3. Dynamics of the optimal disturbance

The temporal evolution of the optimal disturbance's energy at ${\textit {Re}}_s=2000$, ${\textit {A}}=1$ and ${\textit {Wo}}=15$ is shown as a dashed line in figure 1(a). At these parameter values, the optimal disturbance has a helical structure with $(k,m)=(3.24,1)$ and is localized at the outer half of the pipe (exceeding the Stokes-layer thickness), see figure 1(f,g). The optimal point to disturb is during the deceleration phase, at $t_0/T=0.5$, whereas the maximum amplification is reached during the acceleration phase, at $t_f/T=1.2$. The classic $k=0$, $m=1$ optimal disturbance of steady pipe flow is also amplified significantly in this case, albeit an order of magnitude less than the helical one. The classic disturbance initially consists of stream-wise vortices, as shown in figure 1(d,e), and the energy is subsequently transferred to the stream-wise velocity components, while the cross-stream components decay monotonically, see figure 1(b) and supplementary movie 1 available at https://doi.org/10.1017/jfm.2020.940. Overall, the classic perturbation's behaviour appears to be rather insensitive to the change in flow profile throughout the cycle, and the decay is very slow.

Figure 1.

(a) Time series of the kinetic energy $E(t)/E(t_0)$ of the optimal helical, $(k,m)=(3.24,1)$, and classic, $(k,m)=(0,1)$, disturbances at $(Re_s,{\textit {A}},{\textit {Wo}})=(2000,1,15)$. (b,c) Time series of kinetic energy contribution of each velocity component for the classic (b) and helical (c) disturbances. (d,e) Contours of stream-wise vorticity (on an $r$-$\theta$ cross-section) and of stream-wise velocity (on a $z$-$r$ cross-section) of the classic disturbance, and the corresponding base flow profile ${\boldsymbol U}_{b,z}$ at $t_0/T=0.25$ (d) and $t_f/T=1.75$; see supplementary movie 1 for an animation of the disturbance dynamics (e). (f,g) The same as (d,e), but for the helical disturbance at $t_0/T=0.5$ and $t_f/T=1.2$; see also supplementary movie 2. The dashed line denotes the Stokes-layer thickness.

The kinetic energy of the optimal helical perturbation is mostly distributed in the stream-wise and azimuthal components, which self-amplify rapidly during the deceleration phase and a bit slower during the subsequent acceleration phase, see figure 1(c). Initially, the disturbance spirals clockwisely towards the pipe centre while leaning against the background shear profile, see figure 1(f) and supplementary movie 2. As the energy grows, the perturbation switches the spiralling direction and is tilted by the shear until it aligns with it and the disturbance finally decays, see figure 1(g). This is reminiscent of the Orr mechanism (Orr Reference Orr1907). However, approximately $96\,\%$ of the kinetic energy is shared in equal parts between the azimuthal and stream-wise components, which indicates a strong three-dimensional effect, distinct from the two-dimensional Orr mechanism reported for many flows (see e.g. Boyd Reference Boyd1983; Farrell Reference Farrell1988; Maretzke, Hof & Avila Reference Maretzke, Hof and Avila2014; Biau Reference Biau2016).

Von Kerczek (Reference Von Kerczek1982) and Tsigklifis & Lucey (Reference Tsigklifis and Lucey2017) studied the stability of pulsatile channel flows and reported large modal growth of disturbances at low ${\textit {Wo}}$, which also sets in during the deceleration phase and reaches the maximum at the end of the period of pulsation. These authors associated the modal energy growth with the inflectional velocity profiles occurring in the deceleration phase. Here, we found that at the optimal perturbation point $t_0$, the velocity profile has also inflection points, see figure 1(f). We froze this flow profile and did a modal (eigenvalue) analysis to obtain the most dangerous mode. This mode was subsequently used as the initial condition for the linearized Navier–Stokes equations and exhibited large energy growth (albeit an order of magnitude lower than the non-modal optimal). We found similar results in additional runs at ${\textit {Wo}}=5$ and $8$, suggesting that inflection points and modal mechanisms may also play an important role in pulsatile pipe flow.

