2.1 Nonlinear and linearised flow operators

In this section, we review the linear theory of global stability and sensitivity that we apply to the Navier–Stokes equation. Much of this theory can be found in the works by Schmid & Henningson (Reference Schmid and Henningson2001, chap. 7.6.3), Chomaz (Reference Chomaz2005) and Giannetti & Luchini (Reference Giannetti and Luchini2007).

We begin by considering a flow with constant density  $\unicode[STIX]{x1D70C}$ and constant viscosity  $\unicode[STIX]{x1D707}$ . Given a characteristic flow speed $U$ and geometrical length scale $L$ , which we will later specify, we define the Reynolds number $Re=\unicode[STIX]{x1D70C}UL/\unicode[STIX]{x1D707}$ . Let ${\mathcal{U}}$ denote the space of non-dimensional divergence-free velocity fields, and let ${\mathcal{P}}$ denote the space of non-dimensional pressure fields. Choosing the non-dimensional $\boldsymbol{x}$ to reside in some spatial domain  $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$ , we consider velocity fields $\boldsymbol{u}(\boldsymbol{x},t)$ and pressure fields $p(\boldsymbol{x},t)$ evolving over non-dimensional time $t$ , such that $\boldsymbol{u}(\cdot ,t_{0})\in {\mathcal{U}}$ and $p(\cdot ,t_{0})\in {\mathcal{P}}$ at each time instant $t_{0}$ . We can define the Navier–Stokes operator $\mathscr{N}:{\mathcal{U}}\rightarrow {\mathcal{U}}$ by

where $p\in {\mathcal{P}}$ is chosen so that $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\mathscr{N}(\boldsymbol{u})=0$ . In addition, one requires appropriate boundary conditions on $\boldsymbol{u}$ and $p$ , and these are discussed in § A.2. The Navier–Stokes equation then prescribes how a velocity field $\boldsymbol{u}(\boldsymbol{x},t)$ evolves in time, and is given by

A velocity field $\boldsymbol{u}_{0}\in {\mathcal{U}}$ , along with its associated pressure field $p_{0}\in {\mathcal{P}}$ , is an equilibrium or steady-state solution of (2.2) if $\mathscr{N}(\boldsymbol{u}_{0})=\mathbf{0}$ .

The velocity and pressure fields can be written as the perturbations $\boldsymbol{u}^{\prime }(\boldsymbol{x},t):=\boldsymbol{u}(\boldsymbol{x},t)-\boldsymbol{u}_{0}(\boldsymbol{x})$ and $p^{\prime }(\boldsymbol{x},t):=p(\boldsymbol{x},t)-p_{0}(\boldsymbol{x})$ . The linearisation of $\mathscr{N}$ about an equilibrium  $\boldsymbol{u}_{0}$ yields the linearised Navier–Stokes operator $\mathscr{L}:{\mathcal{U}}\rightarrow {\mathcal{U}}$ , given by

where the continuity condition $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\mathscr{L}\boldsymbol{u}^{\prime })=0$ again determines the pressure field $p^{\prime }\in {\mathcal{P}}$ . The linearised problem typically has homogeneous boundary conditions since $\boldsymbol{u}$ and $\boldsymbol{u}_{0}$ generally have the same boundary conditions, as do $p$ and $p_{0}$ . With these conditions, the linearised Navier–Stokes equation is

For the purpose of the stability analysis, it is also helpful to define an adjoint linearised Navier–Stokes operator. We first choose a complex inner product on the space  ${\mathcal{U}}$ . For $\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in {\mathcal{U}}$ , we define

where $\overline{(\cdot )}$ denotes complex conjugation. It should be noted that while the actual velocity field is of course real, we allow for complex velocities because we will be considering complex eigenfunctions of  $\mathscr{L}$ . Denoting the adjoint velocity field by $\hat{\boldsymbol{u}}\in {\mathcal{U}}$ , the adjoint linearised Navier–Stokes operator $\mathscr{L}^{\ast }:{\mathcal{U}}\rightarrow {\mathcal{U}}$ is given by

To clarify the notation, $((\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{v})_{i}:=\sum _{j}(\unicode[STIX]{x2202}u_{j}/\unicode[STIX]{x2202}x_{i})v_{j}$ . As before, the adjoint pressure $\hat{p}\in {\mathcal{P}}$ is determined from the incompressibility condition $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\mathscr{L}^{\ast }\hat{\boldsymbol{u}})=0$ .

In addition, the adjoint problem must satisfy its own boundary condition. Defining

the boundary condition is

where $\boldsymbol{n}$ denotes the unit outward normal vector. To show this, one readily verifies that

integrating over $\unicode[STIX]{x1D6FA}$ , the divergence theorem gives

Thus, when the boundary conditions (2.8) are satisfied, the adjoint relation $\langle \mathscr{L}\boldsymbol{u}^{\prime },\hat{\boldsymbol{u}}\rangle =\langle \boldsymbol{u}^{\prime },\mathscr{L}^{\ast }\hat{\boldsymbol{u}}\rangle$ holds. It should be noted that (2.8) typically simplifies somewhat, since the boundary conditions on $\mathscr{L}$ may make certain terms in (2.7) vanish. The adjoint linearised Navier–Stokes equation is then given by

2.2 Eigendecomposition

We may now consider the eigendecomposition of the linearised Navier–Stokes operator (2.3). Given some eigenvalue $\unicode[STIX]{x1D706}_{j}\in \mathbb{C}$ , the corresponding direct eigenmode $\unicode[STIX]{x1D753}_{j}\in {\mathcal{U}}$ satisfies $\mathscr{L}\unicode[STIX]{x1D753}_{j}=\unicode[STIX]{x1D706}_{j}\unicode[STIX]{x1D753}_{j}$ . These eigenvalues and eigenmodes yield the long-time behaviour of the linearised dynamics. The direct eigenmode corresponding to the most unstable eigenvalue will dominate the linearised dynamics in the infinite time horizon. Furthermore, when an eigenvalue moves in the complex plane from the stable left half to the unstable right half, the corresponding centre eigenmode plays an important role in characterising the local bifurcation and the resulting centre manifold dynamics (Guckenheimer & Holmes Reference Guckenheimer and Holmes1983, chap. 3).

