5.2.1 The vector plots

The dynamics of the filtered VGT for a selected $P$ can be visualized by plotting the fields of the vector $\boldsymbol{V}$ in the $Q$–$R$ plane and using the length of the vector to show the magnitude, which has been used in previous studies (van der Bos et al. Reference van der Bos, Tao, Meneveau and Katz2002; Chu & Lu Reference Chu and Lu2013). The conditional mean trajectories of the vector $\boldsymbol{V}$ are shown in figure 8 for different dilatations. The trajectories show significant deviations for different dilatations. The dynamics of the local flow topology in the incompressible regions describe clockwise cycles around the origin in almost closed trajectories, which is consistent with previous studies (Chong et al. Reference Chong, Soria, Perry, Chacin, Cantwell and Na1998; Martin et al. Reference Martin, Ooi, Chong and Soria1998; Lozano-Durán et al. Reference Lozano-Durán, Holzner and Jiménez2015, Reference Lozano-Durán, Holzner and Jiménez2016), while the conditional mean trajectories of $\boldsymbol{V}$ spiral outwards in the locally compressed regions and inwards in the locally expanded regions. As the level of compression is increased, the outward spiral of $\boldsymbol{V}$ becomes more obvious. By comparing figure 8(a) with figure 8(b), the results based on $P/Q_{0}^{1/2}=0.3$ and $P/Q_{0}^{1/2}=1.0$ are qualitatively consistent with each other, which indicates that the threshold value $P/Q_{0}^{1/2}=0.3$ is enough to reflect the characteristics of the dynamics of the filtered VGT in the locally compressed regions. The magnitudes of $\boldsymbol{V}$ increase with increasing distance from the origin. The directions of $\boldsymbol{V}$ along the Vieillefosse tails are strongly associated with the dilatations. The vectors $\boldsymbol{V}$ point towards the origin in the locally expanded regions and point away from the origin in the locally compressed regions along the Vieillefosse tail, while the vector magnitude remains relatively small along the Vieillefosse tail in the incompressible regions.

Figure 8.

Conditional mean trajectories of $\boldsymbol{V}$ by use of the vectors for demonstration for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$, (c) $P/Q_{0}^{1/2}=1.0$ and (d) $P/Q_{0}^{1/2}=-0.3$. The scales for the vector magnitude are indicated in the upper right corner in each panel. The red lines indicate the streamlines in the $Q$–$R$ plane. Only values where ${\mathcal{P}}(R,Q)>10^{-4}$ are plotted.

Figure 9.

Conditional mean trajectories of the interaction term among the invariants $\boldsymbol{V}_{S}$ by use of the vectors for demonstration for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The scales for the vector magnitude are indicated in the upper right corner in each panel. Only values where ${\mathcal{P}}(R,Q)>10^{-4}$ are plotted.

The components of the evolution of invariants are analysed to understand the results presented above. The conditional mean trajectories of the vector $\boldsymbol{V}_{S}$ in the $Q$–$R$ plane are shown in figure 9. It is seen that $Q$ tends to increase in the enstrophy-production-dominated regions (i.e. $R<0$) and to decrease in the dissipation-production-dominated regions (i.e. $R>0$). Such properties are attributed to the vortex stretching and the self-amplification of the strain-rate tensor (Cantwell Reference Cantwell1992; Chu & Lu Reference Chu and Lu2013). It is seen from (3.13) and (3.14) that, compared with the incompressible regions, $PQ$ and $PR$ are involved in the interaction term among the invariants, which tends to amplify the magnitude of the rate of change of the invariants in the locally compressed regions and to reduce the magnitude in the locally expanded regions.

The conditional mean trajectories of the vector $\boldsymbol{V}_{H}$ in the $Q$–$R$ plane are shown in figure 10. On average, the enstrophy production rate tends to increase in the enstrophy-production-dominated regions and the dissipation production rate tends to increase in the dissipation-production-dominated regions due to the pressure effect, which reveals that the pressure effect term leads to the amplification of $|R|$. Compared with the incompressible regions, $Q$ tends to decrease in the locally compressed regions and to increase in the locally expanded regions on average. Moreover, the pressure effect leads to the local topology of fluid particles changing from the focal regions to the non-focal regions in the locally compressed regions, i.e. from SFC to UFC to UN/S/S or from SFS to SN/S/S, while the pressure effect leads to the local topology of fluid particles changing from the non-focal regions to the focal regions in the locally expanded regions, i.e. from SN/S/S to SFS to UFS or from UN/S/S to UFC.

