Transient behaviors of wall turbulence in temporally accelerating channel flows

International Journal of Heat and Fluid Flow 67 (2017) 13–26

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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff

Transient behaviors of wall turbulence in temporally accelerating channel flows Seo Yoon Jung a, Kyoungyoun Kim b,∗ a b

Korea Atomic Energy Research Institute, Daejeon 34057, Republic of Korea Department of Mechanical Engineering, Hanbat National University, Daejeon 34158, Republic of Korea

a r t i c l e

i n f o

Article history: Received 13 October 2016 Revised 26 June 2017 Accepted 27 June 2017

Keywords: Transient channel flow Direct numerical simulation Bypass-like transition

a b s t r a c t The effects of mean flow acceleration on near-wall turbulent structures were investigated by performing direct numerical simulations of transient turbulent flows in a channel. The simulations were initiated with a fully developed turbulent channel flow and then temporal accelerations were applied. During the acceleration, almost linearly increasing excursions of the flow rate were imposed between the steady initial and final values. The initial Reynolds number (based on the friction velocity) was fixed to Reτ ,i = 180, and four different final Reynolds numbers (Reτ , f = 250, 300, 350, and 395) were selected to show the effects of the Reynolds number ratio (Reτ ,f /Reτ ,i ) on the transient channel flows. To elucidate the effects of the flow acceleration rates on the near-wall turbulence, a wide range of acceleration durations has been examined. Various turbulent statistics and instantaneous flow fields revealed that the rapid increase in the flow rate invokes bypass-transition-like phenomena in the transient flow. In contrast, the flow evolves progressively and the transition does not occur clearly for the mild flow acceleration. When the increase in the Reynolds number is small during the acceleration, distinct bypass-like transition phenomena do not appear in the transient flows, regardless of the acceleration rate. The present study proposed new criteria based on the impulse of the acceleration in order to explain the transition to the new turbulence in the transient channel flow. The bypass-like transition is primarily due to the larger contribution of the impulse to the increase in the flow rate compared with that in viscous friction. © 2017 Elsevier Inc. All rights reserved.

1. Introduction 1.1. Transient turbulent channel flow Transient turbulent flow inside a channel is an unsteady flow generated by temporal changes of the pressure gradient or the flow rate. The transient turbulent channel flows can be encountered in many engineering problems, such as the intake of an engine, heat exchangers, valves, or the starting and stopping operations of power plants. Numerous experimental and numerical studies on the transient turbulent channel flows have been conducted; however, intriguing characteristics of transient flows still remain not fully understood. For instance, the response mechanism of the near-wall turbulence to temporal acceleration is still an open question in fundamental turbulence research. In addition, the conventional turbulence models that are widely applied in engineering cannot yet accurately predict the transient flow in a turbulent channel. ∗

Corresponding author. E-mail addresses: [email protected] (S.Y. Jung), [email protected] (K. Kim).

http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.06.012 0142-727X/© 2017 Elsevier Inc. All rights reserved.

Kataoka et al. (1975) used an electrochemical method to measure the changes in the flow when the flow rate changes inside a pipe. They found that the time for the transition from laminar to turbulent state decreases with increasing Reynolds number. Mizushina et al. (1975) observed a time delay in the response of turbulence, i.e., which reacts to an abrupt change of flow rate slower than the mean flow field, in the transient flow inside a pipe when triggered by a sudden increase of flow rate in the steady state. The delay is more prominent in the center of the pipe than close to the wall. He and Jackson (20 0 0) examined the temporal changes in transient flows, when the flow is either accelerated or decelerated, utilizing the laser Doppler velocimetry (LDV). They observed three delays in the turbulence: a delay in the turbulence production, a delay in turbulence energy redistribution, and a delay in the propagation of turbulence toward the radial direction. Greenblatt and Moss (2004) conducted experiments of transient pipe flows to examine the effects of flow rate which was higher than those considered in the previous studies (Mizushina et al., 1975; He and Jackson, 20 0 0). The authors identified the regimes of the transient flow as the steady, initial, and final states. They also identified the reconstitution of the wake in the final phase.

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Due to the rapid development of supercomputers, numerical studies regarding the transient flow are becoming more popular. Jung and Chung (2012) examined the temporally accelerated flow in the pipe using large-eddy simulation (LES). They numerically confirmed the three delays in the propagation of turbulence in the radial direction, the turbulence production and the redistribution of the turbulence energy, as pointed out in the experiment (He and Jackson, 20 0 0). In addition, Jung and Chung (2012) analyzed the conditionally averaged flow fields associated with Reynolds shear stress producing events. They showed that sweeps and ejections were closely linked to the delays of the turbulence production and of turbulence propagation away from the wall. He and Seddighi (2013) claimed, through the direct numerical simulation (DNS) study on the transient turbulent channel flow, that the abrupt increase of the flow rate causes similar phenomena to the bypass transitions observed in the laminar boundary layer even though the initial flow is turbulent. They explained that the initial turbulence prior to the acceleration acts as a disturbance to create elongated streaks close to the wall, and then these streaky structures are exposed to secondary instabilities which generate a bundle of turbulence vortices, i.e., turbulent spots. In their subsequent study (He and Seddighi, 2015), it was found that, as the difference between the initial and final Reynolds number increases, the bypasslike transition phenomena becomes more prominent while they becomes uncertain as the Reynolds number difference decreases. It has also been reported that a pipe flow following a step increase in flow rate exhibits the transition which is effectively a laminarturbulent transition, similar to channel flow (He et al., 2016).

Jacobs and Durbin (2001) showed that breakdown occurs when the lifted low-speed streaks are buffeted by the high-frequency freestream disturbances. