2.1 Symmetries of the pipe flow

Here we briefly review the symmetries of the problem, and then focus on the properties of the shift-and-reflect flow-invariant subspace, to which we restrict the study that we present in this article. For a detailed discussion of flow-invariant subspaces of the pipe flow, see, for example, the appendix of Willis et al. (Reference Willis, Cvitanović and Avila2013). It will be seen presently that the shift-and-reflect symmetry leads to two dynamically equivalent regions of state space, later observed in simulations.

In pipe flow the cylindrical wall restricts the rotation symmetry to rotation about the $z$ -axis, and translations along it. Let $g(\unicode[STIX]{x1D719},\ell )$ be the shift operator such that $g(\unicode[STIX]{x1D719},0)$ denotes an azimuthal rotation by $\unicode[STIX]{x1D719}$ about the pipe axis and $g(0,\ell )$ denotes the streamwise translation by $\ell$ . Let $\unicode[STIX]{x1D70E}$ denote reflection about the $\unicode[STIX]{x1D703}=0$ azimuthal angle. Thus

The symmetry group of streamwise periodic pipe flow is $\text{SO}(2)_{z}\times \text{O}(2)_{\unicode[STIX]{x1D703}}$ ; in this paper we restrict our investigations to dynamics restricted to the ‘shift-and-reflect’ symmetry subspace

where $g_{z}$ denotes a streamwise shift by $L/2$ , i.e. flow fields (2.2) that satisfy

This requirement couples the streamwise translations with the azimuthal reflection. It is worth emphasising that, by imposing the symmetry $S$ , continuous rotations in $\unicode[STIX]{x1D703}$ are prohibited. Hence we consider only the simplest example of a continuous group for the streamwise translations, i.e. the one-parameter rotation group $\text{SO}(2)_{z}$ , omitting the subscript $z$ whenever that leads to no confusion. In the azimuthal direction only a discrete rotation by half the spanwise periodicity is allowed, i.e. by $\unicode[STIX]{x03C0}/m_{0}$ . We illustrate this property in figure 1. Thus, the symmetry group $G$ of the pipe flow in the shift-and-reflect subspace is

where $g_{\unicode[STIX]{x1D703}}$ denotes the discrete azimuthal shift by $\unicode[STIX]{x03C0}/m_{0}$ and $g_{z}(l)=g(0,l)$ .

Figure 1.

(a) Sketch of a shift-and-reflect symmetric state in the doubly periodic domain $(\unicode[STIX]{x1D703},z)\in [-\unicode[STIX]{x03C0}/m_{0},\unicode[STIX]{x03C0}/m_{0}]\times [0,L]$ (same after shift by $L/2$ in $z$ followed by reflection across the dotted line $\unicode[STIX]{x1D703}=0$ ). (b) In general, if the state in (a) is rotated by an angle $\unicode[STIX]{x1D719}$ , then the shift-and-reflect symmetry is broken. (c) However, the state in (a) rotated by the half-shift $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/m_{0}$ remains in the shift-and-reflect subspace.

Solutions that can be mapped to each other by symmetry operations (2.5) are equivalent, i.e. their physical properties, such as instantaneous energy dissipation rates, are the same. Since the streamwise shift symmetry $g_{z}(l)$ is continuous, one may have ‘relative’ invariant solutions in the state space of the pipe flow. Such invariant solutions that we present in this paper are: (i) travelling waves 

whose sole dynamics is a fixed velocity profile drifting along the axial direction with constant phase speed $c_{TW}$ ; and (ii) relative periodic orbits

which are time-varying velocity profiles that exactly repeat after period $T_{RPO}$ , but shifted streamwise by $l_{RPO}$ . In principle, one also has relative periodic orbits that are also relative with respect to azimuthal half-rotations, such that

These orbits connect two chaotic saddles related by $g_{\unicode[STIX]{x1D703}}$ . As we shall illustrate in the global visualisations of dynamics in § 4, such transitions between the two saddles are quite rare. Therefore, we did not search for relative periodic orbits of the kind in (2.8), and focused instead on one of the chaotic saddles related by azimuthal half-rotation.

2.2 State-space notation

Let $a$ denote the state-space vector which uniquely represents a three-dimensional velocity field $\boldsymbol{u}$ over the given computational domain. While the state-space representation $a$ is technically infinite-dimensional, due to numerical discretisation of the velocity field (spatial discretisation, truncated Fourier expansions, etc.), in practice $a$ is a high- but always finite-dimensional vector.

