4.1. Energy amplification and its mechanism in the presence of weak shear flow

In this section, we focus on the energy amplification in the presence of weak shear flow with a fixed Reynolds number Re = 1, Weissenberg number W = 12 and the viscosity ratio β = 0.75. In these cases, the least stable mode is the centre mode (see figure 5(b)), and the mode type can be identified by the structure of its eigenfunction; see discussion in Khalid et al. (Reference Khalid, Chaudhary, Garg, Shankar and Subramanian2021a). Firstly, we examine the dependence of the largest transient growth G _max on the wavenumbers α and γ. As shown in figure 3(a), it is obvious that spanwise-uniform disturbances (α ≠ 0, γ = 0) reach a larger transient growth than streamwise-uniform disturbances (α = 0, γ ≠ 0), and this interesting phenomenon is opposite to the inertia-dominated flow (Hoda, Jovanović & Kumar Reference Hoda, Jovanović and Kumar2008) where streamwise-uniform disturbances have a greater transient growth than spanwise-uniform disturbances. A similar conclusion was found in a recent investigation based on classical Reynolds decomposition (Hariharan, Jovanović & Kumar Reference Hariharan, Jovanović and Kumar2021) when using a localised measure of disturbance amplification and considering the input stochastic excitations, although this conclusion does not hold when using a square-integrated measure (corresponding to the typical non-modal analysis). Figures 3(b) and 3(c) further display the effects of α and γ on G _max in detail. With an increase in α, G _max monotonically increases, and this growth is mainly manifested in the small-α region (e.g. α ≤ 3), while the growth becomes slow for larger α, indicating that shorter-wave disturbances reach larger energy growth in low-Re flow, and this phenomenon may be eliminated when the diffusion in the constitutive equation is considered (Jovanović & Kumar Reference Jovanović and Kumar2010). On the other hand, when calculating G _max with γ, we scan γ from 0 to 5 with γ having a precision of 0.1, and figure 3(c) further illustrates that the spanwise-uniform disturbances reach the largest energy growth, and spanwise wavenumber always exerts an inhibitory effect.

Figure 4.

(a) Input analysis, (b) output analysis and (c, d) the mapping relationship from input to output disturbances (note that the thickness of the same colour curve reflects the relative contribution magnitude (the thicker the curve, the greater the contribution), while the curves of different colours cannot be compared with each other). Parameters are Re = 1, W = 12, β = 0.75, α = 3 and γ = 0.

Figure 4. Long description

Panel A: A line graph shows the input analysis with the vertical axis labeled G and the horizontal axis labeled t. Multiple lines represent different components of the input analysis, including G, G_i-v,η, G_i-g, G_i-gxx, G_i-gxy, G_i-gxz, G_i-gzz, and G_i-gyz. An inset zooms in on the range from 0 to 1 on the horizontal axis. Panel B: Another line graph displays the output analysis with the vertical axis labeled G and the horizontal axis labeled t. This graph includes lines for G, G_o-v,η, G_o-g, G_o-gxx, G_o-gzz, G_o-gxy, G_o-gxz, and G_o-gyz. An inset zooms in on the range from 0 to 1 on the horizontal axis. Panel C: A diagram illustrates the mapping relationship from input to output disturbances, showing connections between different components labeled as gxx, gyy, gxy, and gyz. Panel D: Another diagram depicts the mapping relationship with connections between components labeled as gxx, gyy, gxy, and gyz, with varying line thicknesses indicating the relative contribution magnitude.

In order to examine the effects of initial conditions on the transient growth and to capture the temporal evolutions of various energy components, we perform an input–output analysis in figure 4. Firstly, we note that a significant transient growth G occurs over an extremely short period of time (t < 1), which is surprising compared with the transient growth in Newtonian fluids (Reddy & Henningson Reference Reddy and Henningson1993). Therefore, to validate this unusual result, we compare the derivative of the G(t) curve at t = 0⁺ and the numerical abscissa of the linear operator, and these two values are equal (i.e. (4.1) holds) when the computation is correct (Schmid & Brandt Reference Schmid and Brandt2014). Specifically, at t = 0⁺, dG/dt = ω _max = 237.7 in this case (note that the meaning of eigenvalue ω in (4.1) is different from that in (2.18), and they correspond to different linear operators, F _out C _out LB _in $\boldsymbol{F}_{in}^{-1}$ and L , respectively), confirming the accuracy of our calculation. Thus, the significant transient growth within a very short time is a distinctive feature in highly elastic fluids with low Re

(4.1) \begin{align} \left.\frac{\text{d}\left(\ln \left(G\left(t\right)\right)\right)}{\text{d}t}\right| _{t={0^{+}}}=\omega _{max }\left(\left(\boldsymbol{F}_{out}\boldsymbol{C}_{out}\boldsymbol{L}\boldsymbol{B}_{in}{\boldsymbol{F}}_{in}^{-1}\right)^{\dagger }+\left(\boldsymbol{F}_{out}\boldsymbol{C}_{out}\boldsymbol{L}\boldsymbol{B}_{in}{\boldsymbol{F}}_{in}^{-1}\right)\right) . \end{align}

Inspecting the energy growth with different input filters displayed in figure 4(a), one observes that the transient energy growth with only the conformation tensor field perturbed initially (input disturbance $\tilde{\boldsymbol{g}}$ ) is very close to the optimal transient growth over all possible initial conditions (input disturbances $\tilde{\text{ v}}$ , $\tilde{\mathit{\eta }}$ , $\tilde{\boldsymbol{g}}$ ), while transient growth with other physical quantities perturbed initially is negligible, which indicates that, physically, to induce this non-modal growth, one needs to perturb the conformation tensor field carefully with other physical fields remaining unperturbed, and mathematically, the non-normality of the linear operator is mainly related to the conformation tensor. For the specific effects of input $\tilde{\boldsymbol{g}}$ components, it can be found that the transient growth is mainly induced by the inputs $\tilde{\text{ g}}$ _yy and $\tilde{\text{ g}}$ _xy (especially, $\tilde{\text{ g}}$ _yy ), and that induced by the input $\tilde{\text{ g}}$ _xx is relatively small, indicating that the y-direction related perturbation input mainly determines the transient growth. For the rapid transient growth of t < 1, all inputs are important, and the input $\tilde{\text{ g}}$ _xy is the most important. In addition, the growth time (which reaches the largest transient growth) with different inputs is different. Specifically, the growth time with the input $\tilde{\text{ g}}$ _xx is shorter (near t = 1), and the growth time with the input $\tilde{\text{ g}}$ _xy is moderate, while the growth time with the input $\tilde{\text{ g}}$ _yy is longer, implying that the response of total energy growth for the input $\tilde{\text{ g}}$ _yy is the slowest, and the fastest for the input $\tilde{\text{ g}}$ _xx .

