3.1 Linear growth and nonlinear energy saturation

The first set of direct numerical simulations start from the laminar base flow profile and a small-amplitude TS wave. In order to ensure that the initial stage is within the linear regime, both the velocity and conformation tensor of the perturbation must be small relative to the mean-flow values. The initial amplitude of TS wave was set to $\unicode[STIX]{x1D716}=0.01\,\%$ relative to the bulk velocity, which guarantees the presence of an initial linear regime during the evolution of the disturbance. The ability to reproduce the linear growth rate of a small-amplitude TS wave using a numerical solution of the full Navier–Stokes equations serves as a validation of our numerical method. In addition, it guarantees that a nonlinear state, once reached, is also a solution of the governing equations and an appropriate starting point for the subsequent transition computations.

The perturbation energy density was evaluated from each of the four simulations,

(3.1) $$\begin{eqnarray}E(t)=\frac{1}{L_{x}\,L_{y}\,L_{z}}\int _{0}^{L_{z}}\int _{0}^{L_{y}}\int _{0}^{L_{x}}\frac{1}{2}({u^{\prime }}^{2}+{v^{\prime }}^{2}+{w^{\prime }}^{2})\,\text{d}x\,\text{d}y\,\text{d}z.\end{eqnarray}$$

Note that the perturbation field is initially monochromatic in $k_{x}$ , and has a discrete spectrum in the nonlinear regime due to the generation of higher harmonics. As such, the perturbation field was obtained by subtracting the streamwise–spanwise average from the total signal. Figure 5 shows the evolution of $E(t)$ from the initial time and through $t=4650$ convective time units. The initial linear amplification of $\log (E)$ corresponds to the exponential amplification of the TS wave in all four simulations. The slopes of the curves from DNS match the predictions of linear stability theory, which are marked in the figure by dashed lines with slopes equal to $2\unicode[STIX]{x1D70E}_{i}$ (values of $\unicode[STIX]{x1D70E}_{i}$ are listed in table 2).

Figure 5.

Evolution of the perturbation energy at $Re=4667$ , $\unicode[STIX]{x1D6FD}=0.9$ and $L_{max}=100$ . (—●—) Newtonian; (—▵—) $Wi=1.83$ ; (—○—) $Wi=4.50$ ; (—♢—) $Wi=6.67$ . Dashed lines mark the prediction from linear stability theory.

Zhang et al. (Reference Zhang, Lashgari, Zaki and Brandt2013) examined the change in the growth rate of the primary TS wave by evaluating the various terms in the perturbation energy budget, and a similar analysis can also be instructive in the nonlinear regime. The evolution equation for Reynolds stress is,

(3.2) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\unicode[STIX]{x2202}t} & = & \displaystyle \underbrace{-\overline{u}_{k}\frac{\unicode[STIX]{x2202}\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\unicode[STIX]{x2202}x_{k}}}_{2{\mathcal{A}}_{ij}}\underbrace{-\left(\overline{u_{j}^{\prime }u_{k}^{\prime }}\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{k}}+\overline{u_{i}^{\prime }u_{k}^{\prime }}\frac{\unicode[STIX]{x2202}\overline{u}_{j}}{\unicode[STIX]{x2202}x_{k}}\right)}_{2{\mathcal{P}}_{ij}}\nonumber\\ \displaystyle & & \displaystyle \underbrace{-\,\frac{\unicode[STIX]{x2202}\overline{u_{i}^{\prime }u_{j}^{\prime }u_{k}^{\prime }}}{\unicode[STIX]{x2202}x_{k}}}_{2{\mathcal{Q}}_{ij}}\underbrace{-\left(\overline{u_{j}^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{i}}}+\overline{u_{i}^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{j}}}\right)}_{2\unicode[STIX]{x1D719}_{ij}}+\underbrace{\frac{\unicode[STIX]{x1D6FD}}{Re}\frac{\unicode[STIX]{x2202}^{2}\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\unicode[STIX]{x2202}x_{k}^{2}}}_{2{\mathcal{D}}_{ij}}\nonumber\\ \displaystyle & & \displaystyle \underbrace{-\,2\frac{\unicode[STIX]{x1D6FD}}{Re}\overline{\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{x_{k}}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{x_{k}}}}_{2\unicode[STIX]{x1D716}_{ij}}+\underbrace{\frac{1-\unicode[STIX]{x1D6FD}}{Re}\left(\overline{u_{j}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D749}_{ik}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}+\overline{u_{i}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D749}_{jk}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}\right)}_{2{\mathcal{W}}_{ij}}.\end{eqnarray}$$

The term ${\mathcal{A}}_{ij}$ is mean advection, ${\mathcal{P}}_{ij}$ is production, ${\mathcal{Q}}_{ij}$ is energy transport by fluctuations, $\unicode[STIX]{x1D719}_{ij}$ is pressure redistribution, ${\mathcal{D}}_{ij}$ is viscous diffusion, $\unicode[STIX]{x1D716}_{ij}$ is dissipation and ${\mathcal{W}}_{ij}$ is polymer work. The energy equation is obtained from (3.2) by setting $i=j$ ,

(3.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}k}{\unicode[STIX]{x2202}t}\equiv \frac{\unicode[STIX]{x2202}\frac{1}{2}\overline{u_{i}^{\prime }u_{i}^{\prime }}}{\unicode[STIX]{x2202}t}=({\mathcal{A}}_{ii}+{\mathcal{Q}}_{ii}+\unicode[STIX]{x1D719}_{ii}+{\mathcal{P}}_{ii}+{\mathcal{D}}_{ii}+\unicode[STIX]{x1D716}_{ii}+{\mathcal{W}}_{ii}). & & \displaystyle\end{eqnarray}$$

