4.1. Shear and rotating motions in viscoelastic turbulence

Figure 1 shows contours of the intensities of shear and rigid-body rotation, $I_S$ and $I_R$ , on two-dimensional planes in the Newtonian case and viscoelastic cases with $\textit{Wi}=2.0$ and 4.6. All figures show regions with $(400\eta )^2$ . These quantities can be used to detect shear layers and vortex tubes in turbulence (Nagata et al. Reference Nagata, Watanabe, Nagata and da Silva2020). Circular patterns of large $I_R$ are the cross-sections of vortex tubes, while their long patterns are related to the axes of in-plane vortex tubes. On the other hand, large $I_S$ is often concentrated in thin layers, which are the shear layers due to turbulent fluctuations. A comparison between the Newtonian and viscoelastic cases shows that the geometry of these structures changes as $\textit{Wi}$ increases. Here, the shear layers in the viscoelastic simulations are longer than in the Newtonian case. Additionally, the vortex tubes appear less frequently in the viscoelastic case than in the Newtonian case. Figure 2 displays the isosurfaces of $I_S$ and $I_R$ in a region $400\eta \times 400\eta \times 150\eta$ . As also shown in the two-dimensional visualisation, the aspect ratio of shear layers becomes large as $\textit{Wi}$ increases. The instability of the shear layers results in the formation of vortex tubes (Vincent & Meneguzzi Reference Vincent and Meneguzzi1994; Watanabe & Nagata Reference Watanabe and Nagata2023; Watanabe Reference Watanabe2024), and their length scales are related to each other. Because of the larger size of the shear layers at higher $\textit{Wi}$ , the size of the vortex tubes also increases in the viscoelastic simulations.

Figure 1. Two-dimensional profiles of intensities of (a–c) rigid-body rotation and (d–f) shear: (a,d) Newtonian case; viscoelastic cases with (b,e) $\textit{Wi}=2.0$ and (c,f) $\textit{Wi}=4.6$ . Only a small part of the computational domain ( $400\eta \times 400\eta$ ) is shown here.

Figure 2. Shear layers (white) and vortex tubes (orange), which are visualised by the isosurfaces of intensities of shear, $I_S=2\langle I_S\rangle$ , and rigid-body rotation, $I_R=4\langle I_R\rangle$ : (a) Newtonian case; viscoelastic cases with (b) $\textit{Wi}=2.0$ and (c) $\textit{Wi}=4.6$ . Only a small part of the computational domain ( $400\eta \times 400\eta \times 150\eta$ ) is shown here.

Figure 3 shows the temporal evolution of shear layers and vortex tubes in a Newtonian fluid over a duration $3\tau _\eta$ . The sequence progresses from figure 3(a) to figure 3(d) at constant intervals $\tau _\eta$ . During this time, the formation of vortices from shear layers is distinctly observable. In figure 3(a), a shear layer, labelled S, already encompasses a vortex V1 that has formed within it. As this layer evolves, additional vortices V2 and V3 emerge from the same layer, as seen in figures 3(b,c). Eventually, this shear layer fragments and partially disappears in figure 3(d). Such vortex formation processes in shear layers are explained by the shear instability, and are also documented in other studies (Vincent & Meneguzzi Reference Vincent and Meneguzzi1994; Watanabe et al. Reference Watanabe, Tanaka and Nagata2020; Watanabe & Nagata Reference Watanabe and Nagata2023).

Figure 3. Temporal evolution of shear layers (white) and vortex tubes (orange) for the reference Newtonian simulation, which are visualised by the same isosurfaces as those in figure 2(a). The progression from (a) to (d) represents the advancement of time in intervals of the Kolmogorov time scale $\tau _\eta$ . Only a small part of the computational domain ( $100\eta \times 100\eta \times 50\eta$ ) is shown here.

Figure 4 presents a similar visualisation but over a longer duration $7\tau _\eta$ for the viscoelastic simulation with $\textit{Wi}=4.6$ . Even in the viscoelastic case, the vortex formation occurs: a shear layer marked as S produces a vortex V. However, this process unfolds more gradually compared to the Newtonian case. Additionally, many shear layers in the viscoelastic fluid do not readily form vortices, in contrast to the vortex generation within shear layers in the Newtonian fluid. Horiuti et al. (Reference Horiuti, Matsumoto and Fujiwara2013) have demonstrated that shear layers in viscoelastic turbulence tend to stabilise due to the tensile forces exerted by polymers on the shear layers by conducting DNS with the Johnson–Segalman model for the polymer stress. This stabilisation inhibits the fragmentation of large shear layers into smaller ones, resulting in shear layers with a low aspect ratio in Newtonian turbulence. Consequently, this inhibited instability in viscoelastic fluids accounts for the observation of larger and flatter shear layers, as shown in figure 2(c).

Figure 4. The same as figure 3 but for the viscoelastic simulation with $\textit{Wi}=4.6$ . Temporal evolution is visualised over $7\tau _\eta$ from (a) to (h).

Figure 5 shows the p.d.f.s of $I_S$ , $I_R$ and $I_E$ normalised by the Kolmogorov time scale. The distributions of the p.d.f.s are similar to those in turbulent jets and HIT of Newtonian fluids (Hayashi et al. Reference Hayashi, Watanabe and Nagata2021a ). The peaks of the p.d.f.s appear at $I_S/\tau _\eta ^{-1}\approx 0.5$ , $I_R=0$ and $I_E/\tau _\eta ^{-1}\approx 0.03$ , indicating that most of the flow is dominated by shearing motion. This is also confirmed in figure 1, where $I_S$ has non-zero value even if it is small, while $I_R\approx 0$ is observed except for regions occupied by vortex tubes. Thus rigid-body rotation and elongation are highly intermittent in space. The insets show the p.d.f.s for moderately large intensities of these motions. As $\textit{Wi}$ increases, the p.d.f.s for large $I_S$ increase, and those for large $I_R$ and $I_E$ decrease. Therefore, shearing motion becomes more dominant in local fluid motion than the other motions in a viscoelastic fluid. The vorticity vector can be decomposed into two components of shear and rigid-body rotation as ${\boldsymbol {\omega }}={\boldsymbol {\omega }}_S+{\boldsymbol {\omega }}_R$ , with $({\boldsymbol {\omega }}_S)_i=\epsilon _{\textit{ijk}}(\boldsymbol {\nabla } {\boldsymbol {u}}_S)_{jk}$ and $({\boldsymbol {\omega }}_R)_i=\epsilon _{\textit{ijk}}(\boldsymbol {\nabla } {\boldsymbol {u}}_R)_{jk}$ (Kolář Reference Kolář2007). The structures related to ${\boldsymbol {\omega }}_S$ and ${\boldsymbol {\omega }}_R$ are shear layers and vortex tubes, respectively (Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015; Nagata et al. Reference Nagata, Watanabe, Nagata and da Silva2020; Fiscaletti et al. Reference Fiscaletti, Buxton and Attili2021). Therefore, the $\textit{Wi}$ dependence of the p.d.f.s suggests that the vorticity is related more to shear layers than vortex tubes in viscoelastic turbulence. The DNS of viscoelastic turbulence have also shown that more sheet-like structures are identified in visualisation with vorticity as $\textit{Wi}$ increases (Watanabe & Gotoh Reference Watanabe and Gotoh2014; Ferreira et al. Reference Ferreira, Pinho and da Silva2016). This tendency is well explained by the dominance of shearing motion over rotating motion at high $\textit{Wi}$ because the shear vorticity ${\boldsymbol {\omega }}_S$ becomes more important in the total vorticity ${\boldsymbol {\omega }}$ as $\textit{Wi}$ increases.

