $P(\tau ^*)$ in the $\mathcal {G}$-filtering shows the most likely state for the axisymmetric contraction, $\tau ^* \to -1$, while in $\mathcal {S}$-filtering, $P(\tau ^*)$ shows a uniform distribution corresponding to a random field with a negligible mean value, $\langle \tau ^* \rangle \approx 0$, as presented in table 2.

Considering the most likely state $\tau ^* \to -1$ for the SFS based on the $\mathcal {G}$-filtering, figure 6(c) implies that the most probable eigenstructure tends to be $\alpha ^*_\tau : \beta ^*_\tau : \gamma ^*_\tau = 2 : -1 : -1$. Therefore, in $\mathcal {G}$-filtering formulation, the descending order based on the absolute values is $|\alpha _\tau |\geq |\gamma _\tau | \geq |\beta _\tau |$ corresponding to extensive–contractive–intermediate eigenvectors.

On the other hand, uniform distribution of $P(\tau ^*)$ for the SFS based on the $\mathcal {S}$-filtering implies $\alpha ^*_\tau : \beta ^*_\tau : \gamma ^*_\tau = 1.7 : 0 : -1.7$. Therefore, the descending order based on the absolute values is $|\alpha _\tau |= |\gamma _\tau |> |\beta _\tau |$, corresponding to contractive–extensive–intermediate eigenvectors. This specific eigenstructure is probably a direct consequence of the symmetrical distribution $\phi \approx \pm 45^\circ$, presented in figure 4.

3.4. Normalized energy flux

The impact of different filtering approaches on the SFEF can be investigated in terms of the normalized energy flux,

(3.4)\begin{equation} \varPi^* = \frac{-\tau^d_{ij}{\mathsf{s}}_{ij}}{(\tau^d_{mn}\tau^d_{mn})^{1/2}({\mathsf{s}}_{kl}{\mathsf{s}}_{kl})^{1/2}}, \end{equation}

which is bounded in the range $[-1,1]$. The positive/negative value of $\varPi ^*$ reflects the forward–backward scatter energy transfer. In the following, the normalized energy fluxes based on the $\mathcal {S}$- and $\mathcal {G}$-filtering methods are denoted by their corresponding subscripts, namely, $\varPi ^*_\mathcal {S}$ and $\varPi ^*_\mathcal {G}$.

Figure 7 displays different statistical distributions of the normalized energy flux based on the $\mathcal {G}$- and $\mathcal {S}$-filters. Figure 7(a) presents the j.p.d.f. based on the $\mathcal {G}$-filtering, $P(\varPi _\mathcal {G}^*,\chi )$. It can be seen that $P(\varPi _\mathcal {G}^*,\chi )$ exhibits three preferential modes corresponding to $(\varPi _\mathcal {G}^*,\chi )\approx (0.29,45^\circ )$, $(0.35,64^\circ )$ and $(0.43 , 90^\circ )$. Two modes $(0.29,45^\circ )$ and $(0.43 , 90^\circ )$ are associated with the most probable Euler angles $45^\circ$ and $90^\circ$, i.e. strong and weak shear conditions, respectively. The third mode $(0.35,64^\circ )$ is related to the average Euler angle $\langle \chi \rangle =64^\circ$, for the $\mathcal {G}$-filtering (see table 1). It can be seen that $P(\varPi _\mathcal {G}^*,\chi )$ is predominantly distributed over the $\varPi _\mathcal {G}^*> 0$ region which corresponds to the forward energy transfer.

Figure 7.

Statistical distribution of the normalized energy flux, $P(\varPi ^*,\chi )$, based on the (a) $\mathcal {G}$-filter, and (b) $\mathcal {S}$-filter. (c) The p.d.f. of $\varPi ^*$ for $\mathcal {G}$-filter and $\mathcal {S}$-filter.