The dominant terms in (3.3) were evaluated at $t=1000$ , which is well within the linear regime according to figure 5. Their wall-normal profiles are plotted in figure 6 normalized by the total perturbation energy, for the case with $Wi=6.67$ as means of example. The production term is the largest contributor to the energy budget. Its peak is reached at $y/h=0.13$ , which is the height of the critical layer, or the location where the base flow speed is equal to the phase speed of the TS wave, $\overline{u}(y/h=0.13)=\unicode[STIX]{x1D70E}_{r}/k_{x}$ . As discussed by Zhang et al. (Reference Zhang, Lashgari, Zaki and Brandt2013), the change in the growth rate of the TS wave tracks the change in this peak production term.

Figure 6.

Dominant terms in the perturbation kinetic energy budget (3.3) for $Wi=6.67$ at $t=1000$ , which is within the linear regime. (——) ${\mathcal{P}}_{ii}/E$ ; (– – –) ${\mathcal{Q}}_{ii}/E$ ; (— ⋅ —) $\unicode[STIX]{x1D716}_{ii}/E$ ; ( $\cdots \cdots$ )  ${\mathcal{W}}_{ii}/E$ .

The integral of the production term at $t=1000$ is reported in table 2 normalized by the total perturbation energy. This quantity increases from the Newtonian to the first viscoelastic case, $Wi=1.83$ , which has the highest TS growth rate. Increasing the Weissenberg number further reduces $\unicode[STIX]{x1D70E}_{i}$ , and the integrated production also reduces. Note that when the TS growth rates of the Newtonian and viscoelastic flows are equal ( $Wi=0.00$ and $4.50$ ), the integrated productions are closest. These trends are therefore in agreement with the results by Zhang et al. (Reference Zhang, Lashgari, Zaki and Brandt2013).

The deviation of the energy curve in figure 5 from the initial linear slope signals the onset of nonlinear effects. In this regime, the perturbation velocities or conformation have reached an appreciable amplitude and, as a result, can generate higher harmonics and distort the base state. The base flow distortion, in turn, alters the growth rate of the TS wave. A saturation of the energy curve in figure 5 signals a balance between the rate at which perturbations extract energy from the distorted base flow and the rate at which they dissipate energy due to viscosity. This argument is based on the Newtonian energy equation which is supplemented by a polymer work term in the polymeric case which can lead to richer behaviour. Indeed only three of the four flow conditions show energy saturation, $Wi=\{0.00,1.83,6.67\}$ , while the viscoelastic simulation at $Wi=4.50$ reaches a periodic state with cycles of energy amplification and decay.

The energy of the saturated state and, as a result, the amplitude of the velocity perturbation in this regime depend on the Weissenberg number. Of the four conditions, the highest Weissenberg number simulation saturates at the lowest perturbation energy, $E=5.59\times 10^{-5}$ . The corresponding maximum amplitudes of the perturbation velocities are $u_{max}^{\prime }=1.6\,\%$ and $v_{max}^{\prime }=0.7\,\%$ . These peak values should be contrasted to the work by Sandham & Kleiser (Reference Sandham and Kleiser1992) who simulated K-type transition in channel flow starting from a TS wave with amplitude equal to 4.5 % of the bulk velocity. A similar initial condition to Sandham & Kleiser (Reference Sandham and Kleiser1992) is therefore not feasible in the present study, since a primary TS wave with such high peak amplitude is not realizable at the highest elasticity.

Figure 7.

Terms in the perturbation kinetic energy budget and mean-velocity profiles in the nonlinear saturated states. (a,d) Newtonian flow at $t=4000$ ; (b,e) $Wi=1.83$ at $t=2980$ ; (c,f) $Wi=6.67$ at $t=4500$ . (a–c) Line types retain the same meaning as in figure 6. (——) ${\mathcal{P}}_{ii}/E$ ; (– – –) ${\mathcal{Q}}_{ii}/E$ ; (— ⋅ —) $\unicode[STIX]{x1D716}_{ii}/E$ ; ( $\cdots \cdots$ ) ${\mathcal{W}}_{ii}/E$ . Panels (d–f) show (——) the velocity profile $U$ and (— ⋅ —) its distortion from the laminar solution, $\overline{u}-\overline{u}_{L}$ .

Terms in the kinetic energy budget were evaluated at the saturated state and are reported in figure 7 which contrasts the Newtonian with the low- and high- $Wi$ flows. In the Newtonian and weak elastic case ( $Wi=1.83$ ), the energy transport by the perturbation field becomes significant across the channel, surpassing the production term which is no longer concentrated in the near-wall region. The respective saturated states from these two simulations are qualitatively similar because the relaxation time at $Wi=1.83$ is short relative to the time scale of the TS wave, and elastic effects on the saturated states are inappreciable. In contrast, the terms in the kinetic energy budget of the highest $Wi$ condition are more akin to the profiles from linear theory, despite the fact that the energy evolution in figure 5 deviates from linear theory. The wall-normal profile of the mean velocity, $\overline{u}$ , is also plotted in figure 7 along with the distortion relative to the reference laminar base state, $\overline{u}_{L}=U(y)$ . In the Newtonian and low $Wi$ cases, the mean-flow distortion is appreciable and switches sign twice, near $y=0.2$ and $y=0.6$ . In contrast, the mean-flow distortion is very weak for the highest $Wi$ condition.

