3.1 Amplitude ratio and phase lag analysis of axial velocity oscillations

First, it is useful to examine the relationship between the centre-line axial velocity oscillation, $\hat{u} _{x,cl}$, and the sinusoidally varying pressure gradient, $\text{d}\hat{p}/\text{d}x$ (equation (2.2)). Following Mao & Hanratty (Reference Mao and Hanratty1986), the balance of oscillatory axial momentum along the pipe centre line under CD VHF forcing is described by the equation

(3.1)$$\begin{eqnarray}\displaystyle \frac{\text{d}\hat{u} _{x,cl}}{\text{d}t}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\text{d}\hat{p}}{\text{d}x}, & & \displaystyle\end{eqnarray}$$

where the centre-line axial velocity oscillation is defined as $\hat{u} _{x,cl}(\unicode[STIX]{x1D719}_{t})\equiv \langle \overline{u}_{x}\rangle (y=R_{0},\unicode[STIX]{x1D719}_{t})-\overline{u}_{x}(y=R_{0})$ (equation (2.6)). A solution of equation (3.1) for the time-harmonic pressure gradient equation (2.2) is

(3.2)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}|\hat{u} _{x,cl}|\sin (\unicode[STIX]{x1D714}t+\unicode[STIX]{x1D6F7}_{cl})=\frac{1}{\unicode[STIX]{x1D70C}}\left|{\displaystyle \frac{\text{d}\hat{p}}{\text{d}x}}\right|\cos (\unicode[STIX]{x1D714}t), & & \displaystyle\end{eqnarray}$$

where $|\hat{u} _{x,cl}|$ and $|\text{d}\hat{p}/\text{d}x|$ are the amplitude of the centre-line axial velocity oscillation and the oscillatory axial pressure gradient, respectively, and $\unicode[STIX]{x1D6F7}_{cl}$ is the phase angle at the pipe centre line. The variation of $\hat{u} _{x,cl}$ throughout the forcing cycle is plotted in figure 1(a), along with data points for each case in table 2. The sinusoidal variation of $\hat{u} _{x,cl}$ indicates that $\unicode[STIX]{x1D6F7}_{cl}=\unicode[STIX]{x03C0}/2$ for both the smooth- and rough-wall data – in line with the form of equation (3.2). Whereas $\hat{u} _{x,cl}$ can be described by a fundamental mode, the validity of this approximation at other wall-normal heights is unclear. To this end, $\hat{u} _{x}$ was expanded as a Fourier series:

(3.3)$$\begin{eqnarray}\displaystyle \hat{u} _{x}(y,\unicode[STIX]{x1D719}_{t})=A_{u}(y)\sin [\unicode[STIX]{x1D714}t+\unicode[STIX]{x1D6F7}(y)]+\mathop{\sum }_{n=2}^{\infty }A_{u}^{n}(y)\sin [n\unicode[STIX]{x1D714}t+\unicode[STIX]{x1D6F7}^{n}(y)], & & \displaystyle\end{eqnarray}$$

where $A_{u}(y)$ is the amplitude of the fundamental mode at each wall-normal location.

Figure 1.

(a) Axial velocity oscillation at the pipe centre-line, $\hat{u} _{x,cl}$, along with sinusoidal waveform $|\hat{u} _{x,cl}|\sin (\unicode[STIX]{x1D714}t)$ (——) for reference. Symbols correspond to DNS data and are defined in table 1. (b) Amplitude normalised by the centre-line amplitude, $A_{u}(y)/A_{cl}$. (c) Phase angle referenced against the centre-line value, $\unicode[STIX]{x1D6F7}(y)-\unicode[STIX]{x1D6F7}_{cl}$. Results for the Womersley (Reference Womersley1955) solution at matched forcing conditions also included for comparison $(\ast )$. Vertical lines on (b) and (c) indicate height of the roughness crests. All data scaled using the GA friction velocity, $u_{\unicode[STIX]{x1D70F}}$.