Figure 5. The p.d.f.s of intensities of (a) shear, (b) rigid-body rotation and (c) elongation, normalised by the Kolmogorov time scale $\tau _\eta$ . The insets show the p.d.f.s with moderately large values of the intensities.

4.2. Mean flow field of small-scale shear layers

This subsection discusses the mean flow field evaluated in the shear coordinate, sketched in figure 6(a). Shearing motion is not usually observed in the reference frame of DNS because the shear layers have no preference in the orientation with respect to $x$ , $y$ and $z$ . However, in the local shear coordinate, a local shear-flow pattern is easily observed in the $\zeta _2$ – $\zeta _3$ plane. This is confirmed in figure 6(b), where a flow around one of the shear layers is visualised on the $\zeta _2$ – $\zeta _3$ plane at $\zeta _1=0$ . The thin shear layer with large $I_S$ is extended in the $\zeta _3$ direction. Flows in the $\pm \zeta _3$ directions are observed on the sides of the shear layer, generating shear in the layer structure. The conditional averages taken as functions of $(\zeta _1, \zeta _2, \zeta _3)$ reveal the mean flow characteristics of the shear layers shown in figure 6(b).

Figure 6. (a) A schematic of a shear coordinate. (b) A shear layer observed in the shear coordinate for $\textit{Wi}=3.0$ . The shear intensity $I_S$ and two-dimensional velocity vectors are shown on the $\zeta _2$ – $\zeta _3$ plane at $\zeta _1=0$ . The velocity vectors relative to the velocity at $(\zeta _1,\zeta _2,\zeta _3)=(0,0,0)$ are shown here. The length of a vector corresponds to the vector magnitude.

Figure 7 shows the mean flow field around the shear layer with the mean shear intensity $\overline {I_S}$ and mean velocity vectors on the visualised planes for the Newtonian case. The three planes intersecting the centre of the shear layers are visualised in each plot. Large $\overline {I_S}$ is observed in a layer that is thin in the $\zeta _2$ direction and long in the other two directions. On the $\zeta _2$ – $\zeta _3$ plane, the mean flows in the $\zeta _3$ and $-\zeta _3$ directions appear for $\zeta _2>0$ and $\zeta _2<0$ , respectively, representing local shearing motion. The mean velocity vectors on the $\zeta _1$ – $\zeta _2$ plane exhibit a biaxial strain with stretching in the $\zeta _1$ direction, and compression in the $\zeta _2$ direction, which are confirmed by $\partial \overline {u_{\zeta _1}}/\partial \zeta _1 >0$ and $\partial \overline {u_{\zeta _2}}/\partial \zeta _2 <0$ . The $\zeta _1$ – $\zeta _3$ plane is parallel to the shear layer, and the $\zeta _3$ dependence of the mean flow on this plane is relatively weak compared to the dependence in the other directions. These mean flow patterns are independent of flows and Reynolds numbers under Kolmogorov normalisation (Watanabe et al. Reference Watanabe, Tanaka and Nagata2020; Fiscaletti et al. Reference Fiscaletti, Buxton and Attili2021; Hayashi et al. Reference Hayashi, Watanabe and Nagata2021a ), and the present results agree with previous studies.

Figure 7. Mean shear intensity $\overline {I_S}$ and mean velocity vectors on (a) the $\zeta _2$ – $\zeta _3$ plane at $\zeta _1=0$ , (b) the $\zeta _1$ – $\zeta _2$ plane at $\zeta _3=0$ and (c) the $\zeta _1$ – $\zeta _3$ plane at $\zeta _2=0$ , for the Newtonian case. The length of a vector corresponds to the vector magnitude.

Figures 8 and 9 present the mean flows near the shear layers in the viscoelastic simulations with $\textit{Wi}=2.0$ and 4.6. The shear and straining flows observed on the $\zeta _2$ – $\zeta _3$ and $\zeta _1$ – $\zeta _2$ planes also appear in the viscoelastic cases. However, the large- $\overline {I_S}$ region is extended for larger $|\zeta _1|$ and $|\zeta _3|$ as $\textit{Wi}$ increases from the Newtonian case. The layer thickness in the $\zeta _2$ direction is thinner in the viscoelastic cases than in the Newtonian case. Therefore, the aspect ratio of the shear layers increases in the viscoelastic turbulence, as also observed for the visualised shear layers in figure 2.

Figure 8. The same as figure 7 but for $\textit{Wi}=2.0$ .

Figure 9. The same as figure 7 but for $\textit{Wi}=4.6$ .

Figures 10 and 11 present the three-dimensional visualisations of the streamlines of mean flow around shear layers for Newtonian and viscoelastic cases, respectively. A shear flow with compressive motion in the direction normal to the layer is observed on either side of the shear layer, as indicated by the streamlines contracting towards the layer. Furthermore, the streamlines within the shear layer are oriented in the $\zeta _1$ direction, where extensive strain facilitates the stretching of shear vorticity. The inherent vorticity of the shear layer with a finite aspect ratio causes the streamlines to exhibit partially rotating patterns at the edges of the shear layers. In fact, the mean profile of the intensity of rigid-body rotation supports the presence of vortices associated with rigid-body rotation at the edges of the shear layer (Watanabe & Nagata Reference Watanabe and Nagata2022). The observed flow pattern, characterised by a combination of shear, stretching in the $\zeta _1$ direction, and compression in the $\zeta _2$ direction, is typical of small-scale shear layers in turbulent flows. This pattern is crucial in defining the structures within turbulent flows in terms of fluid particle motion.

Figure 10. Mean streamlines around the shear layer for the Newtonian case: (a) diagonal view; (b) top view from the $\zeta _1$ direction. The isosurface of $\overline {I_S}/\tau _\eta ^{-1}=1.1$ (white) visualises the shear layer. The streamlines that pass the line connecting $(\zeta _1/\eta, \zeta _2/\eta, \zeta _3/\eta )=(-5, 60, 30)$ and $(5, -60, -30)$ are visualised in a spherical domain with radius $80\eta$ . The arrows indicate the flow direction.

Because of the shear and strain acting on the layer, mean velocity jumps are observed for $\overline {u_{\zeta _1}}$ on the $\zeta _1$ axis, and $\overline {u_{\zeta _2}}$ and $\overline {u_{\zeta _3}}$ on the $\zeta _2$ axis. The corresponding mean velocity profiles are shown in figure 12 for all simulations. For $\overline {u_{\zeta _1}}$ , the mean velocity difference between $\zeta _1>0$ and $\zeta _1<0$ becomes large as $\textit{Wi}$ increases. However, the velocity jump is observed over a longer distance at higher $\textit{Wi}$ , resulting in smaller $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ within the shear layer. For $\overline {u_{\zeta _2}}$ and $\overline {u_{\zeta _3}}$ , the mean velocity differences also increase with $\textit{Wi}$ . However, viscoelasticity has virtually no effect on the mean velocity gradient within the shear layer.

Figure 13 presents an analysis of statistical errors by displaying the mean velocity and shear intensity profiles across the shear layers, as derived from two separate datasets described in § 3.2. Each dataset represents half of the computational domain. The comparison of these profiles reveals minimal differences in their distributions, demonstrating a high degree of statistical convergence. It indicates that even with half the number of statistical samples for the shear layers, the results remain consistent. Therefore, the conclusions drawn from the statistical analysis of the shear layer are robust and not skewed by statistical errors.

Figure 11. The same as figure 10 but for $\textit{Wi}=4.6$ .

Figure 12. Mean velocity profiles across the shear layer: (a) $\overline {u_{\zeta _1}}$ on the $\zeta _1$ axis; (b) $\overline {u_{\zeta _2}}$ on the $\zeta _2$ axis; and (c) $\overline {u_{\zeta _3}}$ on the $\zeta _2$ axis. The profiles are shown along the lines that pass the centre of the shear layer, $(\zeta _1,\zeta _2,\zeta _3)=(0,0,0)$ .

Figure 13. The mean velocity and shear intensity profiles across the shear layer obtained from two distinct datasets, each representing one half of the computational domain (Newtonian case). Their differences provide a measure of statistical errors.

The velocity jumps of shear layers are further examined by plotting the maximum values of the mean velocity gradients against $\textit{Wi}$ in figure 14(a). The error bars indicate the statistical errors by illustrating the values obtained from two separate datasets. Both datasets yield almost identical values for the mean velocity gradients, and the statistical errors are negligible. As shown in figure 12, $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ and $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ become the largest at $\zeta _1=0$ and $\zeta _2=0$ , while $\partial \overline {u_{\zeta _2}}/\partial \zeta _2$ has negative peaks. The mean velocity gradient related to shearing motion, $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ , normalised by the Kolmogorov time scale, is similar for Newtonian and viscoelastic cases, and hardly varies with $\textit{Wi}$ . The extensive strain $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ becomes weak as $\textit{Wi}$ increases. Similarly, the compressible strain $\partial \overline {u_{\zeta _2}}/\partial \zeta _2$ becomes weak as $\textit{Wi}$ increases, although the decrease in $|\partial \overline {u_{\zeta _2}}/\partial \zeta _2|$ with $\textit{Wi}$ is not as significant as that in $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ for high $\textit{Wi}$ .

Figure 14. The Weissenberg number ( $\textit{Wi}$ ) dependence of (a) the mean velocity gradients associated with shear and strain in the shear layer, $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ , $|\partial \overline {u_{\zeta _2}}/\partial \zeta _2|$ and $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ , and (b) the length scales of the shear layer in the $\zeta _1$ , $\zeta _2$ and $\zeta _3$ directions, which are denoted by $\delta _{S1}$ , $\delta _{S2}$ and $\delta _{S3}$ , respectively. The Kolmogorov length ( $\eta$ ) and time ( $\tau _\eta$ ) scales are used for normalisation. The maximum values of the mean velocity gradients near the shear layers are plotted in (a), where $|\partial \overline {u_{\zeta _2}}/\partial \zeta _2|$ is shown instead of $\partial \overline {u_{\zeta _2}}/\partial \zeta _2$ for the usage of the logarithmic scale. In (a), the error bars represent the statistical errors estimated with the two datasets, generated by dividing the computational domain into two equal halves.

The constant values of $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ within the shear layer are observed when normalised by the Kolmogorov time scale, which is defined based on the solvent dissipation rate of kinetic energy. In Newtonian turbulence, the dissipation of kinetic energy is predominantly driven by shearing motion, specifically related to $\partial u_{\zeta _3}/\partial \zeta _2$ in the shear layers (Das & Girimaji Reference Das and Girimaji2020; Nagata et al. Reference Nagata, Watanabe, Nagata and da Silva2020). Mathematically, irrotational strain without shear, represented by $\boldsymbol {\nabla } \boldsymbol {u}_{E}$ in the triple decomposition, can also contribute to dissipation. However, it has been demonstrated that this type of motion does not significantly contribute to dissipation as compared to shearing motion in Newtonian turbulence (Das & Girimaji Reference Das and Girimaji2020; Nagata et al. Reference Nagata, Watanabe, Nagata and da Silva2020). Consequently, $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ normalised by the Kolmogorov time scale tends to remain constant across different Reynolds number flows (Watanabe et al. Reference Watanabe, Tanaka and Nagata2020). The present results indicate that in viscoelastic cases, the physical mechanism underlying the constancy of normalised $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ for different $\textit{Wi}$ values is the predominance of shearing motion in the dissipation by the solvent. Below, the $\textit{Wi}$ dependence of the mean solvent dissipation in turbulence is well explained by changes in dissipation due to shear layers, further confirming that shear layer structures are responsible for solvent dissipation even in viscoelastic turbulence. On the other hand, $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ and $|\partial \overline {u_{\zeta _2}}/\partial \zeta _2|$ show a clear dependence on $\textit{Wi}$ even when normalised by the Kolmogorov time scale. These components are associated with the straining flow around shear layers. In Newtonian turbulence, the configuration analysis of vortex tubes and shear layers has suggested that the straining flow acting on shear layers is partially induced by nearby vortex tubes (Watanabe & Nagata Reference Watanabe and Nagata2022). Therefore, the observed suppression of vortex formations from shear layers in viscoelastic turbulence, as discussed in figures 10 and 11, is likely related to the weakened straining flow observed as the decreases in $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ and $|\partial \overline {u_{\zeta _2}}/\partial \zeta _2|$ at higher $\textit{Wi}$ values. Consistently, the weakened irrotational strain is further confirmed by the $\textit{Wi}$ dependency of $I_E$ in figure 5(c).

