1. Introduction
Pipe flows play a crucial part in our daily life. Consequently, turbulent pipe flows have been extensively studied in turbulence research. Although many studies are readily available on steady turbulent pipe flows, comprehensive investigations of unsteady turbulent pipe flows require further study because turbulent pipe flows in real-life applications are seldom steady. Furthermore, transient turbulent pipe flows display unique flow properties of behaviour compared with steady turbulent pipe flows. Thus, extending our current understanding of unsteady flows is crucial in conceptualising the flow physics of turbulent pipe flows. Uniform momentum zones (UMZs) are hierarchically organised zones of relatively similar momentum within wall-bounded flows, characterised by enhanced vortical activity at their boundaries. Researchers have extensively investigated these coherent structures since their initial inception. Despite the attention, investigations conducted so far on UMZs have been limited to steady turbulent flow cases, with the temporal behaviour of UMZs in unsteady turbulent flow remaining an area open for investigation.
This investigation focuses on the transient behaviour experienced by accelerating turbulent pipe flows from an initially steady turbulent base flow up to a final steady turbulent flow at a higher Reynolds number resulting from a rapid ramp-up acceleration. As a point of entry to understand and investigate transient turbulent flow kinematics from a structural perspective, the current investigation examines UMZs with an emphasis on the temporal evolution of their boundaries, which are regions of high local shear, i.e. internal shear layers (ISLs).
The present paper is organised as follows. Following the introduction section in
$\S$
1, in
$\S$
2, the methodology followed for the current study in terms of the transient direct numerical simulation (DNS) data and the identification techniques of UMZs are provided. The results obtained from the current study are discussed and analysed in
$\S$
3. Finally, a summary of the findings and the overall conclusions arrived at are presented in
$\S$
4.
1.1. Accelerating turbulent flows
Turbulent flows mainly comprise two classifications. These are unsteady turbulent flows and steady turbulent flows (Seddighi et al. Reference Seddighi, He, Vardy and Orlandi2014). Unsteady turbulent flows result from flow perturbations, which could be either constant (static) or time varying. Given their unique flow dynamics, unsteady turbulent flows resulting from time-varying flow perturbations can be further classed into two distinct groups. These are periodic pulsatile flows and non-periodic flows (i.e. accelerating or decelerating flows). Out of the two classifications, periodic pulsatile flows have received more attention due to their practical importance in biological and environmental flows (He & Jackson Reference He and Jackson2000; Manna, Vacca & Verzicco Reference Manna, Vacca and Verzicco2012). To address this gap, the current investigation focuses on the transient behaviour of UMZs during different stages undergone by a rapidly accelerating turbulent pipe flow.
Accelerating turbulent flows have intrigued researchers due to the characteristic transient turbulent properties they possess. Maruyama, Kuribayashi & Mizushina (Reference Maruyama, Kuribayashi and Mizushina1976) examined turbulence statistics in pipes, including the root mean square (r.m.s.) of velocity fluctuations in transient flows resulting from a stepwise increase in flow rate. They identified that a delayed response in turbulence follows the step up in the flow rate. They also noted that the new turbulence generated by the increase in flow rate is produced at the pipe wall and propagates radially to the pipe axis. Furthermore, it was mentioned that the linear decay law applies to the decay in the transient pipe flow, and the radial dependence of the decay rate is proportional to its initial steady value.
Kurokawa & Morikawa (Reference Kurokawa and Morikawa1986) investigated both accelerated and decelerated pipe flows experimentally and showed that the growth of the perturbation boundary layer (or the propagation of viscous effects from the wall towards the pipe centre) is dependent on the nature of the flow excursion imposed. They showed that, for accelerating flow cases, there are distinct differences between the resultant transient turbulent flows when rapid and mild flow excursions were imposed.
Greenblatt & Moss (Reference Greenblatt and Moss2004) investigated rapidly accelerating turbulent pipe flow cases with single-component laser-Doppler velocimeter measurements. Their analysis revealed that coherence was displayed by the resultant flow displacement thickness and the momentum thickness as they developed over time. Furthermore, they were among the first to comment on the turbulent–turbulent interactions of the unsteady flow by pointing out that a rapidly accelerating turbulent pipe flow exhibits three unsteady phases: initial, final and relaxation. Extending this work, He & Seddighi (Reference He and Seddighi2013,Reference He and Seddighi2015) performed DNS on a transient channel flow at relatively low Reynolds numbers following a rapid increase in the flow rate and demonstrated that the flow development undergoes three phases: pre-transition, transition and fully turbulent. Furthermore, they noted that these unsteady flow stages resemble the three boundary layer bypass transition regions: buffeted laminar flow, intermittent flow and fully turbulent flow regions.
Recently, DNS work by Guerrero, Lambert & Chin (Reference Guerrero, Lambert and Chin2021) has provided an extended understanding of the transient dynamics of rapidly accelerating turbulent pipe flows. They investigated rapidly accelerating turbulent pipe flows at low and moderate Reynolds numbers. From their analysis they identified four distinct transient flow stages in all flow cases investigated, characterised by distinctly different flow characteristics. These four stages were the inertial, a stage displaying frozen turbulence and a rapid increase of viscous forces; pre-transition, a rapid reduction of viscous forces with a weak response to turbulence in the near-wall region; transition, a steady increment of turbulent inertia and viscous forces in the inner region; and core relaxation, reconstitution of the wake region due to turbulence transport by diffusion and advection. It was shown that, compared with the time scales of the inertial, pre-transition and transition stages, the reconstitution of the wake region of the flow during the core-relaxation stage requires extended time scales, and this was due to the largest UMZ or, in other words, the quiescent core of the turbulent pipe flow displaying a delayed turbulence response. Furthermore, by comparing the cross-correlation of this large UMZ, they showed that the quiescent core advects without significant change in its geometry, similar to an independent plug flow.
1.2. Uniform momentum zones
Uniform momentum zones are large scales of motion found in wall-bounded flows of relatively uniform streamwise velocity, which are arranged hierarchically from the wall (Chen, Chung & Wan Reference Chen, Chung and Wan2020). Uniform momentum zones were first identified as a result of the pioneering work by Meinhart & Adrian (Reference Meinhart and Adrian1995), and since their inception, UMZs have been extensively investigated in steady turbulent wall-bounded flows. The detection of UMZs within the flow domain has been achieved using different approaches in the literature. The first approach analyses the probability density functions (PDFs) of the instantaneous streamwise velocity, and the respective peaks on the PDFs are detected as the above-mentioned UMZs (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Kwon et al. Reference Kwon, Philip, De Silva, Hutchins and Monty2014; De Silva, Hutchins & Marusic Reference De Silva, Hutchins and Marusic2016; Laskari et al. Reference Laskari, de Kat, Hearst and Ganapathisubramani2018; Chen et al. Reference Chen, Chung and Wan2020).
Apart from the PDF approach, another approach in the literature involves identifying ISLs that correspond to the boundaries of UMZs, as these are local regions with high shear within the flow. Indeed, the ISL method has significant advantages over the PDF method, as it is not sensitive to the streamwise domain length nor the custom local peak isolation thresholds imposed (Chen, Chung & Wan Reference Chen, Chung and Wan2021). The studies by De Silva et al. (Reference De Silva, Philip, Hutchins and Marusic2017) and Chen et al. (Reference Chen, Chung and Wan2021) followed an ISL approach to find UMZs by identifying the peaks of the instantaneous streamwise velocity gradient profiles. Gul, Elsinga & Westerweel (Reference Gul, Elsinga and Westerweel2020) conducted a three-dimensional ISL analysis identifying shear layers, also called vortex sheets, where a strong correlation between high vorticity and strain rate occurred, utilising the
$A_{+}$
criterion proposed by Horiuti & Takagi (Reference Horiuti and Takagi2005). They compared their results from the three-dimensional ISL approach with those from the PDF approach for the same flow fields and obtained similar results. Other approaches have been proposed to identify UMZs or ISLs in the literature, such as a kernel density estimation (KDE) method with a single KDE bandwidth, without the need for multiple user-defined thresholds, unlike in the PDF method (Fan et al. Reference Fan, Xu, Yao and Hickey2019).
Studies of UMZs have shown that zones located at different wall-normal locations exhibit distinct characteristics. For example, in steady turbulent wall-bounded flows, the thickness of UMZs located near the wall is found to be significantly smaller compared with UMZs found further away from the wall (De Silva et al. Reference De Silva, Hutchins and Marusic2016, Reference De Silva, Philip, Hutchins and Marusic2017; Chen et al. Reference Chen, Chung and Wan2020). Kwon et al. (Reference Kwon, Philip, De Silva, Hutchins and Monty2014), in their study of the quiescent core in turbulent channel flows, revealed a distinct jump in turbulence statistics as the core boundary is crossed and stated that the quiescent core exists with a boundary qualitatively similar to the turbulent/non-turbulent interface of boundary layers away from the wall (Kwon et al. Reference Kwon, Philip, De Silva, Hutchins and Monty2014). De Silva et al. (Reference De Silva, Hutchins and Marusic2016) revealed that the average number of UMZs in a boundary layer flow has a log–linear relationship with the Reynolds number, together with individual zone thicknesses showing proportionality to the increasing distance from the wall.
Furthermore, links between dominant quadrant events (Wallace & Brodkey Reference Wallace and Brodkey1977; Wallace Reference Wallace2016) and UMZs have been investigated in order to quantify further the differences between UMZs located within the flow (Laskari et al. Reference Laskari, de Kat, Hearst and Ganapathisubramani2018; Chen et al. Reference Chen, Chung and Wan2020). Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) revealed that flow regions that consist of an extreme number of UMZs are linked to either large-scale
$Q2$
-dominated events, which increase small-scale activity in the log region, or large-scale
$Q4$
-dominated events, which decrease the turbulent activity in the log region. Here,
$Q2$
events corresponds to ejection dominated events, while
$Q4$
events corresponds to sweep dominated events. Furthermore, they showed that the average residence time scales of zones dominated by sweep events were quadruple that of residence times of zones dominated by ejection events (Laskari et al. Reference Laskari, de Kat, Hearst and Ganapathisubramani2018). Following the findings by Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) and Chen et al. (Reference Chen, Chung and Wan2020), UMZs located near the wall are shown to consist of modulations of ejections over sweeps and vice versa in UMZs located near the core of the pipe flow, suggesting a lowered turbulent activity (Chen et al. Reference Chen, Chung and Wan2020). Interestingly, these findings on ejections and sweeps in UMZs by Chen et al. (Reference Chen, Chung and Wan2020) were based on the fluctuations of the wall-normal distances of the UMZs boundaries, not the instantaneous fluctuations of velocities (streamwise and wall normal).
