4.1. Optimal inflow perturbation and its outcome

The unforced base flow ( $\boldsymbol{f}=0$ ) is illustrated in figure 3. Two recirculation bubbles, characterized by negative streamwise velocity, can be observed downstream of the step. The primary bubble is associated with the lower wall and reattaches around $x=11$ , while the secondary bubble is associated with the upper wall, separates around $x=8$ and reattaches around $x=17$ . It can be seen that the flow is almost parallel downstream of the secondary bubble, which supports local, or streamwise-periodic, perturbation developments. This base flow is asymptotically stable but acts as an amplifier of upstream noise and exhibits strong transient energy growth (Blackburn et al. Reference Blackburn, Barkley and Sherwin2008; Marquet et al. Reference Marquet, Sipp and Jacquin2008b ).

The receptivity of this base flow to upstream noise, measured by the gain $K$ , at various spanwise wavenumbers ${\it\beta}$ , final times ${\it\tau}$ and frequencies ${\it\omega}$ is plotted in figure 4. It is seen that inflow perturbations with ${\it\omega}=0.50$ are the most amplified, and the global maximum gain over the parameters considered occurs at ${\it\tau}=150$ , ${\it\beta}=0$ and ${\it\omega}=0.50$ . From the transient receptivity at ${\it\beta}=0$ , as shown in figure 4(a), it is observed that the transient effects vanish at ${\it\tau}=150$ (the gain becomes constant with respect to ${\it\tau}$ ). Since the transient noise amplification can be more clearly illustrated by the receptivity to initial perturbations, in all the following studies of receptivity to inflow noise we will use ${\it\tau}=150$ unless stated otherwise, in order to exclude transient effects. In figure 4(b), one can see that the receptivity is maximized at ${\it\beta}=0$ and decreases almost monotonically with increasing ${\it\beta}$ , except that at small ${\it\omega}$ the optimal value of ${\it\beta}$ is around 1.

Figure 4.

Contours of the gain $K$ at (a)  ${\it\beta}=0$ and (b)  ${\it\tau}=150$ .

In figure 5, the spatial distributions of the optimal inflow perturbation obtained at ${\it\beta}=0$ , ${\it\tau}=150$ and three typical frequencies ${\it\omega}$ are displayed. It shows that the vertical wavenumber of the perturbation increases while the oscillation magnitude decreases with increasing ${\it\omega}$ . For the global optimal inflow perturbation ( ${\it\omega}=0.5$ ), the vertical component is significantly smaller than the streamwise component, and so this global optimal perturbation can be interpreted as a streamwise gust. It is also worth noting that the distribution of the optimal inflow noise is independent of further extensions of the outflow section, whereas for a longer inflow section, it tends to be a uniform streamwise perturbation. Such uniform inflow noise becomes optimal because it induces perturbations around the step with the same magnitude (considering the continuity and streamwise momentum equations), without being diffused in the elongated channel flow upstream of the step.

Figure 5.

Distribution of the optimal inflow velocity $\tilde{\boldsymbol{u}}_{b}$ at ${\it\tau}=150$ and ${\it\beta}=0$ : (a) streamwise velocity component; (b) vertical velocity component. The magnitude is normalized to satisfy $[\tilde{\boldsymbol{u}}_{b},\tilde{\boldsymbol{u}}_{b}]=1$ . The default inflow and outflow lengths are 5 and 50, respectively.

Figure 6 displays the outcomes of the optimal inflow perturbations presented in figure 5. Since the distribution of the outcomes is independent of the domain size as shown in table 2, the optimal outcomes at various domain sizes are not reported here. The wavenumber of the perturbation in the streamwise direction roughly reflects the frequency of the inflow noise. It is worth noting that the outcome of the most energetic inflow perturbation at ${\it\omega}=0.5$ is concentrated in the region downstream of the secondary bubble, with the magnitude of the inflow perturbation velocity being amplified over two orders of magnitude, while the outcomes at ${\it\omega}=0$ and 1.5 are associated with the primary bubble and the fore part of the secondary bubble. For the three outcomes, the structures upstream of the end of the secondary bubble are dominated by strings of vorticity, whose sign changes around the bubble borders, thus manifesting the inflectional point instability associated with shear layers in recirculation bubbles (Marquet et al. Reference Marquet, Sipp, Chomaz and Jacquin2008a ). For the two non-zero-frequency cases, induced perturbations with smaller magnitudes around the upper and lower walls are also observed.