One curious observation in figure 5 is the oscillation of the energy curve when $Wi=4.50$ . Terms in the energy budget and the mean velocity profiles in the linear stage and at the peak and trough of the oscillation are compared in figure 8. At the energy peak, the profiles resemble those extracted from the saturated state in the Newtonian and low- $Wi$ cases (figure 7 a,b). However, the production term is more pronounced near the wall, and the magnitude of the nonlinear transport term is relatively weaker. The associated mean-flow distortion is also small compared to the Newtonian and lower $Wi$ cases. This state is not maintained, but rather gives way to a decay in the kinetic energy and a rapid shift in the budget profiles towards a new configuration that is more akin to the linear stage. For example, the negative polymer work term re-emerges in the near-wall region where the mean-flow distortion vanishes. In addition, the maximum production term increases relative to the remaining terms in the budget and will drive energy amplification at the onset of the subsequent cycle.

That the flow does not reach a constant energy state highlights the rich dynamics that ensues when elasticity is present. As the TS wave amplifies, the mean flow is distorted and, as a result, production, dissipation and polymer work are all modified. At low and high elasticity, similar to the Newtonian case, an equilibrium is reached while at the intermediate $Wi$ such equilibrium is not possible and the flow continues in cycles of amplification of decay.

Figure 8.

Terms in the perturbation kinetic energy budget and mean-velocity profiles for $Wi=4.50$ during the periodic nonlinear state. (a) Results from the linear regime are provided for comparison. The results from the nonlinear state are shown at the local (b,d) peaks and (c,e) troughs. (a–c) Line types retain the same meaning. (——) ${\mathcal{P}}_{ii}/E$ ; (– – –) ${\mathcal{Q}}_{ii}/E$ ; (— ⋅ —) $\unicode[STIX]{x1D716}_{ii}/E$ ; ( $\cdots \cdots$ ) ${\mathcal{W}}_{ii}/E$ . Panels (d,e) show (——) the velocity profile $U$ and (— ⋅ —) its distortion from the laminar solution, $\overline{u}-\overline{u}_{L}$ .

The nonlinear state described above remains two-dimensional and does not break down to turbulence, even though the computational domain is three-dimensional. These simulations are important for two reasons: first, they highlight the propensity of polymers to suppress the disturbance energy in nonlinear regime, which is instrumental to understanding the later stages of transition. Second, these simulations provide an accurate initial base state to which a small-amplitude three-dimensional perturbation can be added in order to study the onset of secondary instability and breakdown to turbulence.

3.2 Secondary instabilities

The simulations of the secondary instability start from the flow fields reported in § 3.1. The flow velocities, pressure and conformation tensor are extracted at $t=t^{\ast }$ which is defined as the time where $E(t^{\ast })=2.25\times 10^{-5}$ . In all simulations, this state precedes the onset of any appreciable deviation from the linear energy curve, or any pronounced nonlinear effects. In the Newtonian case, this energy level corresponds to $u_{max}^{\prime }=0.015U_{b}$ , or $u_{max}^{\prime }=0.01U_{c}$ where $U_{c}$ is the centreline velocity. The fields are supplemented with a zero-mean uncorrelated velocity perturbation with magnitude less than $0.001U_{c}$ . The evolution of the total perturbation energy is shown in figure 9, and is separated into the contributions of the three components of the velocity vector in figure 10. On figure 9, three energy levels are marked and identified as early, mid- and late stages of transition. The energy level and time associated with each of these stages are detailed in table 3, and will serve as landmarks for the discussion.

The initial evolution of the total energy in figure 9 preserves the linear growth of the primary TS wave due to the choice of $t^{\ast }$ and despite the addition of white noise. The same region in figure 10(a) confirms that both the streamwise and wall-normal velocities are amplifying according to linear theory, while the small-amplitude initial white noise that was introduced in the spanwise component simply decays (see inset). After some time, the energy amplification rate in figure 9 increases appreciably, which marks the onset of secondary instability. It is important to note that, while the kinetic energy is lowest for $Wi=6.67$ when secondary instability sets in, the perturbation to the conformation tensor is largest for this case.

The increase in the slope of $E(t)$ at the point of secondary instability resembles the behaviour at the start of nonlinear saturation (§ 3.1), but there the flow remained two-dimensional. Here, on the other hand, three-dimensionality has set in and $E_{z}$ also amplifies as shown in figure 10(a). A helpful way to highlight the change from a two- to three-dimensional perturbation field is to plot the fraction of the total energy due to each velocity component (figure 10 b). In the linear regime, the energy distribution between the streamwise and wall-normal components is constant, as determined by linear theory, and the spanwise component is absent. Once the secondary instability starts to amplify, so does the fraction of the total perturbation energy due to the spanwise velocity. In fact, both $E_{x}$ and $E_{z}$ amplify, followed by a decay in $E_{y}$ , independent of the fluid elasticity. A qualitative difference between the Newtonian and viscoelastic cases is observed at long time, where $E_{x}$ in the Newtonian case represents a smaller fraction of the total perturbation energy in the final state. On the other hand, it is a more appreciable fraction in the viscoelastic flows, and this fraction increases with $Wi$ . A quantitative comparison of the changes in energy and it distribution among the three components is provided in table 3. These trends will be explained with reference to the flow structures that amplify during the transition process.