Figure 10.

Conditional mean trajectories of the pressure effect term $\boldsymbol{V}_{H}$ by use of the vectors for demonstration for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The scales for the vector magnitude are indicated in the upper right corner in each panel. Only values where ${\mathcal{P}}(R,Q)>10^{-4}$ are plotted.

Figure 11 shows the conditional mean trajectories of the vector $\boldsymbol{V}_{V}$. It is found that the viscous effect tends to attract the trajectories to the origin, consistent with previous findings in isotropic turbulence (van der Bos et al. Reference van der Bos, Tao, Meneveau and Katz2002; Chevillard et al. Reference Chevillard, Meneveau, Biferale and Toschi2008; Danish & Meneveau Reference Danish and Meneveau2018) and wall-bounded flows (Chu & Lu Reference Chu and Lu2013). There is no significant dependence of the results on compressibility in the inertial range. It is noted that the magnitude of $\boldsymbol{V}_{V}$ is smaller than other terms in the inertial range, and the viscous effect appears to be negligible at large scales.

Figure 11.

Conditional mean trajectories of the viscous effect term $\boldsymbol{V}_{V}$ by use of the vectors for demonstration for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The scales for the vector magnitude are indicated in the upper right corner in each panel. Only values where ${\mathcal{P}}(R,Q)>10^{-4}$ are plotted.

The conditional mean trajectories of the subgrid effect vectors $\boldsymbol{V}_{G}$ in the $Q$–$R$ plane are shown in figure 12. As expected, the magnitude of $\boldsymbol{V}_{G}$ is larger than $\boldsymbol{V}_{V}$ in the inertial range. Hence, the subgrid effect exerts a non-negligible influence on the filtered velocity gradient dynamics. Significant differences in the direction of the vectors are observed for different dilatations. In the incompressible regions, the subgrid effect term leads to an increase of $R$ and a decrease of $Q$ in the enstrophy-production-dominated regions, while $\boldsymbol{V}_{G}$ is directed to the origin in the fourth quadrant and $\boldsymbol{V}_{G}$ has a small magnitude and ill-defined direction in the first quadrant. In the locally compressed regions, the characteristic of $\boldsymbol{V}_{G}$ is similar to $\boldsymbol{V}_{V}$. That is, $\boldsymbol{V}_{G}$ is directed to the origin in the entire $Q$–$R$ plane and the magnitudes of $\boldsymbol{V}_{G}$ are decreasing with decreasing distance to the origin. In the locally expanded regions, the direction of $\boldsymbol{V}_{G}$ is more irregular and the small magnitude and ill-defined directions of $\boldsymbol{V}_{G}$ are also found in the second quadrant. Moreover, it is noted that the contributions of the subgrid effect appear to be strong along the Vieillefosse tails in the fourth quadrant, where the subgrid effect is similar to the viscous effect, and attract the trajectories to the origin, which indicates that the subgrid effect is mostly dissipative in the inertial range. For the particles along the Vieillefosse tails, the energy transfer mechanism from the inertial scale to the subgrid scale is analogous to the diffusion mechanism, while the ill-defined directions of $\boldsymbol{V}_{G}$ in the incompressible regions and the locally expanded regions are not dissipative, resulting in a difficult mechanism in modelling the SGS tensors.

Figure 12.

Conditional mean trajectories of the subgrid effect term $\boldsymbol{V}_{G}$ by use of the vectors for demonstration for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The scales for the vector magnitude are indicated in the upper right corner in each panel. Only values where ${\mathcal{P}}(R,Q)>10^{-4}$ are plotted.