Figure 6.

Contours of spanwise vorticity for the outcomes of the optimal inflow boundary perturbations at $t=150$ , ${\it\beta}=0$ , ${\it\tau}=150$ and (a)  ${\it\omega}=0$ , (b)  ${\it\omega}=0.5$ or (c)  ${\it\omega}=1.5$ . The inflow perturbation is normalized so that $[\tilde{\boldsymbol{u}}_{b},\tilde{\boldsymbol{u}}_{b}]=1$ . The thick black lines represent the borders of recirculation bubbles.

4.2. Mechanisms of receptivity

To illustrate the mechanisms underlying receptivity, consider as an example the case where ${\it\beta}=0$ , ${\it\tau}=150$ and ${\it\omega}=0.5$ , at which the receptivity reaches a maximum. From the momentum equation of perturbations, we have

In this derivation, the viscous diffusion is neglected and the divergence-free conditions on both the perturbation flow and the base flow are used.

In the region downstream of the secondary bubble, the base flow is almost parallel (see figure 3). Neglecting the vertical component of the base velocity and the streamwise gradient of the streamwise base velocity, we obtain

where $u$ , $v$ and $U$ are the streamwise perturbation velocity, vertical perturbation velocity and streamwise base velocity, respectively.

Note that in the region downstream of the secondary bubble, the perturbations are approximately periodic in the streamwise direction (see figure 6 b), manifesting the local dynamics. Therefore the integrals of the second and fourth terms in (4.1) over a section of the domain covering a period of the perturbations vanish:

since the velocities are zero on the wall boundary and are the same on the inflow and outflow boundaries of the selected domain, whose surface normal $\boldsymbol{n}$ is opposite.

Combining (4.1)–(4.3), we obtain the energy equation of the perturbations,

in a similar way to the derivation for a vortex flow in a cylindrical frame (Pradeep & Hussain Reference Pradeep and Hussain2006). Here $E=\int \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u}/2\,\text{d}V$ is the perturbation energy in this region, whose streamwise extension covers one period of the perturbations.

On the upper part of the channel, $\text{d}U/\text{d}y<0$ , and on the lower part $\text{d}U/\text{d}y>0$ . Therefore, as shown in figure 7, a perturbation with left-headed streamlines (bending to the left) results in $uv(\text{d}U/\text{d}y)<0$ , with energy growing according to (4.4); a perturbation with right-headed streamlines (bending to the right) corresponds to $uv(\text{d}U/\text{d}y)>0$ and decays in time; a rectangular streamline is neutral (i.e. neither grows nor decays, provided that viscous diffusion is neglected).

Figure 7.

Perturbation streamlines that result in energy growth and decay in a parallel base flow. The solid lines with arrows represent the base flow streamwise velocity, and the dashed lines with arrows represent streamlines of the perturbation. The dotted horizontal line is the centreline of the channel, and the solid horizontal and vertical lines delineate the border of a segment of the domain under consideration.

From figure 6(b) it is seen that in the region around $x=10$ , where the primary bubble and the secondary bubble coexist, the perturbation vortex structures can be viewed as a combination of five vortex strings, as schematically plotted in figure 8. The main vortex string is between the two bubbles; owing to the bubble-induced inflectional point instabilities, another two strings with opposite sign, referred to as bubble vortices, are generated around the bubble borders; the interaction of the dominant string of vortices with the wall boundary layer induces another two vortex strings around the upper and lower walls, referred to as wall vortices. In this region, the vortex structures are associated with instability mechanisms and so the five vortex strings are well aligned in the direction normal to the bubble border (shown as vertically distributed in figure 8 to facilitate comparison with downstream structures).

Figure 8.

Sketch of vortex structures in the outcome of the optimal inflow perturbation at ${\it\beta}=0$ , ${\it\tau}=150$ and ${\it\omega}=0.5$ (see figure 6 b).

