2.1. LIF measurements

The experiments were performed in the Environmental Fluid Mechanics Laboratory at the Georgia Institute of Technology. The LIF configuration is illustrated in figure 1 and a detailed description of the system is given in Tian & Roberts (Reference Tian and Roberts2003). The tank has glass walls 6.10 m long $\times$ 0.91 m wide $\times$ 0.61 m deep. The front wall consists of two 3  m long glass panels to enable long unobstructed views. The 5.5 m long pipe was located on the tank floor, and the tank was filled with filtered and dechlorinated water. The pipe was a transparent Lucite tube with radius $R=2.5~\text{cm}$ .

Figure 1.

Schematic of the LIF system of the Georgia Tech Environmental Fluid Mechanics Laboratory (Tian & Roberts (Reference Tian and Roberts2003), see also www.youtube.com/channel/UCg7qksJEB6spCUniij_UKzg).

The pipe was completely submerged in water to avoid refraction and scattering of the emitted light that would occur at the water–lucite–air interface along the pipe walls and downstream at the very end of the pipe when the water flows out with curvy streamlines. With this configuration, we enable unique LIF imaging of the flow structures in a round pipe at high flow rate since the pipe discharge is into ambient water instead of air.

The water was pumped into a damping chamber to calm the flow, and then, after passing through a rigid polyester filter, it flowed into the pipe. Fluorescent dye solution was continuously injected into the flow through a small hole in the pipe wall upstream of the image capture zone of length $20R$ . The solution, a mixture of water and fluorescent dye, is supplied from a reservoir by a rotary pump at a flow rate measured by a precision rotameter. The flow was started and, after waiting a few minutes for the flow to establish, laser scanning began to record the experiment. To acquire high-resolution data, we captured vertical centreline planar fluorescent dye concentration fields $C_{0}(x,y,z=0,t)$ which trace turbulent pipe flow patterns. The pipe Reynolds number $\mathit{Re}=2U_{b}R/{\it\nu}=3200$ , where the bulk velocity (discharge divided by the pipe cross-sectional area) $U_{b}=6.42~\text{cm}~\text{s}^{-1}$ and ${\it\nu}$ is the kinematic viscosity of water. As shown in figure 1, the vertical laser sheet passes through the pipe centreline to focus on flow properties in the central plane ( $z=0$ ). Images of the capture zone ( $2R\times 20R=5\times 50~\text{cm}^{2}$ ) were acquired at 50 Hz for 240 s (see figure 2). The vertical and horizontal image sizes are $65~\text{pixel}\times 622~\text{pixel}$ for a resolution of $0.0794~\text{cm}~\text{pixel}^{-1}$ .

Figure 2.

LIF experiments: snapshot of the planar fluorescent dye concentration field $C_{0}(x,y,z=0,t)$ tracing turbulent pipe flow patterns at Reynolds number $\mathit{Re}=3200$ (bulk velocity $U_{b}=6.42~\text{cm}~\text{s}^{-1}$ , flow from right to left).

2.2. Data analysis

The LIF measurements are planar dye concentration fields $C(x,y,t)=C_{0}(x,y,z=0,t)$ in a vertical slice through the pipe centreline. According to (2.1), at $z=0$ , the field $C$ satisfies

where $\boldsymbol{{\rm\nabla}}_{xy}=(\partial _{x},\partial _{y})$ and $\boldsymbol{v}_{2D}=(U,V)=(U_{0}(x,y,z=0,t),V_{0}(x,y,z=0,t))$ are the in-plane gradient and flow within the 2D slice, and the source

accounts for the out-of-plane transport and diffusion and in-plane source/sinks. The associated in-plane comoving frame, or convective, velocity $U_{d}^{(2D)}$ can be estimated from the measured field $C(x,y,t)$ using (2.5), where the average is performed only in the $x$ and $y$ directions, that is

Clearly, this depends on the in-plane flow and out-of-plane sources/sinks. Similarly, the in-plane comoving frame, or convective, velocity profile $U_{d}^{(2D)}(y,t)$ follows from (2.7) averaging only in the $x$ direction,