We denote the semi-flow induced by the time evolution of the Navier–Stokes equations (2.1) by $f^{t}$ , so that

traces out a trajectory $a(t)$ in the state space. For an infinitesimal time $\unicode[STIX]{x1D6FF}t$ , we can expand (2.9) as $a(t+\unicode[STIX]{x1D6FF}t)=a(t)+v(a(t))\unicode[STIX]{x1D6FF}t+O(|\unicode[STIX]{x1D6FF}t|^{2})$ , where we refer to

as the state-space velocity.

The ordinary differential equation (2.10) has the same symmetry group (2.5) as the Navier–Stokes equations it approximates, i.e. the state-space velocity $v(a)$ and the flow $f^{t}(a)$ commute with the symmetry group actions, $gv(a)=v(ga)$ and $gf^{t}(a)=f^{t}(ga)$ .

3.1 Method of connections

If one were to use the state itself as a template at each instant in time, then a division by zero would always be avoided. This corresponds to projecting out the component parallel to the shift, i.e. the group orbit tangent, at each moment in time. In Rowley & Marsden (Reference Rowley and Marsden2000) this is referred to as the method of connections. While very appealing and sometimes deployed (Kreilos, Zammert & Eckhardt Reference Kreilos, Zammert and Eckhardt2014) to ‘calm’ a turbulent flow, the method of connections is not a symmetry reduction method, in the sense that the dimensionality of the state space is not reduced by one for each continuous symmetry parameter (see e.g. Rowley et al. Reference Rowley, Kevrekidis, Marsden and Lust2003; Cvitanović et al. Reference Cvitanović, Borrero-Echeverry, Carroll, Robbins and Siminos2012). This is illustrated by figure 2. Starting with the same initial condition (large red point), with the method of slices the orbit $\hat{a}(0)=\hat{a}(T)$ closes after one period $T=14.646$ (blue), while with the method of connections the orbit continues filling out the relative periodic orbit torus ergodically, never closing into a periodic orbit. Figure 2(b) shows a projection of the trajectory generated by the two approaches, for three cycles of the periodic orbit. Whereas the orbit closes for the method of slices, for the method of connections the invariant torus remains a torus. In conclusion, nothing is gained by using the method of connections.

The orthogonal unit vectors used in figure 2(b) are the leading components from a principal component analysis (PCA), using 586 symmetry-reduced states equi-spaced in time around the orbit. Further details are given in § 5.

3.2 First Fourier mode slice

The method of slices is local, and the slice border discontinuity is avoided only within a neighbourhood of the template. In our previous work (Willis et al. Reference Willis, Cvitanović and Avila2013), a switching approach was applied to help ensure closeness to the template. Whilst this enabled symmetry reduction of longer trajectories, it proved difficult to switch before reaching a border whilst simultaneously ensuring continuity of $\ell (t)$ .

A different approach was taken by Budanur et al. (Reference Budanur, Cvitanović, Davidchack and Siminos2015) for one-dimensional partial differential equations with SO(2) symmetry, where it was shown that the first term in the Fourier expansion of the flow field can be used as the template for a global slice, the border of which is never visited by generic ergodic trajectories. This method relied on the observation that projections of group orbits on the subspace spanned by the first Fourier mode components (sine and cosine) are non-overlapping circles; hence one can find a unique polar angle in this projection to quotient the SO(2) symmetry. For a scalar field $u(x,t)=u(x+L,t)$ in one periodic space dimension, a slice template of the form $u^{\prime }=a\cos (2\unicode[STIX]{x03C0}x/L)+b\cos (2\unicode[STIX]{x03C0}x/L)$ , where $a$ and $b$ are constants, defines a first Fourier mode slice. In higher dimensions, one has more freedom in choosing first Fourier mode slice templates. For pipe flow, any $\hat{a}^{\prime }$ corresponding to a velocity field of the form

where $\boldsymbol{u}_{c,s}$ are three-dimensional vector fields that depend only on $r$ and $\unicode[STIX]{x1D703}$ , can be a candidate for a first Fourier mode slice template. The vector fields $\boldsymbol{u}_{c,s}$ should be chosen such that the slice border condition $\langle a\vphantom{\hat{a}^{\prime }}|\vphantom{a}\hat{a}^{\prime }\rangle +\text{i}\langle a\vphantom{g_{z}(L/4)\hat{a}^{\prime }}|\vphantom{a}g_{z}(L/4)\hat{a}^{\prime }\rangle =0$ is avoided by generic flow fields $\boldsymbol{u}=\boldsymbol{u}(r,\unicode[STIX]{x1D703},z,t)$ .