However, in the region downstream of the secondary bubble, where the bubble instability vanishes, the bubble vortices are no longer aligned with the dominant vortices. In this region, when whole vortex structures are convected downstream by the base flow, the dominant vortices move faster than the bubble vortices, since the dominant vortices are located in the middle of the channel where the base flow streamwise velocity is maximal. Therefore the dominant vortices and bubble vortices form left-headed structures or streamlines at around $x=20$ , approximately rectangular profiles at $x=28$ and, finally, right-headed structures at around $x=40$ , as shown in figures 8 and 6(b). However, the wall vortices always align with the main vortices, since they are induced by the dominant vortices and are not affected by the convection. This transformation of streamline patterns from left-headed to rectangular and then to right-headed structures was also observed in previous non-normality studies, with the neutral rectangular profile appearing also at around $x=28$ (Blackburn et al. Reference Blackburn, Barkley and Sherwin2008).

Combining figures 6(b), 7 and 8, it can be summarized that the inflow perturbation is first amplified by the bubble-induced inflectional point instabilities, further strengthened by the convection effects, which generate perturbations with left-headed (energy growth) streamlines, and then self-limited by the base flow convection, which eventually converts the perturbations to right-headed structures (energy decay). Therefore the perturbation magnitude grows upstream of the (neutral) rectangular streamlines, which are located at around $x=28$ , and decay downstream of this location. This threshold point for perturbation development is not obvious in figure 6(b) because of the choice of contour levels, but it can be seen more clearly in a later discussion about development of the optimal initial perturbation in figure 23. This amplification effect associated with base flow convection will be referred to as the misalignment mechanism in the following.

This mechanism also explains the receptivity at other frequencies. For the ${\it\omega}=0$ case, there is no alternative vorticity and the misalignment mechanism does not appear, suggesting that the receptivity depends only on the bubble instabilities as shown in figure 6(a). At higher frequencies, e.g.  ${\it\omega}=1.5$ , only the primary bubble induces instabilities and the secondary bubble instability is not activated, as shown in figure 6(c). Therefore the misalignment effects start downstream of the primary bubble and, owing to the relatively short streamwise wavelength of the perturbation, the right-headed streamlines can be generated faster and appear even upstream of the end of the secondary bubble. Therefore neither the bubble instability nor the misalignment mechanism is fully exploited in this case.

4.3. Nonlinear development of the optimal inflow perturbations

The receptivity presented above concerns the linearized development of perturbations. In this subsection, the linear assumption is discarded and the magnitude of the boundary perturbation is introduced to study the development of the perturbed flow (which can be interpreted as the nonlinear evolution of the perturbations) through direct numerical simulations (DNS). The numerical set-up is the same as for the calculation of the base flow, except that the inflow velocity is perturbed by the optimal inflow perturbation:

where the subscript ‘ $\mathit{in}$ ’ stands for the inflow boundary, $\boldsymbol{U}_{\mathit{in}}$ represents the parabolic steady velocity condition used to calculate the base flow, $T$ is a temporal function as defined in (2.6), $\tilde{\boldsymbol{u}}_{b}$ is the global optimal inflow boundary perturbation obtained at ${\it\omega}=0.5$ , ${\it\tau}=150$ and ${\it\beta}=0$ and $s$ denotes the magnitude of the inflow perturbation. To capture the complex fluid dynamics in the nonlinear development of the optimal inflow perturbation, a denser grid consisting of 2317 spectral elements is used in this subsection.

In the spanwise direction, periodic boundary conditions are implemented. The flow past a backward-facing step was reported to lose stability at $\mathit{Re}=748$ to three-dimensional waves with spanwise wavenumber ${\it\beta}=0.91$ (Barkley et al. Reference Barkley, Gomes and Henderson2002), while reaching maximum three-dimensional transient energy growth at ${\it\beta}=0.645$ (Blackburn et al. Reference Blackburn, Barkley and Sherwin2008). Therefore, to roughly accommodate both the three-dimensional instabilities and optimal transient energy growth, the spanwise domain length is set to $L=20.9$ and 64 Fourier modes are calculated with spanwise wavenumbers 0, 0.3, 0.6, 0.9, $\dots ,$ 18.9.