For example, figure 3 shows the comoving frame velocity profiles computed from (2.12) including (a) all spatial scales of the measured $C$ , (b) the small scales (wavelengths $L_{x}<0.2R$ , $L_{y}<0.2R$ ) and (c) the large scales ( $L_{x}>2R$ , $L_{y}>0.4R$ ). Clearly, the small scales advect more slowly than the large scales, in agreement with Krogstad et al. (Reference Krogstad, Kaspersen and Rimestad1998). Moreover, the maximum comoving frame velocity of the large scales ( $=8.52~\text{cm}~\text{s}^{-1}$ ) is close to the centreline mean flow speed ( $=8.78~\text{cm}~\text{s}^{-1}$ ) estimated from the frequency–wavenumber spectrum of $C(x,y=0,t)$ (see figure 4). Further, the frequency spectrum of the comoving frame velocity $U_{d}(t)$ estimated from (2.11) accounting for large scales only is also shown in figure 3(d). It decays approximately as $f^{-5/3}$ , indicating that Taylor’s hypothesis is approximately valid, possibly due to the non-dispersive behaviour of large-scale motions.

Figure 3.

LIF experiments: estimated comoving frame, or convective velocity $U_{d}^{(2D)}(y,t)$ using (2.12): time-average profile (solid line), instantaneous profile (dashed line) and its standard deviations about the mean (thin solid lines) estimated accounting for (a) all spatial scales of the measured $C$ ( $\text{max speed}=6.32~\text{cm}~\text{s}^{-1}$ ), (b) small scales ( $\text{max speed}=2.76~\text{cm}~\text{s}^{-1}$ ), (c) large scales ( $\text{max speed}=8.52~\text{cm}~\text{s}^{-1}$ ); (d) observed noisy (red line) and filtered (black line) frequency spectra of the large-scale comoving frame velocity $U_{d}^{(2D)}$ (see (2.11)). Pipe radius $R=2.5~\text{cm}$ .

Figure 4.

Observed log-values of the frequency–wavenumber spectrum $S(k_{x},f)$ of the fluorescent dye concentration $C(x,y=0,t)$ at the pipe centreline. Estimated mean flow velocity $U_{m}={\it\omega}/k_{x}=2{\rm\pi}f/k_{x}\sim 8.78~\text{cm}~\text{s}^{-1}$ (dashed line). $U_{m}/U_{b}=1.37$ and bulk velocity $U_{b}=6.42~\text{cm}~\text{s}^{-1}$ .

In the fixed frame $(x,t)$ , the space–time evolution of the measured dye concentration $C(x,y=0,t)$ on the pipe centreline is shown in figure 5(a). The associated evolution in the comoving frame $(x-\ell _{d}(t),t)$ is shown in figure 5(b). The shift $\ell _{d}$ is computed by numerically integrating $U_{d}^{(2D)}$ in time, which is estimated from (2.12) accounting for all spatial scales of $C$ . Note the shape-changing dynamics of the passive scalar structures, which still experience a drift in the comoving frame. Moreover, a slowdown or deceleration is observed in the dye concentration peaks, possibly related to the above-mentioned turbulent flow ejections. This is clearly seen in figure 5(c), which depicts the normalized instantaneous peak concentration $C_{peak}$ (normalized to $C_{max}$ ) as a function of the associated peak speed $u$ (normalized to $U_{d}^{(2D)}$ ), with $C_{max}$ denoting the maximum value of $C$ over the whole 2D data set. Further, the peak speed $u$ is approximately 40 % larger than the comoving frame velocity, which is roughly constant during the event ( $U_{d}^{(2D)}=6.32\pm 0.22~\text{cm}~\text{s}^{-1}$ ). Note that in oceanic wave groups, large focusing crests tend to slow down as they evolve within the group, as a result of the natural wave dispersion of unsteady wave trains (Banner et al. Reference Banner, Barthelemy, Fedele, Allis, Benetazzo, Dias and Peirson2014; Fedele Reference Fedele2014a ,Reference Fedele b ). Thus, we argue that the observed slowdown of the passive scalar peaks may be due to the wave-like dispersive nature of small-scale turbulent structures.

Figure 5.