For the same eigenvalue $\unicode[STIX]{x1D706}_{j}$ , the corresponding adjoint eigenmode $\unicode[STIX]{x1D74D}_{j}\in {\mathcal{U}}$ satisfies $\mathscr{L}^{\ast }\unicode[STIX]{x1D74D}_{j}=\bar{\unicode[STIX]{x1D706}}_{j}\unicode[STIX]{x1D74D}_{j}$ . These adjoint eigenmodes are important in assessing the receptivity of the linear operator to initial conditions and external forcing (see, e.g., Schmid & Henningson Reference Schmid and Henningson2001; Chomaz Reference Chomaz2005; Giannetti & Luchini Reference Giannetti and Luchini2007). This feature can be understood if we introduce an external forcing $\boldsymbol{f}(\boldsymbol{x},t)$ in the linearised dynamics; that is,

If we assume that $\mathscr{L}$ has a full set of eigenvectors $\unicode[STIX]{x1D753}_{j}$ , then the solution to this differential equation can be expressed as

where the coefficients $a_{j}(t)$ are governed by

with the initial condition

Therefore, when considering the initial condition and the external forcing, it is their overlap with the adjoint eigenmodes that determines their role in the linearised dynamics.

It is important to note that flows with $Re\gg 1$ tend to have strongly non-normal linearisations, i.e. $\mathscr{L}\mathscr{L}^{\ast }$ and $\mathscr{L}^{\ast }\mathscr{L}$ are very different in some sense, and a number of important consequences follow. Most significantly, the direct eigenmodes tend to be nearly parallel, and the adjoint eigenmodes tend to be nearly parallel as well. As a result, even systems that are stable in the infinite-time sense can experience enormous transient growths that the eigenmodes do not individually predict. The topic of transient growth, which Schmid & Henningson (Reference Schmid and Henningson2001) and Chomaz (Reference Chomaz2005) review, is beyond the scope of this study. Nonetheless, various formulations of the overlap between direct and adjoint eigenmodes are useful in yielding the sensitivity of the linearised dynamics to different perturbations.

2.3 Sensitivity

The sensitivity analysis can be understood as follows. Let us once again consider the eigendecomposition $\mathscr{L}\unicode[STIX]{x1D753}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D753}$ , where we have dropped the index $j$ from above. An arbitrary infinitesimal perturbation $\unicode[STIX]{x1D6FF}\mathscr{L}$ of $\mathscr{L}$  (2.3) will yield the perturbed eigendecomposition

If we take the inner product of this equation with the adjoint eigenmode $\unicode[STIX]{x1D74D}$ , then the expression can be simplified by noting that $\langle \mathscr{L}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D753},\unicode[STIX]{x1D74D}\rangle =\langle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D753},\mathscr{L}^{\ast }\unicode[STIX]{x1D74D}\rangle =\langle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D753},\bar{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D74D}\rangle =\langle \unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D753},\unicode[STIX]{x1D74D}\rangle$ . Thus, equation (2.16) reduces to

At this point, it remains to define the operator perturbation $\unicode[STIX]{x1D6FF}\mathscr{L}$ . Different formulations of $\unicode[STIX]{x1D6FF}\mathscr{L}$ define different types of sensitivity.

Let us first consider the sensitivity to spatially localised feedback by briefly summarising the work of Giannetti & Luchini (Reference Giannetti and Luchini2007). We assume that the perturbation $\unicode[STIX]{x1D6FF}\mathscr{L}$ is an infinitesimal proportional feedback mechanism, in which the local velocity at a point determines the direction and magnitude of a generic force we apply at that point. This response could be, for instance, the action of a collocated actuator and sensor, where the sensor signal determines the actuation signal via a proportional control law. To formulate this idea, let us suppose that $\unicode[STIX]{x1D743}\in \mathbb{R}^{3}$ is the spatial location of the collocated actuator and sensor, $\unicode[STIX]{x1D6FF}_{D}$ is the Dirac delta function on  $\mathbb{R}^{3}$ , $\unicode[STIX]{x1D6FF}s\in \mathbb{R}$ is an infinitesimal feedback gain and $\unicode[STIX]{x1D63E}\in \mathbb{R}^{3\times 3}$ couples the direction of actuation with the direction of sensing. Then, we can model the feedback mechanism by setting the forcing term $\boldsymbol{f}(\boldsymbol{x},t)=\unicode[STIX]{x1D6FF}s\unicode[STIX]{x1D6FF}_{D}(\boldsymbol{x}-\unicode[STIX]{x1D743})\unicode[STIX]{x1D63E}\boldsymbol{u}^{\prime }(\boldsymbol{x},t)$ in the linearised Navier–Stokes equation (2.12). This formulation is equivalent to defining $\unicode[STIX]{x1D6FF}\mathscr{L}=\unicode[STIX]{x1D6FF}s\unicode[STIX]{x1D6FF}_{D}(\boldsymbol{x}-\unicode[STIX]{x1D743})\unicode[STIX]{x1D63E}$ , so that the perturbation evolves according to

Letting $(\cdot )^{\ast }$ denote the Hermitian adjoint, equation (2.17) becomes

Using standard pointwise vector 2-norms and the induced matrix norm, we can observe that

It is the right-hand side of (2.20) that most authors utilise when analysing the sensitivity of a particular flow to spatially localised feedback. This quantity dictates the size of the eigenvalue shift as a function of the feedback location, assuming the worst-case feedback gain direction.

In our study, however, we choose to focus on the special case where $\unicode[STIX]{x1D63E}$ is the $3\times 3$ identity matrix. The limitation of this approach is that it requires the feedback mechanism to be one where the actuator applies forces in the same direction or the opposite direction to the local velocity, depending on the sign of $\unicode[STIX]{x1D6FF}s$ . If $\unicode[STIX]{x1D6FF}s>0$ , then $\unicode[STIX]{x1D6FF}\mathscr{L}$ is a positive feedback mechanism; if $\unicode[STIX]{x1D6FF}s<0$ , then $\unicode[STIX]{x1D6FF}\mathscr{L}$ is a negative feedback mechanism. This feature simplifies the sensitivity formulation by considering a fixed feedback direction, instead of the worst-case direction.

Nevertheless, this special case offers at least two advantages. First, we are able to work with the exact sensitivity

instead of relying on an upper bound (2.20) for which the value of $\unicode[STIX]{x1D63E}$ maximising the eigenvalue sensitivity is not obvious. Second, (2.21) allows us to consider both the real and imaginary parts of $\text{d}\unicode[STIX]{x1D706}/\text{d}s$ more readily than (2.20), which only provides a magnitude. Since $\unicode[STIX]{x1D6FF}s\in \mathbb{R}$ , the complex-valued left-hand side of (2.21) allows us to determine the effect of the operator perturbation on the growth rate and frequency, individually, of each eigenvalue. If $\text{Re}(\text{d}\unicode[STIX]{x1D706}/\text{d}s)>0$ , then local positive feedback is globally destabilising; alternatively, local negative feedback is globally stabilising. On the other hand, $\text{Re}(\text{d}\unicode[STIX]{x1D706}/\text{d}s)<0$ indicates that local positive feedback is globally stabilising. The imaginary parts, as usual, are associated with the frequency of the eigenvalue. For instance, if $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}s)>0$ and $\text{Im}(\unicode[STIX]{x1D706})>0$ , or if $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}s)<0$ and $\text{Im}(\unicode[STIX]{x1D706})<0$ , then positive feedback increases the frequency of the eigenvalue; alternatively, negative feedback decreases the frequency. The opposite occurs if $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}s)<0$ and $\text{Im}(\unicode[STIX]{x1D706})>0$ , or if $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}s)>0$ and $\text{Im}(\unicode[STIX]{x1D706})<0$ .