Five typical values of $s$ are tested: 0.0002, 0.001, 0.005, 0.01 and 0.02. The perturbed flow is stable to three-dimensional perturbations for the first three $s$ values and becomes three-dimensional for the last two. For the first three $s$ values, the two-dimensional perturbed flow is illustrated in figure 9. It is seen that as $s$ increases, the vortex-shedding point moves upstream, since the perturbation is gradually amplified and reaches a maximum at around $x=28$ when convected downstream. At $s=0.0002$ , where the inflow velocity perturbation is less than 0.3 % of the inflow velocity in the base flow, vortex shedding occurs downstream of the bubbles where the perturbations are mostly amplified while the separation and reattachment of the two bubbles are almost intact. At $s=0.001$ , oscillation, or flapping, of the reattachment point of the secondary bubble is observed. At $s=0.005$ , the flapping extends to the reattachment point of the primary bubble and the separation point of the secondary bubble. It is worth noting that for the $s=0.001$ and 0.005 cases, vortex shedding is not maximized at $x=28$ as in the linear simulations, owing to the nonlinear interaction of the perturbations, which limits the linearly predicted amplifications.

Figure 9.

Contours of spanwise vorticity for flow perturbed by the optimal inflow perturbation at ${\it\beta}=0$ , ${\it\omega}=0.5$ and ${\it\tau}=150$ , with the magnitude $s$ of the inflow perturbation being (a) 0.0002, (b) 0.001 or (c) 0.005.

The three-dimensional perturbed flow at $s=0.01$ and $s=0.02$ is shown in figure 10. It is seen that at $s=0.01$ , six periods can be identified in the spanwise direction, corresponding to the dominant wavenumber ${\it\beta}=1.8$ . This wavenumber is much higher than the reported wavenumbers of the most unstable three-dimensional waves ( ${\it\beta}=0.91$ in Barkley et al. Reference Barkley, Gomes and Henderson2002) and the optimal initial perturbations ( ${\it\beta}=0.645$ in Blackburn et al. Reference Blackburn, Barkley and Sherwin2008), since this three-dimensionality is developed based on a two-dimensional vortex-shedding flow rather than an unperturbed steady flow, where the reported most energetic three-dimensional modes are calculated. For $s=0.02$ , the flow becomes more turbulent and no dominant wavenumbers can be identified. As $s$ increases, the secondary bubble moves upstream and its spanwise oscillation can be clearly observed at $s=0.02$ .

Figure 10.

Iso-surfaces of spanwise vorticity $-0.7$ (coloured by streamline velocity) in the flow perturbed by the optimal inflow perturbation at ${\it\beta}=0$ , ${\it\omega}=0.5$ , ${\it\tau}=150$ and (a)  $s=0.01$ or (b)  $s=0.02$ .

The flapping of the bubbles, or the oscillation of the separation and reattachment points, can be studied in a more quantitative manner by calculating the point on the wall where the shear stress changes sign. For three-dimensional flow, the shear stress is averaged across the spanwise direction. The temporal development of the separation and reattachment points is displayed in figure 11. The perturbed flow has been evolved for 150 time units to wash out the transient effects before collecting the separation/reattachment points for another 150 time units. To focus on the flapping of the primary bubble, only the lower wall is considered.

Figure 11.

Oscillation of separation and reattachment points on the lower wall when the flow is perturbed by the optimal inflow perturbation at ${\it\beta}=0$ , ${\it\omega}=0.5$ and ${\it\tau}=150$ , with the magnitude $s$ of the inflow perturbation being (a) 0.001, (b) 0.005, (c) 0.01 or (d) 0.02.

At $s=0.001$ , the reattachment point of the primary bubble oscillates with small magnitude, while three or four smaller bubbles can be observed at a given time, as shown in figure 11(a). These smaller bubbles are associated with the string of wall vortices induced by the main vortices (see figure 8) downstream of the primary bubble. As discussed above, the nonlinear interaction of the perturbations limits vortex shedding downstream of the bubbles, and so these smaller bubbles shrink and eventually vanish at around $x=28$ . Both the primary bubble and the smaller bubbles oscillate at the frequency of the inflow perturbation, i.e.  ${\it\omega}=0.5$ , corresponding to Strouhal number $\mathit{St}=0.08$ . The Strouhal number of bubble flapping in flow past a backward-facing step has been observed to be 0.06 (Le et al. Reference Le, Moin and Kim1997), 0.07 (Métais Reference Métais2001) and $O(0.1)$ (Wee et al. Reference Wee, Yi, Annaswamy and Ghoniem2004) in fully turbulent regimes. This close agreement of the Strouhal number between current and previous work indicates that the flapping can result from receptivity to the energetic inflow noise.