LIF experiments: space–time evolution of the dye concentration $C(x,y=0,t)$ at the pipe centreline in the (a) lab frame ( $x,t$ ) and (b) comoving frame $(x-x_{d}(t),t)$ ; (c) normalized instantaneous concentration peak intensity $C_{peak}/C_{max}$ tracked from the initial time $t/T_{d}=0$ (○) as a function of the observed peak speed $u/U_{d}^{(2D)}$ , with $C_{max}$ denoting the observed maximum value of $C$ over the whole data set. $U_{d}\approx 6.34~\text{m}~\text{s}^{-1}$ and $T_{d}=U_{d}/R$ .

Drawing from differential geometry, the observed excess speed $u-U_{d}^{(2D)}$ of concentration peaks is explained in terms of geometric phases.

4.1. Dynamical and geometric phases

From (4.4), the action of the map ${\it\Pi}_{j}$ is to shift the orbit $\boldsymbol{z}(t)=\{z_{m}(t)\}$ in $\mathscr{P}$ by an amount

and the resulting desymmetrized or sliced orbit

has Fourier components

Note that the desymmetrized orbit $\boldsymbol{Z}=g_{-\ell _{s}}(\boldsymbol{z})$ does not satisfy the same dynamical equation (4.2) for $\boldsymbol{z}$ , i.e. $\text{d}\boldsymbol{z}/\text{d}t=\mathscr{N}_{1}(\boldsymbol{z})$ . Indeed (see appendix A),

where

is the tangent space to the group orbit at $\boldsymbol{Z}$ (see, for example, Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2012).

It is well known that the total drift $\ell _{s}$ is the sum of dynamical ( $\ell _{d}$ ) and geometric ( $\ell _{g}$ ) phase drifts (Simon Reference Simon1983; Samuel & Bhandari Reference Samuel and Bhandari1988)

where

Here, we have defined the associated dynamical ( $U_{d}$ ) and geometric ( $U_{g}$ ) phase velocities and the total drift speed follows as

The decomposition into dynamical and geometric components of the drift $\ell _{s}$ and associated velocity follows from the condition of transversality of the symmetry-reduced trajectory $\boldsymbol{Z}$ to the group orbit $g_{\ell _{s}}(\boldsymbol{Z})$ , that is $\text{d}\boldsymbol{Z}/\text{d}t$ is transversal to the group orbit tangent $T(\boldsymbol{Z})$ (Viswanath Reference Viswanath2007; Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2012). Indeed, multiply both members of (4.13) by $\overline{T(\boldsymbol{Z})}$ as

where

is a weighted scalar product of two vectors with weights $W_{pq}=\overline{W}_{qp}$ . In this work we will use the standard scalar product and the group orbit is sliced orthogonally, i.e. $W_{pq}={\it\delta}_{pq}$ where ${\it\delta}_{pq}$ is the Kronecker symbol.

The rate of change of the total drift $\ell _{s}$ is a real number and it follows from the real part of (4.18) as

Here,

is the so-called dynamical phase velocity (Simon Reference Simon1983; Samuel & Bhandari Reference Samuel and Bhandari1988). Since $\overline{T(\boldsymbol{Z})}\boldsymbol{\cdot }\mathscr{N}_{1}(\boldsymbol{Z})$ and $|T(\boldsymbol{Z})|^{2}$ are invariant under translation symmetry, $\ell _{d}$ can also be determined by replacing $\boldsymbol{Z}$ with the orbit $\boldsymbol{z}$ in $\mathscr{P}$ , which is usually known or observable in applications. Indeed, from (4.2) and (4.21)

It is straightforward to show that $U_{d}$ depends on the evolution of the concentration field $c$ in the fixed laboratory frame $(x,t)$ associated with the orbit $\boldsymbol{z}$ in $\mathscr{P}$ . Indeed, since

and

it follows that

Thus, the dynamical phase velocity $U_{d}$ is the 1D comoving frame, or convective, speed similar to that defined for 2D and 3D concentration fields (see (2.5) and (2.11), respectively). Clearly, $U_{d}$ also follows by minimization of the spatial mean of the material derivative of $c$ as in (2.4). Further, from (4.20) we define the geometric phase velocity as