On the other hand, as shown in figure 4(b), it is noted that, like the input analysis in figure 4(a), among various energy components, the perturbation elastic energy is also the most amplified, with the amplified perturbation kinetic energy related to the wall-normal velocity, G _o-v being moderate and the perturbation kinetic energy related to wall-normal vorticity being always decayed (see the G _o-η curve). In addition, for the components of output elastic energy, G _o-gxx and G _o-gyy are the most important and basically amplified to the same magnitude, while the growth of $\tilde{\text{ g}}$ _xy is relatively very small (see the brown line). The mapping relationship from input disturbances to output disturbances in figures 4(c) and 4(d) further indicates that every input disturbance separately contributes the same proportions as those shown in figures 4(a) and 4(b) to the mainly amplified disturbances $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy , and for the growth of the perturbation velocity, the input disturbance $\tilde{\text{ g}}$ _xy instead of $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy plays a more important role. Based on the input analysis in figure 4(a), we know that the input $\tilde{\text{ g}}$ _xy has important effects on energy growth, but the output $\tilde{\text{ g}}$ _xy is very small. Therefore, the input disturbance $\tilde{\text{ g}}$ _xy is converted into the energy of $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy output, and this process usually involves the interaction between the perturbed conformation tensor field and the perturbed hydrodynamic field; see the non-zero operators in (A3, A5, A6, A8) in Appendix A. Specifically, for the operators in (A8), the input perturbation $\tilde{\text{ g}}$ _xy can be converted into the production of perturbation wall-normal velocity $\tilde{\text{ v}}$ (even if there are other perturbations in production, they are proven to have no promotion effect on the transient growth, evidenced by the Ab and Ag terms in figure 6), and the production $\tilde{\text{ v}}$ can be converted and amplified into the perturbation $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy output, see the operators in (A3, A5, A6). The more related and detailed mechanism can be found in the subsequent perturbation elastic energy analysis and optimal initial conditions/responses.

Figure 5.

(a) Transient energy growth with different modes (note that the G(t) curve of ‘all modes’ is calculated by the entirely operator-based method) and (b) the corresponding eigenspectrum. Parameters are Re = 1, β = 0.75, W = 12, α = 3 and γ = 0.

Figure 6.

(a) Elastic energy analysis; (b) components of the contribution Au and (c) the perturbation kinetic energy growth related to wall-normal velocity G _o-v . Note that all non-displayed components in figure 6(b) can be basically ignored, and the parameters are Re = 1, β = 0.75, W = 12, α = 3 and γ = 0.

Figure 6. Long description

Panel A: A line graph shows contributions to delta epsilon p over time t. The y-axis represents contributions to delta epsilon p, and the x-axis represents time t on a logarithmic scale. The graph includes four lines: Elastic energy (black), Ag (blue), Au (red), and Ab (green). The lines show different trends over time, with notable peaks and troughs. Panel B: A line graph displays Au components over time t. The y-axis represents Au components, and the x-axis represents time t on a logarithmic scale. The graph includes five lines: Au (black), Au_gyyv(red), Au_gxxv (orange), and Au_gxxu (pink). The lines exhibit various trends, with significant peaks and troughs. Panel C: A line graph illustrates the perturbation kinetic energy growth related to wall-normal velocity Go-v over time t. The y-axis represents Go-v, and the x-axis represents time t on a logarithmic scale. The graph shows a single line with a prominent peak and subsequent decline.

Next, we attempt to explain the mechanism of transient growth both mathematically and physically. Firstly, in mathematics, we know that the transient energy growth is caused by the non-orthogonality of the eigenvectors corresponding to different modes (Schmid Reference Schmid2007). Therefore, as shown in figure 5, we present the eigenspectrum and the transient growth caused by the combination of various modes. It can be found that the continuous spectrum (CS) indeed causes obvious transient growth (i.e. the G(t) curve caused by the modes of −0.05 ≥ ω _i ≥ −0.2, where ω _i is the imaginary part of the eigenvalue of the operator L ), while the least stable mode, the centre mode, does not affect this transient growth (see the G(t) curve caused by the modes of ω _i ≥ −0.2). Besides, the remaining modes also play an important role. Specifically, with the addition of the remaining modes (i.e. the decrease of the minimum ω _i from −0.2), the transient growth of smaller t increases rapidly, and the original peak has a small increase and eventually merges into a smooth curve instead of a peak, which leads to a new peak in the G(t) curve with larger G _max and shorter t _max (the time that reaches G _max ). We need to indicate that the remaining discrete modes themselves do not generate transient growth (see the G(t) curve of −5000 ≤ ω _i ≤ −0.2 in figure 5 a), and their real impact is to promote the transient growth produced by the CS, while in Newtonian fluids, these modes with very small growth rates have almost no effect on transient growth (Reddy & Henningson Reference Reddy and Henningson1993; Schmid & Henningson Reference Schmid and Henningson2001). The promotion effect of the remaining discrete modes enhances with an increase of W and Re or a decrease of β, and can cause another peak transition phenomenon of the transient growth even if it is calculated by the operator-based method (including all modes), which will be discussed in § 4.2.

Secondly, in physics, based on the output analysis in figure 4(b), we know that disturbance elastic energy is the most amplified instead of the disturbance kinetic energy, so elastic energy analysis is introduced. In this analysis, the following results are obtained by the optimal initial conditions. Following the investigations of Butler & Farrell (Reference Butler and Farrell1992), Zhang et al. (Reference Zhang, Lashgari, Zaki and Brandt2013) and Buza et al. (Reference Buza, Page and Kerswell2022), we can obtain a new equation by multiplying the governing equations of the perturbation conformation tensor by the corresponding complex conjugate component $g_{ij}^{\dagger}$ , and then another equation is obtained by taking the complex conjugate of this new equation. Then, we average these two equations and multiply the averaged equation by the coefficient (1/2) × (1 − β)/(ReW), which leaves an evolution equation for the perturbation elastic energy ε _p in the control volume Ω (we reiterate that the definition of Ω and the related parameters a and b can be found in § 2.3)

(4.2) \begin{align} \frac{1}{\varOmega }\int _{\varOmega }\frac{\partial \varepsilon _{p}}{\partial t}\text{d}V& =\frac{1}{\varOmega }{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\frac{\partial }{\partial t}\left[\frac{1}{2}\boldsymbol{\cdot }\frac{1-\beta }{ReW}\left(\!\!\begin{array}{c} {g}_{xx}^{2}+{g}_{yy}^{2}+{g}_{zz}^{2}\\ +2{g}_{xy}^{2}+2{g}_{xz}^{2}+2{g}_{yz}^{2} \end{array}\!\!\right)\right]\,\text{d}x\,\text{d}y\,\text{d}z\nonumber\\ & =Ag+Ab+Au , \end{align}

where the term on the left means the growth rate of perturbation elastic energy, the right-hand side terms Au, Ab and Ag represent the contribution to perturbation elastic energy from the perturbation velocity field, the base flow field and the polymeric relaxation, respectively, and their specific forms are

(4.3) \begin{align} Ag&=\frac{2}{\varOmega }\boldsymbol{\cdot }\frac{1-\beta }{ReW}{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\left(-\frac{g_{xx}g_{xx}}{2W}-\frac{g_{yy}g_{yy}}{2W}-\frac{g_{zz}g_{zz}}{2W}-\frac{g_{xy}g_{xy}}{W}\right.\nonumber\\[6pt] &\left.\qquad\qquad \qquad \qquad -\frac{g_{xz}g_{xz}}{W}-\frac{g_{yz}g_{yz}}{W}\right)\,\text{d}x\,\text{d}y\,\text{d}z , \end{align}