All slices are local, but since $\langle g(-\ell )a\vphantom{\boldsymbol{t}^{\prime }}|\vphantom{g(-\ell )a}\boldsymbol{t}^{\prime }\rangle =\langle a\vphantom{g(\ell )\boldsymbol{t}^{\prime }}|\vphantom{a}g(\ell )\boldsymbol{t}^{\prime }\rangle$ , one way to construct slice hyperplanes with larger domains of validity is by picking templates with smoother group orbits. Smoother states (i.e. states dominated by low Fourier modes) tend to be associated with lower dissipation or wall friction. Guided by this intuition, we construct the first Fourier mode slice template (3.7) by taking a low-dissipation solution from the turbulent set and setting all of its components to zero other than those with axial Fourier modes $k=1$ (Willis et al. Reference Willis, Short and Cvitanović2016). For the calculation in figure 2, such a template was capable of capturing the whole orbit.

Table 1.

The list of the invariant solutions reported in this work. Shown are the average rate of dissipation $\bar{D}$ , average downstream phase velocity $\bar{c}$ , dimension of the unstable manifold $d_{U}$ , and real part of the largest stability eigenvalue/Floquet exponent $\unicode[STIX]{x1D707}^{max}$ . The last column corresponds to the imaginary part $\unicode[STIX]{x1D714}$ of the leading stability eigenvalue for travelling waves, and phase $\unicode[STIX]{x1D703}$ of the leading Floquet multiplier for relative periodic orbits. Travelling waves are labelled by their dissipation rate $\bar{D}$ , relative periodic orbits by their period $T_{\text{}}$ . A family of 12 relative periodic orbits which appear to have similar physical properties are grouped together and labelled with subscript $F$ . Solutions marked with ‡ were previously reported in table 1 of Willis et al. (Reference Willis, Short and Cvitanović2016). The six solutions marked with $M$ were obtained by the multiple-shooting Newton method of appendix A.

The slice-fixing shift $\ell (t)$ for a trajectory $a(t)$ is computed from the polar angle in the plane spanned by $(\hat{a}^{\prime },g_{z}(L/4)\hat{a}^{\prime })$ as

where $\text{Arg}$ denotes the argument of the complex number. One can then find the translation symmetry-reduced trajectory $\hat{a}(t)$ by shifting the full state-space trajectory $a(t)$ back to the slice by $\hat{a}(t)=g_{z}(-\ell (t))a(t)$ .

5.1 Choice of the norm

In this work, we use two rather different norms: the standard energy norm, and a hand-crafted ‘low-pass’ norm. In what follows, we show how the choice of the norm can significantly alter the state-space visualisations, and the conclusions drawn from them.

Let $\boldsymbol{u}=\sum _{km}\boldsymbol{u}_{km}(r)\exp (2\text{i}\unicode[STIX]{x1D6FC}kz+\text{i}m_{0}m\unicode[STIX]{x1D703})$ denote the Fourier series of a velocity field $\boldsymbol{u}$ defined in a pipe of axial length $L=\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}$ . The variables $\boldsymbol{u}_{km}$ denote the Fourier coefficients corresponding to the axial and azimuthal directions as functions of the radial distance $r$ . For two velocity fields $\boldsymbol{u}_{1}$ and $\boldsymbol{u}_{2}$ , we define the $L^{2}$ inner product

where $V$ denotes the cylindrical flow domain and $E_{HP}$ is the kinetic energy of the Hagen–Poiseuille flow. In (5.2), we write the integral explicitly in terms of Fourier modes and radial integration, which in practice are approximated numerically. This inner product corresponds to the $L^{2}$ or the kinetic energy norm,

We sometimes find it more informative to use a metric that emphasises larger-scale structures along the continuous symmetry directions. For this reason, we define the ‘low-pass’ metric,

which penalises higher Fourier modes (short wavelengths). In the axial and azimuthal directions this is a variation of a Sobolev $H^{-1}$ norm (Lax Reference Lax2002): the weights are smaller for larger values of $k$ and $m$ , hence shorter wavelengths are de-emphasised.

In the work reported here, we rely primarily on the energy norm (5.2), except for § 5.4 where we contrast state-space visualisations using the energy and the low-pass norms, and comment on their relative effectiveness in our searches for relative periodic orbits.

5.2 Global visualisations: principal component analysis

We begin our investigation of state space with a data-driven method in order to obtain a general qualitative picture. The use of PCA, otherwise known as proper orthogonal decomposition in the context of fluids, has been well documented (see e.g. Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993). Broadly speaking, the method extracts a set of orthogonal vectors that span the data with minimal residual, with respect to some norm.