The wall-normal variation of $A_{u}(y)$ normalised by its centre-line value, $A_{cl}$, is plotted in figure 1(b). For wall-normal locations greater than $y^{+}>100$, the rough- and smooth-wall data follow a horizontal line of unit amplitude. On the other hand, the amplitude ratio exhibits an overshoot in the vicinity of the roughness crests before decaying within the roughness canopy. The wall-normal variation of $\unicode[STIX]{x1D6F7}(y)$ referenced against its centre-line value, $\unicode[STIX]{x1D6F7}_{cl}$, is plotted in figure 1(c). Relative to the centre-line value, a negligible phase difference is observed for wall-normal locations greater than $y^{+}>100$ for both the smooth- and rough-wall configurations. As a result, the phase angle maintains its centre-line value of $\unicode[STIX]{x1D6F7}_{cl}=\unicode[STIX]{x03C0}/2$ (equation (3.2)) in the outer region of the flow, whereas, in the region below the roughness crests, $\unicode[STIX]{x1D6F7}$ decreases with increasing roughness height. A final observation based on figure 1(b,c) is the close match between the smooth-wall data and the Womersley (Reference Womersley1955) laminar solution at matched forcing conditions – a well-documented behaviour in pulsatile smooth-wall turbulent pipe flow, e.g. see recent work by Cheng et al. (Reference Cheng, Jelly, Illingworth, Marusic and Ooi2020).

Overall, figure 1 implies that, for wall-normal locations greater than $y^{+}>100$, axial velocity oscillations are well approximated by the fundamental Fourier mode (equation (3.3)), regardless of the roughness topographies considered here (table 2). Hence, spatially averaged PA axial velocity profiles, $\langle \overline{u}_{x}\rangle \equiv \overline{u}_{x}+\hat{u} _{x}$ (equation (2.6)), are expected to retain their GA distribution in the outer region but with a periodic shift in response to the unsteady forcing. In what follows below, spatially averaged PA data are presented at the four phases marked on figure 1(a), namely, the peak-acceleration $(\unicode[STIX]{x1D719}_{t,A}=0)$, zero-acceleration $(\unicode[STIX]{x1D719}_{t,B}=1/4)$, peak-deceleration $(\unicode[STIX]{x1D719}_{t,C}=1/2)$ and zero-deceleration $(\unicode[STIX]{x1D719}_{t,D}=3/4)$ phases.

3.2 Assessment of Townsend’s outer-layer similarity hypothesis using PA data

Spatially-averaged PA axial velocity profiles for case $h_{20}\unicode[STIX]{x1D706}_{141}$ are plotted in figure 2(a), along with non-pulsatile rough-wall and smooth-wall data for reference. The first notable observation based on figure 2(a) is the preservation of a (nominal) logarithmic layer at each phase – implying that velocity profiles retain their self-similar functional form throughout the forcing cycle. The self-similar nature of the velocity profiles is clearer after recasting them into defect form, which, with reference to the top-left inset on figure 2(a), collapses the PA data onto a single curve, and on top of the results of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018), when plotted against $y/R_{0}$. A second notable observation based on figure 2(a) is that the log-laws of the rough- and smooth-wall flows oscillate with the same phase and amplitude – corroborating the preceding analysis of axial velocity oscillations (figure 1). Spatially-averaged PA axial velocity profiles for cases $h_{40}\unicode[STIX]{x1D706}_{283}$ and $h_{80}\unicode[STIX]{x1D706}_{565}$ are plotted in figures 2(c) and 2(e), respectively, and show the same trends in the outer region. These observations imply that the log-layer remains intact but with a periodic shift due to the unsteady forcing. Hence, the rough-wall log-law can be expressed in PA form as

