3.1 Flow statistics

Figure 4.

Contours of the mean streamwise component of (a) velocity $\overline{u}$ and (b) vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ . In each panel, the left (right) half of the duct is used for the Newtonian (viscoelastic) fluid, respectively. The colour scale ranges from 0 (blue) to $1.4U_{b}$ (red) in (a), and from $-0.5U_{b}/h$ (blue) to $0.5U_{b}/h$ (red) in (b). Panel (a) also reports the in-plane mean velocity with arrows.

Figure 4 shows the contours of (a) the mean streamwise velocity component $\overline{u}$ and (b) the mean streamwise vorticity component $\overline{\unicode[STIX]{x1D714}}_{x}$ , where, in each panel, the left half of the duct reports the data for the Newtonian flow, whereas the right half is for the viscoelastic counterpart. For more clarity, figure 5 shows the difference between the Newtonian and viscoelastic flows. The Newtonian case shows the streamwise velocity contour typical of a duct flow, characterised by the symmetry inherited from the considered geometry. The maximum streamwise velocity is located at the centre of the duct and equals $1.3U_{b}$ . The streamwise vorticity displays the in-plane secondary flow typical of turbulent duct flows: eight regions (four in the half-section shown) can be easily identified with alternate sign of vorticity, located symmetrically with respect to the horizontal and vertical lines passing through the centre and with the two diagonals. In particular, in figure 4 we mark with a filled black circle the location of one of the maxima of the vorticity, which we find at $y=-0.74h$ , $z=-0.9h$ in the Newtonian case. In the polymeric flows, the streamwise velocity and vorticity components are altered (see figure 5): indeed, the two flows differ mainly in the near-wall region, with strong differences in the streamwise velocity close to the corners. The difference in the streamwise vorticity component is mostly concentrated along the edges. In particular, the streamwise velocity contour exhibits higher values close to the corners and, at the same time, the secondary flow is modified, with the locations of the maximum vorticity moving towards the centre, i.e. they are displaced away from the walls. In fact, the coordinates of the selected maximum vorticity become $y=0.23h$ , $z=0.78h$ . The magnitude of the vorticity maxima are also modified, reducing by 20 % from $0.29U_{b}/h$ in the Newtonian fluid to $0.23U_{b}/h$ in the presence of polymers, while the overall integral of the vorticity in each of the eight sectors increases by 17 % from $0.023U_{b}/h$ in the Newtonian flow to $0.027U_{b}/h$ in the viscoelastic one. Thus, although the peak value of the in-plane vorticity is reduced, overall it is enhanced by the polymer addition due to an enlargement of the area with non-zero vorticity. The modification of the secondary flow is further analysed by the maximum of the magnitude of $\overline{v}$ in the domain; indeed, this slightly decreases from approximately $0.019U_{b}$ in the Newtonian flow to $0.018U_{b}$ in the polymer solution.

Figure 5.

Contours of the difference of the mean streamwise component of (a) the velocity $\overline{u}$ and (b) the vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ between the Newtonian and viscoelastic fluids. The colour scale ranges from $-0.2$ (blue) to $0.2U_{b}$ (red) in (a) and from $-0.5U_{b}/h$ (blue) to $0.5U_{b}/h$ (red) in (b).

Figure 6.

Mean streamwise velocity as a function of the wall-normal distance in logarithmic scale and wall units. The blue and red lines are used for the numerical results without and with polymers. The magenta dashed line is the polymer maximum drag reduction (Virk Reference Virk1975; Lvov et al. Reference Lvov, Pomyalov, Procaccia and Tiberkevich2004) while the black dashed line is the Newtonian log law. The panels correspond to different sections: (a)  $z=0$ , (b)  $z=0.3h$ and (c)  $z=0.6h$ .

Figure 6 shows the mean streamwise velocity component $\overline{u}$ as a function of the wall-normal distance $\widetilde{y}$ (both in wall units). Figure 6 shows (a) the mean velocity profile in the midplane ( $z=0$ ), (b) the velocity at $z=0.3h$ and (c) the velocity at $z=0.6h$ . In the midplane, where the secondary flow is weak, we can identify three regions in the velocity profile of the Newtonian fluid (figure 6 a) similarly to what is found for a turbulent channel flow: first, the viscous sublayer for $\widetilde{y}^{+}<5$ where the variation of $\overline{u}^{+}$ with $\widetilde{y}^{+}$ is approximately linear; then, the so-called log-law region, $\widetilde{y}^{+}>30$ , where the variation of $\overline{u}^{+}$ versus $\widetilde{y}^{+}$ is logarithmic; finally, the region between 5 and 30 wall units is called the buffer layer and neither laws hold. As we approach the sidewalls, $x=0.3h$ and $0.6h$ , the range of the equilibrium layer where the velocity profile is logarithmic reduces (figure 6 b), eventually disappearing (figure 6 c). The profiles have similar trends for the viscoelastic fluid, in which case, however, the inertial range has an upward shift, consistent with the observed drag reduction (Warholic et al. Reference Warholic, Massah and Hanratty1999). Note that all the data stay below the MDR curve (Virk Reference Virk1975; Lvov et al. Reference Lvov, Pomyalov, Procaccia and Tiberkevich2004), a limit curve approached only in the case of very high $Wi$ , shown as the magenta dashed line in figure 6.

Figure 7.

Wall-normal mean profile of (a–c) the streamwise r.m.s. velocity fluctuation, (d–f) the wall-normal r.m.s. velocity fluctuation, and (g–i) the cross-term of the Reynolds stress tensor $u^{\prime }v^{\prime }$ . The columns correspond to different sections: (a,d,g)  $z=0$ , (b,e,h)  $z=0.3h$ and (c,f,i)  $z=0.6h$ . The line and colour style is the same as in figure 6.

Figure 7 displays the velocity fluctuation profile as a function of the wall-normal distance $\widetilde{y}$ . In particular, figure 7(a–c) reports the root mean square (r.m.s.) of the streamwise component of the velocity fluctuations $u_{rms}$ , figure 7(d–f) shows the r.m.s. of one of the wall-normal components $v_{rms}$ , and figure 7(g–i) displays the cross-term of the Reynolds stress tensor, $u^{\prime }v^{\prime }$ . The Newtonian case shows a peak of $u_{rms}$ close to the wall, with slightly different values in the three sections, decreasing when moving from the midplane towards the sidewalls. Away from the wall, the streamwise velocity fluctuations reach a plateau with a local minimum whose amplitude is approximately half the peak value in the centreline. On the other hand, the wall-normal velocity fluctuations have a weak maximum close to the wall in the centreline, which then decreases at $z=0.3h$ , and eventually disappears at $z=0.6h$ , when approaching the corners. The flow with polymers displays similar r.m.s. profiles; however, the peaks of both $u_{rms}$ and $v_{rms}$ are located further away from the wall; also, a strong decrease in the wall-normal fluctuation $v_{rms}$ is observed in all sections, whereas an increase in the streamwise component is evident only close to the centreline, as observed in-plane channels. Finally, we consider the cross-term of the Reynolds stress tensor $u^{\prime }v^{\prime }$ ; the profiles show a peak close to the wall and a value approaching zero towards the centreline due to symmetry. The profile at $z=0.3h$ has a lower peak, and still decreases monotonically after its maximum value; on the other hand, the Reynolds stress $u^{\prime }v^{\prime }$ closer to the sidewalls, $z=0.6h$ , goes to zero rapidly and remains approximately zero until ${\tilde{y}}=h$ . In the cases with polymers, the Reynolds shear stress is much weaker than in the Newtonian cases; it even becomes negative away from the wall in the section at $z=0.6h$ . This is due to the transverse shear near the walls leading to additional vorticity (figure 4); this effect is more pronounced in the viscoelastic case than in the Newtonian flow, due to the enhancement of the near-wall vorticity, as shown in figure 5(b). Overall, the velocity fluctuation data indicate that the polymer solution has enhanced streamwise velocity fluctuations, and strongly reduced in-plane components and shear stresses, in good agreement with many previous studies in the literature (see e.g. the collection of data from numerous authors in Escudier et al. (Reference Escudier, Nickson and Poole2009)).

Figure 8.

Contours of four components of the conformation tensor $\unicode[STIX]{x1D60A}_{ij}$ : (a)  $\unicode[STIX]{x1D60A}_{11}$ , (b)  $\unicode[STIX]{x1D60A}_{22}$ , (c)  $\unicode[STIX]{x1D60A}_{12}$ and (d)  $\unicode[STIX]{x1D60A}_{23}$ . Colour scales range from 30 (green) to 800 (red), from 5 (blue) to 50 (red), from $-30$ (blue) to 30 (red), and from $-10$ (blue) to 10 (red), for panels (a) to (d), respectively.

We conclude the analysis of the main flow statistics by showing in figure 8 four components of the conformation tensor $\unicode[STIX]{x1D60A}_{ij}$ . Note that the conformation tensor is strongly related to the added stress term due to the polymers (see (2.2)). In particular, in figure 8 we show the streamwise diagonal component  $\unicode[STIX]{x1D60A}_{11}$ , one of the two diagonal in-plane components  $\unicode[STIX]{x1D60A}_{22}$ , the corresponding off-diagonal term  $\unicode[STIX]{x1D60A}_{12}$ , and the in-plane off-diagonal term  $\unicode[STIX]{x1D60A}_{23}$ . The other components can be deduced from symmetry arguments: in particular, $\unicode[STIX]{x1D60A}_{33}$ and $\unicode[STIX]{x1D60A}_{13}$ can be obtained from $\unicode[STIX]{x1D60A}_{22}$ and $\unicode[STIX]{x1D60A}_{12}$ by symmetry around the bisection of the quadrants I and III. As expected, the streamwise component is the dominant one, with strong peak values located close to the wall, while, interestingly, in the four corners and in the centre of the duct $\unicode[STIX]{x1D60A}_{11}$ reaches its minimum value. Also, $\unicode[STIX]{x1D60A}_{11}$ is the only diagonal component of the conformation tensor showing the full eight symmetries; indeed, $\unicode[STIX]{x1D60A}_{22}$ presents only horizontal and vertical symmetries, with the diagonal ones recovered through the combination with the other component $\unicode[STIX]{x1D60A}_{33}$ (not shown here). In particular, the in-plane diagonal component $\unicode[STIX]{x1D60A}_{22}$ is minimum at the wall, and acts mainly on the horizontal centreline. The cross-term $\unicode[STIX]{x1D60A}_{12}$ has its peak located at the walls, while the component $\unicode[STIX]{x1D60A}_{23}$ has its peaks on the two diagonals. The in-plane shear component $\unicode[STIX]{x1D60A}_{23}$ is not zero close to the corners, and over the two diagonals, where it shows a series of alternating maxima and minima, in a similar fashion to the shear Reynolds stress, which exhibits alternate signs moving from one wall to the other (figure 7). Figures 9(a), 9(b) and 9(c) show the wall-normal profiles of $\unicode[STIX]{x1D60A}_{11}$ (solid line) and $\unicode[STIX]{x1D60A}_{22}+\unicode[STIX]{x1D60A}_{33}$ (dashed line) in the midplane of the duct ( $z=0$ ), at $z=0.3h$ and at $z=0.6h$ , respectively. Note that $\unicode[STIX]{x1D60A}_{22}+\unicode[STIX]{x1D60A}_{33}$ is a measure of the in-plane stretching of the polymers, while its sum with $\unicode[STIX]{x1D60A}_{11}$ is the trace of the conformation tensor. We observe that $\unicode[STIX]{x1D60A}_{11}$ (and the trace) has a peak close to the wall and then decreases to its minimum value in the centre of the duct. Also, the peak value increases moving from the midplane towards the sidewalls; $\unicode[STIX]{x1D60A}_{11}$ starts increasing for all ${\tilde{y}}$ close to the sidewalls ( $z=0.6h$ ) as we are approaching the sidewall peaks (see figure 8 a). The in-plane stretching ( $\unicode[STIX]{x1D60A}_{22}+\unicode[STIX]{x1D60A}_{33}$ ) is more moderate than the streamwise one, and the profiles show minimum values at the wall. Also, the peak values of $\unicode[STIX]{x1D60A}_{22}+\unicode[STIX]{x1D60A}_{33}$ are located around $h/2$ in the midplane and move towards  ${\tilde{y}}=1$ when approaching the sidewalls (see figure 8 b). Figures 8 and 9 show that the polymers mainly elongate in the streamwise direction in the near-wall region and away from the corners; however, in-plane elongation originating from the secondary cross-flow is present as well, especially at a distance from the walls of approximately $h/2$ .

Figure 9.

Mean profile of the streamwise $\unicode[STIX]{x1D60A}_{11}$ (solid line) and in-plane component $\unicode[STIX]{x1D60A}_{22}+\unicode[STIX]{x1D60A}_{33}$ (dashed line) of the conformation tensor $\unicode[STIX]{x1D60A}_{ij}$ . The panels correspond to different sections: (a)  $z=0$ , (b)  $z=0.3h$ and (c)  $z=0.6h$ .

Figure 10.

(a) Cross-correlations of the streamwise (left half) and wall-normal (right half) velocities and polymer force, defined in (3.1). Colour scales range from $-1$ to 1, and from $-0.25$ to 0.25, in the left and right halves, respectively. (b) Contours of the mean first normal difference ${\mathcal{N}}_{1}$ , with the colour scale ranging from 0 (blue) to 0.35 (red).

Next, figure 10(a) shows the cross-correlation $\unicode[STIX]{x1D70C}_{i}$ defined as

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{i}={\displaystyle \frac{\overline{u_{i}\,f_{i}}}{{u_{i}}_{rms}\,{f_{i}}_{rms}}},\end{eqnarray}$$