5.2.2 The flux measurements

To elucidate which terms are responsible for the change of the dynamics of the filtered VGT in statistics with dilatations, the study of the net effects of each term as a function of $P$ is required. The total probability flux calculated across a closed boundary placed in the $Q$–$R$ plane is presented to quantify the effects of each term (Danish & Meneveau Reference Danish and Meneveau2018). As shown in figure 13, a circle centred at the origin is chosen to calculate the probability flux. The total flux crossing the circle can be obtained by

(5.4)$$\begin{eqnarray}{\mathcal{J}}=\oint _{c}\,({\mathcal{P}}\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{n})\,\text{d}S,\end{eqnarray}$$

where $c$ is the closed circle, $\boldsymbol{n}$ is the normal vector outward from the surface and ${\mathcal{J}}$ represents the total flux of probability density. Using the divergence theorem, we have

(5.5)$$\begin{eqnarray}{\mathcal{J}}=\iint \unicode[STIX]{x1D735}\boldsymbol{\cdot }({\mathcal{P}}\boldsymbol{V})\,\text{d}R\,\text{d}Q={\mathcal{J}}_{S}+{\mathcal{J}}_{H}+{\mathcal{J}}_{V}+{\mathcal{J}}_{G},\end{eqnarray}$$

where ${\mathcal{J}}_{S}$, ${\mathcal{J}}_{H}$, ${\mathcal{J}}_{V}$ and ${\mathcal{J}}_{G}$ are fluxes due to the divergence of ${\mathcal{P}}\boldsymbol{V}_{S}$, ${\mathcal{P}}\boldsymbol{V}_{H}$, ${\mathcal{P}}\boldsymbol{V}_{V}$ and ${\mathcal{P}}\boldsymbol{V}_{G}$, respectively. If the flux is positive, the dynamics leads to a net outward trend in the $Q$–$R$ plane, which is denoted as a ‘source-like’ behaviour, while the dynamics leads to a net inward trend and corresponds to a ‘sink-like’ behaviour if the flux is negative. The statistically stationary condition implies that the total flux must be zero for a closed surface in the $P$–$Q$–$R$ space. The total flux is not zero in the $Q$–$R$ plane for a given $P$, while the fluxes in (5.5) can still evaluate the relative importance of the pressure effect, the viscous effect, the subgrid effect and the interaction between the invariants.

Figure 13.

Representative circles with radii equal to 0.75 and 1.25 centred at the origin of the $Q$–$R$ plane.

The fluxes are calculated numerically within the circles in the $Q$–$R$ plane for a given $P$. It is noted that the results depend strongly on the radius of the circle, since ${\mathcal{P}}$ decreases rapidly with increasing radius. In order to minimize the effect of the radius of the circle, the fluxes are normalized as follows:

(5.6)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\mathcal{J}}_{S}^{\ast }=\frac{{\mathcal{J}}_{S}}{\sqrt{{\mathcal{J}}_{S}^{2}+{\mathcal{J}}_{H}^{2}+{\mathcal{J}}_{V}^{2}+{\mathcal{J}}_{G}^{2}}},\quad {\mathcal{J}}_{H}^{\ast }=\frac{{\mathcal{J}}_{H}}{\sqrt{{\mathcal{J}}_{S}^{2}+{\mathcal{J}}_{H}^{2}+{\mathcal{J}}_{V}^{2}+{\mathcal{J}}_{G}^{2}}},\\ \displaystyle {\mathcal{J}}_{V}^{\ast }=\frac{{\mathcal{J}}_{V}}{\sqrt{{\mathcal{J}}_{S}^{2}+{\mathcal{J}}_{H}^{2}+{\mathcal{J}}_{V}^{2}+{\mathcal{J}}_{G}^{2}}},\quad {\mathcal{J}}_{G}^{\ast }=\frac{{\mathcal{J}}_{G}}{\sqrt{{\mathcal{J}}_{S}^{2}+{\mathcal{J}}_{H}^{2}+{\mathcal{J}}_{V}^{2}+{\mathcal{J}}_{G}^{2}}}.\end{array}\right\}\end{eqnarray}$$

The values of the fluxes lie between $-1$ and $+1$ as a result of this normalization.