Despite UMZs being extensively investigated since the pioneering work by Meinhart & Adrian (Reference Meinhart and Adrian1995), most investigations have been limited to steady turbulent wall-bounded flow scenarios. To the authors’ understanding there are no studies available in the literature investigating UMZs in the context of unsteady turbulent pipe flows nor in the context of rapidly accelerating turbulent pipe flows. As a result, the authors aim to provide insight into the temporal evolution of UMZs in a rapidly accelerating flow scenario and motivate future work on addressing the existing gap in understanding how UMZs behave in unsteady, turbulent wall-bounded flows. Furthermore, since UMZs are large-scale structures, investigating how UMZs evolve during the transient process that the rapidly accelerating flow undergoes is expected to provide a better conceptual understanding of large-scale motions and provide further insight into the transient development of the rapidly accelerating flow from a structural perspective. Hence, instantaneous UMZs detected during the transient flow stages of the rapidly accelerating flow are investigated. The present study primarily employs the PDF-based method, alongside custom threshold parameters, for UMZ detection. Briefly, it supplements the collected results with data from the ISL method based on instantaneous streamwise velocity gradient profiles. Furthermore, the findings of the dominant motions of UMZs located at various wall-normal locations by Chen et al. (Reference Chen, Chung and Wan2020) are extended based on the quadrant analysis method by Wallace & Brodkey (Reference Wallace and Brodkey1977) in the context of rapidly accelerating turbulent pipe flows.
2. Methodology
The datasets of transient pipe flow analysed in this paper have been obtained from Guerrero et al. (Reference Guerrero, Lambert and Chin2021), and an extensive insight into the numerical schemes utilised can be found in Guerrero, Lambert & Chin (Reference Guerrero, Lambert and Chin2020) and Guerrero et al. (Reference Guerrero, Lambert and Chin2021).
2.1. Transient data
The spectral element solver Nek5000 (Fischer, Kruse & Loth Reference Fischer, Kruse and Loth2002) was used to generate the DNS datasets with high spectral accuracy, and the transient datasets have been produced from accelerated turbulent flow fields starting from independent and uncorrelated turbulent flow fields. Two such transient turbulent pipe flow datasets were used for the current study to ensure statistical convergence of our findings. The high fidelity simulations were run on a periodic spectral mesh of domain length
$8\pi R$
, where
$R$
is the pipe radius. The spectral mesh consisted of seventh-order Gauss–Labatto–Legendre quadrature points. The velocity field has been resolved using the Helmholtz equation and the pressure field using the
$P_{N}P_N$
solver algorithm of Nek5000 (Fischer et al. Reference Fischer, Kruse and Loth2002). For temporal integration of the momentum and continuity equations, a third-order backward difference scheme was implemented. The simulation parameters used in the DNS simulations herein are summarised in table 1.
Computational parameters from Guerrero et al. (Reference Guerrero, Lambert and Chin2021).
Note that the friction velocity
$u_{\tau } = \sqrt {\tau _w/\rho }$
, where
$\tau _w$
is the mean wall shear stress of the fluid. The friction Reynolds number
$Re_\tau = u_{\tau }R/\nu$
, where
$\nu$
is the kinematic viscosity and
$\rho$
is the density of the fluid. The ‘+’ superscript corresponds to normalisation in viscous units with indices ‘0’ and ‘1’ denoting normalisation in viscous units concerning the initial and the final states of steadiness attained by the transient flow after undergoing acceleration. The ‘ramp’, ‘samp’ and ‘sim’ subscripts correspond to the ramp-up (acceleration) time, the sampling time interval of data stored and the duration of the DNS simulation. Also,
$\gamma$
is a non-dimensional ramp rate parameter which incorporates the bulk velocity
$U_b$
of the flow, where
$\gamma = [{\rm d}{U_b}/{\rm d}t)(1/U_{b0})(D/u_{\tau 0})]$
, as introduced by He & Jackson (Reference He and Jackson2000). A value of
$\gamma \gg 1$
confirms that the flow deviates from any pseudo-steady behaviour as the flow is accelerated and confirms unsteady turbulent behaviour of the flow during the flow excursion. The notations
$z$
,
$\theta$
and
$r$
denote streamwise, azimuthal and radial directions, respectively, and
$y=-r$
is the wall-normal direction. Finally,
$\Delta z^{+}$
,
$\Delta R\theta ^{+}$
and
$\Delta y^{+}$
are the inner normalised grid resolution of the spectral mesh implemented by Guerrero et al. (Reference Guerrero, Lambert and Chin2021).
Temporal evaluation of the skin friction coefficient
$C_{\!{f}}$
. Solid line (
) depicts data from the current study. Time scales of unsteady flow stages from table 2 are depicted by (
) for the reader.
The time scales for each transitional flow stage were identified from the temporal evolution of the skin friction coefficient
$C_{\!{f}}$
as in figure 1 (Guerrero et al. Reference Guerrero, Lambert and Chin2021). The inertial stage starts when the flow is accelerated at
$t^{+0} = 0$
and finishes at the peak attained by
$C_{\!{f}}$
. During the pre-transition stage,
$C_{\!{f}}$
decays, and according to He & Seddighi (Reference He and Seddighi2013), it finishes when
$C_{\!{f}}$
has attained a minimum. The transition stage is defined from its minimum to when
$C_{\!{f}}$
overshoots the final steady state. Finally, the core-relaxation stage is defined for the time scales where
$C_{\!{f}}$
remains quasi-steady while the wake evolves towards its final steady state. The results of the time scales identified for the current flow in figure 1, based on the flow parameters set in table 1 are summarised in table 2.
Time scales for each transitional stage for the current study.
2.2. Uniform momentum zone detection scheme
Uniform momentum zones were detected following a similar approach to Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) and Chen et al. (Reference Chen, Chung and Wan2020). The PDF for the instantaneous streamwise velocity
$U_z$
from rapidly accelerating turbulent pipe flow fields was used to identify the UMZs, where the peaks refer to the zone modal velocities (
$u_m$
) of the UMZs (De Silva et al. Reference De Silva, Hutchins and Marusic2016; Chen et al. Reference Chen, Chung and Wan2020) and the minimum PDF value between two consecutive peaks refers to the boundary of UMZs (
$u_k$
) (Chen et al. Reference Chen, Chung and Wan2020). Window sizes of axial length
$0.2R$
were investigated along the flow domain similar to Chen et al. (Reference Chen, Chung and Wan2020) in order to gain comparable initial steady-state statistics (both the current work and Chen et al. (Reference Chen, Chung and Wan2020) have the same initial steady friction Reynolds number). A three-dimensional streamwise flow data utilisation along the flow domain of
$0.2R$
windows for a given snapshot is indicated in the example flow visualisation in figure 3(b). It should be mentioned that, as UMZ data are extracted based on
$0.2R$
flow windows along each streamwise–wall-parallel plane for
$0\leqslant \theta \leqslant 2\pi$
, at each window, ISLs were treated as two-dimensional flow structures. A uniform bin size for the PDFs was defined with each bin representing 1 %
$U_z/U_{\textit{CL}}$
, where
$U_{\textit{CL}}$
is the global mean centre-line velocity for each instantaneous flow field, and it was used to normalise all
$U_z$
values so that
$U_z/U_{\textit{CL}}$
$\epsilon$
[0,1.1).
Probability density function of the number of UMZs
$N_{\textit{UMZ}}$
detected. Solid line (
) depicts PDFs at
$t^{+0}=-2.8138$
(negative sign for initial steady state).
A peak isolation scheme was then defined similarly to Chen et al. (Reference Chen, Chung and Wan2020) to distinguish UMZs from local peaks of the PDFs with three distinct constraints,
$F_{d}$
,
$F_{h}$
and
$F_{p}$
. The constraint
$F_{d}$
was defined as the minimum gap between adjacent local PDF peaks in scales of bins, the constraint
$F_{h}$
was defined as the minimum PDF value of local PDF peaks to be considered a UMZ, and the constraint
$F_{p}$
was defined as the minimum percentage of prominence compared with the average of 10 % neighbouring PDF values of five bins on each side of the local peak. Since the PDF-based UMZ identification method is highly susceptible to the peak isolation parameters imposed, the threshold values for the current analysis were set similarly to Chen et al. (Reference Chen, Chung and Wan2020) at
$F_{d}$
= 0.03
$U_z/U_{\textit{CL}}$
,
$F_{h}$
= 0.5 and
$F_{p}$
= 25 % in order to generate comparable statistics. In Appendix B
$\S$
4.4, the sensitivity of the threshold parameters of the PDF-based method was then tested, and the UMZ identification method was repeated with the core of the flow removed to supplement further the definitive nature of the identification scheme in identifying UMZs.