where $f_{i}$ is the polymer volume force, i.e. the contribution to the Navier–Stokes equation of the polymeric stress tensor $f_{i}=(1-\unicode[STIX]{x1D6FD})/Re\,\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{ij}/\unicode[STIX]{x2202}x_{j}$ . The left half of figure 10(a) shows the streamwise component  $\unicode[STIX]{x1D70C}_{1}$ , while the right half shows the two identical wall-normal ones $\unicode[STIX]{x1D70C}_{2}=\unicode[STIX]{x1D70C}_{3}$ . In the streamwise direction ( $\unicode[STIX]{x1D70C}_{1}$ ), the polymer contribution is correlated with the corresponding velocity component in the near-wall region of the midplane far from the corners, thus enhancing fluctuations in this particular region. On the contrary, the streamwise polymer body force and velocity become anticorrelated in the bulk of the flow away from the walls, where the polymer contribution opposes the turbulent fluctuations. This is similar to what was found by Dubief et al. (Reference Dubief, Terrapon, White, Shaqfeh, Moin and Lele2005) for a turbulent plane channel flow. However, as the corners are approached, the behaviour drastically changes and the two streamwise components are strongly anticorrelated everywhere, even in the near-wall region, thus indicating that the addition of polymers tends to suppress fluctuation in the corners, as also shown in figure 7(c,f,i). The wall-normal component of the cross-correlation $\unicode[STIX]{x1D70C}_{2,3}$ shows a different behaviour; the wall-normal velocity and polymer force are anticorrelated almost everywhere, especially in the bulk of the flow, while they are strongly correlated in the four corners, where the polymer in-plane stress is acting to enhance the turbulent fluctuations.

Finally, we analyse the mean first normal stress difference ${\mathcal{N}}_{1}=\unicode[STIX]{x1D70E}_{11}-\unicode[STIX]{x1D70E}_{22}$ , with $\unicode[STIX]{x1D70E}_{11}$ and $\unicode[STIX]{x1D70E}_{22}$ being the mean streamwise and wall-normal total stresses, i.e. the sum of the solvent and polymer stresses. The first normal stress difference ${\mathcal{N}}_{1}$ averaged over the whole domain is positive whereas the second one ${\mathcal{N}}_{2}$ is slightly negative, with $|{\mathcal{N}}_{1}/{\mathcal{N}}_{2}|\gg 1$ . The spatial distribution of ${\mathcal{N}}_{1}$ is shown in figure 10(b); ${\mathcal{N}}_{1}$ is greater than zero in the whole domain and its peaks are located at the wall, around $0.5h$ from the sidewalls. Also, the first normal stress difference is close to zero at the centreline and in the four corners. Note that, as expected, the described spatial distribution of ${\mathcal{N}}_{1}$ closely follows that of $\unicode[STIX]{x1D60A}_{11}$ previously discussed in figure 8.

3.2 Flow structures

To further characterise the flow, we study the mean local wall stress along one edge in wall coordinates normalised by the average over the whole wall, similarly to what was proposed by Pinelli et al. (Reference Pinelli, Uhlmann, Sekimoto and Kawahara2010). These authors suggested that an upper bound for the number of wall velocity streaks over one of the four walls can be estimated from the dimension of the square expressed in wall units, since the average distance between streaks of different velocity sign is of the order of $50^{+}$ (Kim et al. Reference Kim, Moin and Moser1987). In our case, we can expect more streaks in the Newtonian case than in the viscoelastic flow with polymers since increasing streak spacing is a well-known effect of polymer drag reduction that has been observed in both experiment and numerical works (e.g. De Angelis, Casciola & Piva Reference De Angelis, Casciola and Piva2002). Indeed, this argument is confirmed by figure 11, which shows that in the Newtonian case each edge of the duct hosts up to five streaks, with two high-velocity streaks close to the sidewalls and one in the centre, and two low-velocity streaks in between, represented in figure 11 by the three maxima and two minima in the wall stress profile. In the drag-reduced case with polymers, the profile presents only two maxima and one minimum, indicating that two high-velocity streaks are close to the sidewalls and one low-velocity streak is located between them at the centre of the edge; also, the two peaks are closer to the centreline than in the Newtonian case. In figure 11, we also report with symbols three curves pertaining to the results by Pinelli et al. (Reference Pinelli, Uhlmann, Sekimoto and Kawahara2010) for a Newtonian duct flow at $Re_{\unicode[STIX]{x1D70F}}=178$ (▴), 150 ( $\ast$ ) and 120 (▾): the first is used as reference, showing the good agreement between our Newtonian results and those from the literature, while the other two are included to provide further insight into the polymeric flow. Indeed, the Newtonian case at $Re_{\unicode[STIX]{x1D70F}}=150$ has approximately the same Reynolds number as our polymeric flow (155), but still shows a local maximum at the centreline, while the Newtonian case at $Re_{\unicode[STIX]{x1D70F}}=120$ exhibits a pattern of minima and maxima similar to the one obtained by our simulation with polymers, even if with different (lower) Reynolds number. Thus, even at a similar friction Reynolds number, the shear stress profile does not resemble that of a Newtonian flow due to the drag-reducing effect of polymers.

Figure 11.

Mean local wall stress along one edge in wall coordinates normalised by the average over the whole wall  $\unicode[STIX]{x1D70F}_{w}$ . The blue and red lines are used to show our numerical results of the Newtonian and polymeric flows, respectively, while the grey lines with symbols are the data taken from the work of Pinelli et al. (Reference Pinelli, Uhlmann, Sekimoto and Kawahara2010) for three Newtonian flows with different Reynolds numbers: $Re_{\unicode[STIX]{x1D70F}}=178$ (▴), 150 ( $\ast$ ) and 120 (▾).