The normalized fluxes as a function of dilatation are presented in figure 14. It is found that the results are quite independent of radius in the range of 0.75 to 1.25, and the change of the fluxes mainly occurs at $-0.3<P/Q_{0}^{1/2}<0.3$. Therefore, the threshold values $P/Q_{0}^{1/2}=\pm 0.3$ are reasonable to distinguish the expanded and compressed regions for the dynamics of the velocity gradients. The flux for the interaction term among the invariants is shown with the solid lines in figure 14. The results indicate that the compressibility effect is directly related to ${\mathcal{J}}_{S}$, which changes from negative to positive values on increasing the value of $P$. Substituting $\boldsymbol{V}_{S}=\langle RS,QS\rangle _{R,Q}=\langle PR,PQ-3R\rangle _{R,Q}$ into (5.5) we can obtain

(5.7)$$\begin{eqnarray}{\mathcal{J}}_{S}=2P+\iint (\boldsymbol{V}_{S}\boldsymbol{\cdot }\unicode[STIX]{x1D735}){\mathcal{P}}\,\text{d}R\,\text{d}Q,\end{eqnarray}$$

which indicates that the flux for the interaction term is measured by $P$.

The flux for the pressure effect term is shown with the dashed lines in figure 14. It is found that ${\mathcal{J}}_{H}$ increases with the increase of $P$ and remains relatively unchanged in the locally compressed regions. The flux for the pressure term is positive and acts as a source, which is related to the fact that $\boldsymbol{V}_{H}$ points away from the origin, leading to an amplification of $|R|$. Moreover, it is identified that the flux for the viscous term is negative and acts as a sink. The flux for the viscous term is relatively small and independent of the dilatation, indicating that the viscous effect is weak in the inertial range and insensitive to the dilatation.

Figure 14.

Normalized fluxes as a function of the dilatation $P$, with radius (a) $r=0.75$ and (b) $r=1.25$. The solid, dashed, dash-dotted and dotted lines represent the flux for the interaction term among the invariants (${\mathcal{J}}_{S}^{\ast }$), the pressure effect term (${\mathcal{J}}_{H}^{\ast }$), the viscous effect term (${\mathcal{J}}_{V}^{\ast }$) and the subgrid effect term (${\mathcal{J}}_{G}^{\ast }$), respectively.

The flux for the subgrid effect term is shown in figure 14 with the dotted lines. It is seen that ${\mathcal{J}}_{G}$ decreases rapidly with the increase of $P$ in the locally expanded regions and remains a constant value in the locally compressed regions. Compared with the other terms, the flux for the subgrid effect term changes most with the dilatation. The flux for the subgrid effect term is negative in the locally compressed regions, which is related to the fact that the characteristic of $\boldsymbol{V}_{G}$ is similar to $\boldsymbol{V}_{V}$, leading to a ‘sink-like’ behaviour; while the flux is positive and acts as a source in the locally expanded regions when $P/Q_{0}^{1/2}<-0.18$ due to the ill-defined directions of $\boldsymbol{V}_{G}$ in the upper part of the $Q$–$R$ plane and mostly outward-pointing. Above all, the change of the flux for the subgrid effect term with the dilatation is offset by the flux for the pressure effect term and the flux for the interaction term among the invariants to establish stationary statistics. The fluxes are directly related to the compressibility of the flow especially in the locally expanded regions.

5.3.1 The effect of compressibility on non-normality

The Schur decomposition of the filtered VGT, $\unicode[STIX]{x1D63C}=\unicode[STIX]{x1D63D}+\unicode[STIX]{x1D63E}$, is used to develop better insights into compressible turbulence in this section. Firstly, the relative importance of the normal and the non-normal tensors is analysed. It is legitimate to approximate the behaviour of the VGT with eigenvalue-based formulations if $\unicode[STIX]{x1D63E}$ is small while the non-normality should be considered in the dynamics of the VGT if $\unicode[STIX]{x1D63E}$ is important. A standardized difference is used to evaluate the importance of non-normal effects in enstrophy and dissipation (Keylock Reference Keylock2018):

(5.8a,b)$$\begin{eqnarray}\unicode[STIX]{x1D705}_{B,C}^{S}=\frac{\Vert \unicode[STIX]{x1D64E}^{B}\Vert -\Vert \unicode[STIX]{x1D64E}^{C}\Vert }{\Vert \unicode[STIX]{x1D64E}^{B}\Vert +\Vert \unicode[STIX]{x1D64E}^{C}\Vert },\quad \unicode[STIX]{x1D705}_{B,C}^{W}=\frac{\Vert \unicode[STIX]{x1D652}^{B}\Vert -\Vert \unicode[STIX]{x1D652}^{C}\Vert }{\Vert \unicode[STIX]{x1D652}^{B}\Vert +\Vert \unicode[STIX]{x1D652}^{C}\Vert }.\end{eqnarray}$$