Figure 9.

Evolution of the perturbation kinetic energy at $Re=4667$ , $\unicode[STIX]{x1D6FD}=0.9$ and $L_{max}=100$ after the addition of background white noise at $t-t^{\ast }=0$ . (—●—) Newtonian; (—▵—) $Wi=1.83$ ; (—○—) $Wi=4.50$ ; (—♢—) $Wi=6.67$ .

Figure 10.

Evolution of the perturbation kinetic energy at $Re=4667$ , $\unicode[STIX]{x1D6FD}=0.9$ and $L_{max}=100$ after the addition of background white noise at $t-t^{\ast }=0$ . (a) Energy due to each velocity component and (b) the respective fractions of the total energy. (—●—) Newtonian; (—▵—) $Wi=1.83$ ; (—○—) $Wi=4.50$ ; (—♢—) $Wi=6.67$ .

Table 3.

Fraction of the total perturbation energy due to each component of the velocity vector at the early, mid- and late stages of transition.

Despite being short lived, the development of secondary instabilities has been studied in detail in Newtonian flows. It is characterized by the generation and amplification of near-wall vortical structures which are initially $\unicode[STIX]{x1D6EC}$ -shaped and selectively develop into hairpin vortices prior to breakdown to turbulence. In Newtonian flows, these dynamics take place near the peak of the energy evolution curve (see e.g. Sandham & Kleiser Reference Sandham and Kleiser1992; Sayadi, Hamman & Moin Reference Sayadi, Hamman and Moin2013).

Figure 11.

Plan views and energy spectra during the early stage of transition. Left panels show contours of $-0.01<v^{\prime }<0.01$ at $y=0.12$ . Right panels show the streamwise-averaged energy spectra, $\overline{E_{x}}$ , as a function of the integer spanwise wavenumber, $n_{k_{z}}\equiv k_{z}L_{z}/2\unicode[STIX]{x03C0}$ . (a) Newtonian; (b) $Wi=1.83$ ; (c) $Wi=4.50$ ; (d) $Wi=6.67$ .

The early stage secondary instability of the TS wave is visualized in figure 11 using contours of the wall-normal velocity perturbation. The two-dimensional waves have been deformed in the span, and their shape already has the appearance of $\unicode[STIX]{x1D6EC}$ -shaped structures. Their arrangement demonstrates that transition in the Newtonian and viscoelastic cases is H-type, as inferred from the staggering in the spanwise direction. The streamwise subharmonic nature of the secondary instability is thus preserved independent of $Wi$ . The results are consistent with the literature, where H-type breakdown is commonly reported when the two-dimensional instability is moderate (Kleiser & Zang Reference Kleiser and Zang1991; White Reference White2006; Sayadi et al. Reference Sayadi, Hamman and Moin2013), and in the present case it is approximately $1\,\%$ of the mean-flow speed. However elasticity alters the spanwise wavelength of the secondary instability and, as a result, of the width of the $\unicode[STIX]{x1D6EC}$ -structures.

Figure 12.

Top view of $\unicode[STIX]{x1D6EC}$ -shaped vortices during the mid-stage of transition. The vortices are visualized using the $\unicode[STIX]{x1D706}_{2}$ vortex-identification criterion with thresholds $\unicode[STIX]{x1D706}_{2}=\{-0.1,-0.14,-0.1,-0.05\}$ . (a) Newtonian; (b) $Wi=1.83$ ; (c) $Wi=4.50$ ; (d) $Wi=6.67$ .

A typical ratio of spanwise to streamwise wavelengths is $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D706}_{x}=0.7$ for H-type transition in Newtonian flow (see e.g. White Reference White2006, and present figure 12). In the present simulations, $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D706}_{x}=0.66$ for the Newtonian flow. As a result three $\unicode[STIX]{x1D6EC}$ structures are formed across the span at every streamwise position, and the same spacing also emerges at the intermediate $Wi=4.50$ . When elasticity is weak ( $Wi=1.83$ ) the secondary instability is slightly narrower, $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D706}_{x}=0.5$ ; and when elasticity is high ( $Wi=6.67$ ) the instability becomes relatively wider, $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D706}_{x}=0.8$ , with only two $\unicode[STIX]{x1D6EC}$ structures in the span. The size of the ensuing secondary instability can be clearly identified from the energy spectra, $\overline{E}_{x}(n_{k_{z}})$ , where $n_{k_{z}}$ is the integer spanwise wavenumber. In figure 11, the spectra were computed from the streamwise velocity perturbation and averaged in the streamwise direction. The peak energy first shifts from the integer wavenumber $n_{k_{z}}=3$ to a higher value ( $n_{k_{z}}=4$ for $Wi=1.83$ ), and then to lower wavenumbers as the $Wi$ is further increased. For $Wi=6.67$ , the peak energy is recorded at $n_{k_{z}}=2$ and its first harmonic due to the elongation of $\unicode[STIX]{x1D6EC}$ -structures in the streamwise direction.