Figure 12.

Contours of the instantaneous (a,c) streamwise velocity fluctuation component $u^{\prime }$ in the $x{-}z$ plane located at $\widetilde{y}\approx 18^{+}$ from the wall and of (b,d) streamwise vorticity $\unicode[STIX]{x1D714}_{x}$ in the $x{-}y$ plane. Panels show the Newtonian (a,b) and polymer (c,d) flows, respectively. The colour scale ranges from $-0.5U_{b}$ (blue) to $0.5U_{b}$ (red).

The previous result is confirmed by figure 12(a), which reports contours of the instantaneous streamwise velocity fluctuation $u^{\prime }$ in an $x{-}z$ section located at a distance $\widetilde{y}\approx 18^{+}$ from one wall. Figure 12(a) clearly shows the instantaneous footprints of the near-wall low- and high-speed streaks. Even from an instantaneous picture of the flow, the difference between the Newtonian and polymeric flows appears evident; indeed, the former shows two long low-speed streaks and more than two high-speed streaks, which are shorter and wider than the low-speed ones, whereas the viscoelastic flow presents just one low-speed streak close to the centreline, bounded by two high-speed streaks. Further, both the low- and high speed streaks are stronger and correlated over longer distances in the streamwise direction in the presence of polymers (Warholic et al. Reference Warholic, Massah and Hanratty1999; Escudier et al. Reference Escudier, Nickson and Poole2009). Pinelli et al. (Reference Pinelli, Uhlmann, Sekimoto and Kawahara2010) showed that the mean streamwise vorticity (secondary flow) strongly depends upon the statistically preferred location of the quasi-streamwise vortices associated with the pair of fast/slow streaks closest to the corner. As just shown in figures 11 and 12, the inclusion of the polymers in the flow strongly modifies these structures, by enhancing the streamwise coherence of the flow and by increasing the size of the near-wall structures (see figure 12 c), thus moving the quasi-streamwise vortices away from the bottom wall and sidewalls: therefore, the secondary motion is characterised by large structures close to the centre of the duct, due to the constraint imposed by the size of the domain.

3.3 Vorticity budget

Gavrilakis (Reference Gavrilakis1992) explains the origins of the mean secondary flow using an equation for the mean streamwise vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ , which we here modify for a fully developed duct flow with polymers and whose derivation has been reported in appendix A. The equation for the streamwise vorticity reads

(3.2) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D714}}_{x}}{\unicode[STIX]{x2202}t} & = & \displaystyle \underbrace{\overline{v}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D714}}_{x}}{\unicode[STIX]{x2202}y}+\overline{w}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D714}}_{x}}{\unicode[STIX]{x2202}z}}_{{\mathcal{C}}}+\underbrace{\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y\unicode[STIX]{x2202}z}(\overline{w^{\prime }w^{\prime }}-\overline{v^{\prime }v^{\prime }})}_{{\mathcal{A}}}+\underbrace{\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}z^{2}}\right)\overline{v^{\prime }w^{\prime }}}_{{\mathcal{D}}}\nonumber\\ \displaystyle & & \displaystyle \underbrace{-\frac{\unicode[STIX]{x1D6FD}}{Re}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}z^{2}}\right)\overline{\unicode[STIX]{x1D714}}_{x}}_{{\mathcal{V}}}\;\underbrace{-\frac{1-\unicode[STIX]{x1D6FD}}{Re}\left(\frac{\unicode[STIX]{x2202}^{2}\overline{\unicode[STIX]{x1D70F}}_{32}}{\unicode[STIX]{x2202}y^{2}}+\frac{\unicode[STIX]{x2202}^{2}\overline{\unicode[STIX]{x1D70F}}_{33}}{\unicode[STIX]{x2202}z\unicode[STIX]{x2202}y}-\frac{\unicode[STIX]{x2202}^{2}\overline{\unicode[STIX]{x1D70F}}_{22}}{\unicode[STIX]{x2202}y\unicode[STIX]{x2202}z}-\frac{\unicode[STIX]{x2202}^{2}\overline{\unicode[STIX]{x1D70F}}_{23}}{\unicode[STIX]{x2202}z^{2}}\right)}_{{\mathcal{P}}}=0.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The first two terms in the equation (denoted as ${\mathcal{C}}$ ) are the convection of the mean vorticity by the secondary flow itself, while the last two terms are a sink of streamwise vorticity due to the viscosity ( ${\mathcal{V}}$ ) and the polymer contribution ( ${\mathcal{P}}$ ), which we found to be a sink as well. The two remaining terms involve the Reynolds stresses: the first is a production term associated with the anisotropy of the in-plane normal stresses ( ${\mathcal{A}}$ ), while the second ( ${\mathcal{D}}$ ) can be both a production and dissipation term due to the cross-Reynolds-stress component $\overline{v^{\prime }w^{\prime }}$ . Figure 13 shows the contribution of the five terms highlighted in (3.2), i.e.  ${\mathcal{C}}$ , ${\mathcal{A}}$ , ${\mathcal{D}}$ , ${\mathcal{V}}$ and ${\mathcal{P}}$ , in one of the eight sectors and averaged among all the sectors. Blue and red colours are used for the Newtonian and polymeric flows, respectively. The only positive contribution in the Newtonian case is provided by the anisotropic terms, while all the others are negative. With polymers, the polymeric contribution ${\mathcal{P}}$ is found to be a sink of vorticity, the anisotropic and viscous terms decrease in magnitude compared to the Newtonian flow, while ${\mathcal{D}}$ changes sign and increases in magnitude, changing from dissipation for the Newtonian flow to production for the flow with polymers. Finally, the convection term ${\mathcal{C}}$ is null for both cases. The budget analysis suggests that, although the polymeric stress is small in amplitude, its presence modifies the flow itself, by reducing the anisotropy of the in-plane velocity fluctuations.