The second type of sensitivity we consider is the sensitivity to base flow modifications, and we briefly summarise the work of Marquet et al. (Reference Marquet, Sipp and Jacquin2008). In this analysis, the operator perturbation $\unicode[STIX]{x1D6FF}\mathscr{L}$ is not a feedback mechanism, but rather an infinitesimal change in the base flow  $\boldsymbol{u}_{0}$ by $\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}\in {\mathcal{U}}$ . The resulting operator perturbation of $\mathscr{L}$  (2.3) is then given by

Using (2.22) in (2.17), we find that

Now, using indicial notation, we let $(\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0})_{i}=\unicode[STIX]{x1D6FF}u_{0i}$ and find the equivalent expression

In the integration by parts above, we have utilised fact that $\unicode[STIX]{x1D753}\in {\mathcal{U}}$ ; hence, $\unicode[STIX]{x1D753}$ is a divergence-free field. Using the divergence theorem, we obtain Therefore, the condition that

for $\boldsymbol{x}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ is sufficient for the eigenvalue perturbation to be expressed as

where we write the sensitivity to base flow modifications as

In this analysis, it is typical (e.g. Marquet et al. Reference Marquet, Sipp and Jacquin2008) to assume that $\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}=\mathbf{0}$ on the boundary – that is, that base flow modifications occur only in the interior of the geometry – so that the boundary condition (2.26) is automatically satisfied. We will see in § A.2 that we do not actually require such an assumption in pipe bifurcation flow.

As in the case of the sensitivity to spatially localised feedback, (2.28) can be decomposed into real and imaginary parts. If $\text{Re}(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}>0$ in a part of the domain, then the base flow perturbation $\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}$ there is globally destabilising. Likewise, if $\text{Re}(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}<0$ at a particular location, then the base flow perturbation there is globally stabilising. We can make a similar argument for the frequency of the same eigenvalue: if $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}>0$ in some part of the domain, then $\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}$ increases the frequency if $\text{Im}(\unicode[STIX]{x1D706})>0$ , and decreases the frequency if $\text{Im}(\unicode[STIX]{x1D706})<0$ . The opposite is true if $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}<0$ in a particular region.

It is also possible to decompose (2.28) into its two additive components (Marquet et al. Reference Marquet, Sipp and Jacquin2008). Let us define

so that $\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0}=(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})_{\text{p}}+(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})_{\text{t}}$ . We can observe that $(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})_{\text{p}}$ is derived from the $-\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}$ term in the linearised Navier–Stokes perturbation (2.22), which dictates the production of disturbances by the base flow modification. On the other hand, $(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})_{\text{t}}$ is derived from the $-\unicode[STIX]{x1D6FF}\boldsymbol{u}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{\prime }$ term in (2.22), which dictates the transport of disturbances by the base flow modification. We make this distinction in § 5.

3.1 Geometry

The pipe bifurcation geometries we employ have a square cross-section. Figure 1 shows the $90^{\circ }$ T-junction used in our computations, along with the associated lengths and axes. The $x$ -axis is aligned with the outlet pipes, and $x=0$ is the centre plane halfway between the two outlets. The $y$ -axis is aligned with the inlet pipe, and $y=0$ is the bottom surface. Finally, the $z$ -axis is the depth direction, and $z=0$ is the plane halfway between the front and back walls of the junction. With a cross-section of $1\times 1$ , the inlet pipe measures five units from the inlet to the bottom of the junction, and each outlet pipe measures five units from the outlet to the centre plane between the two outlets. For the Reynolds numbers we consider here, these dimensions are sufficient for approximating a junction with an infinitely long inlet and infinitely long outlets. In one test at $Re=560$ , extending the outlets to 10 units caused the leading eigenvalue of $\mathscr{L}$ to shift by a distance of $6.41\times 10^{-3}$ in the complex plane. In another, shortening the inlet to three units caused the leading eigenvalue of $\mathscr{L}$ to shift by $1.61\times 10^{-3}$ , and the corresponding eigenvalue of $\mathscr{L}^{\ast }$ to shift by $1.76\times 10^{-3}$ .

The right-angle corners of the junction are rounded with quarter circles. For our analyses, the radius of curvature at these corners is $R=0.4$ to match experiments reported by Vigolo et al. (Reference Vigolo, Radl and Stone2014), unless we state otherwise. The radii of curvature $R=0$ , i.e. a square corner, and $R=1$ are also explored. We used these radii of curvature in our previous work (Chen et al. Reference Chen, Rowley and Stone2015).

The $\unicode[STIX]{x1D703}=70^{\circ }$ arrow-junction is shown in figure 2 and the $\unicode[STIX]{x1D703}=110^{\circ }$ Y-junction in figure 3. The outlet lengths remain at five units long, and the radius of curvature remains at $R=0.4$ . These two figures depict the flows near the Hopf bifurcation of the stable asymmetric branch past the supercritical pitchfork bifurcation.

3.2 T-junction

3.3 Arrow- and Y-junctions

The single-phase, bubble-laden and particle-laden flows through pipe bifurcations with angles ranging from $\unicode[STIX]{x1D703}=30^{\circ }$ to $120^{\circ }$ are described in Ault et al. (Reference Ault, Fani, Chen, Shin, Gallaire and Stone2016). We refer readers to this work for additional details about flows through arrow- and Y-junctions, such as Reynolds-number effects, qualitative and quantitative descriptions of the vortex breakdown and the capture of bubbles and particles in the recirculation zones.

In a rough sense, the flow behaviour for smaller values of $\unicode[STIX]{x1D703}$ intuitively has a natural equivalence with Dean flows with smaller centreline radii of curvature $a$ . Given that the Dean flow with pipe diameter $d$ is characterised primarily by the Dean number $\mathit{Dn}=Re\sqrt{d/(2a)}$ , there is a natural expectation that flow features in curved pipes with smaller $a$ will be exhibited at smaller $Re$ . The same is essentially true in pipe bifurcation flow, where phenomena such as vortex breakdown and local bifurcations occur at lower $Re$ for lower $\unicode[STIX]{x1D703}$ . A description of the onset of breakdown at various junction angles can be found in the article by Ault et al. (Reference Ault, Fani, Chen, Shin, Gallaire and Stone2016).