Figure 7(b) presents the same j.p.d.f. based on the $\mathcal {S}$-filtering, i.e. $P(\varPi _\mathcal {S}^*,\chi )$. In contrast to the $\mathcal {G}$-filtering, it can be observed that the most probable state of $(\varPi _\mathcal {S}^*,\chi )$ shows a single-mode pattern at $(\varPi _\mathcal {S}^*,\chi )\approx (0,90^\circ )$, corresponding to the most probable Euler angle $\chi =90^\circ$ in figure 3. However, the statistical distribution of $\varPi _\mathcal {S}^*$ shows a non-trivial difference compared with that of $\varPi _\mathcal {G}^*$: while the distribution of $P(\varPi _\mathcal {G}^*,\chi )$ is predominantly in the forward (positive) flux region, $P(\varPi _\mathcal {S}^*,\chi )$ shows an almost symmetrical distribution over both forward–backward flux regions.

This different statistical distribution of the normalized energy transfer for the $\mathcal {S}$- and $\mathcal {G}$-filters can be better demonstrated in terms of their overall p.d.f.s, i.e. $P(\varPi _\mathcal {S}^*)$ and $P(\varPi _\mathcal {G}^*)$, which are presented in figure 7(c). First, it can be observed that the filter type does not considerably influence the most probable value of the normalized energy flux, i.e. $\varPi _\mathcal {G}^*=0.38$ and $\varPi _\mathcal {S}^*=0.30$. However, the statistical distribution of the normalized energy transfer, $P(\varPi ^*)$, is significantly affected by the filter type: $P(\varPi _\mathcal {G}^*)$ is skewed towards the positive region $\varPi _\mathcal {G}^*>0$, and is concentrated around its most probable value $\varPi _\mathcal {G}^*=0.38$ with forward mean normalized energy transfer $\langle \varPi _\mathcal {G}^* \rangle = 0.28$. On the other hand, $P(\varPi _\mathcal {S}^*)$ exhibits an almost symmetrical distribution over both positive and negative flux regions $\varPi _\mathcal {S}^* \gtrless 0$. As a result of this nearly symmetrical distribution, significant cancellation occurs between forward and backward energy transfer, leading to a considerably smaller mean normalized energy flux in $\mathcal {S}$-filtering, $\langle \varPi _\mathcal {S}^* \rangle = 0.08$, compared with that in $\mathcal {G}$-filtering.

This issue can be further investigated by considering the forward–backward components of the energy flux,

(3.5)\begin{equation} \varPi^\pm = \tfrac{1}{2}(\varPi \pm |\varPi|). \end{equation}

The contribution of these forward–backward scatter energy fluxes to the overall SFEF can be quantified by $\varepsilon \equiv \varPi /(\varPi ^+ - \varPi ^-)=\varPi /|\varPi |$, which equals $+1(-1)$ in regions with $\varPi >0(\varPi <0)$ (note that this parameter, $\varepsilon$, should not be confused with the dissipation rate, which is mainly denoted by $\epsilon$).

Figure 8 shows the p.d.f. of $\varepsilon$, $P(\varepsilon )$, obtained based on $\mathcal {S}$- and $\mathcal {G}$-filtering approaches. Obviously, for both $\varPi _\mathcal {S}$ and $\varPi _\mathcal {G}$, this parameter shows a double-Delta-function distribution $P(\varepsilon )=f\delta (\varPi -\varPi ^-)+ (1-f)\delta (\varPi -\varPi ^+)$, where $f$ is the probability of the backscatter energy fluxes. It is found that for the $\mathcal {G}$-filter $f=0.17$, while for the $\mathcal {S}$-filter $f=0.42$, i.e. the backscatter fraction in $\varPi _\mathcal {S}$ is almost 2.5 times larger than that in $\varPi _\mathcal {G}$.

Figure 8.

The p.d.f. of $\varepsilon \equiv \varPi /(\varPi ^+ - \varPi ^-)$ based on the $\mathcal {S}$-filter and $\mathcal {G}$-filter.

This relatively large backscatter fraction in $\mathcal {S}$-filtering, which leads to a substantial cancellation among the forward–backward energy transfers, will be discussed in the following sections. However, it is worth comparing this observation with the previously reported results. In a turbulent channel flow, the SFEF based on the $\mathcal {G}$- and $\mathcal {S}$-filters results in backscattering in approximately $30\,\%$ and $50\,\%$ of the grid points, respectively (Piomelli et al. Reference Piomelli, Cabot, Moin and Lee1990). Comparing this reported result for an anisotropic turbulence (channel flow) with the observed results in an isotropic turbulence (current research) prompts the conjuncture that a high backscatter fraction could be an intrinsic property of the $\mathcal {S}$-filtering.

