ABSTRACT

INTRODUCTION
Turbulent pipe flows have been studied for about 140 years, starting from the seminal inves- tigation conducted by Reynolds (1883).  Most of the existing studies concerning turbulent pipe flow in smooth and rough walls have been performed assuming steady turbulence conditions (i.e.  con- stant flow rate).  These studies have provided an essential understanding for estimating essential flow quantities, such as the mean skin friction coefficient. A suitable prediction of the steady-state skin fric- 1 tion coefficient allows quantifying the energy dissi- pation in a fluid transportation system at a constant flow rate.  Nonetheless, fluid transportation through pipes is seldom steady. Indeed, the flow rate within a pipe is constantly modulated by control valves or pumping systems to fulfil the system’s demands.  As a result, industrial pipe flows are constantly expe- riencing accelerations and decelerations.  However, transient turbulent pipe flows have received little attention in comparison.  Hence a more profound understanding of the flow physics in unsteady (i.e. accelerating and decelerating) turbulent pipes be- comes necessary to quantify and predict the tran- sient energy dissipation. In this paper we will focus on accelerating tur- One of the interesting char- bulent pipe flows. acteristics of accelerating turbulent pipe flows is that turbulence generation and propagation expe- riences a delay concerning the increase in the flow rate (Maruyama et al. , 1976).  In fact, during the early flow excursion, turbulence exhibits a 'frozen' behaviour.  Later, there exists a second delay as- sociated with the diffusion of 'new' vorticity from the wall towards the pipe centreline (Maruyama et al. 1976).  In addition to this, there has been observed a third delay in the turbulence energy re- distribution throughout the three orthogonal veloc- ity components, implying anisotropy in the turbu- lence response (He & Jackson, 2000). In this context, He & Seddighi (2013) analysed the mean flow response of an accelerating turbulent channel flow.  That study revealed that the time response of the mean skin friction coefficient ( C f ) followed a bypass-transition-like development. Ad- ditionally, it was determined that turbulence in the flow followed two transient stages.  More recently, (Guerrero et al. , 2021) performed a series of direct numerical simulations of accelerating turbulent pipe flows between two steady Reynolds numbers.  The analysis of the mean flow dynamics of those DNS time series complemented previous investigations and revealed that a turbulent pipe flow following a rapid and linear acceleration exhibits four unam- biguous stages; (I) inertial: a rapid increase in vis- cous forces and frozen turbulence behaviour; (II) pre-transition: a weak turbulence response in the near-wall region with a simultaneous fast reduction in the viscous forces; (III) transition:  a proportional increase in viscous and turbulent forces at the inner region; and (IV) core-relaxation: a slow propagation of turbulence from the wall towards the wake region. Moreover, based on the FIK identity developed by Fukagata et al. (2002), Guerrero and co-workers de- rived several alternative expressions suitable to de- compose the skin friction coefficient of an unsteady pipe flow into its dynamic contributions in integral form (Guerrero et al. , 2021, 2022a,b).  This decom- position helps determine how the different terms of the mean momentum balance influence the mean WSS as a function of time. The studies mentioned previously have provided substantial insight into the mean flow statistics rel- evant to accelerating flows.  Nevertheless, the role of the LSM and the SSM during the flow development and how they contribute to the frictional drag is still elusive. Therefore, this investigation uses high- and low-pass Fourier filters to investigate how the large- (LSM) and small-scale motions (SSM) contribute to the generation and propagation of turbulence dur- ing the different stages undergone by a linearly ac- celerating flow. Furthermore, we use the FIK iden- tity to identify how the large- and small-scale flow dynamics contribute to the temporal evolution of the frictional drag.
NUMERICAL DETAILS
Direct numerical simulations of an accelerat- ing turbulent flow have been conducted using the Navier-Stokes solver Nek5000 (Fischer et al. , 2019). In this work, initially steady turbulent flow fields at a steady friction Reynolds number Re , 0 ≈ 500 from Guerrero et al. (2020) are linearly accelerated by increasing their flow rate until they attain a value Re r,1 ≈ 670.  Subsequently, the flow remains at a constant flow rate until it fully develops and the universal laws of turbulence are attained.  A cylin- drical coordinate system has been adopted where r, θ and z are the radial, azimuthal and streamwise directions, respectively.  The wall-normal direction is denoted as y = R − r .  Similarly, the resolved ve- locity vector fields contain three orthogonal scalar components U r = −U y, U o and U z whose fluctuating components are u r = − u y , u g and u z , respectively. It is noteworthy that the “+0” or “+1” superscripts denote normalisation in viscous units at the initial or final steady Reynolds numbers of the simulations, respectively. The computational grid used in the present study was designed to have a suitable resolution at the highest Reynolds number attained. The compu- tational grid used in the present investigation has a  resolution  Δ z +1 = 7.4,  Δ R θ +1 = 6.3,  y +1 = 0.05 and ycore = 7.5.  Further details about the simula- tion setup can be found in Guerrero et al. (2021).