Figure 10.

(a) Eigenvalues at $Re=170$ for $\unicode[STIX]{x1D703}=70^{\circ }$ , near the supercritical pitchfork bifurcation at $Re=168$ . (b) The symmetric solution at $Re=170$ , which does not contain recirculation; cf. figure 2.

As mentioned above, however, the Dean description is insufficient for pipe bifurcation flow because of the inherent three-dimensionality. In the $70^{\circ }$ arrow-junction (figure 2), the recirculation regions are initially aligned orthogonally to the inflow, but quickly turn towards the outlets because of the upward-sloped bottom wall. In contrast, the $110^{\circ }$ Y-junction (figure 3) allows the vortex core of the recirculation regions to remain essentially orthogonal to the inflow. The counter-rotating vortices form near the bottom of the pipe intersections, and do not encounter the downward-sloping top walls for some distance.

One of the key features that distinguishes arrow- and Y-junction flows from T-junction flow is the supercritical pitchfork bifurcation that renders the stable steady-state solution asymmetric for a certain Reynolds number range. This pitchfork bifurcation occurs at $Re=168$ for $\unicode[STIX]{x1D703}=70^{\circ }$ (figure 10) and $Re=649$ for $\unicode[STIX]{x1D703}=110^{\circ }$ (figure 11), and was not reported by Ault et al. (Reference Ault, Fani, Chen, Shin, Gallaire and Stone2016). Figures 2 and 3 show the stable asymmetric solutions at $Re=190$ for $\unicode[STIX]{x1D703}=70^{\circ }$ and $Re=698$ for $\unicode[STIX]{x1D703}=110^{\circ }$ respectively. In the former case, recirculation is only present in the rear half of the domain. The establishment of the bifurcation as a supercritical pitchfork type is evident from the appearance of stable asymmetric steady states, as well as the rearrangement of the vortex pair in the direction of the antisymmetric centre eigenmode.

Figure 11.

(a) Eigenvalues at $Re=650$ for $\unicode[STIX]{x1D703}=110^{\circ }$ , near the supercritical pitchfork bifurcation at $Re=649$ . (b) The symmetric solution at $Re=650$ ; cf. figure 3.

The supercritical pitchfork bifurcation in arrow- and Y-junctions is reminiscent of the pitchfork bifurcation in cross-slot flow (Haward et al. Reference Haward, Poole, Alves, Oliveira, Goldenfeld and Shen2016), where two opposing sides of a ‘+’ shape are inlets and the opposing sides in the orthogonal direction are outlets. The nature of that bifurcation is different, however. Instead of exhibiting an asymmetric vortex pair the bifurcation is rather the formation of a single spiral vortex. In the experiments, the orientation of the rotation was determined by geometrical imperfections in the apparatus. Although the cross-slot geometry bears close similarity to the T-junction, the key difference is that the T-junction enforces the no-penetration and no-slip condition in what would be the plane dividing the two inlets of the cross-slot. Therefore, the single spiral vortex cannot form around this plane.

In fact, it is noteworthy that the T-junction does not exhibit any pitchfork bifurcations at Reynolds numbers below that of the first Hopf bifurcation. As figure 9 implies, the stable periodic orbit of the Hopf bifurcation displays symmetry not only between the two outlets, but also between the front and back walls. Figure 12(b) also shows this front–back symmetry, and a weaker vortex pair with rotation opposite to the main vortex pair is visible at the top of the pipe. In contrast, the post-bifurcation asymmetry in the arrow- and Y-junctions (figure 12 a,c) causes the main vortex pair to align diagonally across each outlet pipe. Additionally, the secondary vortex pair is either smaller and weaker, or not present at all. Currently, we do not have a rigorous explanation for the absence of the pitchfork bifurcation in the T-junction; this remains a topic for further study. It is reasonable to suppose that for maximal stability, the preferential arrangement of the vortex pair, whether symmetrically or diagonally, can be explained by the physics of the flow turning through the junction.

Figure 12.

Normal slices of an outlet pipe two units of displacement from the junction centre, showing $Q$ (red, $Q>0$ ; blue, $Q<0$ ; colours exaggerated for clarity): (a)  $\unicode[STIX]{x1D703}=70^{\circ }$ and $Re=190$ (see figure 2); (b)  $\unicode[STIX]{x1D703}=90^{\circ }$ and $Re=560$ (see figure 1); (c)  $\unicode[STIX]{x1D703}=110^{\circ }$ and $Re=698$ (see figure 3).

We also briefly remark that our complementary flow experiments (Ault et al. Reference Ault, Fani, Chen, Shin, Gallaire and Stone2016) show an asymmetry between the two outlets under certain conditions. Our simulations do not reveal any such mechanism, but this may be a limitation of the pressure boundary conditions prescribed at the outlet (see § A.2). To investigate such asymmetries, a less rigid pressure condition would probably be required.

At Reynolds numbers higher than that of the supercritical pitchfork bifurcation, the symmetric branch and both asymmetric branches next undergo Hopf bifurcations. For $\unicode[STIX]{x1D703}=70^{\circ }$ , the asymmetric branches are projected to bifurcate at $Re=200$ and the symmetric branch at $Re=184$ . For $\unicode[STIX]{x1D703}=110^{\circ }$ , the asymmetric branches are projected to bifurcate at $Re=699$ and the symmetric branch at $Re=698$ . Only the bifurcation of the asymmetric branches would be directly observed in an experiment or time-resolved simulation. The nature of the centre eigenmodes is briefly discussed in § 4.2.

4.1 T-junction

A common feature of inertia-dominated convective flows (i.e. $Re\gg 1$ ) is that direct eigenmodes tend to be large only in a downstream region of the flow, and that adjoint eigenmodes tend to be large only in an upstream region. To list a few examples, Åkervik et al. (Reference Åkervik, Hœpffner, Ehrenstein and Henningson2007), Giannetti & Luchini (Reference Giannetti and Luchini2007), Qadri et al. (Reference Qadri, Mistry and Juniper2013) and Fani et al. (Reference Fani, Camarri and Salvetti2013) report these features respectively for the flow over a shallow cavity, the flow around a cylinder, a swirling jet with vortex breakdown and a mixing T-junction flow. This response is largely because the base flow transports flow perturbations, as described by the second term in the linearised Navier–Stokes operator (2.3). On the other hand, it is the negative of the base flow that transports adjoint flow perturbations, as described by the second term in the adjoint operator (2.6). As a result, $Re\gg 1$ flows tend to be strongly non-normal. Direct eigenmodes are typically nearly parallel to each other, as are adjoint eigenmodes; see Schmid & Henningson (Reference Schmid and Henningson2001) for a more complete discussion.