(3.4)$$\begin{eqnarray}\displaystyle \langle \overline{u}_{x}^{+}\rangle (y,\unicode[STIX]{x1D719}_{t})=\unicode[STIX]{x1D705}^{-1}\log y^{+}+B-\langle \unicode[STIX]{x0394}U^{+}\rangle (\unicode[STIX]{x1D719}_{t}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ is the von Kármán constant, $B$ is the intercept and $\langle \unicode[STIX]{x0394}U^{+}\rangle \equiv \unicode[STIX]{x0394}U^{+}+\unicode[STIX]{x0394}\hat{U} ^{+}(\unicode[STIX]{x1D719}_{t})$ is the PA Hama roughness function, defined as the difference between the GA smooth-wall and PA rough-wall centre-line velocities at matched flow conditions. For each case, the oscillatory Hama roughness function, $\unicode[STIX]{x0394}\hat{U} ^{+}$, achieves its maximum value at the zero-acceleration phase ($\unicode[STIX]{x1D6F7}_{t,B}=1/4$), following a transient favourable pressure gradient $(\text{d}\hat{p}/\text{d}x<0)$, whereas the opposite occurs at the zero-deceleration phase ($\unicode[STIX]{x1D6F7}_{t,D}=3/4$), following a transient adverse pressure gradient $(\text{d}\hat{p}/\text{d}x>0)$. In order of increasing roughness amplitude, $\unicode[STIX]{x0394}U^{+}=\{6.4,9.0,11.5\}$ for the pulsatile data – matching the past results of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018) to within 3 % for each case in table 2.

Figure 2.

Spatially averaged PA velocity statistics including (a,c,e) axial velocity profiles, $\langle \overline{u}_{x}\rangle$, and (b,d,f) axial r.m.s. profiles, $\langle u_{x,rms}^{\prime }\rangle$. Phase-averaged data are shown at the four phases marked on figure 1(a). Non-pulsatile data from Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018) (symbols) and smooth-wall $(\times )$ data is included. Top-left inset panel shows the PA velocity defect profiles, $\langle \overline{u}_{x,cl}\rangle ^{+}-\langle \overline{u}_{x}\rangle ^{+}$, and top-right inset panels show the oscillatory component of the PA roughness function, $\unicode[STIX]{x0394}\hat{U} ^{+}$. Line and symbol types are given in table 1. Data are scaled by the GA friction velocity, $u_{\unicode[STIX]{x1D70F}}$.

In line with the PA velocity defect profiles inset on figure 2(a,c,e) the profiles of r.m.s. axial turbulence fluctuations shown in figure 2(b,d,f), also exhibit an invariance with respect to temporal phase in the outer region. In addition, the PA r.m.s. profiles closely follow the non-pulsatile data of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018), as well as the smooth-wall data, for each case in table 2. Such frozen behaviour is well-documented in past studies of smooth-wall pulsatile turbulent pipe flows. For instance, Papadopoulos & Vouros (Reference Papadopoulos and Vouros2016) demonstrated that PA turbulent intensity profiles relax onto their non-pulsatile counterpart for forcing frequencies in excess of $\unicode[STIX]{x1D714}^{+}\approx 0.04$, noting that this behaviour demarcates the onset of the VHF regime. In addition, past work by Bhaganagar (Reference Bhaganagar2008) also showed that the modulation of turbulence in the outer region of pulsatile rough-wall channel flow decreases with increasing frequency and asymptotes to zero for a forcing frequency of $\unicode[STIX]{x1D714}^{+}=0.07$.

Overall, two key conclusions can be drawn from figure 2: (i) PA axial velocity profiles retain a self-similar functional form in the outer region, which can be described using a PA extension of the rough-wall log-law (equation (3.4)), and (ii) second-order turbulence statistics appear frozen in the outer region and collapse onto the non-pulsatile and smooth-wall data for all phases. Together, (i) and (ii) provide considerable evidence in support of Townsend’s outer-layer similarity hypothesis.