The $\textit{Wi}$ dependence of $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ and $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ has an important implication for vorticity dynamics. The interaction between the shear and biaxial strain causes large enstrophy production (Watanabe et al. Reference Watanabe, Tanaka and Nagata2020; Hayashi et al. Reference Hayashi, Watanabe and Nagata2021a ). This is explained by the vorticity arising from large $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ of shearing motion, which is stretched by the extensive strain in the $\zeta _1$ direction with large $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ . The vorticity due to shear is unlikely to be influenced by viscoelasticity, as attested by the weak $\textit{Wi}$ dependence of $\partial \overline {u_{\zeta _3}}/\partial \zeta _2$ , whereas the extensive strain $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ becomes weaker with increasing $\textit{Wi}$ . Therefore, the enstrophy production due to shear and surrounding straining motion is weakened as the viscoelastic effects become significant. This is also confirmed below with the enstrophy budget near the shear layer.

Another interesting behaviour is in the balance between $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ and $\partial \overline {u_{\zeta _2}}/\partial \zeta _2$ . The incompressible condition imposes $\partial {u_{\zeta _1}}/\partial \zeta _1+\partial {u_{\zeta _2}}/\partial \zeta _2+\partial {u_{\zeta _3}}/\partial \zeta _3=0$ . In the Newtonian case, $\partial \overline {u_{\zeta _1}}/\partial \zeta _1+\partial \overline {u_{\zeta _2}}/\partial \zeta _2\approx 0$ is satisfied in the shear layer, suggesting that the two-dimensional strain is acting on the shear layer with approximately equal strength of the extension and compression. However, the extensive strain $\partial \overline {u_{\zeta _1}}/\partial \zeta _1$ becomes weaker than the compressive strain $\partial \overline {u_{\zeta _2}}/\partial \zeta _2$ at high $\textit{Wi}$ . This can cause an additional extensive strain $\partial \overline {u_{\zeta _3}}/\partial \zeta _3>0$ in the shear-flow direction. The magnitude of the mean velocity vectors in figures 7–9 is expressed with the lengths of the arrows. In the Newtonian case, the mean velocity in the $\zeta _3$ direction is negligibly small within the shear layer, which is along the $\zeta _3$ axis at $\zeta _2=0$ in figure 7(a). However, in figure 9(a) with $\textit{Wi}=4.6$ , the mean velocities with $\overline {u_{\zeta _3}}>0$ and $\overline {u_{\zeta _3}}<0$ are observed for $\zeta _3>0$ and $\zeta _3<0$ in the shear layer, respectively, confirming that the extensive strain is indeed acting in the $\zeta _3$ direction. As discussed below, this $\textit{Wi}$ dependence of $\overline {u_{\zeta _3}}$ is important for the polymer stress within the shear layers.

Figure 14(b) shows the $\textit{Wi}$ dependence of the size of the shear layers in the three directions evaluated with the mean shear intensity $\overline {I_S}$ . Here, a normalised mean shear intensity $\hat {I}_S$ is defined as $\hat {I}_S(\zeta _1,\zeta _2,\zeta _3)=[\overline {I_S}(\zeta _1,\zeta _2,\zeta _3)-\langle I_S\rangle ]/[\overline {I_S}(0,0,0)-\langle I_S\rangle ]$ . Because $\overline {I_S}=\langle I_S\rangle$ at large $|\zeta _i|$ , $\hat {I}_S$ is equal to 1 at the shear layer centre, and decreases to 0 as $|\zeta _i|$ increases. The length scale of the shear layer in the $\zeta _1$ direction, $\delta _{S1}$ , is quantified as the distance between two points of $\hat {I}_S(\zeta _1,0,0)=0.1$ on the $\zeta _1$ axis at $(\zeta _2, \zeta _3)=(0,0)$ . Similarly, the length scales in the $\zeta _2$ and $\zeta _3$ directions, $\delta _{S2}$ and $\delta _{S3}$ , are quantified with the locations $\hat {I}_S(0,\zeta _2,0)=0.1$ and $\hat {I}_S(0,0,\zeta _3)=0.1$ , respectively. In figure 14(b), $\delta _{S1}$ and $\delta _{S3}$ increase, and $\delta _{S2}$ decreases, as $\textit{Wi}$ becomes large. Thus the shear layers become larger in the layer-parallel directions and thinner in the normal direction as the viscoelastic effects become more important. This geometrical change also agrees with the visualised shear layers in figure 2. The shear layer is one of the smallest-scale vortical structures, manifesting with a sheet-like geometry. In Newtonian turbulence, the thickness of these shear layers scales with $\eta$ (Watanabe et al. Reference Watanabe, Tanaka and Nagata2020; Fiscaletti et al. Reference Fiscaletti, Buxton and Attili2021; Hayashi et al. Reference Hayashi, Watanabe and Nagata2021a ). However, as $\textit{Wi}$ increases, the thickness of these layers, when normalised by $\eta$ , decreases. This observation indicates that the Kolmogorov scale does not adequately characterise the length scale of vortical structures at the smallest scales in viscoelastic turbulence.

The above viscoelastic effects on the mean properties of the shear layers are significant at high $\textit{Wi}$ . However, the shear layers in viscoelastic turbulence at low $\textit{Wi}$ behave similarly to Newtonian turbulence. In figure 14(a), the mean velocity gradients of the biaxial strain hardly change due to the viscoelastic effects up to $\textit{Wi}\approx 2$ , whereas the gradient in the $\zeta _1$ direction begins to decrease for $\textit{Wi}>2$ . Similarly, the shear layer thickness $\delta _{S2}$ is influenced by viscoelasticity for $\textit{Wi}>2$ . These results imply that the shear layers in viscoelastic turbulence are similar to those in Newtonian turbulence until $\textit{Wi}$ becomes sufficiently large. Once $\textit{Wi}$ exceeds a certain criterion at $\textit{Wi}\approx 2$ , viscoelasticity drastically influences the shear layers, as is also observed as the changes in the kinetic energy spectra (Ferreira et al. Reference Ferreira, Pinho and da Silva2016).