3.5. Cancellation effect

In this section, the forward–backward energy flux cancellation is further investigated in terms of the relative eigenframe alignment. To this end, the normalized energy flux (3.4) is decomposed into different forward–backward partial energy fluxes as follows (Higgins et al. Reference Higgins, Parlange and Meneveau2004):

(3.6) \begin{align} \varPi^* &= \frac{1}{6} \{\underbrace{(\boldsymbol{v}^{\boldsymbol{s}}_{\alpha} \boldsymbol{\cdot} \boldsymbol{v}^{\boldsymbol \tau}_{\alpha})^2 (\beta^*_s - \alpha^*_{s}) ( \alpha^*_{\tau} - \beta^*_{\tau} )}_{\varPi^*_1<0} + \underbrace{(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma} \boldsymbol{\cdot} \boldsymbol{v}^ {\boldsymbol{\tau}}_{\gamma})^2 (\beta^*_{s} -\gamma^*_s) (\gamma^*_{\tau} - \beta^*_{\tau})}_{\varPi^*_2<0} \nonumber\\ &\quad+ \underbrace{(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma} \boldsymbol{\cdot} \boldsymbol{v}^{\boldsymbol{\tau}}_{\alpha})^2 (\beta^*_{s} -\gamma^*_s) (\alpha^*_{\tau} - \beta^*_{\tau})}_{\varPi^*_3>0} + \underbrace{(\boldsymbol{v}^{\boldsymbol{s}}_{\alpha} \boldsymbol{\cdot} \boldsymbol{v}^{\boldsymbol{\tau}}_{\gamma})^2 (\beta^*_{s} -\alpha^*_s) (\gamma^*_{\tau} - \beta^*_{\tau})}_{\varPi^*_4>0} \nonumber\\ &\quad + \underbrace{ 3\beta^*_{s}\beta^*_{\tau}}_{\varPi^*_5} \}, \end{align}

in which the eigenvalues are sorted as $\alpha ^* \geq \beta ^* \geq \gamma ^*$. Note that unlike § 2.2 where the eigenframes are parameterized based on the absolute values of eigenvalues, here, the eigenvalues are sorted based on their values. According to this ordering of eigenvalues, the partial energy fluxes $\varPi ^*_1$ to $\varPi ^*_4$ are sign-definite and can be classified into forward ($\varPi ^*_3$ and $\varPi ^*_4$) and backward ($\varPi ^*_1$ and $\varPi ^*_2$ ) energy fluxes. Although the last partial flux, i.e. $\varPi ^*_5 =3\beta ^*_{s}\beta ^*_{\tau }$, is not sign-definite and can be either positive or negative, it is observed that on average, $\langle \varPi ^*_5\rangle <0$, and it contributes to the backward energy flux.

Figure 9 shows the averaged contribution of all partial fluxes, $\langle \varPi ^*_1\rangle$ to $\langle \varPi ^*_5\rangle$, for both filtering formulations. As explained above, it can be seen that only $\langle \varPi ^*_3\rangle$ and $\langle \varPi ^*_4\rangle$ contribute to the forward energy transfer, while the rest of the partial fluxes are responsible for the backward energy transfer. However, the most striking point in figure 9 is the impact of the filter kernel on the partial energy fluxes. It can be seen that the mean partial fluxes $\langle \varPi ^*_1\rangle$, $\langle \varPi ^*_4\rangle$ and $\langle \varPi ^*_5\rangle$ do not show a significant dependence on the type of filter kernel. However, considering $\langle \varPi ^*_2 \rangle$ and $\langle \varPi ^*_3 \rangle$, this figure clearly shows that, compared with the $\mathcal {G}$-filtering, the $\mathcal {S}$-filtering has a hampering effect on the forward partial energy flux $\langle \varPi ^*_{\mathcal {S}3}\rangle <\langle \varPi ^*_{\mathcal {G}3}\rangle$, while augmenting the backward partial energy flux $|\langle \varPi ^*_{\mathcal {G}2}\rangle |<|\langle \varPi ^*_{\mathcal {S}2}\rangle |$ (note that, for example, $\varPi ^*_{\mathcal {G}2}$ denotes the partial energy flux $\varPi ^*_{2}$ based on the $\mathcal {G}$-filtering). Thus, the larger backscatter fraction in $\varPi _\mathcal {S}$ compared with $\varPi _\mathcal {G}$, observed in § 3.4, originates from different weights of the backward–forward partial energy fluxes, i.e. $\langle \varPi ^*_2\rangle$ and $\langle \varPi ^*_3\rangle$.