(4.4) \begin{align} Ab =\frac{2}{\varOmega }\boldsymbol{\cdot }\frac{1-\beta }{ReW}{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\left[\substack{\displaystyle-\frac{1}{2}\overline{U}g_{xx}\frac{\partial g_{xx}}{\partial x}-\frac{1}{2}\overline{U}g_{yy}\frac{\partial g_{yy}}{\partial x}-\frac{1}{2}\overline{U}g_{zz}\frac{\partial g_{zz}}{\partial x}\\[5pt] \displaystyle-\,\overline{U}g_{xy}\frac{\partial g_{xy}}{\partial x}-\overline{U}g_{xz}\frac{\partial g_{xz}}{\partial x}-\overline{U}g_{yz}\frac{\partial g_{yz}}{\partial x}\\[5pt] \displaystyle+\frac{\overline{U}^{\prime}}{\overline{S}}g_{xy}g_{yy}+\frac{\overline{U}^{\prime}}{\overline{S}}g_{xx}g_{xy}+\frac{\overline{U}^{\prime}}{\overline{S}}g_{xz}g_{yz}}\right]\,\text{d}x\,\text{d}y\,\text{d}z , \end{align}

(4.5) \begin{align} Au&=\frac{2}{\varOmega }\boldsymbol{\cdot }\frac{1-\beta }{ReW}\nonumber\\&\!\!\!\!\!\!{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\left[\!\!\!\begin{array}{c} \underset{Au\_ g_{xx}u}{\underbrace{g_{xx}\frac{\partial u}{\partial x}}}{+}\underset{Au\_ g_{xx}v}{\underbrace{\frac{1}{2}\left(\frac{2\overline{C}_{xy}{\overline{C}^{\prime}}_{xy}}{\overline{S}^{2}}-\frac{{\overline{C}^{\prime}}_{xx}}{\overline{S}^{2}}\right)g_{xx}v-\overline{C}_{xy}g_{xx}\frac{\partial v}{\partial x}}}\\ {+}\,\underset{Au\_ g_{yy}v}{\underbrace{\overline{C}_{xy}g_{yy}\frac{\partial v}{\partial x}{+}g_{yy}\frac{\partial v}{\partial y}}}{+}\underset{Au\_ g_{zz}w}{\underbrace{g_{zz}\frac{\partial w}{\partial z}}}{+}\underset{Au\_ g_{xy}u}{\underbrace{\frac{\overline{C}_{xy}}{\overline{S}}g_{xy}\frac{\partial u}{\partial x}{+}\frac{1}{\overline{S}}g_{xy}\frac{\partial u}{\partial y}}}\\ \underset{Au\_ g_{xy}v}{\underbrace{-\,\frac{{\overline{C}^{\prime}}_{xy}}{\overline{S}}g_{xy}v{+}\left(\frac{\overline{C}_{xx}-2\overline{C}_{xy}\overline{C}_{xy}}{\overline{S}}\right)g_{xy}\frac{\partial v}{\partial x}-\frac{\overline{C}_{xy}}{\overline{S}}g_{xy}\frac{\partial v}{\partial y}}}\\ {+}\,\underset{Au\_ g_{xz}u}{\underbrace{\frac{1}{\overline{S}}g_{xz}\frac{\partial u}{\partial z}}}\underset{Au\_ g_{xz}v}{\underbrace{-\frac{\overline{C}_{xy}}{\overline{S}}g_{xz}\frac{\partial v}{\partial z}}}{+}\underset{Au\_ g_{xz}w}{\underbrace{\overline{S}g_{xz}\frac{\partial w}{\partial x}}}{+}\underset{Au\_ g_{yz}v}{\underbrace{g_{yz}\frac{\partial v}{\partial z}}}{+}\underset{Au\_ g_{yz}w}{\underbrace{\overline{C}_{xy}g_{yz}\frac{\partial w}{\partial x}{+}g_{yz}\frac{\partial w}{\partial y}}} \end{array}\!\!\!\right]\!\text{d}x\,\text{d}y\,\text{d}z . \end{align}

We need to indicate that there is an expression ${\int }_{0}^{t}\varepsilon _{p}\text{d}t=G_{o-g}$ , and this can be used as an a posteriori check of the results.

As shown in figure 6(a), we first see that the energy transfer from the perturbation velocity field to the perturbation conformation tensor field (Au term) promotes elastic energy growth, and the polymeric relaxation (Ag term) suppresses this growth, while the direct contribution from the base flow field (Ab term) can be ignored. Among all components of the energy growth term Au, see figure 6(b), the energy exchange between the wall-normal velocity $\tilde{\text{ v}}$ and the normal conformation tensor components $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy , is the most important, and the energy exchange between the perturbation streamwise velocity $\tilde{\text{ u}}$ and $\tilde{\text{ g}}$ _xx has a small promotion effect. Therefore, in the process of energy amplification, even if the perturbation kinetic energy growth related to wall-normal velocity, G _o-v is relatively very small (see figure 4(b)), the perturbation wall-normal velocity $\tilde{\text{ v}}$ still plays a vital role in the amplification of elastic disturbances, and its function is similar to the ‘catalyst’ connecting the input and output elastic disturbances. The evolution of G _o-v in figure 6(c) also supports this conclusion, and specifically, for 0 < t < 1, G _o-v has a significant increase, i.e. there is a significant growth in the number of catalysts, which causes the rapid amplification of perturbation elastic energy for 0 < t < 1 (see figures 4(a) and 4(b)). Furthermore, with an increase in t from t = 1, the change of G _o-v is relatively slow before t = t _max , while after t = t _max , G _o-v begins to have a significant decrease, so the ability to catalyse the perturbation elastic energy output decreases and is less than the elastic energy consumed by the polymeric relaxation, causing the decrease of G for t > t _max . Combining the contributions of different input perturbations to the output perturbation velocity, shown in figure 4(c), the input $\tilde{\text{ g}}$ _xy has a greater promotion effect on the growth of perturbation velocity, so the input $\tilde{\text{ g}}$ _xy causes the elastic energy amplification by providing more catalysts $\tilde{\text{ v}}$ .

To further investigate the behaviour of perturbations in transient growth, we plot in figure 7 the optimal initial conditions and the corresponding optimal responses. We can see that, at t = 0, the optimal initial perturbation conformation tensor $\tilde{\boldsymbol{g}}$ is inclined against the base flow in structure and concentrated near the upper and lower plates, while the direction of optimal initial perturbation velocity $\tilde{\text{ v}}$ is the same as the base flow. Besides, see figure 7(b), we provide the optimal initial perturbation velocity calculated by CS for comparison (the remaining optimal initial conditions and optimal responses are basically the same as those calculated by the linear operator, so they are not shown here), and it can be seen that, in this case, the structure of perturbation wall-normal velocity $\tilde{\text{ v}}$ , inclined against the base flow and concentrated near the upper and lower plates, is significantly different from that calculated by the linear operator. This indicates that the other discrete modes with small growth rates, excluding the CS, significantly promote the transient growth by providing different optimal initial perturbation velocities.

Figure 7.

(a) Optimal initial conditions and optimal responses at α = 3, (b) optimal initial perturbation velocity calculated by CS (using the eigenvalues and eigenvectors-based method) and linear operator at α = 3 and (c) optimal responses at α = 10. Parameters are Re = 1, β = 0.75, W = 12, γ = 0 and λ = 2π/α.