For the large Reynolds numbers of interest here, we observe the same types of non-normal responses in our pipe bifurcation flows. For reference, figure 9(a) shows the eigenvalues of the linearised Navier–Stokes operator with $\unicode[STIX]{x1D703}=90^{\circ }$ , $R=0.4$ and $Re=560$ , near the first Hopf bifurcation at $Re=556$ . Figure 13 shows the two least stable direct and adjoint eigenmodes, which correspond to $\unicode[STIX]{x1D706}_{1}=0.003\pm 2.208\text{i}$ and $\unicode[STIX]{x1D706}_{2}=0.000\pm 1.534\text{i}$ . Since the most unstable eigenmode (figure 13 a) is close to the centre mode of the Hopf bifurcation, this eigenmode is the representation of the helical oscillation that appears after the steady-state base flow loses stability, as described in § 3. The most unstable eigenmode has its largest magnitude in and near the outlet vortex pairs, just downstream of the junction. The axial wavelength of the eigenmode oscillations is approximately two and a half times the spacing between the vortex cores. This feature is in agreement with the axial wavelength of the helical oscillations we observe in the time-resolved flow at $Re=570$ , shown in figure 9(b). This direct eigenmode is also large in a smaller weaker vortex pair at the top of the outlet pipes, which is not directly visible in figure 13 but can be seen in figure 12(b). Time-resolved simulations show that the periodic orbit arising from the Hopf bifurcation involves oscillations of all four vortices in the outlet pipes.

Figure 13.

Direct and adjoint eigenmodes for the T-junction flow at $Re=560$ and $R=0.4$ . The direct eigenmodes are shown as single level sets of the pointwise norms $\Vert \text{Re}(\unicode[STIX]{x1D753})\Vert$ (red) and $\Vert \text{Im}(\unicode[STIX]{x1D753})\Vert$ (cyan); the adjoint eigenmodes are shown as single level sets of $\Vert \text{Re}(\unicode[STIX]{x1D74D})\Vert$ (green) and $\Vert \text{Im}(\unicode[STIX]{x1D74D})\Vert$ (purple). Streamlines (blue, low speed; grey, high speed) and a $Q$ -contour (yellow) are overlaid. (a) The most unstable eigenmode, $\unicode[STIX]{x1D706}_{1}=0.003\pm 2.208\text{i}$ . (b) The second least stable eigenmode, $\unicode[STIX]{x1D706}_{2}=0.000\pm 1.534\text{i}$ .

Whereas the first eigenmode $\unicode[STIX]{x1D753}_{1}$ loses stability at $Re=556$ , the next several modes lose stability at only slightly higher $Re$ (Chen et al. Reference Chen, Rowley and Stone2015). The close proximity of their critical Reynolds numbers suggests that the solution branch rapidly becomes complex as $Re$ increases. As shown in figure 13(b), the second direct eigenmode $\unicode[STIX]{x1D753}_{2}$ is largest in a region that is further downstream than the first eigenmode. We find that the third direct eigenmode is qualitatively very similar to the second; the fourth direct eigenmode is qualitatively very similar to the first; in fact, all further eigenmodes that we investigated resemble the first or second least stable eigenmodes. These results are in line with previous studies at similar Reynolds numbers (e.g. Åkervik et al. Reference Åkervik, Hœpffner, Ehrenstein and Henningson2007, for the flow over a shallow cavity), which have demonstrated that eigenmodes tend to be nearly parallel, occupying almost the same area in space, with very similar physical structures.

The adjoint eigenmodes $\unicode[STIX]{x1D74D}$ , in contrast, are large only in a small upstream region of the flow. This region, which figure 13 shows for the first and second modes, resides in the inlet and the junction, along the front and back walls. Specifically, the region of large $\unicode[STIX]{x1D74D}$ wraps around the counter-rotating vortex pairs in the junction; see figure 14. Typically, the adjoint eigenmodes are larger just outside the vortices than inside. Based on the linear receptivity theory we reviewed in § 2.2, the choice of initial conditions and the placement of external disturbances outside the vortex pairs has a greater effect on the linearised dynamics (2.12) than those inside. In addition, the receptivity region in the inlet pipe bears notable resemblance to adjoint impulse responses and modes that others have computed for the flat plate boundary layer (e.g. Bagheri, Brandt & Henningson Reference Bagheri, Brandt and Henningson2009). In general, the adjoint eigenmodes show even less variation among mode numbers than the direct eigenmodes. The similarity among adjoint eigenmodes hints that the linear receptivity mechanisms in this flow are highly localised only to a small region external to the vortices.

Figure 14.

View from an outlet of the most unstable adjoint eigenmode (green and purple) shown in figure 13(a), for $\unicode[STIX]{x1D703}=90^{\circ }$ , $Re=560$ and $R=0.4$ .

Two additional details are worthy of note. First, as mentioned above, the leading eigenmodes are symmetric about the $x=0$ and $z=0$ planes, which is consistent with the symmetry of the geometry. In fact, at least the first 10 modes are symmetric, and any antisymmetric or asymmetric modes would have eigenvalues far in the complex left half-plane. The symmetry of the leading modes is in contrast to the dynamics in arrow- and Y-junctions, which is discussed in § 4.2. Second, the junction corner radius of curvature does not appear to play a major role in the stability and receptivity characteristics of the flow. Figure 15 shows the least stable direct and adjoint eigenmodes close to the first Hopf bifurcation for $R=0$ and 1. As in the default case of $R=0.4$ , the direct eigenmodes reside primarily in the outlet pipes, either near the inlet–outlet intersection or further downstream, and the adjoint eigenmodes reside in a small region near the walls of the inlet and junction.

Figure 15.

The centre direct (red and cyan) and adjoint (green and purple) eigenmodes close to the first Hopf bifurcation, for different junction corner radii of curvature. Only the right half is shown for brevity: (a)  $R=0$ and $Re=590$ ; (b)  $R=1$ and $Re=550$ .

4.2 Arrow- and Y-junctions

For the pipe bifurcations with $\unicode[STIX]{x1D703}=70^{\circ }$ and $110^{\circ }$ , we briefly comment on the centre eigenmode corresponding to the supercritical pitchfork bifurcations near $Re=170$ and $650$ respectively. The eigenvalues and symmetric solutions are shown in figures 10 and 11, and the stable asymmetric solutions at slightly higher Reynolds numbers are shown in figures 2 and 3. The centre eigenmodes themselves are shown in figures 16 and 17. In both cases, the centre eigenmode remains symmetric about $x=0$ (that is, the plane between the two outlets), but an antisymmetry about $z=0$ (that is, between the front and back walls) is present; refer to figure 1 for the coordinate axes. Specifically, the $x$ - and $y$ -components of $\boldsymbol{u}$ are antisymmetric about $z=0$ , and the $z$ -component is symmetric (figures 16 b and 17 b).