4.3. Polymer-stress distribution near the shear layers

The velocity gradients due to the shear and strain are expected to affect the polymer molecules near the shear layers. The present study examines the polymer-stress tensor expressed in the shear coordinate, ${\mathsf{\sigma}} _{\zeta _i\zeta _j}^{[P]}$ , whose averages taken around the shear layer are $\overline {{\mathsf{\sigma}} _{\zeta _i\zeta _j}^{[P]}}$ . Here, the normalised mean stresses are defined as $\hat {{\mathsf{\sigma}} }_{\zeta _i\zeta _j}^{[P]}=\overline {{\mathsf{\sigma}} _{\zeta _i\zeta _j}^{[P]}}/(\rho u_\eta ^{2})$ . Figure 15 shows six independent components of the mean polymer-stress tensor for $\textit{Wi}=1.3$ and 4.6 across the shear layer. The mean polymer stresses vary significantly near the shear layer, suggesting that the shear layers impact the extension of the polymer molecules. Here, spatial variations near the shear layer are observed for $\hat {{\mathsf{\sigma}} }_{\zeta _1\zeta _1}^{[P]}$ , $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _2}^{[P]}$ , $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ and $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _3}^{[P]}$ , which are strongly influenced by the shear layers, whereas the remaining components are almost zero. The polymer-stress tensor is defined with the conformation tensor evolving according to (2.8), which is helpful to understand the relation between the conformation tensor in the shear coordinate, ${\unicode{x1D60A}}_{\zeta _i\zeta _j}$ , and the velocity jumps around shear layers. The extensive strain with $\partial \overline {u_{\zeta _1}}/\partial \zeta _1>0$ and compressive strain with $\partial \overline {u_{\zeta _2}}/\partial \zeta _2<0$ appear in the governing equations of ${\unicode{x1D60A}}_{\zeta _1\zeta _1}$ and ${\unicode{x1D60A}}_{\zeta _2\zeta _2}$ as ${\unicode{x1D60A}}_{\zeta _1\zeta _1}(\partial {u_{\zeta _1}}/\partial \zeta _1)$ and ${\unicode{x1D60A}}_{\zeta _2\zeta _2}(\partial {u_{\zeta _2}}/\partial \zeta _2)$ , respectively. For ${\unicode{x1D60A}}_{\zeta _1\zeta _1}$ , the extensive strain causes stretching of the polymer molecules, which results in large ${\unicode{x1D60A}}_{\zeta _1\zeta _1}$ . For the same reason, the compressive strain decreases ${\unicode{x1D60A}}_{\zeta _2\zeta _2}$ . These contributions of the straining flow to ${\unicode{x1D60A}}_{\zeta _1\zeta _1}$ and ${\unicode{x1D60A}}_{\zeta _2\zeta _2}$ explain large $\hat {{\mathsf{\sigma}} }_{\zeta _1\zeta _1}^{[P]}$ and small $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _2}^{[P]}$ within the shear layers. Similarly, $\partial \overline {u_{\zeta _3}}/\partial \zeta _2>0$ due to the shear causes the polymer distortion, and contributes to an increase of ${\unicode{x1D60A}}_{\zeta _2\zeta _3}={\unicode{x1D60A}}_{\zeta _3\zeta _2}$ . For ${\unicode{x1D60A}}_{\zeta _2\zeta _3}$ , the compressive strain in the $\zeta _2$ direction may contribute to a reduction of ${\unicode{x1D60A}}_{\zeta _2\zeta _3}$ . However, figure 14(a) has shown that the shear causes a larger velocity gradient than the strain, namely $\partial {u_{\zeta _3}}/\partial \zeta _2>|\partial {u_{\zeta _2}}/\partial \zeta _2|$ . Therefore, an increase of ${\unicode{x1D60A}}_{\zeta _2\zeta _3}$ due to the shear can surpass a decrease due to the compressive strain. Thus ${\unicode{x1D60A}}_{\zeta _2\zeta _3}$ , namely $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _3}^{[P]}$ , is locally large near the shear layer. A comparison between $\textit{Wi}=1.3$ and $\textit{Wi}=4.6$ suggests that the relative importance of $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ compared to the other components depends on $\textit{Wi}$ : $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ within the shear layer ( $\zeta _2\approx 0$ ) is as large as $\hat {{\mathsf{\sigma}} }_{\zeta _1\zeta _1}^{[P]}$ at $\textit{Wi}=4.6$ , whereas $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ is much smaller than $\hat {{\mathsf{\sigma}} }_{\zeta _1\zeta _1}^{[P]}$ for $\textit{Wi}=1.3$ . This $\textit{Wi}$ dependence is explained by $\partial \overline {u_{\zeta _3}}/\partial \zeta _3$ , which increases from zero as $\textit{Wi}$ increases because of the imbalance between the extension and compression of the straining flow in figure 14(a). For $\textit{Wi}>2$ , the extensive strain in the $\zeta _3$ direction with $\partial \overline {u_{\zeta _3}}/\partial \zeta _3>0$ begins to act in the shear layer. This strain causes an increase of ${\unicode{x1D60A}}_{\zeta _3\zeta _3}$ due to the polymer stretching, ${\unicode{x1D60A}}_{\zeta _3\zeta _3}(\partial {u_{\zeta _3}}/\partial \zeta _3)>0$ , which appears in the governing equation for ${\unicode{x1D60A}}_{\zeta _3\zeta _3}$ . This additional extensive strain becomes important only for sufficiently high $\textit{Wi}$ . Therefore, the large peak of $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ within the shear layer is observed at $\textit{Wi}=4.6$ but not at $\textit{Wi}=1.3$ .

Figure 15. Mean polymer stresses $\overline{{{\mathsf{\sigma}}}^{[P]}_{\zeta_{i}{\zeta_j}}}$ near the shear layer for (a) $\textit{Wi}=1.3$ and (b) $\textit{Wi}=4.6$ . The results are shown for the non-dimensionalised stresses $\hat {{\mathsf{\sigma}} }^{[P]}_{\zeta _{i}{\zeta _j}} =\overline {{{\mathsf{\sigma}} }^{[P]}_{\zeta _{i}{\zeta _j}}}/(\rho u_\eta ^{2})$ , and are plotted against $\zeta _2/\eta$ at $(\zeta _1,\zeta _3)=(0,0)$ . The stress tensor is evaluated in the shear coordinate $(\zeta _1, \zeta _2, \zeta _3)$ .

Figures 16 and 17 visualise the two-dimensional profiles of $\hat {{\mathsf{\sigma}} }_{\zeta _i\zeta _j}^{[P]}$ on the $\zeta _2$ – $\zeta _3$ plane at $\zeta _1=0$ for $\textit{Wi}=1.3$ and 3.0. White lines are the isolines of $\overline {I_S}$ , which mark the location of the shear layer. Qualitative differences between the large and small $\textit{Wi}$ cases are observed for $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ and $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _3}^{[P]}$ . The difference in $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ is due to the stretching in the $\zeta _3$ direction arising from the imbalance in the strain, as discussed above. As a result, $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ for $\textit{Wi}=3.0$ becomes large within the shear layer. However, large $\hat {{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ appears on the sides of the shear layer for $\textit{Wi}=1.3$ . A similar difference is found for $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _3}^{[P]}$ , which is large within the shear layer at $\textit{Wi}=3.0$ . In particular, very large $\hat {{\mathsf{\sigma}} }_{\zeta _2\zeta _3}^{[P]}$ appears along the $\zeta _3$ axis at $\zeta _2=0$ above and below the centre of the shear layer. The governing equation of ${\unicode{x1D60A}}_{\zeta _2\zeta _3}$ contains ${\unicode{x1D60A}}_{\zeta _2\zeta _3}(\partial \overline {u_{\zeta _3}}/\partial \zeta _3)$ , which causes an increase of ${\unicode{x1D60A}}_{\zeta _2\zeta _3}$ at high $\textit{Wi}$ because of the $\textit{Wi}$ dependence of $\partial \overline {u_{\zeta _3}}/\partial \zeta _3$ explained above. As discussed below, the $\textit{Wi}$ dependence of ${{\mathsf{\sigma}} }_{\zeta _3\zeta _3}^{[P]}$ is crucial in the vorticity dynamics of shear layers.