When transient growth reaches its maximum, the structure of the perturbation conformation tensor $\tilde{\boldsymbol{g}}$ at t = t _max , tilts into the mean shear direction, and the phenomenon of concentrated disturbances near the plates is alleviated. In two-dimensional Newtonian fluid, the perturbed velocity field has a similar change, and the related kinetic energy growth mechanism is referred to as the Orr mechanism (Orr Reference Orr1907). Specifically, the optimal initial velocity perturbation is inclined against the mean shear flow in structure, and subsequently this structure tilts into the mean shear direction, leading to energy ‘release’ or named transient growth (Schmid & Henningson Reference Schmid and Henningson2001). Therefore, in a viscoelastic fluid, the change in the inclination of the structure of perturbation $\tilde{\boldsymbol{g}}$ can also be understood as the ‘release’ of perturbation elastic energy in the polymer stretch layer near the plates, and the related mechanism of perturbation elastic energy growth can be regarded as the Orr mechanism in the conformation tensor field. A similar phenomenon can be found in high-Re viscoelastic pipe flow (Zhang Reference Zhang2021). Combined with the elastic energy analysis in figure 6, it can be reasonably speculated that, for t < t _max , the opposite inclination between the perturbation wall-normal velocity $\tilde{\text{ v}}$ and the perturbation conformation tensor $\tilde{\boldsymbol{g}}$ causes their positive phase relationship, which makes the Au term positive, and so rapidly promotes transient growth. As time evolves, the positive phase relationship reduces, and for t > t _max , this promotion effect is weaker than the inhibitory effect of the polymeric relaxation. Even for a longer t, this phase relationship can become negative to inhibit energy growth, as evidenced by the Au curve in figure 6(a). Besides, by comparing the two different optimal initial velocity perturbations in figure 7(b) and the optimal initial configuration tensor in figure 7(a), we seem to intuitively observe that there is a larger positive phase relationship between the velocity perturbation and the conformation tensor perturbation calculated by the linear operator, which may explain why the results obtained by the linear operator achieve greater energy amplification over a shorter time interval than those obtained by CS calculation (see figure 5(a)). Furthermore, if some small-scale and near-wall perturbations (e.g. wall roughness in Page & Zaki (Reference Page and Zaki2022)) are additionally considered, the related transient growth may be more significant.

Figure 7(c) additionally displays the optimal responses at a higher streamwise wavenumber (e.g. α = 10) with a larger transient growth, and the typical characteristic of the optimal initial condition, e.g. the structure of $\tilde{\boldsymbol{g}}$ inclined against the base flow, is unchanged (so not shown). Obviously, the Orr mechanism in the conformation tensor field is still at work, while unlike the results in figure 7(a), the optimal responses are more concentrated near the walls, and the symmetry of their structures is contrary to the case of α = 3, e.g. $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy are symmetrical about the channel centreline. These characteristics of optimal responses are unique results at high streamwise wavenumbers, which cannot be observed at low wavenumbers, even with a larger Weissenberg number. In addition, these responses bear some similarities to the instability structures in elastic turbulence (Buza et al. Reference Buza, Page and Kerswell2022; Morozov Reference Morozov2022; Beneitez et al. Reference Beneitez, Page and Kerswell2023) in terms of symmetry and near-wall characteristics, implying that transient growth may play a significant role in the early linear stage of the transition to elastic turbulence.

4.2. Parametric behaviour and scaling laws of energy amplification

In this section, we examine the effects of the dimensionless parameters W, β and Re on the transient growth and the associated scaling laws. Firstly, as shown in figure 8, the increase of W enhances transient growth, and the relationship G _max ∝ W ² holds, which reflects the scaling law relationship between perturbation elastic energy (G ≈ G_o-g under the used parameters, see figure 4(b)) and the Weissenberg number. Using stochastic excitations, the previous investigations of Jovanović & Kumar (Reference Jovanović and Kumar2010) and Hariharan, Jovanović & Kumar (Reference Hariharan, Jovanović and Kumar2021) reported that the conformation tensor component $\tilde{\text{ c}}$ _xx (in the Reynolds decomposition) reaches the O(W ²) growth for streamwise-uniform disturbances, but this relationship does not hold for spanwise-uniform disturbances (i.e. the configuration considered in the present case), and even transient growth reduces as W increases. These discrepancies highlight the importance of new geometric perturbative expansions with a clear physical interpretation and the associated appropriate definition of perturbation elastic energy. As illustrated in figure 8(b), the increase in fluid elasticity not only promotes the growth of perturbation elastic energy but also enhances the growth of perturbation kinetic energy. In the previous experimental studies, Pan et al. (Reference Pan, Morozov, Wagner and Arratia2013) and Shnapp & Steinberg (Reference Shnapp and Steinberg2022) also observed that velocity fluctuations increase as W increases. Besides, the related scaling law approximately satisfies G _{o-v,η max} ∝ W ^2.5. In contrast, Hariharan, Jovanović & Kumar (Reference Hariharan, Jovanović and Kumar2021) only found that this growth of streamwise-uniform perturbation is proportional to W (this relationship is also recovered in the present study, see the green line), while in their study, the kinetic energy growth of spanwise-uniform perturbation decreases with an increase of W. We note that, like the amplification of perturbation elastic energy, spanwise-uniform velocity disturbances also reach the greatest kinetic energy amplification (see the inset in panel (b)), while the difference is that there is a noticeable local maximum near α = 2, indicating that, compared with the growth of elastic energy, short-wave disturbances (i.e. the larger values of α) cannot cause a significant growth in perturbation kinetic energy.

Figure 8.

The effects of W on (a) the maximum perturbation energy growth G _max at β = 0.75 (note that the left inset is the transient growth curve at W = 28 and α = 3 and the right inset is the G _max –W ² relationship for W ≤ 20), (b) the maximum perturbation kinetic energy growth G _{o-v,η max} at β = 0.75 (note that the inset is the contour of G _{o-v,η max} in the α–γ plane at W = 12), (c) G _max and G _{o-v,η max} at β = 0.15 and α = 3 (note that the inset is the G(t) curve at W = 12), and (d) t _max at β = 0.75 and α = 3. Other parameters are Re = 1 and γ = 0.

Figure 8. Long description

Panel A: A line graph shows the maximum perturbation energy growth Gmax at beta = 0.75. The x-axis represents W on a logarithmic scale, and the y-axis represents Gmax on a logarithmic scale. Different symbols and colors represent different alpha values (alpha = 1, 2, 3). The left inset shows the transient growth curve at W = 28 and alpha = 3, with the x-axis representing time t and the y-axis representing G. The right inset shows the Gmax versus W^2 relationship for W less than 20, with the x-axis representing W^2 and the y-axis representing Gmax. Panel B: A line graph shows the maximum perturbation kinetic energy growth Go-v, max at beta = 0.75. The x-axis represents W on a logarithmic scale, and the y-axis represents Go-v, max on a logarithmic scale. Different symbols and colors represent different alpha values (alpha = 1, 2, 3). The inset shows the contour of Go-v, max in the alpha-W plane at W = 12, with the x-axis representing alpha and the y-axis representing W. Panel C: A line graph shows Gmax and Go-v, max at beta = 0.15 and alpha = 3. The x-axis represents W on a logarithmic scale, and the y-axis represents Gmax and Go-v, max on a logarithmic scale. Different symbols and colors represent G and Go-v, max. The inset shows the G(t) curve at W = 12, with the x-axis representing time t and the y-axis representing G. Panel D: A line graph shows tmax at beta = 0.75 and alpha = 3. The x-axis represents W on a logarithmic scale, and the y-axis represents tmax on a logarithmic scale. An annotation indicates a peak transition.