Figure 16.

The centre eigenmode for the $\unicode[STIX]{x1D703}=70^{\circ }$ pitchfork bifurcation at $Re=170$ . (a) Single level sets of the pointwise norms $\Vert \unicode[STIX]{x1D753}\Vert$ (red) and $\Vert \unicode[STIX]{x1D74D}\Vert$ (green). (b) The slice $x=0$ between the two outlets, showing $Q$ (red, $Q>0$ ; blue, $Q<0$ ) and $\unicode[STIX]{x1D753}$ vectors (white) scaled by $\Vert \unicode[STIX]{x1D753}\Vert$ .

Figure 17.

The centre eigenmode for the $\unicode[STIX]{x1D703}=110^{\circ }$ pitchfork bifurcation at $Re=650$ . See the caption to figure 16. (b) An orthogonal slice three units downstream of the junction centre, as viewed from the outlet. Vectors into the plane are obscured by the plane to emphasise the out-of-plane component, which is antisymmetric.

For both angled junctions, the primary effect of the asymmetry is the rearrangement of the outlet-oriented vortices in each outlet pipe, as shown in figures 16(b) and 17(b). The two components of the centre eigenmode normal to the streamwise direction are generally larger than the streamwise component. These two dominant components essentially push one of the large vortices downward while pushing the counter-rotating counterpart upward. Such an effect on the smaller counter-rotating vortices is also significant in figure 16(b). The rearrangements caused by the centre eigenmode can be seen in figure 12(a,c).

It is of particular note that in the $\unicode[STIX]{x1D703}=70^{\circ }$ junction, the asymmetric branch of the pitchfork bifurcation exhibits recirculation in half of the junction at $Re=180$ (figure 2), but no recirculation is observed at the bifurcation point. Ault et al. (Reference Ault, Fani, Chen, Shin, Gallaire and Stone2016) also do not report any recirculation in the symmetric solution at $Re=180$ . Therefore, the appearance of the recirculation in the asymmetric branch can be attributed, at least in part, to the antisymmetric centre eigenmode.

Figure 18.

The eigenmodes corresponding to the Hopf bifurcation on the stable asymmetric branch: (a)  $\unicode[STIX]{x1D703}=70^{\circ }$ and $Re=190$ (cf. figure 2); the $Q$ contour is excluded for clarity; (b)  $\unicode[STIX]{x1D703}=110^{\circ }$ and $Re=698$ (cf. figure 3). See the colour legend in the caption to figure 13.

Another significant difference between the centre eigenmodes at $\unicode[STIX]{x1D703}=70^{\circ }$ and $110^{\circ }$ is the downstream position. At $\unicode[STIX]{x1D703}=70^{\circ }$ (figure 16 a), the eigenmode is largest in the dominant vortex structures in the pipe intersection; at $\unicode[STIX]{x1D703}=110^{\circ }$ (figure 17 a), it is largest substantially further downstream. Intuitively, this effect can probably be primarily attributed to the difference in Reynolds number at the pitchfork bifurcations. The arrangement of the centre eigenmode in the downstream direction dictates the asymmetry arising from the pitchfork bifurcation. Figure 2 shows that the asymmetric branch at $\unicode[STIX]{x1D703}=70^{\circ }$ is strongly asymmetric in and near the pipe intersection, with recirculation present in only one half of the junction. In contrast, the asymmetric branch at $\unicode[STIX]{x1D703}=110^{\circ }$ (figure 3) retains symmetry near the junction centre to a large degree. The vortices are noticeably diagonally arranged only after some distance downstream.

The nature of the adjoint centre eigenmodes (figures 16 a and 17 a) is qualitatively similar to what is seen in the Hopf bifurcation of the T-junction. The primary difference is that the centre eigenmode of the pitchfork bifurcation is real-valued, but the physical arrangement of the adjoint mode in space is largely unchanged. For this reason, we do not comment further on the receptivity of the bifurcating modes in the arrow- and Y-junctions.

We also comment on the centre eigenmodes of the Hopf bifurcations that take place on the stable asymmetric branches. In the arrow-junction (figure 18 a), the centre eigenmode – as in the pitchfork bifurcation (figure 16 a) – remains close to the intersection of the inlet and the outlets. Again, the placement of these vortex oscillations is probably the result of the lower Reynolds number leading to a weaker influence of downstream convection in the stability mechanisms. The centre eigenmode of the Y-junction is shown in figure 18(b), and displays an oscillation of the vortex in a downstream location, as in the T-junction (figure 13). In the arrow- and Y-junctions, however, the oscillations corresponding to the periodic orbit arising from the Hopf bifurcation are asymmetric.

5.1 T-junction

Recently, we described how the sensitivity to spatially localised feedback is connected to the recirculation arising from vortex breakdown (Chen et al. Reference Chen, Rowley and Stone2015). To reiterate these results briefly, figure 19(a) shows the region of sensitivity to spatially localised feedback for the most unstable eigenmode at $Re=560$ , and figure 19(b) shows the sensitivity for the second eigenmode (see (2.21)). We find that all of the regions of sensitivity to spatially localised feedback are qualitatively similar. For all of the eigenmodes we investigated, the overlap between the direct and adjoint eigenmodes is greatest inside, and immediately around, the recirculation regions and the other counter-rotating vortex structures.

Figure 19.

The sensitivity to spatially localised feedback for the most unstable eigenmode at $Re=560$ and $R=0.4$ , shown as the level sets (magenta) (a)  $|\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s|=1.25$ and (b)  $|\text{d}\unicode[STIX]{x1D706}_{2}/\text{d}s|=2.5$ .

Our previous work only considered the magnitude $|\text{d}\unicode[STIX]{x1D706}/\text{d}s|$ of this sensitivity. We now extend this analysis by considering in greater detail the effect that feedback has on the stability and frequency of the linearised dynamics. As discussed in § 2.3, the real and imaginary parts of $\text{d}\unicode[STIX]{x1D706}/\text{d}s$ describe these effects. Figure 20(a) shows level sets of $\text{Re}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)$ , which dictate the regions where collocated proportional feedback, in the same direction or opposite direction to the flow perturbation from the base flow, would have a stabilising or destabilising effect on the flow. An important implication is that negative feedback is stabilising in most areas inside and around the junction vortices. This result is especially the case inside the recirculation regions arising from vortex breakdown. There, the control strategy of infinitesimally forcing against velocity perturbations is particularly effective at dampening the perturbations. Only within small regions around and inside the vortices, by the intersection of the inlet and outlets, would negative feedback actually destabilise the flow. The discussion of the stabilising or destabilising effect of proportional feedback control is a standard topic in classical control theory and can be understood with Nyquist and root-locus plots (see, e.g., Franklin, Powell & Emami-Naeini Reference Franklin, Powell and Emami-Naeini2010).