The value of $\unicode[STIX]{x1D705}_{B,C}^{S}$ and $\unicode[STIX]{x1D705}_{B,C}^{W}$ lies between $-1$ and $+1$, and the normal effect becomes more important as the value approaches 1. Figure 15 shows $\unicode[STIX]{x1D705}_{B,C}^{S}$ and $\unicode[STIX]{x1D705}_{B,C}^{W}$ conditioned on the dilatation $P$. It is seen that the importance of the non-normal effects is related to the compressibility of the flow. As shown in figure 15(a), an increase of the normal effect and a decrease of the non-normal effect of the strain-rate tensor with increasing $P$ are found. The $\unicode[STIX]{x1D705}_{B,C}^{S}$ value is negative in the locally expanded regions and weak compressed regions, and is positive in the strong compressed regions with $P/Q_{0}^{1/2}>0.22$. The value of $\unicode[STIX]{x1D705}_{B,C}^{W}$ increases with $P$ while the change of $\unicode[STIX]{x1D705}_{B,C}^{W}$ is slighter than $\unicode[STIX]{x1D705}_{B,C}^{S}$. The $\unicode[STIX]{x1D705}_{B,C}^{W}$ value is negative for all values of $P$ and distributes around $\unicode[STIX]{x1D705}_{B,C}^{W}=-0.44$. It is noted that $\Vert \unicode[STIX]{x1D652}^{B}\Vert =0$ and $\unicode[STIX]{x1D705}_{B,C}^{W}=-1$ are calculated by definition in the non-focal regions, resulting in a relatively small value of $\unicode[STIX]{x1D705}_{B,C}^{W}$. Hence the dynamics of the rotation-rate tensors is dictated only by the non-normal parts in the non-focal regions.

Figure 15.

Plots of (a) $\unicode[STIX]{x1D705}_{B,C}^{S}$ and (b) $\unicode[STIX]{x1D705}_{B,C}^{W}$ conditioned on the dilatation $P$.

Figure 16.

Conditional mean density of $\unicode[STIX]{x1D705}_{B,C}^{S}$ in the $Q$–$R$ plane for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The contour levels of the solid lines are 0.1, 0.2 and 0.3 and the contour levels of the dashed lines are $-0.1$, $-0.2$ and $-0.3$, respectively.

The conditional mean density of $\unicode[STIX]{x1D705}_{B,C}^{S}$, which is defined as ${\mathcal{P}}(R,Q)\langle \unicode[STIX]{x1D705}_{B,C}^{S}\rangle _{R,Q}$, for different dilatations in the $Q$–$R$ plane is shown in figure 16. The results indicate that the non-normal effect is associated with particular values of the invariants. It is noted that the second invariant $Q$ is important in the distribution of $\unicode[STIX]{x1D705}_{B,C}^{S}$. In the enstrophy-dominated regions where $Q>0$, the conditional mean of $\unicode[STIX]{x1D705}_{B,C}^{S}$ is negative, which indicates that the non-normal effect is strong in these regions, while the conditional mean of $\unicode[STIX]{x1D705}_{B,C}^{S}$ is positive in the dissipation-dominated regions where $Q<0$, which indicates that the normal effect is more important than the non-normal effect. It is found that the cover areas of the contour lines change with the dilatation, i.e. the cover areas of $\unicode[STIX]{x1D705}_{B,C}^{S}=0.3$ increase with the increase of $P$ and the cover areas of $\unicode[STIX]{x1D705}_{B,C}^{S}=-0.3$ decrease with the increase of $P$, indicating a decrease of the non-normal effect with increasing $P$. Moreover, it is noted that the maximal values of the conditional mean density lie along the Vieillefosse tails in the fourth quadrant, which indicates that the normal effect is important along the Vieillefosse tail. Thus, an eigenvalue-based description of the flow is particularly effective near the Vieillefosse tail, consistent with the fact that the eigenvalue-based restricted Euler equation (Cantwell Reference Cantwell1992) is able to describe the dynamics of the VGT along the Vieillefosse tail. The importance of non-normal effects in the enstrophy-dominated regions suggests that a number of properties of vortical structures, such as their topologies and alignments, are poorly described by the eigenvalues.