In addition, we find that with a further increase of W, G _max transitions from the original peak with larger t (named peak 1) to the new peak with smaller t (named peak 2) in the presence of Poiseuille flow at Re = 1 (see the left inset in figure 8(a)), and this phenomenon is easily observed in the viscoelastic fluid with lower viscosity ratio β (see the inset in figure 8(c)) or higher Re (see the inset in figure 9(a) below), while in these cases, the similar peak transition phenomenon does not exists for the perturbation kinetic energy growth G _o-v,η . We need to mention that the peak transition in figures 8(a) and 8(c) is different from the similar phenomenon in figure 5(a), and the current results in figure 8 are all obtained through the entirely operator-based method, while the results in figure 5(a) are calculated using two different methods (the eigenvalues and eigenvectors-based method and the entirely operator-based method), merely to illustrate the influence of discrete modes on the transient growth. Nevertheless, the results shown in figures 5 and 8 are also interrelated. Specifically, based on the discussion in figure 5, we know that the significant amplification at smaller t is related to the promotion effect of discrete modes on CS instead of the individual effect of CS, and the increase of Weissenberg number and the decrease of viscosity ratio further enhance this effect in the presence of weak Poiseuille flow, leading to the peak transition in figure 8. For the scaling law between the G _max with smaller t and W (i.e. the scaling law of new peak 2), shown in figure 8(a) for W>100 and figure 8(c), it can be found that the relationship G _max ∝ W ² also holds, while the largest perturbation kinetic energy growth satisfies G _{o-v,η max} ∝ W ^2.2 in figure 8(c) instead of G _{o-v,η max} ∝ W ^2.5 in figure 8(a). Figure 8(d) further displays the dependence of growth time t _max on the Weissenberg number. Specifically, with an increase in W, t _max increases for the transient growth dominated by peak 1, and this relationship can be approximated as: t _max ∝ W ^0.67, while it decreases slowly for the transient growth dominated by peak 2 and the relationship can be approximated as: t _max ∝ W ^−0.7, indicating that, in the highly elastic fluids, the greater energy amplification occurs in a shorter time interval.

Figure 9.

(a) Effects of Re on transient growth at β = 0.75 and W = 12 (note that the left (right) inset is the elastic (kinetic) energy growth curve at Re = 2.5 (Re = 14)); (b) effects of β on G _max at Re = 0.01, 1, 10 and 100; (c) the scaling law between G _{o-g max} , G _{o-v,η max} and W at Re = 100 and (d) effects of Re and W on t _max . Other parameters are α = 3 and γ = 0.

As shown in figure 9, we consider the effects of Re. With an increase of Re from 1 to 100, both the largest perturbation elastic energy growth G _{o-g max} and kinetic energy growth G _{o-v,η max} increase, and although the elastic energy always dominates, the disparity between elastic energy and kinetic energy progressively narrows. Like the effects of W in figure 8, there remains a similar peak transition phenomenon of the perturbation elastic energy growth G _{o-g max} near Re = 2.5 (see the left inset in figure 9(a)), and the growth of perturbation kinetic energy G _{o-v,η max} also has a peak transition near Re = 14 (see the right inset in figure 9(a)), while contrary to the transition phenomenon of G _{o-g max} , G _{o-v,η max} transitions from the peak with the smaller t to the new peak with the larger t. It should be noted that although the discussion of figure 6 suggests that the magnitude of G _o-v,η reflects the capacity of the catalytic polymer perturbation to induce elastic energy growth, a larger peak of G _o-v,η (at a greater t) in this case does not necessarily result in a more significant transient growth in elastic energy because of the possible larger polymeric relaxation and the rapid decrease of the positive phase relationship between the wall-normal velocity perturbation $\tilde{\text{ v}}$ and the polymer perturbation $\tilde{\boldsymbol{g}}$ for t _max < t < t _{o-v,η max} (t _{o-v,η max} denotes the time when the perturbation kinetic energy growth reaches its maximum value G _{o-v,η max} ). Besides, we show the variation of G _max caused by the CS with Re for comparison in figure 9(a), and the results suggest that shear flow not only fails to promote this transient growth, but also actively suppresses it, indicating that the remaining discrete modes apart from the CS play a more significant role in the presence of a stronger shear flow.

The variation in G _max with viscosity ratio β at different Re is shown in figure 9(b), and an Oldroyd-B fluid is equivalent to a UCM fluid for β → 0 and a Newtonian fluid for β → 1. The first observation is that the energy amplification increases as β decreases, indicating that the presence of non-zero solvent viscosity has a pronounced stabilising effect on the flow, and a similar phenomenon was found in the kinetic energy growth caused by input velocity disturbances (Hoda, Jovanović & Kumar Reference Hoda, Jovanović and Kumar2008). In addition, we note that, for 0.1 < β < 0.8, the effect of β on G _max is very small at Re = 0.01 where the peak 1 with a larger t (see the discussions around figures 8(a) and 8(c) and figure 9(a)) has the largest transient growth, indicating that, compared with the transient growth dominated by peak 1, the viscosity ratio mainly affects the energy growth dominated by peak 2 with a smaller t. Figure 9(c) shows the scaling laws of G _{o-g max} and G _{o-v,η max} at Re = 100, and the G _max caused by the CS is provided for comparison. It can be found that the relationship G _{o-g max} ∝ W ² still holds, and G _{o-v,η max} is proportional to W ^1.7 at Re = 100, which is different from the scaling law G _{o-v,η max} ∝ W ^2.5 at Re = 1. We also calculate the scaling law of G _{o-v,η max} with W at other Re values (the corresponding figures are not shown), and specifically G _{o-v,η max} ∝ W ^2.88 at Re = 0.01, and G _{o-v,η max} ∝ W ^2.31 at Re = 10, implying that the greater shear strength causes the smaller growth rate of perturbation kinetic energy with Weissenberg number. Besides, unlike the effects of Re in figure 9(a), the G _max caused by the CS is greatly affected by the Weissenberg number, and it satisfies G _max ∝ W ², reflecting that the modes in the CS still exhibit pure elasticity even at a higher Re. For the variation of growth time t _max , as shown in figure 9(d), both increasing Re and W can promote transient growth, but their effects on t _max are different. Specifically, like the Re = 1 case, here, the increase of W still reduces the time of transient growth, while the enhancement of shear strength extends the growth time, which respectively correspond to the two ways of enhancing transient growth: one is to increase the growth rate (for W), and the other is to increase the growth time (for Re).

The results above were obtained with the Oldroyd-B model, which allows infinite polymer extension. To assess whether the large transient growth persists under more realistic constitutive models, we repeated selected calculations with the FENE-P model (Bird et al. Reference Bird, Dotson and Johnson1980; Herrchen & Ottinger Reference Herrchen and Ottinger1997). The extended linear perturbation equations for the v-η- g formulation are provided in Appendix B.