Note that $U_{g}$ and $U_{d}$ in (4.25) are not the same since $\text{d}\boldsymbol{Z}/\text{d}t\neq \text{d}\boldsymbol{z}/\text{d}t$ (see (4.2) and (4.13)). Further, in contrast to the dynamical $U_{d}$ , the geometric $U_{g}$ cannot be related to the evolution of the concentration field $c$ in the fixed laboratory frame $(x,t)$ ; it depends only on the shape-changing evolution of the desymmetrized field $c_{D}$ (see (4.6)) in the symmetry-reduced frame $(x-\ell _{d}-\ell _{g},t)$ . Here, we recall that $c_{D}$ is associated with the desymmetrized orbit $\boldsymbol{Z}$ in the quotient space $\mathscr{M}$ , or base manifold. Indeed, (4.26) can be written as

where we have used (4.23) and (4.24) replacing $\boldsymbol{z}$ with $\boldsymbol{Z}$ . Clearly, the geometric phase velocity depends on the arbitrary choice of the Fourier slice $S_{j}$ . Indeed, different slices yield different desymmetrized concentration fields $c_{D}$ , as discussed later. Further, different scalar products in (4.19) could be used to filter out the contribution of large or small flow scales leading to different slice representations. As mentioned above, in this work we only consider the standard scalar product and all flow scales are accounted for.

The comoving orbit

is the orbit seen from a comoving frame drifting at the speed $U_{d}$ . In physical space it corresponds to an evolution of the dye concentration in the comoving frame $(x-\ell _{d},t)$ . Note that in general $\ell _{d}(t)$ is time varying, and constant only for travelling waves. Clearly, the dynamical drift $\ell _{d}$ increases with the time taken by the trajectory $\boldsymbol{z}(t)$ to wander around $\mathscr{P}$ . The geometric drift $\ell _{g}$ instead depends upon the path ${\it\Gamma}=\{Z_{n}(t)\}$ associated with the desymmetrized orbit $\boldsymbol{Z}(t)$ in the quotient space. Indeed,

The desymmetrized orbit $\boldsymbol{Z}$ is obtained by further shifting the comoving orbit $\boldsymbol{Z}_{d}$ in (4.28) by the geometric drift $\ell _{g}$ as

Different slice representations yield different symmetry-reduced frames $(x-\ell _{d}-\ell _{g},t)$ from which one observes distorted shape-changing dynamics of the dye concentration field. Only relative equilibria or travelling waves have null geometric phase, since their shape is not dynamically changing in the base manifold as they reduce to equilibria. The geometric drift $\ell _{g}$ and associated speed $U_{g}$ can be indirectly computed from (4.15) and (4.17) as $\ell _{g}=\ell _{s}-\ell _{d}$ and $U_{g}=U_{s}-U_{d}$ respectively. The pairs $(\ell _{s},U_{s}=\text{d}\ell _{s}/\text{d}t)$ and $(\ell _{d}=\int _{0}^{t}U_{d}\,\text{d}{\it\tau},U_{d})$ are easily estimated from concentration measurements.

The Fourier slice should be properly chosen to provide a physically meaningful symmetry-reduced frame, as discussed in the next section.

4.2. Symmetry reduction of LIF measurements

In this section, we present a symmetry reduction of the acquired LIF measurements of turbulent pipe flow (see § 2.1). In particular, we study their evolution in physical space and in the associated state space $\mathscr{P}$ of dimension $N=40\,430$ equal to the total number of data image pixels ( $65\times 622$ ).

Regarding the choice of the Fourier slice $S_{j}$ , it is in general entirely arbitrary. There is no unique way to quotient out the symmetry. The most likely choice would be $S_{1}$ , but for our measurements this choice will not produce a physically meaningful symmetry reduction. Higher-order slices are required.