At $s=0.005$ , the small bubbles become stronger and move upstream until they reach the reattachment point of the primary bubble, as can be seen in figure 11(b). The reattachment curve consists of two connected segments, one with smaller slope at $10<x<13$ , corresponding to the primary bubble, and the other with greater slope at $13<x<28$ , corresponding to the smaller bubbles. Owing to the upstream extension of the smaller bubbles compared with the $s=0.001$ case, four or five smaller bubbles can be identified at a given time.

At $s=0.01$ , where the maximum inflow velocity perturbation reaches 0.014 (non-dimensionalized by the maximum inflow velocity of the unperturbed flow), another string of bubbles appears inside the primary bubble and vanishes before reaching the primary reattachment point, as shown in figure 11(c). The smaller bubbles downstream of the primary bubble vanish at around $x=25$ , owing to the nonlinear interaction of perturbations, which enhances mixing and prevents the generation of separation bubbles. At a given time, four or five smaller bubbles can be identified, with one upstream and the others downstream of the primary reattachment point.

At $s=0.02$ , where the maximum inflow perturbation reaches 0.028 (2.8 % of the maximum inflow velocity of the unperturbed flow), the flow becomes turbulent, which further prevents separation. Therefore, as can be seen in figure 11(d), only one or two smaller bubbles can be identified downstream of the primary reattachment point, and it vanishes at around $x=15$ . However, two smaller bubbles appear inside the primary bubble and extend upstream until the step. For all the $s$ values studied, the bubble flapping frequency is associated with the frequency of the inflow perturbations.

The flapping of the reattachment point of the primary bubble is extracted from figure 11 by taking the shortest distance in the range $8<x<28$ (to exclude the smaller bubbles inside the primary bubble) into consideration (Schäfer et al. Reference Schäfer, Breuer and Durst2009). As shown in figure 12, the unperturbed ( $s=0$ ) reattachment is steady. At small values of $s$ , e.g.  $s=0.001$ or 0.005, the reattachment point oscillates periodically around the unperturbed point, while the magnitude of the oscillation increases with $s$ . When the flow becomes three-dimensional at $s=0.01$ , the enhanced mixing reduces the size of the primary bubble. At $s=0.02$ , the flow is turbulent and the primary bubble reduces further in size but is still flapping significantly. The flapping frequency of the primary bubble is not clearly altered by the nonlinear development, and even in three-dimensional and turbulent conditions the flapping frequency is still associated with that of the inflow perturbation.

Figure 12.

Flapping of the reattachment point of the primary bubble at ${\it\beta}=0$ , ${\it\omega}=0.5$ and ${\it\tau}=150$ .

Figure 13.

Outcome of the random inflow perturbation at ${\it\beta}=0$ , ${\it\omega}=0.5$ and ${\it\tau}=150$ : (a) contours of spanwise vorticity for the two-dimensional outcome at inflow turbulent intensity 0.01; (b) iso-surface of spanwise vorticity $-0.7$ (coloured by streamline velocity) in the three-dimensional outcome at inflow turbulent intensity 0.05.

Figure 14.

Oscillation of separation and reattachment points on the lower wall when the flow is perturbed by the random inflow perturbation at turbulent intensities (a) 0.01 and (b) 0.05.

Figure 15.

The Reynolds number dependence of the gain to optimal inflow noise at ${\it\tau}=150$ and ${\it\beta}=0$ .

Figure 16.

Contours of spanwise vorticity for the outcome of the optimal inflow boundary perturbation at ${\it\beta}=0$ , ${\it\tau}=t=150$ , ${\it\omega}=0.5$ and (a)  $\mathit{Re}=600$ , (b)  $\mathit{Re}=800$ or (c)  $\mathit{Re}=1000$ . Different contour levels are chosen in each plot to highlight structures. The thick black lines represent the borders of recirculation bubbles.

Figure 17.

Iso-surfaces of spanwise vorticity $-0.7$ (coloured according to the streamline velocity) in the flow perturbed by the optimal inflow perturbation at ${\it\beta}=0$ , ${\it\omega}=0.5$ , ${\it\tau}=150$ , $s=0.001$ and $\mathit{Re}=1000$ .

Figure 18.