3.3 Hydraulic properties of pulsatile rough-wall turbulent pipe flow

The GA friction factor, $\overline{f}$, and bulk Reynolds number, $Re_{b}$, for each pulsatile case is plotted on a Moody (Reference Moody1944) chart in figure 3 and matches the steady data of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018) to within 3 % for each case listed in table 2. As expected, $\overline{f}$ increases as the ratio of mean pipe radius to the roughness height decreases, or, put in other words, $Re_{b}$ decreases with decreasing $R_{0}/h$. The spatially averaged PA friction factor, $\langle f\rangle$, and PA bulk Reynolds number, $\langle Re_{b}\rangle$, are also shown on figure 3 and form point-clouds that orbit the GA data. The former quantity is defined here as $\langle f\rangle =4\langle \unicode[STIX]{x1D70F}_{w}\rangle /(\unicode[STIX]{x1D70C}\langle U_{b}\rangle ^{2}/2)$, where $\langle \unicode[STIX]{x1D70F}_{w}\rangle$ is the total PA wall shear stress (which is composed of viscous and pressure drag components) and $\langle U_{b}\rangle$ is the PA bulk velocity. To aid the analysis of results, the PA data on figure 3 is colour-coded using the accelerating and decelerating intervals of the centre-line axial velocity oscillation, $\hat{u} _{x,cl}^{+}$ (see top-right inset figure 3). For the smooth-wall case, the phase lag between $\langle f\rangle$ and $\hat{u} _{x,cl}^{+}$ (figure 1a) is equal to $\unicode[STIX]{x03C0}/4$ (to within 3 %), indicating that the oscillatory component of the smooth-wall PA friction factor closely follows the Womersley (Reference Womersley1955) solution in the limit of VHF forcing. In addition, the phase lag between $\langle f\rangle$ and $\hat{u} _{x,cl}^{+}$ for cases $h_{20}\unicode[STIX]{x1D706}_{141}$ and $h_{40}\unicode[STIX]{x1D706}_{283}$ matches the smooth-wall value of $\unicode[STIX]{x03C0}/4$ to within 8 %. On the other hand, the phase lag angle for case $h_{80}\unicode[STIX]{x1D706}_{565}$ is equal to $\unicode[STIX]{x03C0}/6$, indicating that the time delay between the PA friction factor and centre-line velocity reduces by one third, relative to the smooth-wall case. A possible explanation for this behaviour is that pressure drag dominates viscous drag for case $h_{80}\unicode[STIX]{x1D706}_{565}$, and, as a result, $\langle f\rangle$ is unlikely to exhibit a Stokes-type behaviour in the fully-rough regime. A notable observation based on figure 3 is that, after computing the oscillatory component of the PA bulk Reynolds number, $\hat{Re}_{b}(\unicode[STIX]{x1D719}_{t})\equiv \langle Re_{b}\rangle -Re_{b}$, and friction factor, $\hat{f}(\unicode[STIX]{x1D719}_{t})\equiv \langle f\rangle -f$, then normalising by their respective r.m.s. values, the PA rough- and smooth-wall data follow a similar near-circular orbit (see $\hat{f}/\hat{f}_{rms}$ versus $\hat{Re}_{b}/\hat{Re}_{b,rms}$ plot, top-left inset panel figure 3). This result implies the orbital trajectory is insensitive to changes in surface topography and only depends on the applied forcing condition, at least for the four wall topographies considered in this study (table 2). It is worth noting, however, that in order of increasing roughness height, $\hat{f}_{rms}$ is equal to 33 %, 37 % and 67 % of $\overline{f}$ and $\hat{Re}_{b,rms}$ is equal to 8 %, 10 % and 13 % of $Re_{b}$ – indicating that increasing the size of the roughness elements leads to increasingly large variations in both $\langle f\rangle$ and $\langle Re_{b}\rangle$.

Figure 3.