Adopting the windowing approach as mentioned previously resulted in a normalised distribution with its peak at
$N_{\textit{UMZ}}\approx 5$
(figure 2). This result was in good agreement with the distribution found in Chen et al. (Reference Chen, Chung and Wan2020), which further solidified the numerical foundation of the current study with UMZs identified in the initial steady turbulent base flow data utilised for the current study. An example instantaneous PDF result with four UMZs detected from the filtering criteria utilised for the current study alongside the corresponding contour is provided in figures 3(b) and 4. In figure 4, the unwrapped cross-section plot for the first quarter (
$0 \leqslant \theta \leqslant \pi /2$
) of the azimuthal span is plotted. It is worth pointing out that, in figure 4, similar to results by Chen et al. (Reference Chen, Chung and Wan2020), low-speed UMZs near the wall contoured by green and blue regions are not characterised as UMZs based on the implemented UMZ detection scheme. The reasons behind this are that, at this friction Reynolds number, the flow separation is not distinct enough for near-wall low-speed UMZ boundaries to separate them vividly and that the PDF-based results are dependent on the isolation parameters imposed. Furthermore, the characteristic properties of UMZs are depicted in figure 4, where
$t_1, t_2, t_3$
and
$t_4$
correspond to the thicknesses of each UMZ captured,
$r_1, r_2, r_3$
and
$r_4$
correspond to the radial distance of the lowest UMZ interface of each UMZ from the core of the pipe and finally,
$R_1, R_2, R_3$
and
$R_4$
correspond to the rank of each UMZ in the snapshot. As mentioned by Chen et al. (Reference Chen, Chung and Wan2020), due to the hierarchical arrangement of UMZs in pipes,
$R_1$
UMZs are always closest to the pipe centre, while the highest rank (in this case
$R_4$
) UMZs are always closer to the wall.
(a) An example PDF of
$U_z$
for a window size of
$0.2R$
. The red dash lines (
) depict the boundaries with 
depicting zone modal velocities of UMZs. (b) The corresponding three-dimensional streamwise volumetric slice in the transient turbulent flow used to generate the PDF. The black solid lines (
) depict the boundaries of UMZs. (a) Example PDF for a window of
$U_z$
. (b) Three-dimensional volumetric flow data slice.
The corresponding
$R-\theta$
contour of the transient turbulent flow from figure 3(a). The solid black lines (
) depict the boundary contours of UMZs.
In Appendix A
$\S$
4.4, the standard deviation inside individual UMZs, normalised by
$U_{\textit{CL,0}}$
, based on the extracted streamwise velocity data is presented in order to validate the accuracy of the UMZ extraction algorithm implemented for the current work.
3. Results and discussion
3.1. The transient nature of the probability density function of
$N_{\textit{UMZ}}$
The behaviour of the PDFs of
$N_{\textit{UMZ}}$
captured during the four transitional stages of the accelerating flow are investigated in this section. Figure 5 provides the results obtained alongside the time scales over which the data were computed. In figure 5, the arrows indicate the major trends seen with progressing time.
The PDF of
$N_{\textit{UMZ}}$
detected during the (a) inertial, (b) pre-transition, (c) transition and (d) core-relaxation stages of the transient turbulent flow. Solid lines () depict PDFs during a stage and dashed lines (
) depict the PDFs of the initial and the final steady states. Accompanied by the dash lines, the blue markers (
) correspond to initial state statistics while black markers (
$\circ$
) correspond to the final steady-state statistics. For each transient stage blue, red, yellow and purple depict statistics during the stage while green depicts statistics of the successive stage the flow transitions to following the directions of the inward and outward arrows. Arrows point towards trends observed with progressing time.
Figure 5(a) illustrates the variation during the inertial flow stage. The peak of the distribution remains unchanged, resulting in the average number of zones seen being five. Flow regions where
$N_{\textit{UMZ}}\gt 4$
are seen less, while flow regions where
$N_{\textit{UMZ}}\leqslant 4$
show an abundance of detection. Despite the peak remaining at
$N_{\textit{UMZ}}=5$
, the PDF value of the peak increases, indicating a slight drop of UMZs with higher
$N_{\textit{UMZ}}$
. This is consistent with the laminarescent trend observed in accelerating flows towards the onset of the pre-transition stage, where complex flow structures are somehow annihilated near the wall as the flow tends to relaminarise itself (Narasimha & Sreenivasan Reference Narasimha and Sreenivasan1973; He & Seddighi Reference He and Seddighi2013). Overall, during the inertial period, regions with higher
$N_{\textit{UMZ}}$
drop, while regions with a lower
$N_{\textit{UMZ}}$
become more common.
Figure 5(b) illustrates the variation during the pre-transition flow stage towards the start of the transition stage. During the early stages of the flow, the PDF peak shifts to the left, resulting in the average number of zones dropping from five to four. This shift is accompanied by the narrowing of the PDF widths, suggesting more UMZs with lesser
$N_{\textit{UMZ}}$
are seen during the pre-transition stage. Later on, the peak recovers back to five. It is worth pointing out that, for the majority of this flow stage, flow regions with
$N_{\textit{UMZ}}\geqslant 6$
are scarce, further supplementing the initial observation of densely packed regions containing UMZs with
$N_{\textit{UMZ}}\geqslant 5$
dropping during the inertial stage.
The result of the average
$N_{\textit{UMZ}}$
dropping to four and then recovering to five during the pre-transition in the current study could be explained as a continued effect of the laminarescent trend observed during the inertial stage of the turbulent flow. As the reduction in the level of turbulence causes the average
$N_{\textit{UMZ}}$
to drop towards the later stages of the flow, the levels of turbulence improve with the radial propagation of new turbulence from the wall (Guerrero et al. Reference Guerrero, Lambert and Chin2021), causing the peak to recover.
Figure 5(c) illustrates the variation during the transition flow stage. The PDF skews to the right, while the average number of zones remains at five. Furthermore, towards the later transition stage, the peak drops at five, and the PDF widens. This suggests that the UMZs are gradually being established during this stage, as flow regions containing highly populated UMZs with
$N_{\textit{UMZ}}\gt 5$
are more common as the flow transitions to its final stage.
The continuation of propagation of the new turbulence spots creates additional complex flow patterns, as seen by He & Seddighi (Reference He and Seddighi2013,Reference He and Seddighi2015) and Guerrero et al. (Reference Guerrero, Lambert and Chin2021), which enhance the turbulence level of the transient flow. This change in turbulence levels is argued to be a significant rationale for the average
$N_{\textit{UMZ}}$
increases and the widening of the PDFs.
Figure 5(d) illustrates the variation during the recorded time scales of the core-relaxation stage. Following the trend from the previous stage, the PDF distribution starts to settle with an average
$N_{\textit{UMZ}}=5$
. These results suggest that the average
$N_{\textit{UMZ}}$
at the end of the core-relaxation period is similar to the initial base flow despite rapidly accelerating the flow. It can be observed that, although the peak returns to five, the trend does not overlap with the initial steady-state data, as at
$t^{+0}\approx 820$
, the flow has not yet fully achieved a state of complete steadiness. This result resonates with the findings of the quiescent core of the pipe by Guerrero et al. (Reference Guerrero, Lambert and Chin2021), where it was concluded that, for a rapidly accelerated turbulent pipe flow, compared with the first three flow stages, the time scales required to reconstitute the core of the flow are noticeably larger during the core-relaxation stage.
3.1.1. Temporal evolution of ISLs
In order to further supplement the fluctuations of
$N_{\textit{UMZ}}$
seen during the four transient stages of the flow, the temporal behaviour of the average number of ISLs (
$N_{\!\textit{ISL}}$
) seen at each temporal instance was briefly investigated. The ISLs are detected based on the methodology followed by De Silva et al. (Reference De Silva, Philip, Hutchins and Marusic2017) and Chen et al. (Reference Chen, Chung and Wan2021), where the peaks and valleys of the instantaneous streamwise velocity gradient profiles generated at each streamwise grid point/slice of the flow domain are identified as regions of intensified local shear. It should be noted that the instantaneous streamwise velocity gradient profiles were normalised with respect to initial steady-state viscous units (
${\partial {U^{+0}}}/{\partial y^{+0}}$
), prior to identifying ISLs and that at each temporal instance, the averaged
$N_{\!\textit{ISL}}$
was rounded off to the nearest whole number to extract a realistic, yet averaged, result.
Once the average number of ISLs (
$N_{\!\textit{ISL}}$
) at each temporal instance was investigated on a temporal scale, as shown in figure 6, it was seen that the drops and recoveries in
$N_{\textit{UMZ}}$
are reflected in the fluctuations in
$N_{\!\textit{ISL}}$
, especially during the pre-transition stage (
$10\leqslant t^{+0} \leqslant 100$
). It is observed that
$N_{\!\textit{ISL}}$
remains relatively unchanged during the inertial stage, reflecting the characteristic frozen turbulence behaviour in rapidly accelerating turbulent flows (Maruyama et al. Reference Maruyama, Kuribayashi and Mizushina1976; He & Seddighi Reference He and Seddighi2013; Guerrero et al. Reference Guerrero, Lambert and Chin2021), and attains its lowest value as it progresses towards the pre-transition stage. Later on, towards the onset of the transition stage, it begins to increase until it recovers as the flow enters the transition stage. Afterwards,
$N_{\!\textit{ISL}}$
continues to increase and finally settles at a relatively higher
$N_{\!\textit{ISL}}$
as the flow settles at
$Re_{\tau } = 670$
during the core-relaxation stage. Apart from the core of the flow requiring extended time scales to recover, the relatively high number of ISLs seen towards the end of the flow excursion could explain why more flow regions of
$N_{\textit{UMZ}}=5$
are seen in the final steady-state distribution of
$N_{\textit{UMZ}}$
compared with the initial steady-state distribution of
$N_{\textit{UMZ}}$
.
Finally, it is worth noting that not all ISLs correspond to UMZ boundaries but rather to regions of high local shear, hence why directly relating the temporal evolution of
$N_{\!\textit{ISL}}$
to the trend seen in
$N_{\textit{UMZ}}$
would not be accurate. However, this result can be used to supplement our conclusions by providing insight into how strong layers of internal shear behave in the current unsteady flow scenario and that UMZs undergo a dynamic reconfiguration during the flow excursion.
Temporal evolution of the average number of ISLs (
$\langle N_{\!\textit{ISL}} \rangle$
), rounded off to the nearest whole number.