Figure 9.

Contribution of different partial energy fluxes, i.e. $\langle \varPi ^*_1\rangle$–$\langle \varPi ^*_5\rangle$ to the overall energy transfer based on the $\mathcal {S}$-filter and $\mathcal {G}$-filter.

The impact of different filter kernels on these forward–backward partial energy fluxes is further elaborated by considering their relations:

(3.7) \begin{equation} \left.\begin{gathered} \varPi^*_2 = \tfrac{1}{6} \underbrace{(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma} \boldsymbol{\cdot} \boldsymbol{v}^{\boldsymbol{\tau}}_{\gamma})^2}_{{term~1}} \underbrace{(\beta^*_{s} -\gamma^*_{s})}_{{term~2}} \underbrace{(\gamma^*_{\tau} - \beta^*_{\tau})}_{{term~3}}, \\ \varPi^*_3 = \tfrac{1}{6}\underbrace{ (\boldsymbol{v}^{\boldsymbol{s}}_{\gamma} \boldsymbol{\cdot} \boldsymbol{v}^{\boldsymbol{\tau}}_{\alpha})^2}_{{term~1}} \underbrace{(\beta^*_{s} -\gamma^*_s)}_{{term~2}} \underbrace{(\alpha^*_{\tau} - \beta^*_{\tau})}_{{term~3}}.\end{gathered}\right\} \end{equation}

It can be seen that these partial energy fluxes are functions of eigendirection alignments, i.e. term 1, and eigenstructures of FSR and SFS tensors, i.e. term 2 and term 3, respectively. Considering term 2 in these partial fluxes, it can be easily seen that this term is not affected by the filter type: the SFS eigenstructure is exclusively a function of the strain-rate parameter $s^*$ (see (3.3)), and $P(s^*)$ is trivially affected by the filter type (see figure 6a). Therefore, the filter kernel can effectively modify $\langle \varPi ^*_2\rangle$ and $\langle \varPi ^*_3\rangle$ through the other two parameters, namely, term 1 and term 3.

Figure 10 shows the impact of the filter type on term 1 and term 3. The p.d.f.s of term 3 in $\varPi _2^*$ and $\varPi _3^*$, viz. $(\gamma ^*_{\tau } - \beta ^*_{\tau })$ and $(\alpha ^*_{\tau } - \beta ^*_{\tau })$, are presented in figures 10(a) and 10(b). The statistical distributions of these two parameters clearly show that compared with $\mathcal {G}$-filtering, $(\gamma ^*_{\tau } - \beta ^*_{\tau })$ has a statistically larger value in $\mathcal {S}$-filtering, while $(\alpha ^*_{\tau } - \beta ^*_{\tau })$ shows the opposite statistical behaviour. Therefore, through the SFS eigenstructure, i.e. term 3, the $\mathcal {S}$-filtering augments the backward energy flux $\langle \varPi ^*_2\rangle$ while having a hampering effect on the forward energy flux $\langle \varPi ^*_3\rangle$.

Figure 10.

Statistical distributions of the influential terms for two partial energy fluxes $\varPi ^*_2$ and $\varPi ^*_3$ based on the $\mathcal {G}$-filtering and $\mathcal {S}$-filtering. p.d.f.s of (a) the eigenvalue difference $(\gamma ^*_{\tau } - \beta ^*_{\tau })$, (b) the eigenvalue difference $(\alpha ^*_{\tau } - \beta ^*_{\tau })$, (c) the alignment between contractive–contractive eigenvectors $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma })$ and (d) the alignment between contractive–extensive eigenvectors $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$.