Here we apply the method to extract principal components, relative to the mean $\bar{\hat{a}}$ of the data set of $N$ states $\hat{a}_{i}$ in the symmetry-reduced state space, symmetrised with respect to $g_{\unicode[STIX]{x1D703}}$ . Singular value decomposition is applied to the matrix of inner products of the deviations $\tilde{\hat{a}}_{i}=\hat{a}_{i}-\bar{\hat{a}}$ ,

from which the $j$ th principal component is calculated,

Each principal component $\hat{\boldsymbol{e}}_{j}$ has the property that the root-mean-square of the projection $p_{j}=\langle \tilde{\hat{a}}_{i}(t)\vphantom{\hat{\boldsymbol{e}}_{j}}|\vphantom{\tilde{\hat{a}}_{i}(t)}\hat{\boldsymbol{e}}_{j}\rangle$ (taking the mean over $i$ ) equals the $j$ th singular value, $\unicode[STIX]{x1D61A}_{jj}$ , of the correlation matrix.

Figure 3.

Relative periodic orbits (closed curves, different colours), five travelling waves (marked with dark dots) and two turbulent trajectories (light grey curves) of the pipe flow projected onto the first three principal components (5.6), at two different viewing angles.

Principal components were extracted from $2000$ uncorrelated symmetry-reduced states obtained from ergodic trajectories. Figure 3 shows all our relative periodic orbits, five travelling waves and two turbulent trajectories projected onto the principal components, computed as above. There are some notable observations about figure 3. Firstly, periodic orbits appear to be localised on two sides of the $p_{1}=0$ plane, and the turbulent trajectories rarely switch from one side to the other. The first Fourier mode slice reduces the continuous translation symmetry of pipe flow but the discrete half-rotation symmetry $g_{\unicode[STIX]{x1D703}}$ still remains; the two sides of figure 3 are related to each other by this discrete symmetry operation. Also note that the highly symmetric $N4$ travelling wave $\text{TW}_{N4U/3.28}$ , which is invariant under $g_{\unicode[STIX]{x1D703}}$ , appears to lie at the origin of the $(p_{1},p_{3})$ plane.

It is clear from figure 3 that the principal component $\hat{\boldsymbol{e}}_{1}^{\prime }$ is aligned along the symmetry direction. This makes half of the information contained along this direction redundant, since each solution with $p_{1}>0$ has a copy with $p_{1}<0$ . As we are most interested in the details of the turbulent set and invariant dynamical behaviour of the system, our next step is to reduce the discrete $g_{\unicode[STIX]{x1D703}}$ symmetry. For this purpose, we define the ‘fundamental domain’ (Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2017) as $p_{1}>0$ and bring all our data from turbulence simulations and invariant solutions to this half of the state space. With the desymmetrised turbulence data, we recompute principal components $\tilde{\boldsymbol{e}}_{j}^{\prime }$ , which we will refer to as ‘fundamental domain principal components’.

Figure 4.

(a) Relative periodic orbits (closed curves of various colours) and five turbulent trajectories (light grey curves) projected onto the first three fundamental domain principal components. (b) Eleven orbits, which appear to fill out a region of the state space.

Figure 4 shows relative periodic orbits and turbulent trajectories projected onto the three dominant fundamental domain principal components. In figure 4, we no longer have two ‘clouds’ and relative periodic orbits are mostly located in the region of state space where turbulent trajectories spend most of their time. One striking observation from figure 4(a) is that a subset of relative periodic orbits seems to be located close to one another and their projections also qualitatively resemble each other. These are the relative periodic orbits labelled with subscript $F$ in table 1. We have already noted that the periods of these relative periodic orbits are approximately integer multiples of the shortest one. Their qualitative similarities in the state-space projections of figure 4 provide further evidence that these orbits are related to one another, possibly through sequences of bifurcations at other values of the $Re$  number (Short & Willis Reference Short and Willis2017).

In order to develop more intuition about the state-space geometry, we reduce the flow further to a Poincaré section defined by

In the projections of figure 4, the Poincaré section (5.7) corresponds to the $\tilde{p}_{3}=0$ plane, and for visualisations of figure 5, we project intersections onto the $(\tilde{\boldsymbol{e}}_{1}^{\prime },\tilde{\boldsymbol{e}}_{2}^{\prime })$ plane. However, it should be noted that this Poincaré section is a codimension-one hyperplane in the symmetry-reduced state space. Figure 5(a) shows $8560$ intersections (grey) of turbulent trajectories with the Poincaré section (5.7) that were obtained from $147$ individual runs, along with those (red and green) of relative periodic orbits.