As the secondary instability intensifies, $\unicode[STIX]{x1D6EC}$ -shaped vortical structures start to take form during the mid-stage of transition. They are visualized in figure 12 using the $\unicode[STIX]{x1D706}_{2}$ vortex identification criterion. The $\unicode[STIX]{x1D706}_{2}$ criterion is defined as the second eigenvalue of the tensor, $\unicode[STIX]{x1D61A}_{ik}\unicode[STIX]{x1D61A}_{kj}+\unicode[STIX]{x1D619}_{ik}\unicode[STIX]{x1D619}_{kj}$ , where $\unicode[STIX]{x1D61A}_{ij}$ and $\unicode[STIX]{x1D619}_{ij}$ are the symmetric and antisymmetric parts of the velocity gradient tensor (Jeong & Hussain Reference Jeong and Hussain1995). Elasticity affects both the spanwise scale of the vortices and also their strength. For example, the threshold for the vortex identification criterion in figure 12 was initially increased relative to the Newtonian value when plotting the results for $Wi=1.83$ and subsequently reduced at higher elasticities. It is important to note that the perturbation energy level is equal among all four conditions at this stage in the transition process ( $E=1.91\times 10^{-3}$ ), although it is distributed differently among the three velocity components.

Figure 13.

End views of the perturbation velocity field near the $\unicode[STIX]{x1D6EC}$ -structures during the mid-stage of transition. The streamwise locations of the planes are marked by dashed lines in figure 12. (a) Newtonian; (b) $Wi=1.83$ ; (c) $Wi=4.50$ ; (d) $Wi=6.67$ . Thick green line is a cut through the $\unicode[STIX]{x1D6EC}$ -structures; flood contours are $-0.5\leqslant u^{\prime }\leqslant 0.5$ and arrows denote the in-plane $(\boldsymbol{v}^{\prime },\boldsymbol{w}^{\prime })$ vectors.

A cross-flow plane which intersects the legs of the $\unicode[STIX]{x1D6EC}$ -shaped structures is shown in figure 13. When the legs of the $\unicode[STIX]{x1D6EC}$ -structures are wider in the span, the ejection event between the long legs is less intense, as shown by the velocity vectors in the cross-flow plane. Both the $\boldsymbol{v}$ - and $\boldsymbol{w}$ -vectors are thus shortest at $Wi=6.67$ – a trend which is consistent with lower $E_{y}$ and $E_{z}$ in figure 10(a). The outcome is a weaker negative $u$ -perturbation field between the legs of the $\unicode[STIX]{x1D6EC}$ -structures, although $E_{x}$ reaches a commensurate level among all flow conditions (figure 10 a). Therefore, the positive $u$ -perturbation associated with sweep motion is enhanced at the highest Weissenberg number.

The streamwise conformation field, $\unicode[STIX]{x1D624}_{xx}^{\prime }$ , at the same instance as figure 13 is shown in figure 14. A pronounced sheet of positive and negative $\unicode[STIX]{x1D624}_{xx}^{\prime }$ is observed near the wall, and undulates in the spanwise direction. The sheet itself is due to the primary Tollmien–Schlichting wave, which is a spanwise vorticity perturbation superimposed onto the mean shear. This connection was explained by Page & Zaki (Reference Page and Zaki2015) in the more canonical setting of a spanwise Gaussian vortex in homogeneous shear. Two key effects lead to this phenomenology: (i) the action of the spanwise vorticity perturbation onto the large mean $\unicode[STIX]{x1D60A}_{xx}$ and (ii) the action of the mean shear on the perturbation in the polymer conformation (for details see Page & Zaki Reference Page and Zaki2015). The sheet of $\unicode[STIX]{x1D624}_{xx}^{\prime }$ is subsequently distorted by the secondary instability as shown in figure 14, being displaced outward with ejections and closer to the wall with sweep events.

Figure 14.

End views of the perturbation in the conformation field, $\unicode[STIX]{x1D624}_{xx}^{\prime }$ near the $\unicode[STIX]{x1D6EC}$ -structures during the mid-stage of transition. The streamwise locations of the planes are marked by dashed lines in figure 12. (a) Newtonian; (b) $Wi=1.83$ ; (c) $Wi=4.50$ ; (d) $Wi=6.67$ . Thick green line is a cut through the $\unicode[STIX]{x1D6EC}$ -structures; flood contours are $-400\leqslant \unicode[STIX]{x1D624}_{xx}^{\prime }\leqslant 400$ and contour lines denote $u^{\prime }$ with spacing equal to $4\times 10^{-1}$ .

Since the base state and the TS wave are two-dimensional, the only contributions to $\unicode[STIX]{x1D624}_{xz}$ and $\unicode[STIX]{x1D624}_{yz}$ are due to the secondary instability. For example, figure 15 shows contours of conformation field in the cross-flow plane, and clearly captures the increase in $\unicode[STIX]{x1D624}_{yz}^{\prime }$ at the higher Weissenberg number. The anti-symmetry around the legs of the $\unicode[STIX]{x1D6EC}$ -structures is consistent with a kinematic interpretation of the sweep event: at the outer boundaries of both legs, the perturbation to the conformation is a sweep towards the wall. However, that sweep is in the negative $z$ -direction near the right leg and in the positive direction at the opposite side.

Figure 15.

End views of the perturbation in the conformation field, $\unicode[STIX]{x1D624}_{yz}^{\prime }$ near the $\unicode[STIX]{x1D6EC}$ -structures during the mid-stage of transition. The streamwise locations of the planes are marked by dashed lines in figure 12. (a) Newtonian; (b) $Wi=1.83$ ; (c) $Wi=4.50$ ; (d) $Wi=6.67$ . Thick green line is a cut through the $\unicode[STIX]{x1D6EC}$ -structures; flood contours are $-100\leqslant \unicode[STIX]{x1D624}_{yz}^{\prime }\leqslant 100$ and contour lines denote $u^{\prime }$ with spacing equal to $4\times 10^{-1}$ .