3.2. Evolution of UMZs during transient stages
The aim of the current section is to provide insight into the behaviour of UMZs from a structural point of view. To conduct a systematic study, all instantaneous UMZs identified are categorised into groups based on two distinct criteria (tables 3 and 4). Similar to Chen et al. (Reference Chen, Chung and Wan2020), the first criterion was the magnitude of the zone modal velocity of the UMZs (
$u_m$
). Here,
$u_m$
corresponds to the characteristic velocity of UMZs (the peaks of the PDFs). The UMZs classified under this criterion are denoted as
$M_i$
, where the subscript i = 1 always corresponds to the fastest group, characterised by the highest zone modal velocities. In steady turbulent pipe flow, this group contains the UMZs closest to the pipe centre. Furthermore, in steady turbulent flows, as the subscript i of a group increases, the UMZs represented by that group exhibit slower modal velocities. They are located progressively closer to the pipe wall (Chen et al. Reference Chen, Chung and Wan2020).
Uniform momentum zone groups grouped based on the magnitude of
$u_m$
.
Uniform momentum zone groups grouped based on the rank of
$u_m$
.
The second classification criterion was based on Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) and Chen et al. (Reference Chen, Chung and Wan2020), where the identified instantaneous UMZs were grouped based on the rank of the magnitude of the zone modal velocity
$u_m$
. The UMZs grouped under this criterion are denoted by
$R_i$
, where the subscript i = 1 always represents UMZs nearest to the pipe centre in steady turbulent pipe flow. In contrast, the highest rank will represent UMZs closest to the pipe wall.
Conditionally averaged characteristics of UMZ based on rank. (b) Lower bound UMZ boundary
$y_k$
and (c) thickness between the upper and lower bound boundaries of UMZs
$t_k$
at
$t^{+0}=-2.8138$
. Readers are referred to table 4 for symbols used.
The conditionally averaged characteristic statistical identities of UMZs are investigated first, based on UMZ ranking groups
$R_i$
as depicted in figure 7. The characteristic statistical identities investigated are the zone modal velocities of UMZs
$u_m$
(figure 7
a), the wall-normal locations of UMZ boundaries
$y_k$
(figure 7
b) and the thickness of UMZs
$t_k$
(figure 7
c). For the current study, the parameter definitions imposed are similar to the definitions considered by Chen et al. (Reference Chen, Chung and Wan2020), where
$t_k$
is defined as the difference between the lower boundary (the boundary that is closest to the wall) and the upper boundary (the boundary that is closest to the core) of a given UMZ, and
$y_k$
is defined as the wall-normal location to the lowest boundary of a given UMZ. The median quartile data were investigated by extracting the centred 50 % of the PDF data of each statistical entity (
$u_m$
,
$t_k$
and
$y_k$
).
Figure 7 also provides the span of the centred conditionally averaged statistics extracted during the initial steady state and the mean of each centred conditionally averaged statistic (denoted by
$\bigtriangledown$
). These results are in agreement with Chen et al. (Reference Chen, Chung and Wan2020), with the upper and lower limits of certain UMZ identities overlapping or extending within or beyond each other in instantaneous
$u_m$
,
$y_k$
and
$t_k$
results, as expected. A similar behaviour was observed in the statistics by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000), where they also considered the centred 50 % of their data.
Furthermore, following the steady turbulent UMZ results from Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) and Chen et al. (Reference Chen, Chung and Wan2020), it is mentioned that, as a newer zone is present, the existing UMZs speed up and re-locate themselves onto higher wall-normal locations and significantly reduce their thicknesses. Results from figure 7 of the current study computed prior to the flow excursion are in agreement with this phenomenon. Following the acceptable results obtained, these statistics were further extended to gain proper insight into the temporal nature of the UMZ characteristics during the transient stages of the flow. In the subsequent portion of this section, the mean of each rank of
$R_i$
groups from
$N_{\textit{UMZ}} = 2$
to
$5$
were investigated on a temporal scale.
Temporal variation of the conditionally averaged statistics of
$u_m$
based on zone modal velocity rank groups
$R_i$
during (a) inertial, (b) pre-transition, (c) transition and (d) core-relaxation phases.
Temporal variation of the conditionally averaged
$y_k$
statistics of zone modal velocity rank groups
$R_i$
during (a) inertial, (b) pre-transition, (c) transition and (d) core-relaxation phases.
Temporal variation of the conditionally averaged
$t_k$
statistics of zone modal velocity rank groups
$R_i$
during (a) inertial, (b) pre-transition, (c) transition and (d) core-relaxation phases.
Figures 8(a), 8(b), 8(c) and 8(d) depict the overall temporal evolution of the mean conditionally averaged zone modal velocity
$u_m$
statistics from each
$R_i$
group as time scales progress. Here,
$u_m$
is normalised with respect to the initial base flow centre-line velocity
$U_{CL_{0}}$
in order to provide quantitative evidence of how UMZs speed up alongside the flow acceleration on the temporal scale. Statistics from figure 8 reveal that, indeed, the flow acceleration causes UMZs to ramp up in their convection velocities up to the early pre-transition stage. From these results, it is observed that, similar to UMZs in steady turbulent flows, once a zonal group disappeared, the pre-existing zonal groups slowed down (
$t^{+0}\approx 55$
), despite the flow acceleration causing all the UMZs to accelerate. It is worth pointing out that, at
$t^{+0}\approx 55$
, a sudden drop in the mean conditionally averaged
$u_m$
statistics was detected. This coincides with the
$N_{\textit{UMZ}}$
peak shifting from four to three for the first time, as seen in figure 5(b) during the pre-transition stage.
Figures 9(a), 9(b), 9(c) and 9(d) depict the overall temporal evolution of the mean conditionally averaged
$y_k$
statistics from each
$R_i$
group with time and figures 10(a), 10(b), 10(c) and 10(d) depict the overall temporal evolution of mean conditionally averaged
$t_k$
statistics for each
$R_i$
group with time. Similar to
$y_k$
statistics from steady state, once a zonal group disappears (
$t^{+0}\approx 55$
), the pre-existing zonal groups are observed much closer to the wall in their wall-normal location. They are seen to expand their zonal thickness to accommodate the newly available domain space.
Furthermore, from figure 9(a), it was noticed that, at
$t^{+0}\approx 10$
during the inertial stage, UMZs closest to the wall are located slightly away from the wall. The annihilation of near-wall flow structures during this stage is speculated to be the leading cause of this upward shift. However, as the flow enters the pre-transition stage, as shown in figure 9(b), UMZs start being present at wall-normal locations closer to the wall until
$t^{+0}\approx 55$
and they began being present at wall-normal locations towards the core again until
$t^{+0}\approx 100$
. Despite the ongoing relaminarisation of the flow near the wall, the reduction of
$N_{\textit{UMZ}}$
causes the existing zones to move closer to the wall. After UMZs are present at wall-normal locations closer to the wall, they become subject to the lowered near-wall turbulence and cause UMZs to be seen at higher wall-normal locations. These time instances again coincide with the statistics from figure 5(b), where the peak shifts from four to three. After that, as depicted in figure 9(c), all the UMZs are seen to be at wall-normal locations close to the wall until the end of the transition stage. As the transition stage improves, the lowered levels of turbulence near the wall result in more zones being visible near the wall. During the core-relaxation stage, as plotted in figure 9(d), the combined effect of increasing
$N_{\textit{UMZ}}$
and the improving levels of turbulence causes UMZs to be seen settling at their initial steady-state wall-normal locations away from the wall.
Unlike mean
$u_m$
and
$y_k$
statistics, the temporal evolution of mean conditionally averaged
$t_k$
is seen to exhibit highly unique behaviour. During the inertial stage, as depicted in figure 10(a), except for the zonal thickness of ranking group
$R_6$
initially dropping to
$t_k\approx 0.1$
prior to spiking to
$t_k\approx 0.2$
, the rest of the zonal thicknesses remain relatively unchanged. This spike in zonal thickness occurs as the nearest wall zones begin to be seen at locations towards the core, as depicted in figure 9(a) at
$t^{+0}\approx 10$
. Towards the end of the pre-transition stage, an overall increment of the thicknesses of zones is observed. This increase is due to zones being seen more often near the core. Towards the end of the transition stage, it is observed that, despite the zonal thicknesses of near-wall zones beginning to drop, the zonal thickness of the nearest core UMZs continues to increase. The authors speculate that this increase in thickness is closely related to the core region of the flow becoming wider as the Reynolds number increases (He & Seddighi Reference He and Seddighi2013,Reference He and Seddighi2015; Guerrero et al. Reference Guerrero, Lambert and Chin2021), alongside zones being abundantly visible closer to the wall, allowing the core to exhibit signs of expansion. The most near-core UMZs continue to grow in thickness towards the later core-relaxation stage. Overall, during the flow excursion, the mean conditionally averaged thickness of
$R_1$
UMZs remained within
$0.3\leqslant {t_k}\leqslant 0.6$
, and
$0.05\leqslant {t_k}\leqslant 0.3$
for all the remaining UMZs. Based on this quantitative finding, a key observation is that, despite the rapidly accelerated flow,
$R_1$
UMZs remain the thickest UMZs, similar to those in steady-state conditions.
Overall, it is observed that, during the transient stages of turbulent flow, the conditionally averaged UMZ characteristics tend to exhibit temporal coherence. Tracking the mean conditionally averaged UMZ statistics of all ranking groups over time has provided an understanding that, as the flow accelerates, so do UMZs. As
$N_{\textit{UMZ}}$
drops, existing flow structures tend to be located near the wall, and the annihilation of flow structures causes the existing UMZs to be located away from the wall. As UMZs are shifting their wall-normal locations, the thicknesses of the zones fluctuate. These results are complemented by the results extracted corresponding to the shifts of the PDF peak of
$N_{\textit{UMZ}}$
(figure 5).