Figure 16. Non-dimensionalised mean polymer stresses $\hat {{\mathsf{\sigma}} }^{[P]}_{\zeta _{i}\zeta _{j}}$ near the shear layer on the $\zeta _2$ – $\zeta _3$ plane at $\zeta _1=0$ for $\textit{Wi}=1.3$ : (a) $\hat {{\mathsf{\sigma}}}^{[P]}_{\zeta _1\zeta _1}$ , (b) $\hat {{\mathsf{\sigma}}}^{[P]}_{\zeta _2\zeta _2}$ , (c) $\hat {{\mathsf{\sigma}}}^{[P]}_{\zeta _3\zeta _3}$ and (d) $\hat {{\mathsf{\sigma}}}^{[P]}_{\zeta _2\zeta _3}$ .

Figure 17. The same as figure 16 but for $\textit{Wi}=3.0$ .

4.4. The relevance of shear layers to vorticity dynamics and kinetic energy dissipation

The transport equation of enstrophy for viscoelastic turbulence is written as

(4.1) \begin{equation} \frac{{\rm D} \omega^2/2}{{\rm D} t} =\underbrace{ \omega_i{\unicode{x1D61A}}_{\textit{ij}}\omega_j }_{P_{\omega}} +\underbrace{ \nu^{[S]}\,\frac{\partial^2 \omega^2/2}{\partial x_j\,\partial x_j} }_{D_{\omega}} - \underbrace{ \nu^{[S]}\,\frac{\partial \omega_i}{\partial x_j}\, \frac{\partial \omega_i}{\partial x_j} }_{\varepsilon_{\omega}} +\underbrace{ \omega_i\epsilon_{nji}\,\frac{\partial^2 {\mathsf{\sigma}}^{[P]}_{mj}}{\partial x_m\,\partial x_n} }_{V_{\omega}}, \end{equation}

where $P_\omega$ is the production term, $D_\omega$ is the viscous diffusion term, $\varepsilon _\omega$ is the viscous dissipation term, and $V_\omega$ represents an enstrophy change due to the interaction between vorticity and polymer stresses. The last term is absent in Newtonian fluids. The average of ${\rm D}(\omega ^2/2)/{\rm D} t$ , i.e. $\langle {\rm D}(\omega ^2/2)/{\rm D}t\rangle$ , is zero in HIT because the statistical stationarity and homogeneity assume $\langle \partial f/\partial t\rangle =0$ and $\langle \partial f/\partial x_i\rangle =0$ for any variable $f$ . However, when the average of ${\rm D}(\omega ^2/2)/{\rm D}t$ is taken conditionally for certain regions in HIT, such as shear layers or vortex tubes, the conditional average of ${\rm D}(\omega ^2/2)/{\rm D}t$ is not necessarily zero because the enstrophy amplification or attenuation may occur in a specific region in HIT.

Figure 18(a) shows the enstrophy budget across the shear layer in Newtonian turbulence, with the conditional averages of each term plotted against $\zeta _2/\eta$ at $(\zeta _1,\zeta _3)=(0,0)$ . The enstrophy production $P_\omega$ becomes large within the shear layer because of the stretching of vorticity arising from the shear by the extensive strain in the $\zeta _1$ direction. The enstrophy is locally large within the shear layer. Therefore, the diffusion term $D_\omega$ reduces and increases the enstrophy inside and outside the shear layer, respectively, transferring the enstrophy from the shear layer to the outside. The dissipation $\varepsilon _\omega$ is also large near the shear layer and has negative peaks at the locations where the enstrophy growth due to the viscous diffusion occurs. These results of the enstrophy budget in HIT are consistent with those in a turbulent planar jet (Hayashi et al. Reference Hayashi, Watanabe and Nagata2021b ).

Figure 18. The enstrophy budget near the shear layer for (a) the Newtonian case, (b) $\textit{Wi}=1.3$ and (c) $\textit{Wi}=3.0$ . The averages of (4.1), $\overline {P_\omega }$ , $\overline {D_\omega }$ , $\overline {\varepsilon _\omega }$ and $\overline {V_\omega }$ , normalised by the Kolmogorov time scale $\tau _\eta$ , are plotted against $\zeta _2/\eta$ at $(\zeta _1,\zeta _3)=(0,0)$ .

Figures 18(b,c) present the enstrophy budget in viscoelastic cases. The negative peak of the diffusion term $D_\omega$ at $\zeta _2=0$ for $\textit{Wi}=3.0$ is larger than in the Newtonian case. The diffusion is related to the enstrophy gradient on both sides of the shear layer. Because the shear layer thickness decreases with $\textit{Wi}$ , the local enstrophy gradient in the $\zeta _2$ direction also becomes large, resulting in the large negative peak of $\overline {D_\omega }$ in the shear layer. On the other hand, the peak of the production term $P_\omega$ decreases as $\textit{Wi}$ increases from the Newtonian case. Large $P_\omega$ within the shear layer is attributed to shear $\partial {u_{\zeta _3}}/\partial \zeta _2$ and extensive strain $\partial {u_{\zeta _1}}/\partial \zeta _1$ . The shear contributes to the vorticity vector in the $\zeta _1$ direction, $\omega _{\zeta _1}$ . Production $P_\omega$ has a term $\omega _{\zeta _1}{\unicode{x1D61A}}_{\zeta _1\zeta _1}\omega _{\zeta _1}$ to which the shear and extensive strain contribute in the form $(\partial {u_{\zeta _3}}/\partial \zeta _2)^2(\partial {u_{\zeta _1}}/\partial \zeta _1)$ . Thus the large velocity gradients $\partial {u_{\zeta _3}}/\partial \zeta _2$ and $\partial {u_{\zeta _1}}/\partial \zeta _1$ of the shear layer cause the large enstrophy production. Figure 14(a) has already shown that as $\textit{Wi}$ increases, the extensive strain $\partial {u_{\zeta _1}}/\partial \zeta _1$ becomes weak, especially when $\textit{Wi}$ exceeds approximately 2. For this reason, the enstrophy production becomes small at high $\textit{Wi}$ . In addition to this change in the enstrophy production, the vorticity/polymer interaction term $V_\omega$ changes its role in the vorticity dynamics within the shear layer. At $\textit{Wi}=1.3$ , $\overline {V_\omega }$ is negative in the shear layer and acts as a destruction term of enstrophy. However, it has a large positive peak comparable to $P_\omega$ at $\textit{Wi}=3.0$ , and acts as an additional production term. This transition implies that shearing motion changes from an inertia-dominated state to an inertio-elasticity dominated state. These observations are consistent with those reported by Abreu et al. (Reference Abreu, Pinho and da Silva2022), where a similar transition from a sink to a source role for the term $V_\omega$ was noted with increasing values of $\textit{Wi}$ , even though their analysis was not focused exclusively on shear layers. In wall-bounded turbulent shear flows, the interaction between vorticity and polymers introduces a sink term for the streamwise component of enstrophy, leading to a reduction in streamwise vorticity (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007). Conversely, when considering enstrophy defined for the mean velocity, namely the squared mean vorticity, this interaction acts as a source term for its streamwise component near the wall (Song et al. Reference Song, Lin, Liu, Lu and Khomami2021).