Figure 13.

Contributions to the budget of the mean streamwise vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ : convection  ${\mathcal{C}}$ , production due to anisotropy of the Reynolds stress  ${\mathcal{A}}$ , production/dissipation source term  ${\mathcal{D}}$ , viscous dissipation  ${\mathcal{V}}$ , and polymeric contribution  ${\mathcal{P}}$ . Blue and red colours are used for the Newtonian and viscoelastic flows, respectively.

Figure 14.

Contributions to the streamwise Reynolds-stress tensor $\overline{u^{\prime }u^{\prime }}$ equation for the Newtonian (a,c) and viscoelastic case with polymers (b,d). On each side (a,c) and (b,d), the contours represent the terms ${\mathcal{A}}_{ij}$ , ${\mathcal{Q}}_{ij}$ , ${\mathcal{R}}_{ij}$ , ${\mathcal{P}}_{ij}$ , ${\mathcal{D}}_{ij}$ , $\unicode[STIX]{x1D716}_{ij}$ , ${\mathcal{W}}_{ij}$ and the total balance (see (3.3)) in order from top left to bottom right. The ranges of the colour scales are $\pm 0.002$ , $\pm 0.008$ , $\pm 0.005$ , $\pm 0.025$ , $\pm 0.01$ , $\pm 0.02$ , $\pm 0.003$ and $\pm 0.001$ , respectively, from top left to bottom right, with colours going from blue (negative values) to red (positive value) and with the zero coloured in green.

3.4 Energy budget

To better understand the effect of polymers on the streamwise velocity fluctuations $\overline{u^{\prime }u^{\prime }}$ , which are related to streaks, we consider the perturbation energy budget. The transport equation for the Reynolds stress in a viscoelastic flow is given by

(3.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\unicode[STIX]{x2202}t} & = & \displaystyle \underbrace{-\overline{u}_{k}\frac{\unicode[STIX]{x2202}\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\unicode[STIX]{x2202}x_{k}}}_{{\mathcal{A}}_{ij}}\underbrace{-\overline{u}_{k}\frac{\unicode[STIX]{x2202}\overline{u_{i}^{\prime }u_{j}^{\prime }u_{k}^{\prime }}}{\unicode[STIX]{x2202}x_{k}}}_{{\mathcal{Q}}_{ij}}\underbrace{-\left(\overline{u_{j}^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{i}}}+\overline{u_{i}^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{j}}}\right)}_{{\mathcal{R}}_{ij}}\underbrace{-\left(\overline{u_{j}^{\prime }u_{k}^{\prime }}\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{k}}+\overline{u_{i}^{\prime }u_{k}^{\prime }}\frac{\unicode[STIX]{x2202}\overline{u}_{j}}{\unicode[STIX]{x2202}x_{k}}\right)}_{{\mathcal{P}}_{ij}}\nonumber\\ \displaystyle & & \displaystyle +\,\underbrace{\frac{\unicode[STIX]{x1D6FD}}{Re}\frac{\unicode[STIX]{x2202}^{2}\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\unicode[STIX]{x2202}x_{k}^{2}}}_{{\mathcal{D}}_{ij}}\underbrace{-\frac{2\unicode[STIX]{x1D6FD}}{Re}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}}_{\unicode[STIX]{x1D716}_{ij}}+\underbrace{\frac{1-\unicode[STIX]{x1D6FD}}{Re}\left(\overline{u_{j}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{ik}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}+\overline{u_{i}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{jk}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}\right)}_{{\mathcal{W}}_{ij}}=0,\end{eqnarray}$$