Figure 10 compares the maximum transient growth G _max and the corresponding optimal time t _max as functions of the extensibility parameter L for two representative cases: spanwise-uniform perturbations at low Reynolds number (Re = 1, β = 0.75, α = 3, γ = 0) and oblique perturbations at high Reynolds number (Re = 1000, β = 0.75, α = 1.6, γ = 1.25, these are the core parameters used in § 4.3, where the dominant amplified energy is perturbation kinetic energy). Based on the previous investigations (Zhang Reference Zhang2021; Buza et al. Reference Buza, Page and Kerswell2022; Khalid et al. Reference Khalid, Badoni, Dutta, Naidu, Subramanian and Shankar2025), we consider the range of L, 40 ≤ L ≤ 10⁴. It can be observed from figure 10 that, for L ≥ 250, the FENE-P model recovers the Oldroyd-B prediction, indicating that the infinite extensibility approximation is acceptable for strongly extensible polymers. Besides, when L is reduced to values commonly encountered in numerical studies (40 ≤ L ≤ 250), G _max decreases monotonically, and the corresponding growth time t _max also diminishes, while even at L = 40, the great limit of polymer extensibility, significant transient growth still exists. Therefore, the Oldroyd-B model can capture the core non-modal elastic instability characteristics and the associated mechanisms, and the consideration of polymer finite extensibility only quantitatively changes the value of transient growth. Except for the discussion here on the effects of finite polymer extensibility, the rest of this work focuses solely on the Oldroyd-B model. It should also be reiterated that, although the largest energy amplification G _max under certain parameters in this section (e.g. see figures 8(a–c), 9(c) and 10(c)) can reach magnitudes of the order of 10⁴ or even larger levels that may readily trigger nonlinear effects, all results presented herein are derived exclusively from linear analysis.

Figure 10.

Effects of L on the largest transient growth G _max and growth time t _max for (a,b) spanwise-uniform perturbation at low Re and (c, d) oblique perturbation at high Re. The parameters in (a, b) and (c, d) are, respectively, the same as those in figure 4 and figure 12, and the blue line denotes the corresponding results of Oldroyd-B fluid.

Figure 11.

At β = 0.75, (a) effects of Re on transient growth at α = 3 and γ = 0; contours of (b) G _max , (c) G _{o-g max} and (d) G _{o-v,η max} at Re = 1000 and W = 12 and contours of G _max for (e) Re = 2000 and W = 12, (f) Re = 3000 and W = 12, (g) Re = 1000 and W = 24 and (h) Re = 3000 and W = 36.

Figure 11. Long description

Panel A: A line graph shows the effects of Reynolds number (Re) on transient growth with three different maximum growth values: Gmax (black line), Go-g max (red line), and Go-v, max (blue line). The x-axis represents Re ranging from 100 to 1000, and the y-axis represents Gmax ranging from 0 to 10000. An energy transition point is marked on the graph. Panel B: A contour plot shows Gmax with alpha (α) on the x-axis ranging from 0.1 to 4 and gamma (γ) on the y-axis ranging from 0 to 4. The color scale ranges from 0 to 20000. Panel C: A contour plot shows Go-g max with the same axes and color scale as Panel B. Panel D: A contour plot shows Go-v, max with the same axes and color scale as Panel B. Panel E: A contour plot shows Gmax for Re = 2000 and W = 12 with alpha (α) on the x-axis ranging from 0.5 to 3 and gamma (γ) on the y-axis ranging from 1 to 3. The color scale ranges from 0 to 40000. Panel F: A contour plot shows Gmax for Re = 3000 and W = 12 with the same axes and color scale as Panel E. Panel G: A contour plot shows Gmax for Re = 1000 and W = 24 with the same axes and color scale as Panel E. Panel H: A contour plot shows Gmax for Re = 3000 and W = 36 with the same axes and color scale as Panel E.

Figure 12.

(a) Input analysis and (b) output analysis for the oblique disturbances with α = 1.6 and γ = 1.25. Other parameters are Re = 1000, β = 0.75 and W = 12.

4.3. Energy amplification and its mechanism in the presence of strong shear flow

With a further increase of Re from 100 to 1000, see figure 11(a), it can be found that the main amplified energy output transitions from perturbation elastic energy to perturbation kinetic energy, while they still have the same order of magnitude, and within this Re range, the least stable mode at α = 3 remains the centre mode. In elasto-inertial pipe turbulence, Kumar & Graham (Reference Kumar and Graham2026) also suggested that the velocity fluctuation and polymer stretch fields have similar magnitude. Besides, we display the dependence of energy amplification on wavenumber in figure 11(b), and it can be seen that, like the case of Re = 1, the energy amplification of spanwise-uniform disturbances is also greater than that of streamwise-uniform disturbances, while unlike the Re = 1 case that spanwise-uniform disturbances reach the largest energy amplification, here, the oblique disturbance (near α = 1.6 and γ = 1.25) reaches the largest energy amplification rather than the spanwise-uniform disturbances in the presence of strong shear flow, and the least stable mode in streamwise is the TS mode rather than the centre mode. We also examine the wavenumber dependence of the transient growth using the high-Re parameters (Re = 1500, W = 20 and β = 0.97) set by Shekar et al. (Reference Shekar, Mcmullen, Wang, McKeon and Graham2019), and similarly, the oblique disturbance with a quasi-two-dimensional characteristic reaches the largest transient growth (not shown). Besides, in addition to the total energy growth G _max , see figures 11(c) and 11(d), both elastic energy growth G _{o-g max} and kinetic energy growth G _{o-v,η max} reach their maximum in the oblique disturbances. Therefore, the previous investigations (Hoda, Jovanović & Kumar Reference Hoda, Jovanović and Kumar2008; Zhang et al. Reference Zhang, Lashgari, Zaki and Brandt2013) that showed that the input streamwise-uniform velocity perturbation reaches the maximum perturbation kinetic energy growth are incomplete. We know that streamwise-uniform inertial perturbation reaches the maximum energy amplification in high-Re Newtonian Poiseuille flow (Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993), and spanwise-uniform elastic perturbation reaches the maximum transient growth in the highly elastic flow with low Re, as evidenced by figure 3. Therefore, it is reasonable to infer that, in this case, the optimal oblique disturbance may be due to the competition or synergy of the above two mechanisms, which will be elaborated on later. Figure 11(e–h) further displays the effects of Re and W on G _max , and we surprisingly find that the optimal wavenumbers (α _opt , γ _opt ) to achieve the largest transient growth are minimally affected. Specifically, the increase of Re (1000 ≤ Re ≤ 3000) and W (12 ≤ Re ≤ 36) slightly decreases α _opt (α _opt ≈ 1.6 to α _opt ≈ 1.3) and γ _opt (γ _opt ≈ 1.25 to γ _opt ≈ 1.1), respectively, and especially, the behaviour of the optimal spanwise wavenumber γ _opt being independent of Re is the same as that of Newtonian fluids (Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993).

To further study this perturbation kinetic energy growth-dominated flow at Re = 1000, we conduct an input–output analysis again near the optimal wavenumber of the largest energy growth in figure 12, and unlike the Re = 1 case in figures 4(a) and 4(b), here, the very rapid transient growth at t<1 disappears due to the larger Re and the smaller α. As shown in figure 12(a), input analysis indicates that, like the low-Re case of Re = 1, the transient energy growth with only the conformation tensor field perturbed initially is also very close to the optimal transient growth over all possible initial conditions, while the transient growth with other physical quantities perturbed initially is negligible, implying that, even if the Weissenberg number is not large enough (W = 12), the polymer perturbations still play a dominant role in transient growth. In the analysis of self-sustenance in elasto-inertial turbulence, Shekar et al. (Reference Shekar, Mcmullen, Mckeon and Graham2020, Reference Shekar, Mcmullen, McKeon and Graham2021) also indicated that even a relatively low W can cause the existence of viscoelastic nonlinear TS attractors, leading to the bypass transition to elasto-inertial turbulence. Besides, among all input polymer disturbances, spanwise-uniform components ( $\tilde{\text{ g}}$ _xx , $\tilde{\text{ g}}$ _yy and $\tilde{\text{ g}}$ _xy ) are the most important even if the input disturbance is oblique, which indicates that the origin of transient growth is two-dimensional and purely elastic in the presence of a strong shear viscoelastic flow. Similarly, the recent modal instability analysis of Khalid et al. (Reference Khalid, Shankar and Subramanian2021b) and Buza et al. (Reference Buza, Page and Kerswell2022) suggested that the instability of the centre mode can also be two-dimensional, and its origin can be viewed as purely elastic.