The $\textit{Wi}$ dependence of the production and destruction of enstrophy due to the polymers, $V_\omega$ , is examined by considering the following decomposition based on the vorticity vector in the shear coordinate:

(4.2) \begin{equation} V_\omega = \underbrace{\omega_{\zeta_1}\epsilon_{\zeta_n\zeta_j\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_m\zeta_j}}{\partial x_{\zeta_m}\,\partial x_{\zeta_n}} }_{V_{\omega_{\zeta1}}}+\underbrace{\omega_{\zeta_2}\epsilon_{\zeta_n\zeta_j\zeta_2}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_m\zeta_j}}{\partial x_{\zeta_m}\,\partial x_{\zeta_n}} }_{V_{\omega_{\zeta2}}}+\underbrace{\omega_{\zeta_3}\epsilon_{\zeta_n\zeta_j\zeta_3}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_m\zeta_j}}{\partial x_{\zeta_m}\,\partial x_{\zeta_n}} }_{V_{\omega_{\zeta3}}},\end{equation}

which represents the interaction of the vorticity in the $\zeta _i$ direction with the polymer stresses. Figures 19(a,b) show the averages of these decomposed terms for $\textit{Wi}=1.3$ and 3.0. The $\zeta _1$ vorticity component has a dominant contribution to this term, whereas the averages of the other terms are negligibly small. Thus the polymer stress interaction with the $\zeta _1$ vorticity, that is, a vorticity due to shear, destroys and produces enstrophy for $\textit{Wi}=1.3$ and 3.0, respectively.

Figure 19. The mean profiles of decomposed vorticity/polymer interaction terms, (4.2), in the enstrophy transport equation across the shear layer for (a) $\textit{Wi}=1.3$ and (b) $\textit{Wi}=3.0$ . (c) The mean profile of a component of the interaction between $\omega _{\zeta _1}$ and ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}$ , the third term $\overline {V_{\omega _{\zeta 1}}}({\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3})$ in (4.3), across the shear layer. The results taken at $(\zeta _1,\zeta _3)=(0,0)$ are plotted against $\zeta _2/\eta$ .

To further examine the $\textit{Wi}$ dependence of $V_\omega$ , $V_{\omega _{\zeta 1}}$ is decomposed into six terms with different components of the polymer-stress tensor, as

(4.3) \begin{align}V_{\omega_{\zeta1}} &= \omega_{\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_1\zeta_3}} {\partial x_{\zeta_1}\,\partial x_{\zeta_2}}+\omega_{\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_2\zeta_3}} {\partial x_{\zeta_2}\,\partial x_{\zeta_2}}+\omega_{\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_3\zeta_3}} {\partial x_{\zeta_3}\,\partial x_{\zeta_2}}\nonumber\\ &\quad-\omega_{\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_1\zeta_2}} {\partial x_{\zeta_1}\,\partial x_{\zeta_3}}-\omega_{\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_2\zeta_2}} {\partial x_{\zeta_2}\,\partial x_{\zeta_3}}-\omega_{\zeta_1}\,\frac{\partial^2{\mathsf{\sigma}}^{[P]}_{\zeta_3\zeta_2}} {\partial x_{\zeta_3}\,\partial x_{\zeta_3}}.\end{align}

The distributions of the stresses near the shear layer in figure 15 suggest that the terms with ${\mathsf{\sigma}} ^{[P]}_{\zeta _2\zeta _2}$ , ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}$ and ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _2}$ have dominant contributions to $V_{\omega _{\zeta 1}}$ . We have examined all the decomposed terms near the shear layer for all simulations with different $\textit{Wi}$ , and noticed that the third term with ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}$ , denoted by $V_{\omega _{\zeta 1}}({\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3})$ , is responsible for the $\textit{Wi}$ dependence of $V_{\omega }$ observed in figure 18. Figure 19(c) presents the mean profile of $V_{\omega _{\zeta 1}}({\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3})$ across the shear layer. For small $\textit{Wi}$ , this term is small even within the shear layer. However, a large positive peak appears for $\textit{Wi}=3.0$ and 4.6. Because of this $\textit{Wi}$ dependence of the interaction between the vorticity in the $\zeta _1$ direction with the normal polymer stress in the $\zeta _3$ direction, the role of $V_\omega$ changes with $\textit{Wi}$ . The shear expressed as $\partial {u_{\zeta _3}}/\partial \zeta _2>0$ contributes to large positive $\omega _{\zeta _1}$ . Therefore, the sign of $V_{\omega _{\zeta 1}}({\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3})$ depends on $\partial ^2 {\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}/\partial x_{\zeta _3}\partial x_{\zeta _2}$ . The distributions of ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}$ for low and high $\textit{Wi}$ have been presented in figures 16(c) and 17(c), respectively. For high $\textit{Wi}$ , large $\overline {{\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}}$ distributes from the first to third quadrants along the shear layer, which is slightly tilted from the $\zeta _3$ direction. Therefore, an instantaneous profile of ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}$ often has a positive peak in the first or third quadrant, which can cause positive $\partial ^2 {\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}/\partial x_{\zeta _3}\partial x_{\zeta _2}$ at $(\zeta _2, \zeta _3)=(0,0)$ , namely the viscoelastic enstrophy production at high $\textit{Wi}$ . On the other hand, large $\overline {{\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}}$ at low $\textit{Wi}$ appears in the second and third quadrants, resulting in negative $\partial ^2 {\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}/\partial x_{\zeta _3}\partial x_{\zeta _2}$ with the enstrophy destruction. As discussed above, these differences related to ${\mathsf{\sigma}} ^{[P]}_{\zeta _3\zeta _3}$ are attributed to the viscoelastic effects on the mean flow field around the shear layer, which changes significantly once $\textit{Wi}$ exceeds approximately 2.