Figure 20.

(a) Decomposition of the sensitivity to spatially localised feedback (figure 19 a) into regions where negative feedback is stabilising ( $\text{Re}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)=1.2$ ; red) and destabilising ( $\text{Re}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)=-1.2$ ; cyan). (b) Decomposition of the sensitivity to spatially localised feedback into regions where negative feedback decreases the oscillation rate ( $\text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)=0.6$ ; green) and increases the oscillation rate ( $\text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)=-0.6$ ; purple).

Next, we examine the effect of feedback control on the oscillation rate associated with the most unstable eigenmode. Figure 20(b) shows level sets of $\text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)$ , which dictate the regions where feedback increases or decreases the oscillation rate of the given eigenmode. Since we have arbitrarily chosen to analyse the eigenvalue $\unicode[STIX]{x1D706}_{1}=0.003+2.208\text{i}$ instead of its complex conjugate, negative feedback decreases oscillation rates where $\text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)>0$ , and increases oscillation rates where $\text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)<0$ . We observe that the application of negative feedback control to velocity perturbations in the smaller vortex pairs above the main pair would decrease the frequency of the mode. The general relation between the location of negative feedback and the associated change in frequency, however, is not immediately clear. The $\text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s)$ field oscillates spatially in multiple directions. Figure 20 does reveal, however, that the regions of growth rate sensitivity and frequency sensitivity essentially coincide.

The sensitivity of the linearised dynamics to base flow modifications is a similar principle to the sensitivity to spatially localised feedback. In both cases, we introduce a perturbation to the linearised operator that we model in a pointwise sense, and we observe the effect of the eigenvalues of the linearisation. Therefore, it is perhaps unsurprising that the sensitivity of the T-junction to base flow modification bears a close resemblance to the sensitivity to spatially localised feedback. Figure 21 shows the sensitivity to base flow modifications with a level set of the pointwise norm $\Vert \text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0}\Vert$ (see (2.28)); one can note the resemblance to figure 19. Much like the case of spatially localised feedback, the modification of the base flow in and near the vortex structures produces the largest effect on the eigenvalues of the linearisation.

Figure 21.

The sensitivity to base flow modifications for the most unstable eigenmodes at $Re=560$ and $R=0.4$ , shown as the level sets (magenta) (a)  $\Vert \text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0}\Vert =35$ and (b)  $\Vert \text{d}\unicode[STIX]{x1D706}_{2}/\text{d}\boldsymbol{u}_{0}\Vert =60$ .

The close relation between the two types of sensitivity is similar in spirit to the cylinder wake analysis reported by Marquet et al. (Reference Marquet, Sipp and Jacquin2008). In Marquet et al.’s analysis, however, the sensitivity to base flow modifications is largest inside the recirculation region, whereas the sensitivity to spatially localised feedback is largest in a more downstream region that includes both the edge of the recirculation as well as the passing free stream. We do not observe this distinction in the T-junction flow.

We may now split the base flow sensitivity into real and imaginary components to analyse the effects of base flow modification on growth rates and frequencies. Figure 22(a) shows level sets of $\Vert \text{Re}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})\Vert$ and $\Vert \text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})\Vert$ , which spatially oscillate with a small wavelength compared with the vortex spacing and the pipe width. The actual vector directions of $\text{Re}(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})$ and $\text{Im}(\text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}_{0})$ , however, are highly complex. Although Marquet et al. (Reference Marquet, Sipp and Jacquin2008) were able to describe the exact base flow modification that had the greatest effect on the eigenvalue growth rate or frequency, we are unable to provide a general description for the T-junction flow.

Figure 22.

(a) Decomposition of the sensitivity to base flow modifications (figure 21 a) into the level sets $\Vert \text{Re}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})\Vert =35$ (where base flow modifications change the growth rate; red) and $\Vert \text{Im}(\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})\Vert =35$ (where base flow modifications change the frequency; cyan). (b) Decomposition of the sensitivity to base flow modifications into the level sets $\Vert (\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})_{\text{p}}\Vert =35$ (the contribution from production; green) and $\Vert (\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})_{\text{t}}\Vert =35$ (the contribution from transport; purple); see (2.29).

Following the discussion in § 2.3, we can also decompose the base flow sensitivity into a different set of parts: the contributions from the production and transport of flow perturbations by the base flow modification. Figure 22(b) shows level sets of $\Vert (\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})_{\text{p}}\Vert$ and $\Vert (\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0})_{\text{t}}\Vert$ to make this distinction. Conceptually, the production and transport terms can be respectively related to local absolute and convective instability (Marquet et al. Reference Marquet, Sipp and Jacquin2008). The contribution from production dominates in the main vortex pair close to the outlet stagnation point, whereas the contribution from transport is larger near the plane between two outlets. The distinction between the regions of large production and the regions of large transport, however, is not large. Both mechanisms are important where there is large swirl in the base flow. This observation is in contrast to the cylinder wake analysis of Marquet et al. (Reference Marquet, Sipp and Jacquin2008), who observed that the contribution from production dominated the contribution from transport nearly everywhere in the flow.

We remark that at $Re=320$ , near the onset of vortex breakdown, the sensitivities to localised feedback and base flow modifications remain strongest near the large vortical structures of the junction; see figure 23. Thus, even when recirculation is very weak or non-existent, perturbations in the linearised dynamics of the horizontal vortices still play a critical role in the location of the linearisation eigenvalues. The connection between the vortices and the sensitivities, however, is not as clear-cut as at higher Reynolds numbers where the recirculation is strong (e.g. figures 19 and 21).

Figure 23.

The sensitivities (magenta) of the most unstable mode at $Re=320$ and $R=0.4$ , at the onset of vortex breakdown. (a) The sensitivity to localised feedback, shown as the level set $|\text{d}\unicode[STIX]{x1D706}_{1}/\text{d}s|=2$ . (b) The sensitivity to base flow modifications, shown as the level set $\Vert \text{d}\unicode[STIX]{x1D706}_{1}/\text{d}\boldsymbol{u}_{0}\Vert =45$ .

Figure 24.