In particular, figures 6 and 7 illustrate the space–time evolution of a passive scalar structure and concentration profiles. Figure 6(a) shows the dye concentration $c(x,t)$ at the pipe centreline in the fixed frame $(x,t)$ (see also figure 7). A drift in the streamwise direction $x$ is observed. The corresponding orbit $\boldsymbol{z}(t)$ in the subspace $\{\text{Re}(z_{11}),\text{Im}(z_{13}),\text{Re}(z_{15})\}$ of $\mathscr{P}$ is shown in figure 8(a,d). Note that the excursion of the orbit while the concentration $c$ lingers above the threshold $0.95c_{max}$ is complicated (bold line) since it wanders around its group orbit as a result of the drift induced by the translation symmetry. Figure 6(c) shows the space–time evolution in the comoving frame $(x-\ell _{d},t)$ . Note that the dye concentration still experiences a significant drift (see also figure 7). As a result, the associated orbit in state space still wanders around the group orbit. A proper choice of the Fourier slice can provide a physically meaningful symmetry-reduced frame. For example, if we choose the first Fourier mode slice $S_{1}$ , figure 6(a) depicts the associated evolution in the symmetry-reduced frame $(x-\ell _{d}-\ell _{g},t)$ . Clearly, the symmetry is quotiented out, but in the Fourier slice $S_{1}$ we observe a distorted shape-changing dynamics of the dye concentration. Instead, if we choose the Fourier slice $S_{25}$ the drift almost disappears in the corresponding symmetry-reduced frame, as shown in figure 6(c) (see also figure 7). Here, this slice is sufficient to symmetry-reduce the orbit $\boldsymbol{z}$ over the analyzed time span as its Fourier components $z_{k},$ with $k\sim 20{-}30$ , never linger near zero, whereas smaller or larger wavenumber modes can be small. The corresponding symmetry-reduced orbits $\boldsymbol{Z}(t)$ associated with $S_{1}$ and $S_{25}$ are computed from (4.4). Their time evolutions within the subspace $\{\text{Re}(Z_{11}),\text{Im}(Z_{13}),\text{Re}(Z_{15})\}$ of $\mathscr{M}$ are shown in figure 8(b,e) and figure 8(c,f) respectively. Here, the excursion of the orbits while the concentration $c$ is high ( ${>}0.95c_{max}$ ) is marked as a bold line. Similar dynamics is also observed when projecting the orbits onto the subspace of their respective most energetic proper orthogonal decomposition (POD) modes, as shown in figure 9. The POD projection of the symmetry-reduced orbit $\boldsymbol{Z}$ is performed within the corresponding symmetry-reduced space. Note that any two POD mode amplitudes are statistically uncorrelated by construction as are any two components $Z_{p}$ and $Z_{q}$ chosen at random. Clearly, this does not imply that they are stochastically independent since they evolve on the quotient manifold $\mathscr{M}$ , which is unknown. As an example, consider two random variables $X$ and $Y$ that satisfy $X^{2}+Y^{2}-1=0$ . They are uncorrelated but not independent and POD projections will not help in revealing the intrinsic manifold structure. Local linear embedding techniques may be more appropriate and appealing (Roweis & Saul Reference Roweis and Saul2000), but they are beyond the scope of our work.

Figure 6.

Symmetry reduction of LIF measurements: space–time evolution of a passive scalar structures: (a,c) measured concentration $C(x,y=0,t)$ at the pipe centreline in the (a) fixed frame, (c) comoving frame $(x-\ell _{d},t)$ ; (b,d) symmetry-reduced frame $(x-\ell _{d}-\ell _{g},t)$ using Fourier slices (b) $S_{1}$ and (d) $S_{25}$ ; time average $\overline{U}_{d}\approx 6.74~\text{cm}~\text{s}^{-1}$ , $\overline{U}_{g}\approx 0.4U_{d}$ and $T_{d}=\overline{U}_{d}/R$ .

Figure 7.

Symmetry reduction of LIF measurements: (a,c) concentration profiles at increasing instants of time of the measured concentration $c(x,y=0,t)$ at the pipe centreline in the (a) fixed frame, (c) comoving frame $(x-\ell _{d},t)$ ; (b,d) symmetry-reduced frame $(x-\ell _{d}-\ell _{g},t)$ using Fourier slices (b) $S_{1}$ and (d) $S_{25}$ . In each plot time increases from bottom to top. Associated 2D patterns are shown in figure 6).

Figure 8.