On the other hand, the results of output analysis are shown in figure 12(b), and it can be found that the growth of input oblique disturbances is different from that of spanwise-uniform disturbances. Specifically, for the growth of perturbation kinetic energy, the perturbation wall-normal velocity has a moderate amplification, and the perturbation wall-normal vorticity is the most amplified, which naturally promotes the formation of oblique streaks with alternating positive and negative regions (e.g. see the similar optimal oblique perturbation cases in a stably stratified shear flow (Jose et al. Reference Jose, Roy, Bale, Iyer and Govindarajan2018)), also evidenced by the non-zero perturbation spanwise velocity $\tilde{\text{ w}}$ of the optimal flow response in figure 14(c). For the growth of perturbation elastic energy, the oblique disturbance causes the tangential components, $\tilde{\text{ g}}$ _xz and $\tilde{\text{ g}}$ _yz , (especially $\tilde{\text{ g}}$ _xz ) to have a greater growth instead of the normal components $\tilde{\text{ g}}$ _xx and $\tilde{\text{ g}}$ _yy , greatly promoting the stretching of the polymer in the spanwise direction. For the specific mapping relationship between input and output disturbances, like the effects in figures 4(c) and 4(d), here, for the separate growth of disturbances $\tilde{\text{ v}}$ , $\tilde{\mathit{\eta }}$ , $\tilde{\text{ g}}$ _xz and $\tilde{\text{ g}}$ _yz , the input disturbances $\tilde{\text{ g}}$ _xx , $\tilde{\text{ g}}$ _yy and $\tilde{\text{ g}}$ _xy contribute the same proportions as those shown in figures 12(a) and 12(b) (and so are not shown), reflecting the proportionality between the total perturbation input/output and the certain perturbation input/output in transient growth.

To further study the physical mechanisms of the most amplified perturbation kinetic energy, we conduct a kinetic energy analysis. Following the previous investigations (Zhang et al. Reference Zhang, Lashgari, Zaki and Brandt2013), we have the following equation (for convenience, the specific derivation is not presented here):

(4.6) \begin{align} &\frac{1}{\varOmega }\int _{\varOmega }\frac{\partial \varepsilon _{k}}{\partial t}\text{d}V=\frac{1}{\varOmega }{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\frac{\partial }{\partial t}\left(\frac{u^{2}+v^{2}+w^{2}}{2}\right)\text{d}V\nonumber\\ &=\underset{Prod}{\underbrace{\frac{1}{\varOmega }{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\left(-uv\frac{d\overline{U}}{\text{d}y}\right)\,\text{d}z\,\text{d}x\,\text{dy}}}\nonumber\\ &\quad+\underset{Vd}{\underbrace{\frac{1}{\varOmega }{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\left(-\frac{\beta }{Re}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial u_{i}}{\partial x_{j}}\right)\,\text{d}z\,\text{d}x\,\text{d}y}}\nonumber\\ &\quad+\underset{Pd}{\underbrace{\frac{1}{\varOmega }{\int }_{\!\!\!-1}^{1}{\int }_{0}^{a}{\int }_{0}^{b}\left[-\frac{1-\beta }{ReW}\left(Qg\right)_{ij}\frac{\partial u_{i}}{\partial x_{j}}\right]\,\text{d}z\,\text{d}x\,\text{dy}}}, \end{align}

where the term on the left means the growth rate of perturbation kinetic energy, the right terms Prod, Vd and Pd represent the contribution to perturbation kinetic energy growth from the base flow field, the viscous dissipation and the perturbed conformation tensor field, respectively.

Figure 13 displays the contribution of each physical field to the growth of perturbation kinetic energy, and the case of elastic energy analysis is additionally provided in panel (b) to compare with the results at Re = 1 (see the discussions of figure 6). As shown in figure 13(a), we can find that the energy transfer from the base flow field to the perturbed hydrodynamic field, i.e. Prod term, is the main means of kinetic energy growth, the energy transfer from the perturbed conformation tensor field to the perturbed hydrodynamic field, i.e. Pd term, is the secondary means of energy growth for the larger t and some of these kinetic energies are dissipated by the fluid viscosity, i.e. Vd term. Therefore, in this case, viscoelasticity mainly promotes the perturbation kinetic energy growth by enhancing both the Orr mechanism and the lift-up mechanism, the kinetic energy growth mechanism of spanwise-uniform and streamwise-uniform disturbances in Newtonian fluids, respectively (see Schmid & Henningson (Reference Schmid and Henningson2001) for details), which naturally leads to the rapid formation of oblique streaks. At the same time, see figure 13(b), the perturbed velocity field also transfers energy to the perturbed conformation tensor field, i.e. Au term, which remains the main path for the growth of perturbation elastic energy at high Re, like the Re = 1 case. The specific energy transfer paths in the Au term mainly involve the interaction between the perturbation conformation tensor components, $\tilde{\text{ g}}$ _xz and $\tilde{\text{ g}}$ _yz , and the perturbation spanwise velocity $\tilde{\text{ w}}$ , while the kinetic energy growth paths in the Pd term (shown in figure 13(a)) mainly involve the energy transfer from the components $\tilde{\text{ g}}$ _xy and $\tilde{\text{ g}}$ _yy to the perturbation streamwise velocity $\tilde{\text{ u}}$ , revealing two energy conversion paths between the perturbed hydrodynamic field and the perturbed conformation tensor field. Besides, unlike the Re = 1 case, the perturbed conformation tensor field at high Re can directly absorb energy from the base flow to further promote elastic energy growth. It should be clarified that, after our verification, this difference is not caused by the presence of high Re, but by the non-zero spanwise wavenumber. The elastic energy analysis is only a byproduct of our studies at high Re, and the subsequent discussions mainly focused on the perturbation kinetic energy with the greater amplification.

Figure 13.

At Re = 1000, β = 0.75 and W = 12, (a) kinetic energy analysis and (b) elastic energy analysis for oblique disturbances and effects of (c) wavenumbers and (d) Re and W on kinetic energy analysis.