The black thick line in figure 2(a) depicts the maximum energy amplification $G(t_f)$ over all $(k,m)$ and initial disturbances time $t=t_0$. The maximum amplification is reached during the acceleration phase via helical disturbances, as shown in figure 2(b), whereas the classic disturbance achieves larger growth only during the second half of the deceleration phase. The colourmap of figure 2(c) shows that the optimal time to perturb the flow is during the middle of the deceleration phase ($t_0/T\approx 0.5$); perturbing during the acceleration phase leads to much lower growth (yielded by the classic disturbance during the deceleration phase). Clearly, the helical mechanism is efficient only in the deceleration phase. The time needed to reach the maximum growth is $\tau \approx (3/4)T$, which explains why the maximum growth occurs during the acceleration phase.

Figure 2.

Transient growth at $(Re_s,{\textit {A}},{\textit {Wo}})=(2000,1,15)$: (a) The red dashed lines denote the temporal evolution of the optimal classic disturbances $(k,m)=(0,1)$ for four different initial times $t_0$, whereas the blue solid lines correspond to the optimal helical disturbances $(k,m)=(3.24,1)$ initialized at the same $t_0$. The thick black line is the maximum gain $G(t_f)$ that can be achieved at a given time $t_f$ (optimized over $k$, $m$ and $t_0$ disturbances). (b) Dependence of the optimal axial wavenumber $k$ (associated with the thick line of (a)) on $t_f$. (c) Colourmap of the maximum gain $G(t_0,\tau )$ (optimized over $k$ and $m$) in the $t_0$–$\tau$ plane. The black cross marks the maximum of $G$.

4. Parametric study of transient growth

We show in figure 3(a) that keeping $({\textit {A}},{\textit {Wo}})=(1,15)$ fixed and increasing the Reynolds number leads to an approximately exponential increase of the energy gain $G_{max}$. Independently of the Reynolds number, the maximum energy amplification occurs always during flow acceleration with helical disturbances introduced during the deceleration phase, see figure 3(b). As shown in figure 3(c), an asymptotic behaviour is approached for ${\textit {Re}}_s\gtrsim 5000$, with $k\approx 3.4$, $t_f/T\approx 1.15$ and $t_0/T\approx 0.35$.

Figure 3.

Optimal transient growth as a function of ${\textit {Re}}_s$ for $({\textit {A}},{\textit {Wo}})=(1,15)$. (a) The black circles show the computed maximum energy gain. The black solid line $G_{max}=\exp (({\textit {Re}}_s/232)-2.12)$ is a fit to the data for ${\textit {Re}}_s\geq 4000$. The orange dotted line $G_{max}=2.72\,{\textit {Re}}_s^2$ is a fit to data for $({\textit {A}},{\textit {Wo}})=(1,25)$. (b,c) Optimal initial and final disturbance times and axial wavenumber.

The effect of the pulsation amplitude ${\textit {A}}$, while keeping $({\textit {Re}}_s,{\textit {Wo}})=(2000,15)$ fixed, is illustrated in figure 4. For steady pipe flow (${\textit {A}}=0$), Meseguer & Trefethen (Reference Meseguer and Trefethen2003) obtained $G_{max}\approx 288.7$, $\tau \approx 24.5$ and $(k,m)=(0,1)$, which is in line with our results for pulsatile flow of low amplitude $A\lesssim 0.55$. Helical disturbances dominate thereafter (see the black line in figure 4a), but their energy gain does not grow monotonically (exponentially) with the amplitude until ${\textit {A}}\gtrsim 4$. For intermediate amplitudes, there is competition between two distinct types of helical disturbances, as shown in figure 4(b). For $0.55\lesssim {\textit {A}}\lesssim 2.5$, the dominant helical disturbance is similar to that examined in detail in figures 2 and 3, whereas for ${\textit {A}}\gtrsim 2.5$ another helical disturbance, with much shorter axial length and shorter evolution time $\tau =t_f-t_0$, takes over. Both types of helical disturbances are triggered in the deceleration phase. Similar results were obtained for lower frequency, at $({\textit {Re}}_s,{\textit {Wo}})=(2000,10)$, see the green lines in figures 4(a) and 4(c). Here, the maximum growth $G$ is over two orders of magnitude larger and helical disturbances dominate earlier (already for $A\gtrsim 0.41$). From the data in figures 3(a) and 4(a), we conclude that the maximum energy gain of helical disturbances scales exponentially with the Reynolds number of the oscillatory flow component ${\textit {Re}}_o = U_o D /\nu =A {\textit {Re}}_s$.