Further information on the flow non-normality can be gleaned from their joint behaviours. The j.p.d.f.s of $Q_{B}^{S}$ and $Q_{C}^{S}$ for different dilatations are shown in figure 17. It is found that $Q_{C}^{S}$ is negative and the distribution of $Q_{C}^{S}$ is not associated with the dilatation distinctly, while the distribution of $Q_{B}^{S}$ is skewed towards negative values and is related to the dilatation. The probability of $Q_{B}^{S}$ reaches its maximum value at $Q_{B}^{S}<0$ in the locally compressed regions and at $Q_{B}^{S}>0$ in the locally expanded regions. The results indicate that the compressibility mainly plays a role in the distribution of the normal part of the VGT. Since $\text{tr}(\unicode[STIX]{x1D64E}^{B})=-P$, the normal effect of the VGT is directly related to the compressibility.

Figure 17.

J.p.d.f.s of $Q_{B}^{S}$ and $Q_{C}^{S}$ on a logarithmic scale for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The contour levels of the lines are $-2$, $-1$, 0 and 1, respectively. The dots represent where the j.p.d.f.s reach their maximal values.

The j.p.d.f.s of $R_{B}^{S}$ and $Q_{B}^{S}$ for different dilatations are shown in figure 18. It is seen that the behaviours of the j.p.d.f.s are related to the dilatation. Compared with the j.p.d.f.s of $Q$ and $R$ shown in figure 5, the normal part of the strain-rate tensor is associated with the flow properties such as the teardrop shape around the origin and the Vieillefosse tail in the fourth quadrant. It is seen that the j.p.d.f.s of $R_{B}^{S}$ and $Q_{B}^{S}$ are more symmetrical with respect to $R=0$ in the locally compressed regions and are more skewed towards the null discriminant curve (i.e. $\unicode[STIX]{x1D6E5}=0$) in the locally expanded regions. Furthermore, the j.p.d.f.s have short tails along the $R<0$ part of the null discriminant curves, which is not observed in figure 5. The short tails along the $R<0$ part of the null discriminant curves are related to the normal effect and should be considered in the models for the VGT dynamics.

Figure 18.

J.p.d.f.s of $R_{B}^{S}$ and $Q_{B}^{S}$ on a logarithmic scale for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The contour levels of the lines are $-3$, $-2$, $-1$ and 0, respectively.

5.3.2 Non-normal effects on the SGS energy dissipation

The subgrid effects can be quantified by considering the local SGS energy dissipation, i.e. $-\unicode[STIX]{x1D70F}_{ij}\unicode[STIX]{x1D61A}_{ij}$ (Piomelli et al. Reference Piomelli, Cabot, Moin and Lee1991). The conditional mean density of the SGS energy dissipation in the $Q$–$R$ plane is used to study the possible trends of the SGS energy dissipation upon particular values of $Q$ and $R$, which is defined as

(5.9)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}(R,Q)={\mathcal{P}}(R,Q)\langle -\unicode[STIX]{x1D70F}_{ij}\unicode[STIX]{x1D61A}_{ij}\rangle _{R,Q}.\end{eqnarray}$$

Figure 19 shows the conditional mean of the SGS energy dissipation $\unicode[STIX]{x1D6F7}(R,Q)$ measured from the filtered data. The SGS energy dissipation is normalized by its average in the statistics region. It is seen that the dominated contributions arise from the fourth quadrant along the Vieillefosse tails. The maximal values of $\unicode[STIX]{x1D6F7}(R,Q)$ are located at the Vieillefosse tails rather than the origin, which is caused by the high value of strain rate along the Vieillefosse tails (Chertkov et al. Reference Chertkov, Pumir and Shraiman1999). Moreover, the results show that the dilatation plays an important role in the energy cascade from the inertial scales to subgrid scales. It is noted that in the locally expanded regions shown in figure 19(c), the conditional mean of the SGS energy dissipation around the origin is negative, which indicates a backscattering region around the origin; while in the incompressible regions and locally compressed regions, the conditional mean of the SGS energy dissipation is positive in the entire $Q$–$R$ plane, which indicates that backscatter is not strongly associated with particular values of the invariants.