Figure 13. Long description

Panel A: A line graph shows contributions to kinetic energy over time (t). The x-axis represents time (t) and the y-axis represents contributions to kinetic energy. The graph includes multiple lines representing different components: kinetic energy (black), Prod (red), Vd (blue), and Pd (green). Symbols indicate specific contributions: Pd_gxx (triangles), Pd_gxy (circles), and Pd_gyy (stars). Panel B: A line graph shows contributions to elastic energy over time (t). The x-axis represents time (t) and the y-axis represents contributions to elastic energy. The graph includes lines for elastic energy (black), Ag (blue), Ab (green), and Au (red), with symbols for Au_gxz (circles) and Au_gyz (stars). Panel C: A line graph shows contributions to kinetic energy over time (t) for different wavenumbers. The x-axis represents time (t) and the y-axis represents contributions to kinetic energy. The graph includes lines for different wavenumbers and parameters: light blue (alpha = 1.6, gamma = 0), pink (alpha = 3, gamma = 0), and orange (alpha = 1.6, gamma = 1.25). Solid thick lines represent kinetic energy, solid thin lines represent Prod, and dash-dot lines represent Pd. Panel D: A line graph shows contributions to kinetic energy over time (t) for different Reynolds numbers (Re) and Weissenberg numbers (W). The x-axis represents time (t) and the y-axis represents contributions to kinetic energy. The graph includes lines for different Re and W values: light blue (Re = 3000, W = 12), orange (Re = 1000, W = 36). Solid thick lines represent kinetic energy, solid thin lines represent Prod, and dash-dot lines represent Pd.

Figure 14.

At t = t _max, the optimal responses of (a, c) perturbation velocity; (b, d) perturbation conformation tensor $\tilde{\text{ g}}$ and the spatial structure of (e) perturbation streamwise velocity $\tilde{\text{ u}}$ and (f) perturbation conformation tensor component $\tilde{\text{ c}}$ _xx in the cross-span plane where α = 1.6 and γ = 1.25. Other parameters are Re = 1000, β = 0.75 and W = 12.

Figure 13(c) displays the effects of wavenumbers on the kinetic energy analysis, and specifically, for the spanwise-uniform perturbation of α = 3 and γ = 0, we can observe that the second means of perturbation kinetic energy growth, the Pd term, discussed in figure 13(a), is more obvious, leading to two distinct growth stages (see the pink thick solid line). For the first stage, similar to the oblique disturbance, the Prod term remains the most important means to cause perturbation kinetic energy growth, while the perturbed polymer conformation tensor field can absorb energy from the perturbed hydrodynamic field to inhibit its growth. For the second stage, the perturbed conformation tensor field directly transfers energy to the perturbed hydrodynamic field, causing kinetic energy growth, whereas the original contribution from base flow is even negative to suppress the growth of perturbation kinetic energy. By comparing the three cases in the second stage, this direct energy transfer mechanism from the perturbed conformation tensor field is enhanced with an increase in streamwise wavenumber, while the influence of the spanwise wavenumber is very small. Besides, the results in figure 13(d) further indicate that the increase in W and the decrease in Re can also effectively promote this direct energy transfer mechanism from the perturbed conformation tensor field.

Next, we display the flow structures of optimal responses at t = t _max . For the optimal initial conditions (not shown), the perturbation velocity closely resembles that of a Newtonian fluid (see Schmid & Henningson (Reference Schmid and Henningson2001) for details), evidenced by the dominant Prod term of perturbation kinetic energy growth in figure 13(a), and additionally, the perturbation conformation tensor is similar to that in figure 6 due to the similar growth mechanisms of perturbation elastic energy (see the discussions in figures 6(b) and 13(b)). As shown in figures 14(a) and 14(c), the optimal response of perturbation velocity $\tilde{\text{ u}}$ and $\tilde{\text{ w}}$ is concentrated near the upper and lower plates and displays a sharp localisation, and $\tilde{\text{ v}}$ is basically concentrated near the channel centreline. We also find that, by comparing the case of α = 1.6 and γ = 0 and the case of α = 1.6 and γ = 1.25, the introduction of spanwise wavenumber additionally increases the growth of perturbation streamwise and spanwise velocity to promote the growth of perturbation kinetic energy. Besides, by comparing the case of α = 1.6 and γ = 0 and the case of α = 3 and γ = 0, it can be observed that the main flow structures are not changed, even though there exists another significant kinetic energy growth mechanism in the second case (see the discussions in figure 13(c)), and the perturbation streamwise velocity $\tilde{\text{ u}}$ has a significant increase as α increases. Therefore, even the least stable mode at α = 3 is the centre mode, a similar TS wave-like structure of perturbation velocity can also be formed through transient growth. For the optimal response of the perturbation conformation tensor, as shown in figures 14(b) and 14(d), all components are also concentrated near the boundary, and like the effects in perturbation velocity, the introduction of spanwise wavenumber additionally amplifies the conformation tensor components related to the spanwise direction, whereas the variation of components related to the streamwise direction is limited. We further display the spatial structures of the perturbation streamwise velocity $\tilde{\text{ u}}$ and the streamwise component of the perturbation conformation tensor in the classical Reynolds decomposition $\tilde{\text{ c}}$ _xx for comparison with previous investigations. As shown in figures 14(e) and 14(f), it can be found that the streamwise velocity fluctuations have a TS wave-like structure, and polymer fluctuations $\tilde{\text{ c}}$ _xx are highly inclined and antisymmetric in this location, which bear a certain resemblance to the nonlinear DNS results (Sid et al. Reference Sid, Terrapon and Dubief2018; Shekar et al. Reference Shekar, Mcmullen, Wang, McKeon and Graham2019; Kumar & Graham Reference Kumar and Graham2024), e.g. the tilted sheet-like structures of polymer stretch, that are the hallmark of elasto-inertial turbulence.

Based on the investigations of Shekar et al. (Reference Shekar, Mcmullen, Wang, McKeon and Graham2019, Reference Shekar, Mcmullen, Wang, McKeon and Graham2020), Beneitez et al. (Reference Beneitez, Page, Dubief and Kerswell2024) and Zhang et al. (Reference Zhang, Cheng, Wang, Zhang, Li and Li2024), they indicated that the self-sustaining mechanisms in elasto-inertial turbulence seem to be related to the near-wall velocity and polymer layers instead of the centre mode instability and its arrowhead characteristic. In addition, Page & Zaki (Reference Page and Zaki2014) and Shekar et al. (Reference Shekar, Mcmullen, Mckeon and Graham2020) suggested that the two-dimensional transient growth induced by the appropriate perturbation spanwise vortex can achieve sufficient energy amplification (reaching a finite amplitude) to cross the edge state, and cause the subsequent formation of a viscoelastic nonlinear TS attractor and the associated self-sustaining mechanisms. In our results, the transient growth induced by the perturbed conformation tensor field is greater than that induced by the perturbed velocity and vortex field, and also causes the formation of highly inclined polymer sheets. Furthermore, the optimal three-dimensional oblique perturbation (α ≈ 1.6 and γ ≈ 1.25) in our results additionally leads to the formation of oblique streaks, which also promote the self-sustaining process (Waleffe Reference Waleffe1997; Schmid Reference Schmid2007; Duriez et al. Reference Duriez, Aider and Wesfreid2009; Jovanović Reference Jovanović2021). This finding implies the importance of three-dimensional investigations in elasto-inertial turbulence, and Lellep et al. (Reference Lellep, Linkmann and Morozov2023) also indicated that the properties of elasto-inertial flows can only be accessed by three-dimensional simulations. Therefore, our results offer a new perspective on transient growth to understand the physical mechanisms of energy amplification and conversion in the linear stage, and may provide a simple mechanistic explanation of this viscoelastic excitation of the TS wave, connecting the production of viscoelastic nonlinear attractor and self-sustaining mechanisms in elasto-inertial turbulence.