Figures 10(c) and 10(d) present the impact of the filter kernel on the relevant eigenframe alignments in $\varPi ^*_2$ and $\varPi ^*_3$, i.e. term 1. In figure 10(c), it can be seen that the alignment between the most contractive eigenvectors, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma })^2$, shows an almost identical statistical distribution in both $\mathcal {S}$- and $\mathcal {G}$-filtering. Therefore, the augmentation effect of the $\mathcal {S}$-filter on this backscatter partial flux, i.e. $|\langle \varPi ^*_{\mathcal {G}2}\rangle |<|\langle \varPi ^*_{\mathcal {S}2}\rangle |$, effectively originated from the increasing effect of this filter kernel on the eigenvalue difference $(\gamma ^*_{\tau } - \beta ^*_{\tau })$.

Figure 10(d) shows the alignment between contractive–extensive eigenvectors, $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$, based on the $\mathcal {S}$- and $\mathcal {G}$-filtering. Unlike eigendirection alignment $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma })^2$, it can be observed that the filter kernel has a non-trivial impact on the alignment of contractive–extensive eigenvectors, $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$. This figure shows that in $\mathcal {S}$-filtering, the most probable angle between the contractive–extensive eigenvectors is close zero, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })\approx 0$, while the statistical distribution of this angle in $\mathcal {G}$-filtering shows a pronounced peak around $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol{\cdot} \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2\approx 0.6$. Therefore, in addition to the larger eigenvalue difference $(\alpha ^*_{\tau } - \beta ^*_{\tau })$, $\mathcal {G}$-filtering can produce a more pronounced alignment between the contractive–extensive eigenvectors, resulting in a stronger partial forward flux compared with the $\mathcal {S}$-filtering, i.e. $\langle \varPi ^*_{\mathcal {S}3}\rangle < \langle \varPi ^*_{\mathcal {G}3}\rangle$. This impact of the filter type on the eigenframe alignment is further investigated in the next section.

3.6. Alignments of the contractive–extensive eigenvectors

In analysing the partial energy flux $\varPi ^*_3$, it is observed that the most probable value of $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$ in $\mathcal {G}$-filtering is considerably larger than that in $\mathcal {S}$-filtering. Here, we analyse the most preferential alignment between $\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$ in terms of the most probable state of Euler angle–axis parameters, i.e. $(\theta,\phi,\chi )$. However, there is an obstacle to analysing this eigenvector alignment based on the triplet angles $(\theta,\phi,\chi )$: the eigenvectors in the Euler angle–axis representation are parameterized based on the $|\alpha | \geq |\beta | \geq |\gamma |$ ordering, while eigenvectors in the partial energy flux $\langle \varPi ^*_3\rangle$, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }\boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$, are ordered as $\alpha \geq \beta \geq \gamma$. Therefore, it is not possible to directly analyse the behaviour of $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }\boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$ based on the behaviour of $(\boldsymbol{v}^{\boldsymbol{s}}_{|\gamma |}\boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{|\alpha |})$ (in the following discussions, to distinguish between different eigenvector sortings, $\boldsymbol{v}^{\boldsymbol{s}}_{|\alpha |}$ is used, for example, for the eigenvector based on the absolute eigenvalue sorting, while $\boldsymbol{v}^{\boldsymbol{s}}_{\alpha }$ is used for the eigenvector sorting based on the eigenvalue itself).

Despite this discrepancy, a statistical relation between two different eigenvector orderings can be established as follows. First, considering $\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }$, it is observed in § 3.3 that the FSR eigenstructure in both $\mathcal {S}$-filtering and $\mathcal {G}$-filtering is $\alpha _s : \beta _s : \gamma _s = 1 : 1 : -2$, which implies that the most contractive eigenvector, i.e. $\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }$, has the largest absolute eigenvalue. Therefore, in terms of the Euler rotation angle, instead of $\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }$, the eigenvector associated with the largest absolute eigenvalue, i.e. $\boldsymbol{v}^{\boldsymbol{s}}_{|\alpha |}$, has to be considered. The same line of reasoning applies to the most extensive eigenvector of the SFS, based on the $\mathcal {G}$-filter: the eigenstructure $\alpha _\tau : \beta _\tau : \gamma _\tau = 2 : -1 : -1$ implies that the most extensive eigenvector of the SFS, i.e. $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$, has the largest absolute eigenvalue. Therefore, for the $\mathcal {G}$-filtering and in terms of the Euler angle–axis representation, in which eigenvalues are ordered based on their absolute magnitudes, the contractive–extensive eigenvector alignment is statistically equivalent to the alignment between eigenvectors with the largest absolute eigenvalues, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }) \equiv (\boldsymbol{v}^{\boldsymbol{s}}_{|\alpha |} \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{|\alpha |})$ .