Top views of the perturbation field at $Wi=6.67$ are shown in figure 16. The three panels show the contours of $\unicode[STIX]{x1D624}_{xx}^{\prime }$ , $\unicode[STIX]{x1D624}_{yz}^{\prime }$ and $\unicode[STIX]{x1D624}_{xz}^{\prime }$ and are overlaid with the $\unicode[STIX]{x1D6EC}$ -structures. The first two components were discussed above, and are included in this figure for completeness. The third component, $\unicode[STIX]{x1D624}_{xz}^{\prime }$ , exhibits anti-symmetry relative to each leg and also across the centre of the entire $\unicode[STIX]{x1D6EC}$ -structure. At the outer edges, the sign of $\unicode[STIX]{x1D624}_{xz}^{\prime }$ mirrors the high-speed streaks which are aligned with the $\unicode[STIX]{x1D6EC}$ -structures towards negative- $z$ at right and positive- $z$ at left. Within the ejection zone, the sign of the conformation field is reversed because it is effected by the local streaks which are low-speed relative to the base flow.

Figure 16.

Top views ( $y=0.12$ ) of the perturbation in the conformation field near the $\unicode[STIX]{x1D6EC}$ -structures during the mid-stage of transition at the highest elasticity, $Wi=6.67$ . Thick green line is a cut through the $\unicode[STIX]{x1D6EC}$ -structures, and line contours denote $u^{\prime }$ with spacing equal to $4\times 10^{-1}$ . Flood contours denote (a) $-400\leqslant \unicode[STIX]{x1D624}_{xx}^{\prime }\leqslant 400$ , (b) $-100\leqslant \unicode[STIX]{x1D624}_{yz}^{\prime }\leqslant 100$ and (c) $-4\leqslant \unicode[STIX]{x1D624}_{xz}^{\prime }\leqslant 4$ .

Since the $\unicode[STIX]{x1D6EC}$ -structures become elongated in the streamwise direction, it is important to examine the associated streamwise vorticity, $\unicode[STIX]{x1D714}_{x}^{\prime }$ and in the viscoelastic cases the streamwise polymer torque, $\unicode[STIX]{x1D712}_{x}^{\prime }$ . The term ‘polymer torque’ refers to the polymeric contribution to the vorticity evolution equation, and is derived by taking the curl of the polymer force in the momentum equation, $\unicode[STIX]{x1D712}_{i}^{\prime }\equiv \unicode[STIX]{x1D716}_{ijk}\,\unicode[STIX]{x2202}f_{k}^{\prime }/\unicode[STIX]{x2202}x_{j}$ where $f_{k}^{\prime }=\unicode[STIX]{x2202}\unicode[STIX]{x1D749}_{km}^{\prime }/\unicode[STIX]{x2202}x_{m}$ . It is directly proportional to the toque exerted on a fluid element by the polymer forces (Page & Zaki Reference Page and Zaki2015). In the streamwise direction, the evolution equation for $\unicode[STIX]{x1D714}_{x}^{\prime }$ is,

(3.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}^{\prime }}{\unicode[STIX]{x2202}t}+\overline{u}_{i}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}^{\prime }}{\unicode[STIX]{x2202}x_{i}}=\unicode[STIX]{x1D714}_{i}^{\prime }\frac{\unicode[STIX]{x2202}\overline{u}_{x}}{\unicode[STIX]{x2202}x_{i}}+\overline{\unicode[STIX]{x1D714}}_{i}\frac{\unicode[STIX]{x2202}u^{\prime }}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x1D6FD}}{Re}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D714}_{x}^{\prime }}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{i}}+\underbrace{\left(\frac{\unicode[STIX]{x2202}f_{z}^{\prime }}{\unicode[STIX]{x2202}y}-\frac{\unicode[STIX]{x2202}f_{y}^{\prime }}{\unicode[STIX]{x2202}z}\right)}_{\unicode[STIX]{x1D712}_{x}^{\prime }},\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D714}}_{i}\equiv \unicode[STIX]{x1D716}_{ijk}\,\unicode[STIX]{x2202}\overline{u}_{k}/\unicode[STIX]{x2202}x_{j}$ is the mean vorticity.

Figure 17(a) shows the results from the Newtonian case. The sign of the streamwise vorticity within the leg of the $\unicode[STIX]{x1D6EC}$ structure conforms to the direction of rotation, and an opposite vorticity is induced between the legs and the wall. Also the vorticity contours are concentrated within the $\unicode[STIX]{x1D6EC}$ -structure with similar orientation. Figure 17(b–d) show the viscoelastic cases with $Wi=\{1.83,4.50,6.67\}$ . The vorticity contours gradually weaken with increasing elasticity, and become less concentrated with the $\unicode[STIX]{x1D6EC}$ -structure. The figures are also supplemented with the perturbation polymer torque, $\unicode[STIX]{x1D712}_{x}^{\prime }$ , which is shown by the flood contours. The torque is primarily resistive and its sheet-like appearance (Page & Zaki Reference Page and Zaki2015) along the boundary between positive and negative vorticity is responsible for the elongation of the vorticity field in the span.