RESULTS AND DISCUSSION
Time dependence of the Reynolds shear stress Figure 1 depicts the Reynolds shear stress {uruz}+0  decomposed into the large- and small-scale components using a wavelength threshold λ ≈ 2R. This decomposition was performed using a high/low pass Fourier filter similar to Chin et al. (2014). The results show that during the inertial stage (figure 1a), the turbulence related to LSM and SSM is completely frozen. Throughout the pre-transitional stage (figure 1 b ) it is observed a stronger response in the large-scale turbulent motions, showing good agreement with the elongated streaks observed by He & Seddighi (2013) during the pre-transitional stage.  During the early pre-transitional period, it is also interesting to note that there is a decay in the contribution of the small-scale structures to the Reynolds shear stress. During the transitional stage (figure 1c), growth in both LSM and SSM turbulence is observed.  How- ever, a careful examination shows that within the inner region of the flow y+0 < 100, the SSM turbu- lence develops at a faster rate. It should be noted that within the core region of the flow, both SSM and LSM turbulence is nearly frozen. Finally, dur- ing the core-relaxation stage (figure 1 d ), the LSM and SSM contributions are nearly frozen at the near-wall region. However, similar rate of develop- ment in the LSM and SSM turbulence at t he core region is observed.
Energy growth in the Reynolds shear stress
Figure 2 depicts the temporal evolution of the Reynolds shear stress’ pre-multiplied co-spectra φ i , u , during a particular time location at the initial steady-state (figure 2 a ), the four transient stages (figures 2b – e) and the final steady-state(figure 2f). It is noteworthy that the co-spectra by itself does not provide an accurate qualitative and quan- titative indication in the energy growth. Thus, the difference in two consecutive spectrograms (i.e. φ + - (t + ∆ t) − φ + o - (t)) provides a clearer picture of what scales of motion gain energy throughout the transient process. Figure 3 exhibits the results of that subtraction.  For instance figure 3( a ) is the result of subtracting φ i , u , (t + 0 = 4.6) − φ i , u , (t + 0 = −1.2) (i.e.  figure 2 b - figure 2 a ).  Moreover, it should be noted that the spectrogram has been subdivided in six quadrants to better understand the regions and the length scales at which the energy growth occurs. In that context, the two vertical dashed lines subdivide the near-wall (y+0 < 30), overlap (30 < y + 0 < 200) and wake ( y + 0 > 200) regions of the Similarly, the horizontal line subdivide the flow. scale ranges into into small- ( λ + 0 ≤ 1000) and large- scale motions, where λ +0 > 1000 (i.e. λ / R > 2). The results observed in figure 3( a ) show that while the mean kinetic energy of the fluid increases during the flow excursion, the turbulent energy does not exhibit any significant changes throughout the pipe domain due to the delay in the turbulence re- sponse characteristic of the inertial period. These results agree with the 'frozen' turbulence observed during the early flow excursion observed in previ- ous studies (Maruyama et al. , 1976; He & Jack- son, 2000). Figure 3( b ) shows that during the pre- transition, large-scale motions exhibit a consider- able energy growth at the buffer region. This is in good agreement with the study by He & Seddighi (2013); Guerrero et al. (2022a) who observed the for- mation of elongated streaks at the near-wall region in accelerated channel and pipe flows, respectively. The energy growth in the Reynolds shear stress throughout the transitional period is observed in fig- ure 3( c ). This figure reveals that most of the energy growth and the turbulence production at the inner region of the flow (i.e.  y+0 < 100) occurs during the transition stage. Within this period, primarily small-scale structures are produced, agreeing with the production of 'new' turbulence occurring during this stage.  Finally, figure 3( d ) reveals substantial growth in the energy density at the wake region of the flow within the small and large-scale spectrum. Moreover, it is noted that there is a minor energy growth at the near-wall region during the core re- laxation period.
Contribution of LSM and SSM into the wall fric-
tion
The FIK identity has been applied to the nu- merical data set used in this investigation follow- ing the alternative expression derived by Guerrero et al. (2021) for unsteady pipe flow. However, that expression was multiplied by 0.5ρ[U h ( t )]² to express the frictional drag in terms of the wall shear stress rather than the skin friction coefficient. It is note- worthy that the turbulent component has been split into large- and small-scale contributions. The results exhibited in figure 2( a ) reveal that the LSM and SSM components of the turbulent term exhibit a substantially different response, es- pecially during the inertial and the pre-transitional stages. The turbulent contribution associated with the small-scale components τ w,ss seems nearly un- changed during the inertial and the pre-transitional stage. However, the large-scale turbulent contribu- tion τ w,LS only exhibits a frozen behaviour during the first stage and undergoes linear growth during the pre-transitional period. This is better observed in the zoomed view depicted in 2( b ), where it is shown that Tw,SS experience decay at 20 < t + 0 < 60, thereby suggesting a "laminarescent" behaviour of the flow during the pre-transitional stage.
Conclusions
Direct numerical simulation data sets of a rapidly accelerating turbulent pipe flow have been used to analyse the large- and small-scale contribu- tions to the Reynolds shear stress throughout the four transitional stages, characteristic of these flows. During the inertial period, frozen turbulence be- haviour is observed in both the LSM and SSM con- tributions. The pre-transitional period exhibits a signifi- cant energy growth in the LSM range ( λ +o ≥ 1000) across the buffer region. These results are in good agreement with the observations by (He & Seddighi, 2013) who observed elongated streaks throughout this period.  Moreover, the Reynolds shear stress and the small scale turbulent contribution of the wall shear stress τ w,ss exhibit a slight temporary decay, suggesting a laminarescent trend during the By the end of this period, early pre-transition. small-scale turbulence starts gaining energy, consis- tent with the formation of new turbulent spots. The transitional stage reveals an energy growth in both the large- and the small-scale range. How- ever, the energy of small-scale turbulence grows at a higher rate during this period, which is consistent with the increase in the Reynolds number. Finally, the core-relaxation period shows slight growth in large- and small-scale turbulence. The Reynolds shear stress co-spectra shows that most of this energy growth occurs at y + 0 > 200.