It is clear in figure 5(a) that the turbulence visits the region containing relative periodic orbits more often than the rest of the state space. In figure 5, the ‘first family’ of relative periodic orbits shown in figure 4 are marked red, except for $\text{RPO}_{F/6.668}$ , the shortest one, which is coloured black. Figure 5(b) is a close-up view of the region containing relative periodic orbits, enclosed by the dashed-rectangle in figure 5(a) and similarly, figure 5(c) is a close-up of the region containing the first family, marked with the dashed rectangle on figure 5(b). In figure 5(d), we show orbits which approximate the three-dimensional unstable manifold of $\text{RPO}_{F/6.668}$ overlaid over figure 5(c). (The computational aspects of tracing out the unstable manifold are discussed in § 5.3.)

Figure 5.

(a) Turbulent trajectories (light grey) and relative periodic orbits (black, $\text{RPO}_{F/6.668}$ ; red, other members of the first family; green, the rest) on the Poincaré section (5.7). (b) Zoom of the region enclosed by the dashed rectangle in (a). (c) Zoom of the region enclosed by the dashed rectangle in (b). (d) Trajectories (cyan) on the unstable manifold of $\text{RPO}_{F/6.668}$ added to (c).

The global visualisations of dynamics we presented above provide us with insights about the state-space structure: the most important observation is that turbulent dynamics frequently visits the neighbourhoods of the relative periodic orbits found in this work. In addition, the ‘first family’ orbits – the set sharing similar physical and stability properties – appear close to each other in all projections of figures 4 and 5. Furthermore, the unstable manifold of the shortest-period member of this family visits the intersections of the other members of the family on figure 4, which provides additional evidence that these orbits emerge from a common bifurcation sequence (Short & Willis Reference Short and Willis2017).

Note that in the zoomed-in projection of the Poincaré section, figure 5(c), some parts are often visited by the turbulent trajectories (as indicated by the overlapping markers), while there are other regions, even the ones that appear close to the frequently visited regions, that tend to remain empty. This illustrates why measuring distances in the state space of a turbulent fluid is a hard problem. Two points in state space that are seemingly very close to each other in $L_{2}$ or a similar norm may be completely separated dynamically, since the geometry of the manifold on which the turbulence takes place can be highly convoluted.

While the global visualisations give us a qualitative view of the state space, they cannot tell us much about the finer structure of the turbulent state space. This is not surprising since the state space in question is very high-dimensional; and it is very unlikely that we can obtain a complete picture of dynamics from two- and three-dimensional global visualisations. For a better understanding of the state-space geometry, one must study the neighbourhoods of important invariant solutions individually, as illustrated in the next section.

5.3 Local visualisations

Global visualisations of the state space in figures 4 and 5 support the earlier suggestion that the members of the first family of relative periodic orbits, embedded in turbulence and listed separately in table 1, may be dynamically related to each other. The shortest-period member $\text{RPO}_{F/6.668}$ of first family has three unstable ( $|\unicode[STIX]{x1D6EC}_{i}|>1$ ) Floquet multipliers:

This renders the associated unstable manifold of $\text{RPO}_{F/6.668}$ three-dimensional even after the symmetry reduction of the space and time translation directions. Leading complex conjugate Floquet multipliers imply spiral-out dynamics in the associated neighbourhood, while the negative-real third Floquet multiplier implies that locally there exists a topologically Möbius band-shaped dynamics such as the one observed in period-doubling bifurcations. In the following, we numerically approximate and visualise these one- and two-dimensional unstable submanifolds.

Budanur & Cvitanović (Reference Budanur and Cvitanović2017) numerically approximated the one- and two-dimensional unstable manifolds of relative periodic orbits in a Kuramoto–Sivashinsky system. In those computations, a local Poincaré section was constructed in the neighbourhood of a periodic orbit where they initiated orbits whose dynamics approximately covered the linear unstable manifold; hence their forward integration approximated the unstable manifold away from the linearised neighbourhood. This strategy was adapted for calculating the one-dimensional unstable manifold of the localised ‘edge state’ relative periodic orbit of the pipe flow in Budanur & Hof (Reference Budanur and Hof2017). Here we apply this method to a relative periodic orbit embedded in turbulence, with a three-dimensional unstable manifold, a case that was not considered in the aforementioned studies.

To this end, we first define a local Poincaré section in the neighbourhood of $\text{RPO}_{F/6.668}$ as the half-hyperplane

where $\hat{a}_{p}$ is a point on $\text{RPO}_{F/6.668}$ , which we have arbitrarily chosen as its intersection with the global Poincaré section (5.7). The relative periodic orbit $\text{RPO}_{F/6.668}$ is a fixed point of the Poincaré map on the section (5.9) with stability multipliers equal to its Floquet multipliers. The associated Floquet vectors, however, need to be projected onto this section. This is a two-stage process since we compute the Floquet vectors as eigenfunctions $V_{i}$ of the eigenvalue problem

in full state space. First we project these vectors onto the slice as

where $\hat{V}_{i}$ denotes the projected vector on the slice. Then we project translation-symmetry-reduced Floquet vectors onto the Poincaré section as

Projections (5.11) and (5.12) onto the slice hyperplane (3.4) and onto the Poincaré section hyperplane (5.9) follow the same geometrical principle as in the appendix of (Budanur & Hof Reference Budanur and Hof2017).