Another important observation from figure 17 is the formation of the head of the $\unicode[STIX]{x1D6EC}$ -structure, which is lifted away from the wall. The same tendency is noted in figure 17 for $Wi=4.50$ although to a lesser extent due to the slightly wider spanwise size of the structure. In both cases, however, it is anticipated that the lifted head of the vortex would roll up and generate hairpin vortices, as described by Sandham & Kleiser (Reference Sandham and Kleiser1992) for the Newtonian case. At the highest Weissenberg number, however, the spanwise size of the legs of the $\unicode[STIX]{x1D6EC}$ -structure are widely separated and the lifted head that is a prerequisite for roll up and final breakdown is not clear in the figure.

Figure 17.

(Left) $\unicode[STIX]{x1D6EC}$ -shaped vortices for (a) Newtonian, (b) $Wi=1.83$ , (c) $Wi=4.50$ , (d) $Wi=6.67$ at the mid-stage of transition. (Right) End views at three downstream locations that cut through the $\unicode[STIX]{x1D6EC}$ -structures. Thick green line is the outline of the $\unicode[STIX]{x1D6EC}$ -structure; flood contours are $-1\leqslant \unicode[STIX]{x1D712}_{x}^{\prime }\leqslant 1$ and contour lines denote $\unicode[STIX]{x1D714}_{x}^{\prime }$ with spacing equal to $4\times 10^{-1}$ .

Figure 18.

Visualization of vortical structures in (a,c,e,g) late-stage transition and (b,d,f,h) near the onset of fully turbulent flow. (a,b) Newtonian; (c,d) $Wi=1.83$ ; (e,f) $Wi=4.50$ ; (g,h) $Wi=6.67$ . The vortical structures are identified using (a,c,e,g) $\unicode[STIX]{x1D706}_{2}=\{-1.0,-0.2,-0.2,-0.1\}$ and (b,d,f,h) $\unicode[STIX]{x1D706}_{2}=\{-10.0,-2.0,-2.0,-1.0\}$ .

3.3 Late stage and breakdown to turbulence

The time shift in transition onset is mirrored by a shift in its final stage when the perturbation energy reaches a peak (figure 9). The slope of the curve $E(t)$ does not change appreciably among the various configurations, although the transition process requires longer time at the highest elasticity because it starts at a lower energy level.

In the final stage of transition in the Newtonian case, the spanwise vorticity at the head of the $\unicode[STIX]{x1D6EC}$ -structures lifts up thus forming an elevated shear layer that becomes the site of the ultimate breakdown to turbulence. This process starts by spawning an $\unicode[STIX]{x1D6FA}$ -shaped structure, known as a hairpin vortex. A series of hairpin vortices are further generated upstream and together form a hairpin packet (Sandham & Kleiser Reference Sandham and Kleiser1992). Figure 18 is a visualization of the vortical structures that are formed during the transition process. The Newtonian flow follows the above description, and the figure shows that the early hairpin structures spread and cover the bottom wall fully, marking the end of intermittency and the onset of the fully turbulence regime. Breakdown to turbulence in the viscoelastic flows at $Wi=1.83$ and $4.50$ are qualitatively similar. At $Wi=1.83$ , the hairpin packets have caused rapid breakdown to turbulence in half of the span, while transition in the rest of the domain lags slightly but ultimately becomes fully turbulent. While breakdown at $Wi=4.50$ most closely resembles the Newtonian case, the final state has a lower population density of vortical structures, which suggests a weaker turbulent activity and potentially lower drag.

At the highest Weissenberg number, the final stage of transition is qualitatively different from the Newtonian case. The $\unicode[STIX]{x1D6EC}$ -structures from the early stage continue to elongate and become progressively more aligned in the streamwise direction. Since the spanwise separation is large and the ejection event is weaker in this configuration, hairpin vortices are not formed in this case. Instead the elongated vortices and their accompanying streaks undergo secondary instability, akin to the process of streak breakdown in bypass transition (Vaughan & Zaki Reference Vaughan and Zaki2011; Hack & Zaki Reference Hack and Zaki2014). Turbulence is not spawned from localized breakdown packets, but in stripes due to the secondary instability of the entire elongated structure. Also, the strength of the vortical structures within the turbulent regime is weaker than in all the other conditions, again pointing to a weaker turbulence activity and potentially a drag-reduced turbulent state.

The evolution of transitional flow structures can be viewed as a microcosm of the dynamics in the more complex fully turbulent configurations. In a similar vein, Kim et al. (Reference Kim, Adrian, Balachandar and Sureshkumar2008) and Kim & Sureshkumar (Reference Kim and Sureshkumar2013) studied the evolution of isolated, conditionally averaged vortical structures extracted from turbulent fields. While the scales of the transitional and fully turbulent structures are different (the $\unicode[STIX]{x1D6EC}$ structures and the hairpin packets in our simulations are much wider than those observed in fully turbulent conditions and the mean shear distribution is different), in both configurations the polymer torque acts to oppose the vorticity perturbation, the disturbance field becomes more elongated in the streamwise direction and the hairpin packets are suppressed. These similarities are beneficial in building our understanding of the influence of elasticity on shear flows.

Ultimately all four simulations reach a fully turbulent state. Figure 19 shows the time evolution of the friction Reynolds number, $Re_{\unicode[STIX]{x1D70F}}\equiv u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}$ , where $u_{\unicode[STIX]{x1D70F}}\equiv \sqrt{\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)_{wall}}$ is the friction velocity evaluated from the streamwise- spanwise-averaged velocity profile. Beyond the sharp rise in $Re_{\unicode[STIX]{x1D70F}}$ in the transition region, it levels off to a relatively steady value in the fully turbulent regime. This plateau in $Re_{\unicode[STIX]{x1D70F}}$ corresponds to the saturation of the energy curves after secondary instability in figure 9.