where ${\mathcal{A}}_{ij}$ is the advection by the mean flow, ${\mathcal{Q}}_{ij}$ the transport by the velocity fluctuations, ${\mathcal{R}}_{ij}$ the pressure term, ${\mathcal{P}}_{ij}$ the production against the mean shear, ${\mathcal{D}}_{ij}$ the viscous diffusion, $\unicode[STIX]{x1D716}_{ij}$ the dissipation, and ${\mathcal{W}}_{ij}$ the polymer work. The evolution equation for the perturbation energy is obtained by setting $i=j$ , and in particular we consider here the streamwise component $i=j=1$ . Figure 14 shows the contours of these seven contributions, and their sum, for the Newtonian (a,c) and polymeric flow (b,d). First, we note that, as expected, the sum of all the contribution is in both cases almost zero (up to $10^{-3}$ ), since the mean Reynolds stress $\overline{u^{\prime }u^{\prime }}$ has reached steady state.

We first discuss the Newtonian flow, reported in figure 14(a,c); the advection ${\mathcal{A}}_{ij}$ and transport ${\mathcal{Q}}_{ij}$ terms display strong gradients close to the walls. In particular, the distribution of ${\mathcal{A}}_{ij}$ reveals that the maximum advection occurs in regions close to the maximum in-plane mean vorticity, i.e. where the secondary flow is maximum (see figure 4). On the other hand, the turbulent transport, ${\mathcal{Q}}_{ij}$ , is observed to be approximately uniform along the edges; in particular, it is zero at the wall, reaches a positive peak value in the near-wall region, followed by a minimum, and finally goes back to zero away from the wall, thus indicating that the largest gradients of the high-order statistics are located mainly in the near-wall region, all along the edges. The pressure and production terms, ${\mathcal{R}}_{ij}$ and ${\mathcal{P}}_{ij}$ , exhibit opposite behaviour: the former is always negative and the latter always positive, reaching a minimum/maximum and then approaching zero again in the bulk of the duct. The viscous diffusion ${\mathcal{D}}_{ij}$ displays a strong peak at the wall and then decreases to zero, while the dissipation term $\unicode[STIX]{x1D716}_{ij}$ has a minimum at the wall, and then increases to its maximum value in the centre of the duct. Figure 14(a,c) shows that all the contributions have an almost uniform distribution along the edges, with only a weak dependence on the distance from the corners, except for the advection ${\mathcal{A}}_{ij}$ and transport ${\mathcal{Q}}_{ij}$ terms.

The behaviour of the various terms in the kinetic energy budget is different for the viscoelastic flow, as shown in figure 14(b,d). In this case, all the terms are not uniform along the edge and strongly depend on the distance from the sidewalls; this is related to the reduced friction Reynolds number of the flow and the consequent increase in size of the near-wall structures previously discussed. In particular, the positive and negative peaks of the advection ${\mathcal{A}}_{ij}$ term are found close to the centreline far from the corners, similarly to the transport term ${\mathcal{Q}}_{ij}$ , which exhibits strong negative and positive peaks in the same region. Also, the pressure ${\mathcal{R}}_{ij}$ , production ${\mathcal{P}}_{ij}$ and dissipation $\unicode[STIX]{x1D716}_{ij}$ terms exhibit a similar trend, with maxima and minima located away from the corner. Finally, the polymer contribution ( ${\mathcal{W}}_{ij}$ ) is not null, but its amplitude is small compared to the other terms, and its distribution across the whole duct is almost uniform. These results clearly indicate that the polymers strongly modify the flow; in particular, their presence induces a less uniform flow in the duct, with most of the streamwise flow fluctuations displaced towards the centreline and away from the corner, as we have already noticed in figure 10. The location where on average the low- and high-speed streaks are present, as previously shown in figures 11 and 12, coincides with the location of major activity of the flow where strong peaks of advection, transport, production and dissipation are present.

3.5 Elasticity effect

We now assess the effect of the parameters defining the viscoelastic behaviour of the fluid. Table 1 reports all the different simulations that we have performed and the drag reduction $DR$ obtained in each case. In particular, starting from the reference polymeric case discussed before (P), we have varied independently the Weissenberg number $Wi$ (P- $Wi\uparrow$ ), the dumbbell extensibility $L$ (P- $L\downarrow$ and P- $L\uparrow$ ) and the viscosity ratio $\unicode[STIX]{x1D6FD}$ (P- $\unicode[STIX]{x1D6FD}\uparrow$ ). From table 1 we can observe that, starting from the drag reduction reported for the reference polymeric case (29 %), a relevant reduction or increase of $L$ brings about only small variations in the overall drag reduction (27 % and 29 %), while even smaller variations of $\unicode[STIX]{x1D6FD}$ are sufficient to alter the solution, as demonstrated by a drag reduction of 25 % for the case with higher viscosity ratio P- $\unicode[STIX]{x1D6FD}\uparrow$ . However, the parameter that has the greatest effect on the solution appears to be the Weissenberg number $Wi$ . Indeed, when $Wi=3$ (P- $Wi\uparrow$ ) the drag reduction increases to 43 %. We will now analyse in more detail this polymeric case with $Wi=3$ and compare the flow with the one discussed above, i.e. the Newtonian and the polymeric with $Wi=1.5$ .