Figure 4.

Optimal transient growth as a function of ${\textit {A}}$ for $({\textit {Re}}_s,{\textit {Wo}})=(2000,15)$. (a) The black circles show the computed maximum energy gain, the green squares and the orange diamonds are for $({\textit {Re}}_s,{\textit {Wo}})=(2000,10)$ and $({\textit {Re}}_s,{\textit {Wo}})=(2000,20)$, respectively. (b,c) Optimal initial and final disturbance times and axial wavenumber. The maximum gain as a function of $t_f/T$ is shown in the inset in (b) for ${\textit {A}}=2.4$ and $2.6$ and illustrates the competition between two distinct different disturbances.

In what follows, we examine the influence of the pulsation frequency on the dominant mechanism of transient growth in more detail. We begin by focusing on ${\textit {Re}}_s=2000$ and ${\textit {A}}=1$. The circles in figure 5(a) show that for sufficiently large ${\textit {Wo}}\gtrsim 20$ the classic disturbance dominates and the optimal gain $G$ of steady pipe flow is recovered (exactly as for the data shown as orange lines in figures 3 and 4). As shown in figure 5(b), the perturbation growth time $\tau \approx 25$ is also indistinguishable from that of steady pipe flow. It can be concluded that in the limit ${\textit {Wo}}\rightarrow \infty$ the disturbance response is solely governed by ${\textit {Re}}_s$. The dynamics of the steady pipe flow is also recovered in the quasi-steady limit ${{\textit {Wo}}\rightarrow 0}$, where the classic perturbation dominates as well. In this limit, the maximum transient growth is governed by the maximum Reynolds number ${\textit {Re}}_\text {max} = (1+{\textit {A}}) {\textit {Re}}_s={\textit {Re}}_s+{\textit {Re}}_o$ and occurs for $t_0,t_f\rightarrow T/4$, see the inset in figure 5(b). For instance, for ${\textit {Re}}_s=2000$, ${\textit {A}}=1$ and ${\textit {Wo}}=3$, the maximum gain is $G\approx 1100$, which is slightly less than the gain for steady pipe flow $G\approx 1155$ at ${\textit {Re}}_s=4000$. As shown in figure 5(c), helical disturbances dominate for intermediate $4\lesssim {\textit {Wo}}\lesssim 18$ and exhibit a sharp maximum in gain at ${\textit {Wo}}\approx 7$, see figure 5(a). The results for ${\textit {A}}=4$ (shown as green squares) are qualitatively similar to those for ${\textit {A}}=1$, but the energy gain is much larger.

Figure 5.

Effect of the Womersley number ${\textit {Wo}}$ on the transient growth. (a) Maximum energy gain for $({\textit {Re}}_s,{\textit {A}})=(2000,1)$ (black circles), $({\textit {Re}}_s,{\textit {A}})=(2000,4)$ (green squares) and oscillatory flow at ${\textit {Re}}_o=8000$ $({\textit {Re}}_s,{\textit {A}})=(0,\infty )$ (orange diamonds). The hollow symbols show the optimal growth of classic disturbances $(k,m)=(0,1)$ in the regime where the helical disturbance dominates. (b) Optimal initial and final disturbance times at $({\textit {Re}}_s,{\textit {A}})=(2000,1)$. The pink dashed line shows $t_f/T = (\tau _{steady} + t_0)/T$, where $\tau _{steady}$ is the optimal evolution time of the classic disturbance in steady pipe flow at ${\textit {Re}}_s=2000$. The inset shows a zoom for ${\textit {Wo}}\leqslant 18$. (c) Optimal axial wavenumber.