Once the statistically stationary state was reached in each simulation, in addition to streamwise and spanwise averaging, sampling was also performed in time during $160$ – $170$ convective units. Flow quantities are then decomposed as $\unicode[STIX]{x1D703}=\langle \unicode[STIX]{x1D703}\rangle +\unicode[STIX]{x1D703}^{\prime }$ , where the angle brackets denote the steady mean. The friction velocity of the fully turbulent stage is therefore $u_{\langle \unicode[STIX]{x1D70F}\rangle }\equiv \sqrt{\unicode[STIX]{x1D708}(\text{d}\langle u\rangle /\text{d}y)_{wall}}$ ; the corresponding friction Reynolds numbers are $Re_{\langle \unicode[STIX]{x1D70F}\rangle }\equiv u_{\langle \unicode[STIX]{x1D70F}\rangle }h/\unicode[STIX]{x1D708}=\{278,230,202,187\}$ for the Newtonian and three viscoelastic cases with increasing $Wi$ . A lower drag state is established in the viscoelastic flows relative to the Newtonian case, even at $Wi=1.83$ which is more unstable and breaks down to turbulence fastest. The reduction in the turbulent drag is more pronounced at higher elasticity. Expressed in terms of the change in $\langle Re_{\unicode[STIX]{x1D70F}}\rangle$ , drag reduction is defined as,

(3.5) $$\begin{eqnarray}DR\equiv \frac{u_{\langle \unicode[STIX]{x1D70F}\rangle ,N}^{2}-u_{\langle \unicode[STIX]{x1D70F}\rangle }^{2}}{u_{\langle \unicode[STIX]{x1D70F}\rangle ,N}^{2}}=\left[1-\left(\frac{Re_{\langle \unicode[STIX]{x1D70F}\rangle }}{Re_{\langle \unicode[STIX]{x1D70F}\rangle ,N}}\right)^{2}\right],\end{eqnarray}$$

where subscript ‘ $N$ ’ marks the Newtonian value. For $Wi=1.83,4.50$ and $6.67$ , the drag reduction is 31.5 %, 47.2 % and 54.7 %. The viscoelastic flows are therefore in the low-, intermediate- and high-drag-reduction regimes, in comparison to the maximum drag-reduction (MDR) state where $DR\approx 70\,\%$ (Warholic, Massah & Hanratty Reference Warholic, Massah and Hanratty1999).

Figure 19.

Time evolution of the friction Reynolds number, $Re_{\unicode[STIX]{x1D70F}}$ , beyond the addition of background white disturbances at $t-t^{\ast }=0$ . (—●—) Newtonian; (—▵—) $Wi=1.83$ ; (—○—) $Wi=4.50$ ; (—♢—) $Wi=6.67$ .

Figure 20.

Statistics from fully turbulent state at $Re=4667$ , $\unicode[STIX]{x1D6FD}=0.9$ and $L_{max}=100$ . (a) Mean velocity profiles in wall units and the Virk MDR asymptote in grey; (b) Reynolds shear stress; (c) production of streamwise velocity perturbation; (d–f) Reynolds normal stresses. The lines correspond to the four conditions: (—●—) Newtonian; (—▵—) $Wi=1.83$ ; (—○—) $Wi=4.50$ ; (—♢—) $Wi=6.67$ .

The mean velocity profile, the Reynolds stresses and turbulence production are plotted in figure 20. As elasticity increases, the mean velocity profile progressively approaches the Virk asymptote, which marks the MDR state (Virk & Mickley Reference Virk and Mickley1970). The peak in the Reynolds streamwise stress, $\langle u^{\prime }u^{\prime }\rangle$ , shifts away from the wall, and becomes observably weaker at $Wi=6.67$ . Since $\langle u^{\prime }u^{\prime }\rangle$ is predominantly due to the streaks, its peak location is determined by the lift-up process, which is encapsulated in the production term, $-\langle u^{\prime }v^{\prime }\rangle \text{d}\langle u\rangle /\text{d}y$ . In the Newtonian case, the mean shear is localized in the near-wall region, thus drawing the peak production and $\langle u^{\prime }u^{\prime }\rangle$ closer to the wall than in the viscoelastic conditions. It should also be noted that, if normalized by the friction velocity, the streamwise stress appears to increase with elasticity due to the appreciable decrease in $u_{\unicode[STIX]{x1D70F}}$ in the drag-reduced state. This trend is consistent with earlier studies of polymer drag reduction (Dimitropoulos, Sureshkumar & Beris Reference Dimitropoulos, Sureshkumar and Beris1998; Warholic et al. Reference Warholic, Massah and Hanratty1999; Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004; Dimitropoulos et al. Reference Dimitropoulos, Dubief, Shaqfeh, Moin and Lele2005). On the other hand, the cross-flow components, $\langle v^{\prime }v^{\prime }\rangle$ and $\langle w^{\prime }w^{\prime }\rangle$ , are appreciably suppressed whether scaled by outer or inner units. Their significant attenuation with increasing $Wi$ is consistent with viscoelasticity effectively weakening streamwise vortical motions that lead to the amplification of streaks.