In order to visualise the one-dimensional unstable submanifold of $\text{RPO}_{F/6.668}$ , we initiate trajectories from the points

These initial conditions approximately cover the locally linear one-dimensional piece of the unstable manifold in $\hat{V}_{3,{\mathcal{P}}}$ direction such that the first return of $\tilde{a}_{{\mathcal{P}}}(0)$ coincides with the initial location of $\tilde{a}_{{\mathcal{P}}}(1)$ . We discretised (5.13) by choosing four equidistant points in $\unicode[STIX]{x1D6FF}$ and set $\unicode[STIX]{x1D716}=10^{-3}$ (Floquet vectors are normalised such that $\Vert a_{p}\Vert _{L^{2}}=\Vert V_{i}\Vert _{L^{2}}$ ). We forward-integrate these initial conditions while recording their intersections with the Poincaré section (5.9). Figure 6(a,b) shows the first $25$ intersections of these orbits with the Poincaré section on two-dimensional projections and figure 6(c,d) shows one of these orbits in a three-dimensional projection along with $\text{RPO}_{F/6.668}$ and $\text{RPO}_{F/13.195}$ . The origin of the projections in figure 6 is $\hat{a}_{p}$ and the projection coordinates are

where $\hat{V}_{6,\bot }$ is the symmetry-reduced Floquet vector in the least stable direction (comes after marginal axial and temporal translation directions), and subscript $\bot$ indicates that these vectors are Gram–Schmidt orthonormalised.

Figure 6.

(a,b) One-dimensional submanifold in the unstable manifold of $\text{RPO}_{F/6.668}$ capturing locally linear dynamics in the $\hat{V}_{3,{\mathcal{P}}}$ direction in the Poincaré section, as two projections onto local coordinates (5.14). The fixed point corresponding to $\text{RPO}_{F/6.668}$ is at the origin (red), and the two-cycle $\text{RPO}_{F/13.195}$ (larger blue dots). (c,d) Three-dimensional projections of one ( $\unicode[STIX]{x1D6FF}=0$ ) of the eight trajectories in panels (a,b).

Figure 6(a,b) shows that the one-dimensional shape of locally linear dynamics is preserved as it is extended far away from the origin. The trajectories in figure 6(a,b) spread in a higher-dimensional manifold once they reach the neighbourhood of period-doubled $\text{RPO}_{F/13.195}$ . For additional comparison, in figure 6(c,d) we plot different three-dimensional projections of $\text{RPO}_{F/6.668}$ , the $\unicode[STIX]{x1D6FF}=0$ in (5.13) orbit, and $\text{RPO}_{F/13.195}$ in different three-dimensional projections. The qualitative similarities between the shape of the unstable manifold and $\text{RPO}_{F/13.195}$ are remarkable. For a quantitative conclusion, one should search for a heteroclinic connection from the three-dimensional unstable manifold of $\text{RPO}_{F/6.668}$ to the relative periodic orbit  $\text{RPO}_{F/13.195}$ . That, however, is beyond the scope of the current work.

For further comparison, we visualise the streamwise velocity and vorticity isosurfaces of $\text{RPO}_{F/6.668}$ , $\text{RPO}_{F/13.195}$ and three snapshots on the unstable manifold of $\text{RPO}_{F/6.668}$ in figure 7. All panels of figure 7 correspond to their respective intersections with the Poincaré section (5.9), and only one-eighth (one-quarter in azimuthal and one-half in axial directions) of the pipe is shown. The one-eighth visualisation suffices, since we work in the subspace with four-fold symmetry in the azimuthal direction, and the other half of the pipe in the axial direction can be obtained from the first half by the shift-and-reflect symmetry (2.4).

While all panels of figure 7 appear similar, they differ in details. Figure 7(d), the initial point on the unstable manifold, is virtually indistinguishable from $\text{RPO}_{F/6.668}$ in figure 7(a). Flow structures of $\text{RPO}_{F/13.195}$ at its two intersections with the Poincaré section, figure 7(b,c), differ from $\text{RPO}_{F/6.668}$ and from each other only in minute details; one has to compare the streak and roll sizes one-by-one. These nuances are reflected on the selected points on the unstable manifold shown in figure 7(d–f), although only identifiable after a careful inspection. These difficulties illustrate the power of state-space visualisation (cf. figure 6), without which the relation of $\text{RPO}_{F/6.668}$ to $\text{RPO}_{F/13.195}$ would have been very hard to elucidate.

Figure 7.

Streamwise velocity isosurfaces at $u=\pm 0.5U_{b}$ (red and blue) and streamwise vorticity isosurfaces at $\unicode[STIX]{x1D714}_{z}=\pm 2.4(U_{b}/D)$ (purple and green) of (a)  $\text{RPO}_{F/6.668}$ , (b,c)  $\text{RPO}_{F/13.195}$ and (d–f) the unstable manifold of $\text{RPO}_{F/6.668}$ at discrete times $n=0,23,24$ .

A set of initial conditions that approximately covers the linearised dynamics in the plane $(\text{Re}\,\hat{V}_{1,{\mathcal{P}}},\text{Im}\,\hat{V}_{1,{\mathcal{P}}})$ is given by

We discretise (5.15) by choosing four equidistant points in $\unicode[STIX]{x1D6FF}$ and $36$ points in $\unicode[STIX]{x1D719}$ and set $\unicode[STIX]{x1D716}=10^{-3}$ . The first three intersections of these initial conditions with the Poincaré section (5.9) are visualised in the projection in figure 8(a) in different colours, where black points correspond to the initial conditions. This figure illustrates the motivation for the particular approximation: initial conditions (5.15) define an elliptic band in the $(\text{Re}\,\hat{V}_{1,{\mathcal{P}}},\text{Im}\,\hat{V}_{1,{\mathcal{P}}})$ plane, such that the inner ellipse is mapped to the outer one by the linearised dynamics on the Poincaré section. The totality of these initial conditions captures well the linearised dynamics in this neighbourhood.

Figure 8(a) also illustrates the validity of linearised dynamics as each initial condition simply expands and rotates according to the real and imaginary parts of $V_{1}$ , when their distance to $\hat{a}_{p}$ is of the order of $10^{-4}$ . In figure 8(b), we show the same projection for $15$ intersections of these orbits on the Poincaré section as they leave the neighbourhood of the relative periodic orbit. At this stage, the shape is no longer an ellipse but it is starting to develop corners, possibly due to being distorted by a stable manifold. It should be noted, however, that the submanifold associated with the linearised dynamics on the plane $(\text{Re}\,\hat{V}_{1,{\mathcal{P}}},\text{Im}\,\hat{V}_{1,{\mathcal{P}}})$ is still two-dimensional. Note that the scales of axes in figure 8(b) are approximately two orders of magnitude larger than those in figure 8(a), and also that they are comparable to the scales of figure 5(d). In other words, a relative periodic orbit not only guides the dynamics in its immediate neighbourhood, but it indeed guides, through its unstable manifold, nearby motions at considerable finite distances.

Figure 8.

Two-dimensional submanifold in the unstable manifold of $\text{RPO}_{F/6.668}$ capturing locally linear dynamics in the $(\text{Re}\,\hat{V}_{1,{\mathcal{P}}},\text{Im}\,\hat{V}_{1,{\mathcal{P}}})$ plane in the Poincaré section projected onto local coordinates (5.14). (a) The initial conditions (black) and their orbits’ first three intersections with the Poincaré section, in different colours. (b) Fifteen intersections illustrating the global shape of the two-dimensional submanifold.

Figure 9.

(a) Minimum distance (5.16) between a long turbulent trajectory $\hat{a}_{turb}(t^{\prime })$ and the relative periodic orbit  $\text{RPO}_{F/6.668}$ . Inset: a time interval containing the closest approach. (b) $\text{RPO}_{F/6.668}$ (red) and a shadowing segment (corresponding to the inset of panel (a)) of a turbulent trajectory (dotted line) visualised as a projection onto (5.14). The closest approach of the turbulent trajectory to $\text{RPO}_{F/6.668}$ , $d<0.15$ , is highlighted yellow.