Moody chart. Global-averaged friction factor, $\overline{f}$, and bulk Reynolds number, $Re_{b}$, shown as the symbols defined in table 1. Phase-averaged data is colour-coded using the accelerating ($\text{d}\hat{u} _{x,cl}^{+}/\text{d}\unicode[STIX]{x1D719}_{t}>0$ —— (red), —— (magenta)) and decelerating ($\text{d}\hat{u} _{x,cl}^{+}/\text{d}\unicode[STIX]{x1D719}_{t}<0$ —— (blue), —— (green)) intervals of the centre-line axial velocity oscillation (top-right inset). Top-left inset panel shows $\hat{f}/\hat{f}_{rms}$ plotted against $\hat{Re}_{b}/\hat{Re}_{b,rms}$. Laminar friction factor, $f=64/Re$ ——(grey), Blasius’ law, $f=0.079/Re^{0.25}$ ——, a line of $Re_{\unicode[STIX]{x1D70F}}=540$ (- - -) and Nikuradse’s smooth-wall data $(\ast )$ are included, along with smooth-wall nonpulsatile pipe flow DNS data at $Re_{\unicode[STIX]{x1D70F}}=\{180,360,540,1002,2003\}$ from Chin, Monty & Ooi (Reference Chin, Monty and Ooi2014) and Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018) $(\times )$.

Figure 4.

Analysis of PA pressure and viscous forces for case $h_{80}\unicode[STIX]{x1D706}_{565}$. (a) Phase diagram of $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ versus $\langle \overline{F}_{p}\rangle$, normalised by the total GA drag force, $\overline{F}_{tot}$. The elliptical point-cloud is centred around $(\overline{F}_{p},\overline{F}_{\unicode[STIX]{x1D708}})/\overline{F}_{tot}=(0.79,0.21)$ (○). (b) Ratio of PA pressure and viscous forces $\langle \overline{F}_{p}\rangle /\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ across the forcing cycle. Shaded grey rectangle denotes region where viscous shear stress becomes negative, i.e. $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle <0$. Data is colour-coded in accordance with figure 3.

Figure 5.

Isosurface of triple PA reverse flow $(\langle u_{x}^{+}\rangle =-0.1)$ for case $h_{80}\unicode[STIX]{x1D706}_{565}$ at the (a) minimum ($\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.05$) and (b) maximum ($\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.55$) value of the PA friction factor, $\langle f\rangle$. Data is shown on a $(\unicode[STIX]{x1D719}_{x}\times \unicode[STIX]{x1D719}_{s})/2\unicode[STIX]{x03C0}=(3\times 2)$ doubly periodic subdomain. Data is scaled by the GA friction velocity, $u_{\unicode[STIX]{x1D70F}}$.

A further observation based on figure 3 is that $\langle f\rangle$ becomes negative for phases on the range $0.50\lesssim \unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}\lesssim 0.60$ for case $h_{80}\unicode[STIX]{x1D706}_{565}$ (not visible on figure 3 due to the log-log format of the Moody chart). In order to associate specific mechanisms to this behaviour, the total PA drag force was decomposed as $\langle \overline{F}_{tot}\rangle =\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle +\langle \overline{F}_{p}\rangle$, where $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ and $\langle \overline{F}_{p}\rangle$ are the spatially averaged PA viscous and pressure forces, respectively. Note that terms $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ and $\langle \overline{F}_{p}\rangle$ both appear on the right-hand side of the PA hydrodynamic force balance equation, which can be expressed as