However, compared with the $\mathcal {G}$-filtering, the situation in $\mathcal {S}$-filtering is more delicate. The eigenstructure $\alpha _\tau : \beta _\tau : \gamma _\tau \approx 1.7 : 0 : -1.7$ in $\mathcal {S}$-filtering implies that both the extensive and contractive eigenvectors of SFS, namely $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\gamma }$, have the largest absolute eigenvalue with equal probability. Therefore, in $\mathcal {S}$-filtering, the contractive–extensive eigenvectors alignment in terms of the absolute eigenvalue ordering is statistically equivalent to the average of $(\boldsymbol{v}^{\boldsymbol{s}}_{|\alpha |} \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{|\alpha |})$ and $(\boldsymbol {v}^{\boldsymbol{s}}_{|\alpha |} \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{|\gamma |})$ with equal probability. Therefore, in the following analysis, $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$ is estimated based on its aforementioned statistical equivalent.

To this end, $(\chi,\theta )$ is considered as their most probable values, which allows one to express $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })$ as an exclusive function of the longitude angle $\phi$. In this framework, figure 11 presents an analysis of $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ based on the statistics of the longitude angle $P(\phi )$.

Figure 11.

Distribution of $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ as a function of the longitude angle $\phi$ against the statistical distribution of $\phi$, conditioned at (a) $\theta =65^{\circ }$ and $\chi =90^{\circ }$ for $\mathcal {G}$-filter, and (b) $\theta =80^{\circ }$ and $\chi =90^{\circ }$ for $\mathcal {S}$-filter.

Figure 11(a) shows $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ and $P(\phi )$ based on the $\mathcal {G}$-filtering, where the most probable rotation and colatitude angles are $(\chi,\theta )\approx (90^\circ,65^\circ )$. Overall, it can be seen that both functions, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ and $P(\phi )$, exhibit a Gaussian-like distribution with respect to $\phi$. It can be clearly observed that the maximum of $P(\phi )$ coincides with a well-pronounced alignment between $\boldsymbol{v}^{\boldsymbol{s}}_{\gamma }$ and $\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha }$: at the most probable longitude angle, $\phi \approx 17^\circ$, it can be seen that $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2\approx 0.6$, which is well matched with the peak observed for $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ in figure 10(d).

Figure 11(b) provides the same analysis for $\mathcal {S}$-filtering, in which the most probable state of the rotation and colatitude angles are $(\chi,\theta )\approx (90^\circ,80^\circ )$. In this case, both $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ and $P(\phi )$ exhibit a bimodal behaviour: the former shows a sinusoidal pattern, while the latter exhibits a symmetrical distribution with respect to $\phi$. Similar to the $\mathcal {G}$-filtering, it can be seen that the extrema of $P(\phi )$ and $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ almost coincide: at the most probable state $\phi \approx -45^\circ$, there is a relatively good alignment between these two eigenvectors, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2\approx 0.42$, while at the other most probable state $\phi \approx +45^\circ$, these two eigenvectors are almost perpendicular, $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot }\boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2\approx 0$. The combined effect of this bimodal distribution results in a smaller value for $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2$ in $\mathcal {S}$-filtering compared with that in $\mathcal {G}$-filtering.

Therefore, to recapitulate, the symmetrical distribution of the rotation angle in $\mathcal {S}$-filtering leads to a lower alignment between the contractive–extensive eigenvectors compared with that in $\mathcal {G}$-filtering, i.e. $(\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2|_{\mathcal {S}} < (\boldsymbol{v}^{\boldsymbol{s}}_{\gamma } \boldsymbol {\cdot } \boldsymbol {v}^{\boldsymbol {\tau }}_{\alpha })^2|_{\mathcal {G}}$. As a result of this lower eigenframe alignment, $\langle \varPi ^*_3\rangle _\mathcal {S} < \langle \varPi ^*_3\rangle _\mathcal {G}$, which in turn results in larger forward–backward energy transfer cancellation in $\mathcal {S}$-filtering compared with that in the $\mathcal {G}$-filtering.