Table 1.

Summary of all the DNS performed at a fixed bulk Reynolds number equal to $Re=2800$ and with different viscoelastic fluids.

Figure 15 shows the contours of the mean streamwise components of the velocity $\overline{u}$ and vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ for the two different Weissenberg numbers considered. As $Wi$ increases, the maximum streamwise velocity is only slightly affected, while the maximum vorticity strongly decreases by 40 %. However, the integral of the vorticity keeps increasing, thus confirming a progressive increase of the in-plane motion with the Weissenberg number $Wi$ . The location of maximum vorticity, marked with a black dot in figure 15, moves further away from the walls, displacing towards the centre; however, this displacement is smaller than what was previously observed when comparing the Newtonian case with the polymeric one with $Wi=1.5$ (see figure 4 b), thus indicating a nonlinear trend with  $Wi$ .

Figure 15.

Contours of the mean streamwise component of (a) velocity $\overline{u}$ and (b) vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ . In each panel, the left (right) half of the duct is used for the polymeric cases with $Wi=1.5$ ( $Wi=3$ ), respectively. The colour scale ranges from 0 (blue) to $1.4U_{b}$ (red) in (a), and from $-0.5U_{b}/h$ (blue) to $0.5U_{b}/h$ (red) in (b). Panel (a) also reports the in-plane mean velocity with arrows.

Figure 16.

Mean streamwise velocity as a function of the wall-normal distance in logarithmic scale and wall units. The blue, red and brown lines are used for the numerical results without and with polymers with $Wi=1.5$ and 3, respectively. The magenta dashed line is the polymer maximum drag reduction (Virk Reference Virk1975; Lvov et al. Reference Lvov, Pomyalov, Procaccia and Tiberkevich2004) while the black dashed line indicates the Newtonian log law. The panels correspond to different sections: (a)  $z=0$ , (b)  $z=0.3h$ and (c)  $z=0.6h$ .

Next, we study the mean streamwise velocity profiles in wall units reported in figure 16. Here, we compare the profiles of the Newtonian fluid and the two viscoelastic cases with different Weissenberg number $Wi$ . In all the considered sections, we observe a progressive departure from the Newtonian velocity profile as $Wi$ increases, with the slope of the inertial range of scales growing with $Wi$ ; the curves for $Wi=3$ approach the maximum drag reduction limit.

Figure 17.

Wall-normal mean profile of (a–c) the streamwise r.m.s. velocity fluctuation, (d–f) the wall-normal r.m.s. velocity fluctuation, and (g–i) the cross-term of the Reynolds stress tensor $u^{\prime }v^{\prime }$ . The columns correspond to different sections: (a,d,g)  $z=0$ , (b,e,h)  $z=0.3h$ and (c,f,i)  $z=0.6h$ . The line and colour style is the same as in figure 16.

Figure 17 shows the profiles of the r.m.s. of the velocity fluctuations and of the Reynolds shear stress component as a function of the wall-normal distance  ${\tilde{y}}$ . We observe that the streamwise velocity fluctuation $u_{rms}$ is only slightly altered by the increase of $Wi$ , except its peak value, which displaces further away from the wall. On the other hand, both the in-plane fluctuation $v_{rms}$ and the shear Reynolds stress component $\overline{u^{\prime }v^{\prime }}$ are strongly affected by the increase of $Wi$ . In particular, both $v_{rms}$ and $\overline{u^{\prime }v^{\prime }}$ are reduced across the whole domain, as already found for the turbulent channel flows with polymers (Ptasinski et al. Reference Ptasinski, Boersma, Nieuwstadt, Hulsen, Van den Brule and Hunt2003; Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004).

Figure 18.

Contours of the mean first normal difference ${\mathcal{N}}_{1}$ , with the colour scales ranging from 0 (blue) to 0.8 (red) for the polymeric cases with $Wi=1.5$  (a) and $Wi=3$  (b).

Finally, we report in figure 18 the effect of the Weissenberg number on the mean first normal stress difference ${\mathcal{N}}_{1}$ . This strongly increases with $Wi$ , with its maximum value growing from approximately 0.3 for $Wi=1.5$ (figure 18 a) to 0.8 in the high-Weissenberg-number case (figure 18 b). We also note that the spatial distribution remains similar for the different $Wi$ studied here, with the peak values of ${\mathcal{N}}_{1}$ close to the walls and decreasing to almost zero in the centre of the duct and in the four corners. However, while for $Wi=1.5$ the region close to the centreline showed small values of the first normal stress difference, with a peak located at a distance of approximately $0.5h$ from the sidewalls, the case with $Wi=3$ exhibits a more uniform distribution around the wall bisectors, except in the corners, where it remains almost zero.