Square integral of the gradient of the Lagrangian with respect to the body force, i.e. $(\boldsymbol{{\rm\nabla}}_{\boldsymbol{f}}\mathscr{L},\boldsymbol{{\rm\nabla}}_{\boldsymbol{f}}\mathscr{L})$ , at ${\it\tau}=150$ and ${\it\beta}=0$ .

4.4. Nonlinear development of random inflow noise

In physical experiments, the inflow boundary condition is perturbed by random noise. Using the notation introduced in appendix A, a random inflow noise can be decomposed as a linear combination of the eigenvectors of a joint operator $\mathscr{M}^{\ast }({\it\tau})\mathscr{M}({\it\tau})$ . Since the optimal inflow perturbation is the leading eigenvector of this operator, it can be considered a component of the random noise. Therefore the development of the optimal inflow perturbation is expected to be observed in the flow perturbed by random inflow noise. Kaiktsis et al. (Reference Kaiktsis, Karniadakis and Orszag1996) used inflow noise with random spatial distributions but a prescribed frequency to perturb the backward-facing step flow and observed that the flow response reaches a maximum with respect to inflow noise at Strouhal number 0.05–0.104. In this subsection, we use spatially and temporally random inflow noise to explore the footprint of the optimal boundary perturbation in a real flow.

The randomly perturbed flow at two typical values of the inflow turbulent intensity, namely 0.01, where vortex shedding is clearly observed, and 0.05, where the flow becomes three-dimensional, is shown in figure 13. It is seen that the vortex shedding downstream of the secondary bubble observed in the optimally perturbed flow in figure 9 is also present in the randomly perturbed flow. However, since only a fraction of the random inflow noise is in the form of the optimal inflow perturbation, the random noise perturbs the flow and activates vortex shedding much less efficiently than the optimal perturbations. In the three-dimensional flow induced by random inflow noise illustrated in figure 13(b), six periods can be identified in the spanwise direction, similar to what was observed in the optimally perturbed case (see figure 10 a).

To further illustrate the role of optimal inflow perturbations in the randomly perturbed flow, the oscillation of separation and reattachment points on the lower wall due to the random noise is plotted in figure 14. It can be observed that there are approximately 12 periods for the small bubbles downstream of the main bubble contained in a time interval of length 150, similar to what is seen in figure 11. This indicates that the bubbles oscillate at a frequency around ${\it\omega}=0.5$ , at which the receptivity is maximized, and therefore amplifications of the optimal inflow perturbations are manifested in this randomly perturbed flow.

4.5. Receptivity at higher Reynolds number

In this subsection, the Reynolds number dependence of the receptivity to inflow noise is studied. Since the recirculation bubble in the base flow extends downstream at higher Reynolds numbers and the perturbation is expected to reach a maximum at further downstream locations, the outflow section is extended to 100 in this subsection. The spanwise wavenumber and final time are fixed at ${\it\beta}=0$ and ${\it\tau}=150$ , respectively.

From figure 15 it can be seen that the gain increases dramatically as the Reynolds number is increased from 500 to 1000, while the maximum appears at ${\it\omega}=0.5$ for all the cases considered. The linear outcomes of the optimal inflow perturbations at different Reynolds numbers are presented in figure 16, where the borders of the recirculation bubbles in the base flow are plotted together. It is seen that at a higher Reynolds number, the upper bubble extends downstream, inducing a longer ‘growth’ section associated with the inflectional point instability, and the rectangular structure appears further downstream. These observations suggest that the mechanism of noise amplification at higher Reynolds numbers is similar to that discussed in § 4.2, where the Reynolds number was fixed at 500.

The nonlinear evolution of the optimal perturbation is further investigated through DNS of the base flow perturbed by the inflow noise at $\mathit{Re}=1000$ , ${\it\beta}=0$ , ${\it\tau}=150$ and ${\it\omega}=0.5$ . Owing to the large magnitude of noise amplifications, as shown in figure 15, even at a perturbation magnitude of $s=0.001$ the flow develops into a three-dimensional and turbulent pattern before reaching the end of the upper bubble, as displayed in figure 17. This observation confirms that the flow at $\mathit{Re}=1000$ is much more sensitive to inflow noise than that at $\mathit{Re}=500$ , and indicates that the ‘growth’ section is not fully exploited at this magnitude of inflow noise.