3.2.1. Three-dimensional evolution of UMZs during transient stages
So far within this section, the structural evolution of UMZs has been discussed quantitatively based on conditionally averaged statistics extracted from UMZs (
$u_m$
,
$y_k$
and
$t_k$
). The aim of this subsection is to qualitatively investigate the structural evolution of UMZs from a three-dimensional perspective, where the three-dimensional structure of the UMZ envelope is visualised at different temporal instances during the four unsteady flow stages. The fastest UMZ, also known as the quiescent core (defined here as UMZ 1), and the second fastest UMZ (defined here as UMZ 2) are investigated three-dimensionally. The reason for isolating and investigating UMZ 1 and UMZ 2 was due to them being the two largest UMZs seen within the pipe and the abundance of these UMZs during the four transitional stages of the flow, where it was found that a significant reduction of UMZs occurs, on average, as the flow recovers from the flow perturbation (refer to figure 5). As these two UMZs are the largest zones seen within the pipe, three-dimensional visualisations of these structures are expected to provide a better understanding of the structural evolution of UMZs. Here, UMZ 2 corresponds to
$R_2$
ranked UMZs, and UMZ 1 corresponds to
$R_1$
ranked UMZs, based on the
$R_i$
ranking group criteria (refer to table 4).
Three-dimensional visualisations of UMZ 1 and UMZ 2: (a) and (b) during the inertial phase (
$0\lt t^{+0}\lesssim 10$
), (c) and (d) during the pre-transition phase (
$10\lt t^{+0}\lesssim 100$
), (e) and ( f) during the transition phase (
$100\lt t^{+0}\lesssim 250$
) and (g) and (h) during the core-relaxation phase (
$t^{+0}\gtrsim 250$
). Panels show (a) UMZ 1 at
$t^{+0}\approx 5.2$
, (b) UMZ 2 at
$t^{+0}\approx 5.2$
, (c) UMZ 1 at
$t^{+0}\approx 53.74$
, (d) UMZ 2 at
$t^{+0}\approx 53.74$
, (e) UMZ 1 at
$t^{+0}\approx 161.76$
, (f) UMZ 2 at
$t^{+0}\approx 161.76$
, (g) UMZ 1 at
$t^{+0}\approx 688.89$
and (h) UMZ 2 at
$t^{+0}\approx 688.89$
.
Figure 11 provides the three-dimensional visualisations of UMZ 1 and UMZ 2, captured at four temporal instances corresponding to the four transitional stages of the rapidly accelerating turbulent pipe flow. Here, the iso-surfaces corresponding to the boundaries (envelope) of each UMZ (
$u_k$
) are depicted three-dimensionally. Finally, all iso-surfaces are coloured to visualise their radial extent from the centre up to the pipe wall. To provide clearer visualisations, up to ten consecutive investigation windows (
$0 \leqslant z/D \leqslant 1$
) along the streamwise direction are depicted. As a result of this approach, the visualisations tend to look as if smaller regions of the flow (or chunks of the flow) are stitched together.
During the inertial phase, as shown in figures 11(a) and 11(b) at
$t^{+0}\approx 5.2$
, UMZ 1 and UMZ 2 are seen to remain unchanged with their spatial extent along all directions remaining relatively uniform. This behaviour remained until
$t^{+0}\approx 53.74$
of the pre-transition stage. This result agrees with the current findings in figures 9 and 10, of the conditionally averaged
$y_k$
and
$t_k$
results.
Towards the later stages of the pre-transition phase of the flow, as depicted by figure 11(c) and 11(d) at
$t^{+0}\approx 53.74$
, it is seen that some flow regions of the UMZ 1 and UMZ 2 expand towards the wall, in response to the perturbation turbulent boundary layer growth from the wall towards the core of the pipe and the emerging turbulent spots within the flow (He & Seddighi Reference He and Seddighi2015; Mathur et al. Reference Mathur, Gorji, He, Seddighi, Vardy, O’Donoghue and Pokrajac2018; Guerrero et al. Reference Guerrero, Lambert and Chin2021). As the perturbation boundary layer grows, the streamwise Reynolds shear stresses increase near the wall, and the spatial extent of these zones expands to preserve the angular momentum of the large-scale flow structures. Compared with UMZ 1, more regions of UMZ 2 appear to exhibit expansions, despite both UMZ 1 and UMZ 2 showing delayed responses to the flow excursion. This is mainly due to UMZ 2 wrapping UMZ 1, and as a result, UMZ 2 responds earlier than UMZ 1. Furthermore, the drop in the number of UMZs and ISLs observed during this phase of the flow (refer to figures 5 and 6) is attributed to the expansion of regions of the UMZ 1 and UMZ 2, which begin to push near-wall, slower UMZs located in the vicinity of the wall, closer towards the wall, causing them not to be seen abundantly. Again, these results are in agreement with the current findings in figures 9 and 10, of the conditionally averaged
$y_k$
and
$t_k$
results.
During the transition phase, as shown in figures 11(e) and 11(f) at
$t^{+0}\approx 161.76$
, more regions of both UMZ 1 and UMZ 2 are seen to expand significantly towards the wall, with UMZ 2 expanding almost to the wall (the limits of the flow domain) in a few instances. This is due to the turbulence propagation from the wall to the core, characteristic of this stage (He & Seddighi Reference He and Seddighi2015; Mathur et al. Reference Mathur, Gorji, He, Seddighi, Vardy, O’Donoghue and Pokrajac2018; Guerrero et al. Reference Guerrero, Lambert and Chin2021). Furthermore, it is likely that the perturbation boundary layer superimposition in the turbulent base flow makes the core thicker, causing UMZ 2 to become significantly thinner, despite both UMZ 1 and UMZ 2 spatially expanding during this phase. This result is in agreement with the current findings in figures 9 and 10, where it is observed that the conditionally averaged thicknesses of
$R_1$
UMZs increase significantly. In contrast, the conditionally averaged thicknesses of all other
$R_i$
UMZs drop.
Towards the end of the core-relaxation phase, it is seen that both UMZ 1 and UMZ 2 show signs of relaxation, as depicted in figures 11(g) and 11(h), captured at
$t^{+0}\approx 688.89$
. This is due to the propagation of new turbulence from the wall towards the core region during this phase (Guerrero et al. Reference Guerrero, Lambert and Chin2021). They gradually attain comparable spatial extents to their visualisations seen during the inertial stage. Again, these visualisations are in agreement with the findings in figures 9 and 10.
Based on instantaneous three-dimensional visualisations of UMZ 1 and UMZ 2, it is clear that the UMZs evolve structurally during the four unsteady flow stages, and the qualitative results of the current investigation reflect the quantitative results of the UMZ 1 and UMZ 2 visualisations.
3.3. Transient turbulence statistics within and outer bounds of UMZs
The previous sections have provided insight into the behaviour of UMZs from a structural point of view and shown that UMZs at different wall-normal locations consist of different turbulent characteristics. In order to provide a better understanding from the perspective of UMZ kinematics, the conditionally averaged low-order turbulence statistics (mean velocity profiles) and high-order turbulence statistics (turbulence intensity profiles) based on the
$M_i$
grouping criteria of UMZs (table 3) are investigated in this section. Here, the fluctuations in turbulence statistics are calculated relative to the zone’s mean velocity. To conduct a robust analysis, only the five fastest
$M_i$
groups are analysed (
$M_{1}$
–
$M_{5}$
).
An illustration of flow regions inside and outside UMZs. The red arrows depict the flow region considered to be inside UMZ 1, while the yellow arrows depict the flow region considered to be outside UMZ 1. The green arrows depict the flow region considered to be inside UMZ 2, while the purple arrows depict the flow region considered to be outside UMZ 2. Finally, the blue arrows depict the flow region considered to be inside UMZ 3. Here, UMZ 3 is the third fastest UMZ detected within the pipe.
Statistics with a hat superscript
$(\,\,\widehat {\ }\,\,)$
denote turbulent statistics inside UMZs, while statistics with a tilde superscript
$(\,\,\widetilde {\ }\,\,)$
denote turbulent statistics outside UMZs. Similar to Kwon et al. (Reference Kwon, Philip, De Silva, Hutchins and Monty2014) and Chen et al. (Reference Chen, Chung and Wan2020), flow regions inside and the outside of a UMZ within the pipe are defined as the flow region from the lower bounding interface up to the centre line of the pipe and the flow region from the lower bounding interface down to the wall of the pipe, respectively. Figure 12 provides an example illustration of flow regions considered to be inside and outside UMZs for an instance of three zones (
$ \mathrm{UMZ}_1$
,
$ \mathrm{UMZ}_2$
and
$ \mathrm{UMZ}_3$
). Here, the flow regions denoted by the red arrow correspond to the flow region inside
$ \mathrm{UMZ}_1$
and the yellow arrow to the flow region outside
$ \mathrm{UMZ}_1$
. The flow regions denoted by the green arrow correspond to the flow region inside
$ \mathrm{UMZ}_2$
and the purple arrow to the flow region outside
$ \mathrm{UMZ}_2$
. Finally, the flow region denoted by the blue arrow corresponds to the flow region inside
$ \mathrm{UMZ}_3$
. The global mean turbulence statistics are denoted by a solid line superscript (
$\,\,\bar {\ }\,\,$
). Table 5 provides a summary of the notations used alongside the Reynolds decomposing of both inside and outside UMZs.
Summary of notations used in figures 13, 14 and 15.
(a) The conditionally averaged mean velocity profiles inside and outside UMZs and (b) the turbulence intensity distribution inside and outside UMZs at
$t^{+0}=-2.8138$
. Lines (
) and (
) depict profiles inside and outside of UMZs, respectively. Solid line (
) depicts the global mean profiles at a given time instance. The cross-over point for
$M_5$
is depicted with a vertical red dash line in (b).
Figure 13(a) depicts the initial steady-state statistics (i.e.