The sensitivity (a) to spatially localised feedback and (b) to base flow modifications for the pitchfork centre eigenmode at $\unicode[STIX]{x1D703}=70^{\circ }$ and $Re=170$ , shown as the level sets (magenta) (a)  $\text{d}\unicode[STIX]{x1D706}/\text{d}s=1$ and (b)  $\Vert \text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}\Vert =10$ .

In § 4, we commented on the lack of a notable effect from the junction corner radius of curvature on the direct and adjoint eigenmodes. In our previous work (Chen et al. Reference Chen, Rowley and Stone2015), we also described the same lack of an effect on the sensitivity to spatially localised feedback. We find that the flow separation at the junction corners does not exhibit any particular connection with stability, receptivity, or either type of sensitivity that we have considered. Therefore, the flow separation at the corners does not interact significantly with factors that modify the dynamics or otherwise introduce disturbances. It is presently unclear why the large horizontally oriented vortices produced by the inlet flow are closely connected to the stability and sensitivity of the flow, but the small vertically oriented vortices produced by the flow separation at the corners are not. A possible explanation is that the horizontal vortices exhibit tightly spiralling streamlines, with recirculation appearing at sufficiently high Reynolds numbers; on the other hand, the vortices at the corners do not exhibit as much swirl, and no recirculation is present. This question remains a topic for future study.

5.2 Arrow- and Y-junctions

Next, we describe the sensitivity of the flow in the $\unicode[STIX]{x1D703}=70^{\circ }$ and $110^{\circ }$ junctions. We focus the discussion on the sensitivities corresponding to the supercritical pitchfork bifurcation. Therefore, the sensitivity can be interpreted as regions where passive or feedback flow control can best stabilise the symmetric solution, or, alternatively, destabilise it and trigger the onset of the asymmetric branch.

The sensitivities to spatially localised feedback and base flow modification are shown in figure 24 for $\unicode[STIX]{x1D703}=70^{\circ }$ and $Re=170$ , near the supercritical pitchfork bifurcation. We observe that the base flow does not exhibit recirculation at this Reynolds number and junction angle, which is in agreement with Ault et al. (Reference Ault, Fani, Chen, Shin, Gallaire and Stone2016). The sensitivity regions for this flow are only weakly correlated with the outlet-oriented vortices. Similarly to the discussion of the eigenmodes in the arrow-junction flow, this placement of the sensitivity region can probably be attributed to the lower Reynolds number. With a greater effect of viscosity, the swirl is probably not significant enough to exhibit a strong effect of feedback forcing or base flow modification on the stability. Rather, modifications to the linearised Navier–Stokes operator in the counter-rotating vortices may only have a notable influence on the downstream flow, since the streamlines would not be expected to feed the effects of the modifications back upstream. A greater upstream sensitivity can be seen in figure 24 than in the T-junction flow at $Re=560$ (figures 19 and 21), where the recirculation effects dominate the sensitivity.

Following the asymmetric branch of the pitchfork bifurcation, the sensitivity of the Hopf mode at $Re=190$ shows a different picture. In figure 25, it can be seen that the asymmetry has a significant effect on the sensitivity. Most notably, the sensitivity is much larger on the side of the junction containing the vortex breakdown (see figure 2). In addition, a greater sensitivity can be seen in the recirculation region than in the equivalent position at the pitchfork bifurcation (figure 24), where recirculation is not present. This observation further supports the idea that recirculation can lead to high eigenvalue sensitivity.

Figure 25.

The sensitivity (a) to spatially localised feedback and (b) to base flow modifications for the Hopf centre eigenmode on the asymmetric branch at $\unicode[STIX]{x1D703}=70^{\circ }$ and $Re=190$ , shown as the level sets (magenta) (a)  $\text{d}\unicode[STIX]{x1D706}/\text{d}s=0.5$ and (b)  $\Vert \text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}\Vert =8$ . Compare with figure 2.

Figure 26.

The sensitivity (a) to spatially localised feedback, showing $\text{d}\unicode[STIX]{x1D706}/\text{d}s=1.5$ , and (b) to base flow modifications, showing $\Vert \text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}\Vert =40$ , for the pitchfork centre eigenmode at $\unicode[STIX]{x1D703}=110^{\circ }$ and $Re=650$ .

Figure 26 shows the sensitivity regions for $\unicode[STIX]{x1D703}=110^{\circ }$ at $Re=650$ , again near the supercritical pitchfork bifurcation. A notable difference between these sensitivity regions and those of the T-junction (see figures 19 and 21) is that the former include regions on the $z=0$ plane between the front and back walls. The inclusion of this region can be partially attributed to the antisymmetry of the bifurcating mode. It is clear from figure 17(b) that the $z$ -component of the Y-junction centre eigenmode is large on the $z=0$ plane, since the pitchfork bifurcation introduces solutions that are asymmetric across this plane. In contrast, the centre eigenmode of the T-junction Hopf bifurcation (figure 13 a) is fully symmetric across $z=0$ , and the $z$ -component of the eigenmode is zero on this plane. The function of this centre eigenmode, in light of the Hopf bifurcation, is to cause vortices on either side of $z=0$ to undergo symmetric corkscrew oscillations, as shown in figure 9(b). It should not be surprising from (2.21) and (2.28) that to stabilise or destabilise the symmetry about $z=0$ in the Y-junction, it is effective to implement feedback or a base flow modification on that plane. On the other hand, to trigger or delay the transition from symmetric base flow to the symmetric periodic orbit in the T-junction, it is not as effective to modify the linearised operator on that symmetry plane.

We also note that the sensitivities of the Y-junction pitchfork bifurcation are not as connected to the recirculation regions as the T-junction Hopf bifurcation. This may be because the Y-junction pitchfork bifurcation primarily induces asymmetry downstream of the recirculation region, whereas the T-junction Hopf bifurcation primarily induces oscillatory motion in the recirculation regions themselves.

Finally, we also very briefly examine the Hopf bifurcation of the asymmetric branch of the Y-junction. In figure 27, the sensitivity clearly shows an asymmetry between the front and back walls of the junction, which is reflective of the arrangement of the vortices in the junction. Furthermore, the bulk of the sensitivity is further downstream from the intersection, as with the pitchfork bifurcation (figure 26).

Figure 27.

The sensitivity (a) to spatially localised feedback, showing $|\text{d}\unicode[STIX]{x1D706}/\text{d}s|=1$ , and (b) to base flow modifications, showing $\Vert \text{d}\unicode[STIX]{x1D706}/\text{d}\boldsymbol{u}\Vert =35$ , for the eigenmode corresponding to the Hopf bifurcation on the asymmetric branch, at $\unicode[STIX]{x1D703}=110^{\circ }$ and $Re=698$ . Compare with figure 3.