Figure 19.

Conditional mean of the normalized SGS energy dissipation $\unicode[STIX]{x1D6F7}(R,Q)$ for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The contour levels of the solid lines are 0.1, 0.4, 0.7 and 1.0 and the contour levels of the dashed lines in (c) are $-0.1$, $-0.4$ and $-0.7$, respectively.

In order to investigate the normal effects and the non-normal effects on the SGS energy dissipation in detail, the flow domain is divided into the normal-effect-dominated regions and the non-normal-effect-dominated regions by the sign of $\unicode[STIX]{x1D705}_{B,C}^{S}$, and the SGS energy dissipation related to the two parts are defined as

(5.10)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{+}(R,Q)=\left\{\begin{array}{@{}cc@{}}{\mathcal{P}}(R,Q)\langle -\unicode[STIX]{x1D70F}_{ij}\unicode[STIX]{x1D61A}_{ij}\rangle _{R,Q} & (\unicode[STIX]{x1D705}_{B,C}^{S}\geqslant 0),\\ 0 & (\unicode[STIX]{x1D705}_{B,C}^{S}<0),\end{array}\right.\end{eqnarray}$$

and

(5.11)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{-}(R,Q)=\left\{\begin{array}{@{}cc@{}}0 & (\unicode[STIX]{x1D705}_{B,C}^{S}\geqslant 0),\\ {\mathcal{P}}(R,Q)\langle -\unicode[STIX]{x1D70F}_{ij}\unicode[STIX]{x1D61A}_{ij}\rangle _{R,Q} & (\unicode[STIX]{x1D705}_{B,C}^{S}<0),\end{array}\right.\end{eqnarray}$$

respectively. The sum of $\unicode[STIX]{x1D6F7}^{+}$ and $\unicode[STIX]{x1D6F7}^{-}$ gives the total conditional mean of the SGS energy dissipation in figure 19.

Figure 20.

Conditional mean of the SGS energy dissipation in the normal-effect-dominated regions $\unicode[STIX]{x1D6F7}^{+}(R,Q)$ for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The contour levels of the lines are 0.2, 0.4, 0.6 and 0.8, respectively.

The conditional mean of $\unicode[STIX]{x1D6F7}^{+}$ is shown in figure 20. The teardrop shapes around the origin are observed and the shapes of $\unicode[STIX]{x1D6F7}^{+}$ shown in figure 20 are similar to the j.p.d.f.s of $Q$ and $R$ shown in figure 5, indicating that ${\mathcal{P}}$ dominates the characteristic of the SGS energy dissipation in the normal-effect-dominated regions. Moreover, compared with the results shown in figure 19(c), the backscattering region is not observed in the locally expanded regions, which means that the backscatter is not associated with the invariants in the normal-effect-dominated regions.

Figure 21.

Conditional mean of the SGS energy dissipation in the non-normal-effect-dominated regions $\unicode[STIX]{x1D6F7}^{-}(R,Q)$ for (a) $P=0$, (b) $P/Q_{0}^{1/2}=0.3$ and (c) $P/Q_{0}^{1/2}=-0.3$. The contour levels of the solid lines are 0.1, 0.4, 0.7 and 1.0 and the contour levels of the dashed lines in (c) are $-0.1$, $-0.4$ and $-0.7$, respectively.

The conditional mean of $\unicode[STIX]{x1D6F7}^{-}$ is shown in figure 21. The SGS energy dissipation in the non-normal-effect-dominated regions shows a stronger tendency to be dominated by the particles along the Vieillefosse tails in the fourth quadrant. The sign of the SGS energy dissipation is conditioned on the invariants in the locally expanded regions, which indicates that the forward and backscatter of the dissipation are conditioned on the invariants. Positive contributions of the SGS energy dissipation are dominated by the Vieillefosse tail and negative contributions appear around the origin. Furthermore, by comparing the results in figures 19 and 21, it is found that the characteristics of the SGS energy dissipation conditioned on the invariants are determined in the non-normal-effect-dominated regions and $\unicode[STIX]{x1D705}_{B,C}^{S}$ is considered to be a key quantity in the SGS energy dissipation.