$t^{+0} \lt 0$
) of the global mean velocity profile alongside the conditionally averaged mean velocity profiles inside and outside the respective zonal
$M_i$
groups. These profiles are normalised with respect to the initial steady-state centre-line velocity (
$U_{\textit{CL,0}}$
). The initial steady-state statistics from the current study are in agreement with the steady turbulent statistics from Chen et al. (Reference Chen, Chung and Wan2020). They show that the mean velocity profiles inside UMZs are always higher than those outside UMZs. The mean velocity profile outside UMZs overlaps the global mean velocity profile in the near-wall region and gradually deviates from it when approaching the core. These characteristics are evident in the kinematic statistics presented in this work. It is detected that the mean velocity profiles inside groups
$M_1$
,
$M_2$
,
$M_3$
,
$M_4$
and
$M_5$
extended up to normalised wall-normal locations of
$y/R\approx 0.212, 0.135, 0.062, 0.027$
and
$0.014$
respectively, while the mean velocity profiles outside groups
$M_1$
,
$M_2$
,
$M_3$
,
$M_4$
and
$M_5$
extended up to normalised wall-normal locations of
$y/R\approx 0.775, 0.759, 0.791, 0.628$
and
$0.518$
respectively. Profiles shown in figure 13(a) provide evidence of UMZs being seen in close vicinity of the wall, as reported by Chen et al. (Reference Chen, Chung and Wan2020), where the mean velocity profiles representing inner regions of UMZs (
) extend from the centre line (
$y/R\approx 1$
) to the close vicinity of the wall of the pipe (
$y/R\approx 0.01$
).
Figure 13(b) depicts the steady-state statistics of the global turbulence intensity profile alongside the conditionally averaged turbulence intensity profiles inside and outside the respective zonal
$M_i$
groups. These profiles are inner normalised with respect to the friction velocity of the initial steady turbulent flow (
$u_{\tau ,0}$
). It is observed that the turbulence intensities inside groups
$M_1$
,
$M_2$
,
$M_3$
,
$M_4$
and
$M_5$
are always less than the respective turbulence intensities outside the groups. This result from the current study agrees with the results by Kwon et al. (Reference Kwon, Philip, De Silva, Hutchins and Monty2014) and Chen et al. (Reference Chen, Chung and Wan2020), where they reported that the turbulence intensity inside a UMZ is always less than the turbulence intensity outside a UMZ. Corresponding to figure 13(a), the turbulence intensity profiles of figure 13(b) exist only within the normalised wall-normal locations where the inner and outer mean velocity profiles extended. Furthermore, it is observed that the cross-over point, which is defined as the point at which the turbulence intensity inside the zone exceeds the turbulence intensity outside the zone, tends to move towards the wall when considering groups
$M_1$
–
$M_5$
. The cross-over points for groups
$M_1$
,
$M_2$
,
$M_3$
,
$M_4$
and
$M_5$
from the current work are detected to be
${y}/R\approx 0.743, 0.736, 0.371, 0.265$
and
$0.127$
, respectively. The reason for these points to move closer to the wall when considering faster to slower
$M_i$
groups was that thinner near-wall UMZs contained intense levels of turbulence compared with the turbulence intensity inside a thicker UMZ further away from the wall (Chen et al. Reference Chen, Chung and Wan2020).
As mentioned earlier, the initial steady-state kinematic statistics from the current study are in agreement with the steady turbulent pipe statistics from Chen et al. (Reference Chen, Chung and Wan2020). Similarities between the statistics from the current study and statistics from Chen et al. (Reference Chen, Chung and Wan2020) are further complimented as the mean velocity profile inside group
$M_5$
extended near the wall up to
$y/R\approx 0.01$
, similar to Chen et al. (Reference Chen, Chung and Wan2020), and the cross-over point for group
$M_2$
from the current study is detected to be similar to the value from Kwon et al. (Reference Kwon, Philip, De Silva, Hutchins and Monty2014) and Chen et al. (Reference Chen, Chung and Wan2020) (
$\approx 0.74$
). Despite the similarities, the extents of the mean profiles outside the UMZ groups are detected to be shorter than the reported profiles by Chen et al. (Reference Chen, Chung and Wan2020), which the current authors believe to have been impacted by the post-processing techniques utilised, as mentioned in
$\S$
2.2.
Figures 14 and 15 depict the temporal evolution of these statistics and, for added clarity, different marker styles are used on top of the line styles defined in table 5. The marker styles that correspond to each time instance are mentioned in the caption of each figure.
3.3.1. Zonal mean velocity
Figure 14(a) provides the overall temporal evolution of the global mean velocity profile alongside conditionally averaged mean velocity profiles both inside and outside UMZs based on
$M_i$
grouping criteria during the inertial unsteady flow stage. As expected, a significant response to the flow excursion is observed towards the end of the inertial stage. This result agrees with the frozen turbulence-like behaviour characteristic of rapidly accelerating wall-bounded flows, as reported by Maruyama et al. (Reference Maruyama, Kuribayashi and Mizushina1976), He & Seddighi (Reference He and Seddighi2013) and Guerrero et al. (Reference Guerrero, Lambert and Chin2021). The global mean velocity profiles up to the centre region (
$y/R\approx 1$
) begin to shift upwards. The conditionally averaged profiles inside and outside UMZs shift upwards alongside the global mean velocity profiles. This shift is caused by the increase in bulk velocity (or, in other words, the mean streamwise velocity gradient) near the wall and the increase of wall shear stress within the near-wall region as the flow begins to accelerate (Guerrero et al. Reference Guerrero, Lambert and Chin2021). During this stage, significant transient behaviour is seen in slower UMZ groups closest to the wall (
$M_4$
and
$M_5$
) as the mean velocity profiles of these groups display considerable deflections, unlike the overlapping profiles of rest of the
$M_i$
groups plotted. This result suggests that UMZs nearest the wall are the first to respond to the flow acceleration. In contrast, faster UMZ groups
$M_1$
,
$M_2$
and
$M_3$
primarily located in proximity to the core remain relatively incoherent during this stage.
Temporal evolution of the conditionally averaged mean velocity profiles during (a) inertial, (b) pre-transition, (c) transition and (d) core-relaxation phases. Solid lines () depict the global mean velocity profile while (
) depict zonal mean inside zones and (
) depict zonal mean outside zones. For added clarity, different marker styles as shown in each panel legend are overlayed over the line styles for each temporal instance plotted.
Temporal evolution of the conditionally averaged turbulence intensity during (a) inertial, (b) pre-transition, (c) transition and (d) core-relaxation phases. Solid lines (
) depict the global mean velocity profile while (
) depict zonal mean inside zones and (
) depict zonal mean outside zones. For added clarity, different marker styles as shown in each panel legend are overlayed over the line styles for each temporal instance plotted.
Figure 14(b) provides the overall temporal evolution of the global mean velocity profile alongside conditionally averaged mean velocity profiles both inside and outside UMZs based on
$M_i$
groups during the pre-transition stage of the flow. The global mean velocity profile drops near the wall during this stage. Accordingly, group profiles
$M_4$
demonstrated significant temporal coherence with the fluctuations observed in the mean velocity profiles. However, the global mean profiles and the conditionally averaged mean velocity profiles inside UMZs for groups
$M_1$
,
$M_2$
and
$M_3$
remain unhindered as they are primarily located towards the outer regions of the pipe. The global mean velocity profile drop is seen up to the flow region of
$y/R\approx 0.1$
from the wall. The reduction of the mean streamwise velocity gradient causes this downward shift of the global mean velocity profile as a result of the growth of the perturbation boundary layer from the wall towards the core of the pipe (He & Seddighi Reference He and Seddighi2013; Mathur et al. Reference Mathur, Gorji, He, Seddighi, Vardy, O’Donoghue and Pokrajac2018). Furthermore, this drop in the velocity gradient is complemented by the drop of viscous forces near the wall due to significant momentum redistribution within the flow region during the pre-transition stage (Guerrero et al. Reference Guerrero, Lambert and Chin2021). It is worth pointing out that, during the pre-transition stage, the mean velocity profiles inside and outside group
$M_5$
are not seen as prominently as they were during the inertial stage. The reasoning behind this is due to the drop of flow regions of
$N_{\textit{UMZ}}\gt 5$
, as discussed earlier during the pre-transition stage (figure 5
b).
Figure 14(c) provides the overall temporal evolution of the global velocity profile alongside conditionally averaged mean velocity profiles inside and outside UMZs based on
$M_i$
groups during the transition flow stage. An abrupt drop in the global mean velocity profile is detected in the logarithmic region (
$0.03\lt y/R\lesssim 0.5$
) of the flow as the flow within this region begins to slow down and recover following the flow excursion. Simultaneously, an increase in the mean streamwise velocity gradient is seen near the wall (
$y/R\lt 0.03$
), which causes the profile to move upwards within the region. This increase is due to the higher levels of turbulence near the wall as new turbulence generated by the flow excursion is propagating from the wall towards the core (He & Seddighi Reference He and Seddighi2013; Guerrero et al. Reference Guerrero, Lambert and Chin2021). As a consequence of this turbulence propagation, turbulent spots within the near-wall region grow and merge, causing the levels of turbulence to increase (Guerrero et al. Reference Guerrero, Lambert and Chin2021). It is seen that the profiles of group
$M_5$
become prominent again as more flow regions with
$N_{\textit{UMZ}}\gt 4$
become abundant. Profiles of groups
$M_3$
display fluctuations, indicating that UMZs nearest to the core are starting to be influenced by the new turbulence propagating from the wall. Finally, the profiles of groups
$M_4$
and
$M_5$
begin overlapping towards the end of the transition stage, indicating that the UMZs near the wall have begun recovering. It is worth noting that
$M_5$
was the first to attain recovery by the end of this stage.
The temporal variations of the mean global profile results from the current study during the pre-transition and transition stages are in agreement with the ensemble-averaged mean velocity plot results from He & Seddighi (Reference He and Seddighi2013), where a sudden drop was detected in the logarithmic velocity profile as the pre-transition stage began, which was followed by a gradual increase of the velocity profile until the onset of the transition stage. As the transition stage began, they detected that the velocity profile of the core dropped similar to that of a boundary layer transition, and it continued to drop until the end of the transition stage (He & Seddighi Reference He and Seddighi2013).