Symmetry reduction of LIF measurements: (a,d) orbit trajectory $\boldsymbol{z}$ in the subspace $\{\text{Re}(z_{11}),\text{Im}(z_{13}),\text{Re}(z_{15})\}$ of the state space $\mathscr{P}$ associated with the passive scalar dynamics in the lab frame of figure 7 (see also figure 6); and corresponding symmetry-reduced orbits $\boldsymbol{Z}$ in the subspace $\{\text{Re}(Z_{11}),\text{Im}(Z_{13}),\text{Re}(Z_{15})\}$ of the base manifold $\mathscr{M}$ associated with Fourier slices (b,e) $S_{1}$ and (c,f) $S_{25}$ . The bold line indicates the excursion of the orbit while the concentration $c$ lingers above the threshold $0.95c_{max}$ (○  $=$ initial time, $\times =$ final time).

Figure 10(a,c) shows that geometric drifts associated with Fourier slices $S_{1}$ and $S_{25}$ are different and so are the respective geometric phase velocities (see figure 10 b,d). Clearly, the dynamical component $U_{d}$ is the same since it does not depend on the symmetry-reduction scheme or slice. Note that $\ell _{g}$ is not the drift seen by an observer in the symmetry-reduced frame. If it were, the geometric phase velocity associated with the slice $S_{25}$ would be zero since the desymmetrized dye concentration field does not drift. If the same observer drifts by $\ell _{g}$ he will observe the dynamics in the comoving frame. This explains why the geometric phase velocity $U_{g}$ associated with the slice $S_{1}$ is negative in the time span $0.5<t/T_{d}<1$ . With reference to figure 7, in that time interval an observer in the symmetry-reduced frame needs to decelerate in order to follow the dye concentration evolution seen in the comoving frame.

Figure 9.

Symmetry reduction of LIF measurements: (a,c) orbit trajectories $\boldsymbol{z}$ associated with the passive scalar dynamics in the lab frame (see figure 6 a) projected onto the subspace ( $a_{1},a_{2},a_{3}$ ) of the most energetic POD modes; (b,d) corresponding desymmetrized orbit $\boldsymbol{Z}$ in the symmetry-reduced frame associated with the Fourier slice $S_{25}$ . The bold line indicates the excursion of the orbit while the concentration $c$ lingers above the threshold $0.95c_{max}$ (○  $=$ initial time, $\times =$ final time).

Figure 10.

Symmetry reduction of LIF data using Fourier slice $S_{1}$ (a,b) and $S_{25}$ (c,d); (a,c) total, dynamical and geometric drifts and (b,d) corresponding velocities $U_{s}$ , $U_{d}$ and $U_{g}$ associated with the orbit in state space of figure 8.

Figure 11.

LIF experiments: (a) observed normalized dye concentration peak speed $u/U_{d}$ as a function of the amplitude peak $c/C_{max}$ , and (b) associated probability density function, with $C_{max}$ denoting the observed maximum value of dye concentration over the whole data set.

The observed speed $u$ of dye concentration peaks is approximately 40 % larger than the comoving frame velocity $U_{d}$ , which changes slightly during the event. The excess speed ${\it\delta}u=u-U_{d}$ is fairly well explained by the geometric phase velocity $U_{g}\approx 0.4U_{d}$ associated with slice $S_{25}$ , as seen in figure 10(d). This appears to be a general trend of the flow as can be seen in figure 11, which shows the observed normalized speed $u/U_{d}$ of dye concentration peaks tracked in space as a function of their amplitude $c/C_{max}$ , and the associated probability density function, where $C_{max}$ denotes the observed maximum value of dye concentration over the whole data set. As the peak amplitude increases, their speed $u$ tends to $1.43U_{d}$ . Furthermore, in the symmetry-reduced frame, we observe the shape-changing dynamics of passive scalar structures (see figure 7 d). This induces the ‘self-propulsion velocity’ $U_{g}$ of the flow structures similar to that of the motion of a swimmer at low Reynolds numbers (Shapere & Wilczek Reference Shapere and Wilczek1989). Only when the geometric velocity $U_{g}\ll U_{d}$ is Taylor’s approximation valid and as a result the flow structures slightly deform as they are advected at the comoving frame or dynamical phase velocity $U_{d}$ , which is close to the mean flow $U_{m}$ .