5. Transient growth in oscillatory pipe flow

The dependence of the energy gain $G$ on the frequency for the specific case of oscillatory flow at ${\textit {Re}}_o=8000$ is shown as orange diamonds in figure 5(a) and follows the trend of pulsatile flow. In figure 6(a), these data are shown as a function of the Reynolds number of the Stokes layer ${\textit {Re}}_{\delta }=U_o\delta /\nu ={\textit {Re}}_o/(\sqrt {2}{\textit {Wo}})$, together with three additional sets for ${\textit {Wo}}=10$, $15$ and $20$ covering wide ranges of ${\textit {Re}}_o$. At low ${\textit {Re}}_{\delta }$, the maximum gain $G=1$ is reached for $t_0=t_f$, implying that all perturbations decay monotonously, and the disturbance with ($k=0, m=0$) is the least damped, see figure 6(b,c). This can also be seen in figure 5(a), where the limit $G\rightarrow 1$ is reached by the orange data points as ${\textit {Wo}}\rightarrow \infty$ (and hence ${\textit {Re}}_{\delta }\rightarrow 0$ for constant ${\textit {Re}}_o=8000$). For ${\textit {Re}}_{\delta } \gtrsim 100$, the energy gain increases exponentially with ${\textit {Re}}_{\delta }$ and depends only weakly on the Womersley number. All data sets shown in figure 6(a) exhibit excellent collapse, except for the data set at ${\textit {Re}}_o=8000$, which deviates from the exponential scaling for ${\textit {Re}}_\delta \gtrsim 630$ (corresponding to ${\textit {Wo}}\lesssim 9$, i.e. near and below the peak in figure 5a). The optimal point to disturb the flow is during the deceleration phase at $t_0/T\approx 0.35$ (as in figure 3b at sufficiently high ${\textit {Re}}_o$). The maximum energy gain is attained toward the end of the deceleration phase at low ${\textit {Re}}_{\delta }$, and moves progressively into the acceleration phase as ${\textit {Re}}_{\delta }$ increases.

Figure 6.

Transient growth of disturbances in oscillatory pipe flow for ${\textit {Wo}}=10$, $15$ and $20$ as a function of ${\textit {Re}}_{\delta }$: (a) energy amplification, (b,c) optimal initial and final disturbance times and optimal wavenumbers, where the axial wavenumber is scaled with $\delta ^{-1}$. (d–g) Contours of spanwise vorticity (on a $z$–$r$ cross-section) of the optimal helical disturbance ($m=1$) for ${\textit {Wo}}=10$ and ${\textit {Re}}_\delta =530$ at $t_0/T=0.35$ (d), $t/T= 0.4$ (e), $t/T=0.5$ (f) and $t_f/T=0.75$ (g); see supplementary movie 3 for an animation. (h–k) The same as (d–g), but for the optimal axisymmetric disturbance ($m=0$) for ${\textit {Wo}}=15$ and ${\textit {Re}}_\delta =589$ at $t_0/T=0.35$ (h), $t/T= 0.4$ (i), $t/T=0.5$ (j) and $t_f/T=0.75$ (k); see supplementary movie 4.

Snapshots of the spanwise vorticity of the optimal perturbation are shown in figure 6(d–g) for ${\textit {Wo}}=10$ and ${\textit {Re}}_\delta =530$. Initially, the perturbation leans against the background shear and then tilts to align with the streamwise direction as it propagates radially inwards (see supplementary movie 3). The helical structure and dynamics of the disturbance are very similar to that shown in figure 1(f), but here the disturbance is initially more confined to the Stokes layer. In figure 6(h–k) and supplementary movie 4, we show that at higher ${\textit {Wo}}=15$, the optimal disturbance is very similar in dynamics, but is axisymmetric (see also figure 6c). The energy growth occurring here is as in the classic Orr mechanism, with the only difference that the perturbation travels radially inwards as it changes the tilt direction. We stress that despite this difference in the disturbance geometry (helical versus axisymmetric), the energy gain, the point of disturbance and the point of maximum gain are indistinguishable (also for larger ${\textit {Wo}}=20$, see figure 6a–c).