Finally, as depicted in figure 14(d), the global mean profiles are seen to be establishing up to the outer region of the pipe (
$y/R\lt 0.5$
), with the mean global profile in the wake region shifting upwards and hence showing signs of increment of its gradient. This increment is first seen towards the end of the transition stage, as shown within the outer region of figure 14(c). Faster UMZ
$M_i$
groups (
$M_1$
and
$M_2$
) display signs of longer time scales to fully recover alongside the global mean velocity profile as their profiles remained fluctuating towards the end of the core-relaxation stage. Unlike faster UMZ
$M_i$
groups, the groups
$M_3$
,
$M_4$
and
$M_5$
displayed considerable signs of recovery. Again, regarding the mean velocity profiles inside and outside UMZ groups, this has shown that the flow core requires extended time scales for the newly generated turbulence to propagate from the wall and recover. Based on the temporal results from all four stages, it is evident that the UMZs continue to maintain their hierarchical flow arrangement, with slower, near-wall UMZs showing earlier signs of recovery compared with faster, near-core UMZs.
3.3.2. Zonal mean streamwise turbulence intensity
Figure 15 provides the overall temporal evolution of the global mean streamwise turbulence intensity profiles alongside conditionally averaged mean turbulence intensity profiles inside and outside UMZs based on
$M_i$
groups during the four unsteady flow stages. Despite the rapid flow acceleration, the peak of the turbulence intensity profile (globally and outside UMZs) is always detected near the wall throughout all transient flow stages.
In figure 15(a), the global turbulence intensity profiles remain unchanged during the inertial stage of the flow. Correspondingly, the conditionally averaged turbulence intensity profiles inside and outside UMZs remain unchanged for all UMZs during this stage. Further adding to this frozen behaviour, it is seen that, during the inertial stage, the cross-over points remain relatively unchanged as well. This behaviour of turbulence intensities is in agreement with the findings by Maruyama et al. (Reference Maruyama, Kuribayashi and Mizushina1976), He & Jackson (Reference He and Jackson2000), He & Seddighi (Reference He and Seddighi2013) and Guerrero et al. (Reference Guerrero, Lambert and Chin2021), where they emphasised the frozen turbulence behaviour of rapidly accelerating turbulent flows during the inertial stage.
Figure 15(b) provides the overall temporal evolution of the global mean turbulence intensity profile alongside the conditionally averaged mean turbulence intensity profiles inside and outside UMZs based on
$M_i$
groups during the pre-transition stage. The peak of the global turbulence intensity profile increases rapidly near the wall, indicating an increment of near-wall levels of turbulence. Guerrero et al. (Reference Guerrero, Lambert and Chin2021) investigated the temporal evolution of the Reynolds shear stresses
$ \left \langle {u_{z}u_{r}}\right \rangle$
within the flow domain during this stage, and they were able to see that there is a weak growth in Reynolds shear stresses
$ \left \langle {u_{z}u_{r}}\right \rangle$
during the pre-transition stage near the wall. This increase in Reynolds shear stress is closely linked to the streamwise Reynolds stress
$ \left \langle {u_{z}u_{z}}\right \rangle$
, or to be precise, the streamwise turbulence intensity (i.e. fluctuations of the streamwise velocity
$u_z$
). This increase in streamwise Reynolds shear stresses drives a spike in energy growth. As a result, the global turbulence intensity peak near the wall grows. In contrast, the global turbulence intensity away from the wall remains relatively unchanged due to the growth in streamwise Reynolds stresses being predominately limited to the near-wall region. As a result, it is observed that, compared with the conditionally averaged turbulence intensities outside the UMZs, the conditionally averaged turbulence intensities inside the UMZs remain nearly unchanged by the end of the pre-transition stage, indicating different turbulent characteristics within and outside of regions on both sides of the ISL that bounds the UMZ. This result is also in accordance with the findings by Greenblatt & Moss (Reference Greenblatt and Moss1999) and He & Seddighi (Reference He and Seddighi2013), where the streamwise turbulence fluctuations are seen to be increased in intensity near the wall. In contrast, the turbulence response at the overlap and the outer regions of the flow remain unchanged.
During the transition stage, the peak of the global turbulence intensity continues to grow, overshooting its final steady state, as depicted in figure 15(c). However, the peak begins to drop as the flow progresses through the final stages of the transition stage. As the turbulence intensity drops near the wall, the turbulence intensity away from the wall begins to increase, indicating propagation of the newly generated near-wall turbulence towards the core. The observation of the turbulence intensity away from the wall starting to increase during this stage agrees with the findings by Guerrero et al. (Reference Guerrero, Lambert and Chin2021), where they revealed that faster growth of Reynolds shear stresses
$\left \langle {u_{z}u_{r}}\right \rangle$
is seen at regions away from the wall during this stage. The global turbulence intensity propagation behaviour is reflected in both the inside and outside intensities of UMZs captured in this stage as the newly generated turbulence propagated towards the core. The turbulence intensity inside UMZs tends to increase, while the opposite trend is seen outside the UMZs.
Finally, towards the end of the core-relaxation stage, it is observed that the inner flow region (
$y/R \lt 0.2$
) becomes established, as depicted in figure 15(d). It is detected that the newly generated turbulence due to the flow excursion continues to be transported towards the core. This propagation is reflected by the dropping turbulence intensities outside UMZs and the increasing turbulence intensities inside UMZs. Another indication of this propagation is the fluctuation of cross-over points for near-core UMZs, unlike for near-wall UMZs, where they remain relatively steady. Overall, it is observed that the temporal evolution of the conditionally averaged turbulent intensity profiles displays temporal coherence both inside and outside UMZs.
3.3.3. Regions of quadrant Reynolds stress within UMZ
We next investigate the dominant quadrant behaviour by conducting quadrant analysis within UMZs, located from the wall towards the core of the pipe, following the methodology proposed by Wallace & Brodkey (Reference Wallace and Brodkey1977). This analysis aims to investigate how the dominance of sweep and ejection events would behave within UMZs as the flow transitions through the four unsteady stages. Here, the fluctuations in turbulence statistics are calculated relative to the zone’s mean velocity. Figure 16 depicts the composition of dominant quadrant behaviour within the UMZs in the case of
$N_{\textit{UMZ}}=2$
based on initial steady flow data. Following the group ranking notions from previous sections based on the rank of
$u_m$
(table 4),
$\mathrm{UMZ}_{c}$
from figure 16 corresponds to the nearest core UMZ (rank 1 UMZs). At the same time,
$\mathrm{UMZ}_{nw}$
represents the UMZ nearest the wall from figure 16. It is detected that UMZs near the core of the flow are always
$Q4$
dominated up to the penultimate UMZ near the wall. In contrast, UMZs nearest to the wall (most near-wall UMZ) are always
$Q2$
dominated. Starting from the core where the UMZs are
$Q4$
dominated, when moving towards the wall, the
$Q4$
dominance periodically drops in the UMZs, and
$Q2$
motions eventually dominate the nearest wall UMZ. The switch from being
$Q2$
dominant to
$Q4$
is detected to occur across the upper boundary of the most near-wall UMZ. These initial base flow statistics follow steady turbulent flow findings by Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) and Chen et al. (Reference Chen, Chung and Wan2020).
The percentage compositions of quadrant behaviour inside UMZs at the initial steady state in the case of two UMZs. Here,
$\mathrm{UMZ}_{c}$
corresponds to the nearest core UMZ while
$\mathrm{UMZ}_{nw}$
corresponds to the UMZ nearest to the wall.
Once investigated on the temporal scale, it is noted that, despite the rapidly accelerating turbulent flow, the near-wall UMZ remained
$Q2$
ejection motion dominated throughout the flow excursion and that the remaining UMZs remained
$Q4$
sweep motion dominated. This result is in agreement with the numerical findings by He & Seddighi (Reference He and Seddighi2013) of the temporal evolution of the dominant quadrant behaviour of accelerating turbulent channel flow at different wall-normal locations. Four wall-normal locations were investigated at
$y^{+0}=0.16, 5.1, 15.6$
and
$54.7$
, and they showed that the late pre-transition and early transition stages are characterised by significantly increased inward sweep events (
$Q4$
) in close vicinity to the wall. At the same time, they showed that ejection events (
$Q2$
) in the outer region increased significantly. The ejection events (
$Q2$
) in the near-wall region and the sweep events (
$Q4$
) in the outer regions were detected to remain relatively unchanged. Although the flow acceleration significantly increased inward sweep events (
$Q4$
) close to the wall and increased ejection events (
$Q2$
) in the outer region, it was shown that the near-wall region remained overall sweep event (
$Q4$
) dominant and the outer regions remained overall ejection event (
$Q2$
) dominant during the transitional stages. Since UMZs are characterised as large-scale coherent structures within the flow, the dominant events not changing near the wall and within the outer regions of the flow are considered the reasons for the dominant events not changing within UMZs.
4. Summary and conclusions
The current study provides insight into the behaviour of UMZs in an accelerating turbulent pipe flow scenario, where the flow undergoes a rapid acceleration. Based on the results obtained, evidence of the existence of UMZs within the flow despite the flow instability and insights into the unsteady response of UMZs from a structural and kinematics point of view are presented in the current study.
4.1. Most probable zonal configurations of transient UMZs
During the four transitional stages of the rapidly accelerating turbulent pipe flow, the average
$N_{\textit{UMZ}}$
within flow regions of the pipe remains unchanged during the inertial stage, but undergoes a drop during the pre-transition stage. This drop is later recovered as the pipe flow shows signs of recovery past the flow excursion.
Correlations between theses structural changes of
$N_{\textit{UMZ}}$
and the average number of ISLs (
$\langle N_{\!\textit{ISL}} \rangle$
) at each temporal instance were investigated, providing insight into how strong layers of internal shear behave in the current unsteady flow scenario.
The main mechanisms that cause this are speculated to be the re-laminarisation trend of the flow during the inertial and the pre-transition stages, which annihilate flow structures and the subsequent fluctuations in viscous and Reynolds shear forces, reflected in the average
$N_{\!\textit{ISL}}$
. The improvement of turbulence levels during the transition and the core-relaxation stages due to propagation of complex flow structures toward the core from the wall causes
$N_{\textit{UMZ}}$
to increase.