In viscoelastic turbulence, kinetic energy dissipation rates due to viscous and polymer stresses are written as $\varepsilon _\nu =2\nu {\unicode{x1D61A}}_{\textit{ij}}{\unicode{x1D61A}}_{\textit{ij}}$ and $\varepsilon _P={\unicode{x1D61A}}_{\textit{ij}}{\mathsf{\sigma}} ^{[P]}_{\textit{ij}}$ , respectively. Figures 20(a,b) show the profiles of the mean dissipation rates $\overline {\varepsilon _\nu }$ and $\overline {\varepsilon _P}$ near the shear layer, which are normalised by the volume-averaged total dissipation rate $\langle \varepsilon \rangle =\langle \varepsilon _\nu +\varepsilon _P\rangle$ . Both dissipation rates have peaks at the centre of the shear layer. The viscous and polymer dissipation rates tend to decrease and increase, respectively, as $\textit{Wi}$ increases. As a result, the polymer stress contributes more to kinetic energy dissipation than the viscous stress for $\textit{Wi}\geq 2.0$ .

Figure 20. The mean kinetic energy dissipation rates due to (a) viscous stress $\overline {\varepsilon _\nu }$ , and (b) polymer stress $\overline {\varepsilon _P}$ , near the shear layer. (c) Contributions of different polymer-stress components to the dissipation rate (4.4) at $\textit{Wi}=3.0$ . The dissipation rates are normalised by the volume-averaged total dissipation rate $\langle \varepsilon \rangle =\langle \varepsilon _\nu +\varepsilon _P\rangle$ . The results taken at $(\zeta _1,\zeta _3)=(0,0)$ are plotted against $\zeta _2/\eta$ .

The dissipation rate due to the polymer can be decomposed into the contributions from each component of the polymer-stress tensor in the shear coordinate, as

(4.4) \begin{equation}\varepsilon_{P}=\sum_{\alpha=1}^{3}\sum_{\beta=1}^{3}{\mathsf{\varepsilon}}_{P_{\alpha\beta}}, \quad \mathrm{with}\ {\mathsf{\varepsilon}}_{P_{\alpha\beta}}={\unicode{x1D61A}}_{\zeta_{\alpha}\zeta_{\beta}}{\mathsf{\sigma}}^{[P]}_{\zeta_{\alpha}\zeta_{\beta}},\end{equation}

where the summation is not applied to the Greek indices when each component ${\mathsf{\varepsilon}}_{P_{\alpha\beta}}$ is considered. The mean profiles of velocity and polymer stress around the shear layer indicate that ${\mathsf{\varepsilon}} _{P_{11}}$ , ${\mathsf{\varepsilon}} _{P_{22}}$ , ${\mathsf{\varepsilon}} _{P_{33}}$ and ${\mathsf{\varepsilon}} _{P_{23}}$ are dominant because of the large velocity gradients and significant variations of the polymer stresses within the shear layer. Figure 20(c) shows the mean profiles of these dissipation components near the shear layer for $\textit{Wi}=3.0$ . The large dissipation rate by the polymer within the shear layer is caused by ${\mathsf{\varepsilon}} _{P_{11}}$ and ${\mathsf{\varepsilon}} _{P_{23}}$ . These components become large because of the extensive strain and shear, which also contribute to the amplification of the corresponding components of the polymer-stress tensor.

For the present case, $\overline {\varepsilon _P}$ is always positive even at large $\zeta _2$ , where $\overline {\varepsilon _P}$ asymptotically approaches the volume average of $\varepsilon _P$ . Thus the polymer stress leads to the dissipation of the turbulent kinetic energy. The large positive ${\mathsf{\varepsilon}} _{P_{11}}$ in figure 20(c) is attributed to the three-dimensional flow around the shear layers, specifically an extensional strain in the normal direction of the shear plane ( $\zeta _2$ – $\zeta _3$ plane). In DNS of viscoelastic planar jets, an energy conversion to turbulent kinetic energy was observed in the potential core region, where the flow is characterised by a two-dimensional mean shear (Guimarães et al. Reference Guimarães, Pinho and da Silva2023). These studies suggest that the three-dimensionality of small-scale shear layers plays an important role in the turbulent kinetic energy dissipation associated with $\varepsilon _P$ .

The above results are obtained locally with conditional averages for the shear layers. Their contributions to the global characteristics of turbulence are discussed by comparing the conditional averages with volume averages in the entire computational domain. Figure 21(a) shows the $\textit{Wi}$ dependence of the mean production and polymer-stress terms of enstrophy in (4.1). The mean values are evaluated with the volume averages $\langle P_\omega \rangle$ and $\langle V_\omega \rangle$ , or the conditional averages at the centre of shear layers $\overline {P_\omega }$ and $\overline {V_\omega }$ . At large $\textit{Wi}$ , both terms contribute to the enstrophy growth. Therefore, these averages are normalised by the sum of two terms, $\langle P_\omega +V_\omega \rangle$ or $\overline {P_\omega +V_\omega }$ . For the shear layer, $\overline {P_\omega }$ and $\overline {V_\omega }$ respectively decrease and increase with $\textit{Wi}$ , as discussed above. The same trend is observed for the volume averages. Figure 21(b) provides a similar comparison for the kinetic energy dissipation rates due to the viscous and polymer stresses. The averages of these dissipation rates normalised by the total dissipation rate are plotted against $\textit{Wi}$ . For both averages, the viscous dissipation rate decreases with $\textit{Wi}$ , whereas the dissipation rate due to the polymer stress increases with $\textit{Wi}$ . The $\textit{Wi}$ dependence is also similar for both averages. The agreement between these two averages indicates that the $\textit{Wi}$ dependence of the volume-averaged quantities is well explained by the changes in the shear layers at high $\textit{Wi}$ , and that the shear layers are important in the global characteristics of viscoelastic turbulence. In figure 21, the error bars illustrate the range of values obtained from two separate datasets. The narrow range of these error bars suggests that the statistical errors are negligible.

Figure 21. (a) The Weissenberg number ( $\textit{Wi}$ ) dependence of averages of the production term $P_\omega$ and vorticity/polymer interaction term $V_\omega$ in the enstrophy transport equation (4.1). The averages are calculated with the conventional volume averages $\langle P_\omega \rangle$ and $\langle V_\omega \rangle$ , or the conditional averages $\overline {P_\omega }$ and $\overline {V_\omega }$ , at the centre of the shear layer $(\zeta _1, \zeta _2, \zeta _3)=(0,0,0)$ . The sum of the two terms normalises each average. (b) The $\textit{Wi}$ dependence of averages of the viscous dissipation rate $\varepsilon _\nu$ and polymer dissipation rate $\varepsilon _P$ , which are calculated with the volume averages or conditional averages at the centre of the shear layer. Each average is normalised by the average of the total dissipation $\varepsilon =\varepsilon _\nu +\varepsilon _P$ . The error bars represent the statistical errors estimated with the two datasets, generated by dividing the computational domain into two equal halves.