The above Poincaré sections illustrate the ways in which a relative periodic orbit shapes the geometry of its immediate neighbourhood. However, in the example at hand, as the unstable manifold Poincaré section is three-dimensional, it is hard to discern any structure in the ergodic sea in two-dimensional projections such as figure 5: in all our Poincaré sections the ergodic sea appears to be a structureless cloud, exhibiting no foliation typical of – let’s say – a Lorenz attractor or a Kuramoto–Sivashinsky attractor (Budanur Reference Budanur2015). The influence of a relative periodic orbit is here easier to visualise by studying shadowing episodes, i.e. the turbulent trajectory’s visits to a given relative periodic orbit’s neighbourhood, such as figure 9(a). Here we have defined the minimum distance between trajectories labelled ‘ $turb$ ’ and ‘ $RPO$ ’ in the fully symmetry-reduced state space (continuous symmetry reduced by (3.8), the discrete half-rotation symmetry $g_{\unicode[STIX]{x1D703}}$ (2.5) reduced to the fundamental domain), measured in the energy norm (5.3), as

Compared to the typical $\text{RPO}_{F/6.668}$ linearised neighbourhood scales (see figure 6), the yellow shadow in figure 9(b) is a considerable distance away, but it still completes one co-rotating shadowing period very nicely. Such shadowing episodes offer further support to our main thesis, that relative periodic orbits, together with their stable/unstable manifolds, shape the state-space dynamics within their local neighbourhoods.

5.4 Local visualisation: energy norm versus low-pass norm

As our final example of a local visualisation of the state space, we examine the local unstable manifold of a travelling wave. The primary goal here is to show with this example how profoundly the choice of the inner product (or norm) can affect the visualisation, and the conclusions drawn from it. Since in the slice the travelling waves reduce to equilibria, there is no need for a Poincaré section. Therefore, visualising low-dimensional unstable manifolds of travelling waves is more straightforward compared to those of relative periodic orbits discussed above.

Figure 10 shows a two-dimensional unstable submanifold of $\text{TW}_{1.968}$ . Note that the unstable manifold of this travelling wave is nine-dimensional ( $d_{U}=9$ in table 1). The visualised two-dimensional submanifold corresponds to its most unstable subspace characterised by the largest linear stability exponent of the travelling wave. This dominant exponent is complex-valued, with a complex eigen-direction $V_{1}$ which defines a two-dimensional subspace $(\text{Re}\,V_{1},\text{Im}\,V_{1})$ . The three-dimensional visualisations of figure 10 are obtained by projecting each state to the subspace formed by $(\text{Re}\,V_{1},\text{Im}\,V_{1},V_{2})$ , where $V_{2}$ is the third eigen-direction, with a real linear stability exponent.

Figure 10.

Low-dimensional state-space visualisation of the travelling wave  $\text{TW}_{1.968}$ (origin) and the relative periodic orbit  $\text{RPO}_{11.696}$ (blue). A two-dimensional unstable submanifold of the travelling wave is approximated by perturbations (light grey curves) around the travelling wave (see the text for the details). Both panels show the same objects projected to the same three-dimensional subspace. Two types of inner products are used for the projections: (a) low-pass inner product (5.4), (b)  $L^{2}$ inner product (5.2).

All computations are carried out in a slice with the travelling wave  $\text{TW}_{1.968}$ used as the template for symmetry reduction, and placed at the origin of the plots. The unstable submanifold (grey curves) is approximated by forward-integrating several small perturbations to the travelling wave in the direction $\text{Re}\,V_{1}$ . Because of the instability of the travelling wave, the trajectories spiral away from the origin. The spiralling nature of the trajectories is due to the complex stability exponent. In a small neighbourhood of the travelling wave, the ensemble of the trajectories approximates the two-dimensional unstable submanifold that is tangent to the plane $(\text{Re}\,V_{1},\text{Im}\,V_{1})$ . Away from the travelling wave, this approximation fails and the trajectories diverge.

Also shown in figure 10 is the relative periodic orbit  $\text{RPO}_{11.696}$ (blue curve). The two panels show the same objects projected to the same subspace $(\text{Re}\,V_{1},\text{Im}\,V_{1},V_{2})$ . The difference is that in figure 10(a) the low-pass inner product (5.4) is used for the projection while in (b) the $L^{2}$ inner product (5.2) is used.

In the low-pass projection, the relative periodic orbit sits near the travelling wave and appears to be shaped by its unstable manifold. In the $L^{2}$  projection, however, the relative periodic orbit appears to lie rather far from the travelling wave, and there is no hint that their shapes are related. We attribute this to the fact that the low-pass norm filters small-scale features, assessing the distance between fluid states based on their large-scale structures. In the $L^{2}$ norm, on the other hand, even minute small-scale differences between two states contribute to the computed distance.

We close by noting that the low-pass norm was also used to detect near recurrences of ergodic trajectories. These near recurrences then served as the initial Newton iteration guesses for obtaining the relative periodic orbits reported in table 1. We find that the recurrences measured in the low-pass norm tend to converge to relative periodic orbits more frequently than the recurrences measured in the energy norm. A similar observation was reported by Willis et al. (Reference Willis, Cvitanović and Avila2013).