4.2. Transient UMZ characteristics
The temporal evolution of conditionally averaged zonal characteristics (zone modal velocities
$u_m$
, the wall-normal locations of UMZ boundaries
$y_k$
and the thickness of UMZs
$t_k$
) based on the ranks of UMZs were investigated. Results indicate that UMZs seen in rapidly accelerating turbulent pipe flow behave similarly to UMZs seen in steady turbulent flow cases. Added to these characteristics were the speeding up and the settling down of zone modal velocities of the UMZs as the bulk flow accelerates and settles. As the flow began to recover, the zones near the wall recovered first, with extended time scales required for UMZs located at the core. This requirement of extended time scales for the core to recover is in agreement with findings in the literature (Guerrero et al. Reference Guerrero, Lambert and Chin2021). The hierarchical zonal arrangement similar to steady flows was retained in the transient UMZs, with faster zones meandering near the core and slower UMZs meandering near the wall.
Uniform momentum zones are observed to be shifting between wall-normal locations, both closer to and farther away from the wall, during the transitional stages of the flow. These results are reflected with in the instantaneous three-dimensional flow visualisations of UMZ 1 and UMZ 2. Thicker UMZs were seen as zones away from the wall and the opposite, where the thicknesses dropped as they were seen near the wall towards the end of the flow excursion where they settled. Quantitatively, similar to steady-state UMZs, UMZs located at the pipe’s core remained the thickest despite the flow being rapidly accelerated. The shifts in wall-normal locations are linked to the combined effects of dropping
$N_{\textit{UMZ}}$
of flow regions, the annihilation of flow structures and the propagation of the newly generated turbulence from the wall.
4.3. Evolution of turbulence statistics across boundaries of UMZs
The conditionally averaged first-order and second-order turbulence statistics inside and outside UMZs, based on the magnitudes of the zone modal velocity
$u_m$
of UMZs, were investigated. The results provided significant insight into the different turbulent kinematic characteristics on either side of the UMZ boundary (ISLs). The turbulence statistics investigated included the mean velocity and turbulence intensity profiles inside and outside UMZs, as well as the global mean velocity and global mean turbulence intensity profiles.
It was recorded that the slowest and the most near-wall UMZs were the first to react to the flow instability. As the wall shear stress within the near-wall region increased alongside the flow acceleration during the inertial stage, the velocity profiles shifted up. Despite the upward shift in the velocity profiles (both global and UMZs), the turbulence intensity remained unchanged (both global and UMZs). This agrees with the frozen turbulence by Maruyama et al. (Reference Maruyama, Kuribayashi and Mizushina1976) and He & Jackson (Reference He and Jackson2000) during the inertial stage. During the pre-transition stage, the global mean velocity profile near the wall dropped due to the perturbation boundary layer growth. The UMZs closer to the wall remained the only regions reacting to the flow perturbation. Correspondingly, it was detected that the global turbulence intensity near the wall increased towards the end of the pre-transition stage due to the growing Reynolds shear stresses within the region. With this global increase, the intensity outside the UMZs displayed a significant increase, while the intensities inside the zones were relatively unchanged. During the transition stage, it was observed that the mean global profile in the logarithmic region decreased, indicating that the flow had begun to slow down and recover. This behaviour was reflected in the UMZs, as the conditionally averaged mean velocity profiles of the near-wall UMZ began to overlap over time scales, and the conditionally averaged turbulence intensity inside the UMZs began to increase. The propagation of turbulence from the wall towards the core is seen during this stage. At the same time, the intensity outside UMZs near the wall remained relatively unchanged. Finally, during the core-relaxation stage, the mean velocity profiles and turbulent intensity statistics were established, indicating that the flow is showing signs of recovery but has not fully recovered.
The overall results collected from these statistics align with the findings presented in Guerrero et al. (Reference Guerrero, Lambert and Chin2021), where it was observed that, as the flow accelerated, it initially exhibited a frozen turbulence response, followed by the newly generated turbulence propagating from the wall towards the core. These results were further supplemented by findings of He & Seddighi (Reference He and Seddighi2013) of both the ensemble-averaged mean velocity profiles and the ensemble-averaged r.m.s. profiles during the transition stages. Furthermore, the finding that the core of the pipe required extended time scales to recover during the core-relaxation stage was evident based on multiple statistical results from the current study.
4.4. Regions of quadrant Reynolds stress within UMZs
Based on the results obtained from the current study, the authors were able to provide quantified evidence of the dominant nature of the
$Q$
quadrant behaviour within UMZs, as indicated by fluctuations in velocities along the streamwise and wall-normal directions within the flow structures. It was found that, always, the nearest wall, slowest UMZ, was
$ Q2$
ejection motion dominated, while all other UMZs were
$ Q4$
sweep motion dominated, suggesting a reduced level of turbulence between near-wall UMZs and near-core UMZs in initial steady turbulent pipe flow data. When moving from the core towards the wall, the
$Q4$
dominance was detected to decrease periodically. In contrast, the
$Q2$
dominance increased, and the switch from
$Q4$
dominant UMZs to
$Q2$
dominant UMZs was detected to always occur at the upper boundary of the most near-wall UMZ. By findings by He & Seddighi (Reference He and Seddighi2013), this behaviour was detected to hold on the temporal scale as the flow transitioned through the four unsteady flow stages, suggesting that this might be a characteristic behaviour of UMZs in rapidly accelerating turbulent pipe flow based on the flow parameters considered in the current study.
Acknowledgements
The authors are grateful for the transient turbulent pipe flow datasets provided by Dr B. Guerrero and the HPC services at the University of Adelaide. The authors acknowledge the financial support from the Australian Research Council (grant no: DE180100157).
Declaration of interests
The authors report no conflict of interest.
Appendix A. Validation of the UMZ detection scheme
It is crucial to validate the precision of the UMZ extraction scheme developed for the current work. By definition, UMZs are flow regions with similar streamwise velocity, and the standard deviation of the streamwise velocities inside UMZs are fairly low compared with the bulk flow (Meinhart & Adrian Reference Meinhart and Adrian1995). Hence, a standard deviation measure of streamwise flow data within each individual UMZ detected is an acceptable benchmark to validate the UMZ detection scheme developed (De Silva et al. Reference De Silva, Hutchins and Marusic2016). Similar to the ranking done based on the magnitude of zone modal velocity
$u_m$
of all UMZs identified in previous sections, UMZ 1 corresponds to the first (and the fastest) UMZ at the core, while UMZ 5 corresponds to the fifth UMZ from the core of the pipe. The standard deviation of streamwise velocity inside each individual zone is calculated and conditionally averaged. As shown in figure 17, relatively low standard deviations within the streamwise flow data are present in the flow regions within UMZs, consistent with the verification conducted by De Silva et al. (Reference De Silva, Hutchins and Marusic2016).
The standard deviation of the streamwise velocity within individual UMZs extracted, based on the currently implemented extraction algorithm at
$t^{+0}=-2.8138$
(negative sign for initial steady state).
Appendix B. Sensitivity of the UMZ detection scheme
Sensitivity of the UMZ identification scheme.
In order to investigate the sensitivity of the parameter set, first the UMZ detection scheme was repeated while lowering the isolation parameters. Figure 18 shows that the choice of parameters can affect the detection of UMZs. The sensitivity of the
$F_h$
parameter to the UMZ detection scheme was tested, while keeping all other parameters as defined in Chen et al. (Reference Chen, Chung and Wan2020). Here,
$F_h$
parameter corresponds to the minimum PDF value of a local peak to be considered a UMZ;
$F_h$
is lowered from 0.5 to 0.4 and 0.2. The same effect is seen when making the interrogation windows smaller. It was observed that making the interrogation window smaller or lowering threshold values such as
$F_h$
amplified the sensitivity of the PDF method to peaks at slower
$U_{z}/U_{\textit{CL}}$
values, i.e. the detection of UMZs closer to the wall. This result provides evidence to the sensitivity of the UMZ detection scheme implemented and agrees with the discussions by De Silva et al. (Reference De Silva, Hutchins and Marusic2016), Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) and Chen et al. (Reference Chen, Chung and Wan2020).
Temporal evolution of
$N_{\textit{UMZ}}$
with the core removed.
The authors further investigated the extent to which the UMZ statistics would change during the flow transitional stages once the core region of the flow was removed from the flow domain, with all the velocity fields, where
$U_{z}/U_{\textit{CL}} \geqslant 1$
, having been removed from the streamwise flow field. The ‘quiescent core’ is the largest UMZ observed in turbulent wall-bounded flows (Kwon et al. Reference Kwon, Philip, De Silva, Hutchins and Monty2014). Guerrero et al. (Reference Guerrero, Lambert and Chin2021) showed that this zone predominantly exists within the core of the flow throughout the four transitional stages of an accelerating flow, despite the flow excursion imposed. Hence, removing the core would allow us to see local changes in the UMZs more clearly. Figure 19 depicts the PDF of
$N_{\textit{UMZ}}$
where the core of the flow has been removed. The results reveal that the
$N_{\textit{UMZ}}$
drops are similar to the current findings observed in figure 5, without significant changes. With the core of the flow removed, the average
$N_{\textit{UMZ}}$
remained at four (inertial stage), then decreased from four to three and then returned to four (pre-transition stage), before settling at four (transition and core-relaxation stages).
This result reflects the fact that the non-turbulent free-stream region (flow region above the turbulent/non-turbulent interface) in the internal flow scenario, as focused on in the current study, is not as significant as it is in turbulent boundary layer flows. Based on the findings of Kwon et al. (Reference Kwon, Philip, De Silva, Hutchins and Monty2014) of the ‘quiescent core’, the flow regions where
$U_{z}/U_{\textit{CL}} \approx 0.9$
are enveloped to form the largest UMZ within the internal wall-bounded flow, and only removing the core flow would result in a change by one zone. This result provides further insights to the robustness of the UMZ identification scheme utilised in the current work.
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