(3.5)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\frac{\text{d}\langle U_{b}\rangle }{\text{d}\unicode[STIX]{x1D719}_{t}}=\unicode[STIX]{x1D70C}\langle \overline{F}_{x}\rangle V+\underbrace{\left\langle \iint _{\unicode[STIX]{x2202}S}-\overline{p}n_{x}\,\text{d}S\right\rangle }_{\langle \overline{F}_{p}\rangle (\unicode[STIX]{x1D719}_{t})}+\underbrace{\left\langle \iint _{\unicode[STIX]{x2202}S}\left(\unicode[STIX]{x1D708}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\overline{\boldsymbol{u}}\right)\,\text{d}S\right\rangle }_{\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle (\unicode[STIX]{x1D719}_{t})}, & & \displaystyle\end{eqnarray}$$

where $\langle \overline{F}_{x}\rangle$ is the streamwise driving PA pressure gradient, as in equation (2.2), $V$ is the volume occupied by the fluid, $\unicode[STIX]{x2202}S$ is the surface described by the roughness distribution (equation (2.1)) and $\boldsymbol{n}$ denotes the unit-normal vector. The phase diagram of $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ versus $\langle \overline{F}_{p}\rangle$ normalised by the total GA drag force, $\overline{F}_{tot}(=\overline{F}_{\unicode[STIX]{x1D708}}+\overline{F}_{p})$, for case $h_{80}\unicode[STIX]{x1D706}_{565}$ is shown in figure 4(a). The data show that both $\langle \overline{F}_{p}\rangle$ and $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ both attain negative values, and, as a result, the PA value of the total wall shear stress is also negative, i.e. $\langle \unicode[STIX]{x1D70F}_{w}\rangle <0$. One complication arising from this scenario is that the PA friction velocity, $\langle u_{\unicode[STIX]{x1D70F}}\rangle \equiv \sqrt{\langle \unicode[STIX]{x1D70F}_{w}\rangle /\unicode[STIX]{x1D70C}}$, is a complex number and, hence, the rationale behind inner scaling using the local friction velocity falls apart at these particular phases. The ratio of the PA pressure and viscous force, $\langle \overline{F}_{p}\rangle /\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$, is plotted in figure 4(b) and reveals a rich array of unsteady flow physics not observed in the GA data (or in the past results of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2018)). For instance, whereas the ratio of the GA forces is approximately four-to-one in favour of pressure drag, the ratio of the PA forces shows an altogether different picture. Notably, $\langle \overline{F}_{p}\rangle /\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ exhibits two discontinuities as the PA viscous force passes through a pair of zero-crossings – the first occurring at $\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.48$, when $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ switches from positive to negative, and the second occurring at $\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.75$, when $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle$ switches from negative to positive. The zero-crossings bound a region of negative viscous stress, i.e. $\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle <0$, which, as shown below, is related to flow recirculation within the roughness canopy.

In order to associate particular flow events to the variation of the PA pressure and viscous forces (figure 4), reverse-flow motions for case $h_{80}\unicode[STIX]{x1D706}_{565}$ were examined. Isosurfaces of triple PA reverse flow at the phases corresponding to the maximum and minimum values of the total PA drag force are compared in figure 5. Whereas small reverse-flow pockets are visible within the roughness troughs at the former phase ($\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.05$, figure 5a), a significantly larger reverse-flow region engulfs the leeward-facing slopes at the latter phase ($\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.55$, figure 5b). A simple interpretation of this behaviour is as follows. As the flow accelerates, high-speed fluid rushes past the canopy, creating high-pressure regions on windward-facing slopes and (positive) viscous shear that hinders recirculation behind the crests (figure 5a). On the other hand, relatively low-speed fluid passes over the canopy as the flow decelerates, resulting in negative viscous stress ($\langle \overline{F}_{\unicode[STIX]{x1D708}}\rangle <0$), see shaded region on figure 4b), which promotes massive flow separation and recirculation downstream of the crests (figure 5b). Despite the large volume of reverse flow that develops within the canopy of case $h_{80}\unicode[STIX]{x1D706}_{565}$, and the fact that the total PA drag force becomes negative at $\unicode[STIX]{x1D719}_{t}/2\unicode[STIX]{x03C0}=0.55$ (figure 4a), it is worth noting that the PA bulk velocity remains positive for this phase (and all other phases), i.e. $\langle U_{b}\rangle >0$, due to the CD nature of the flow.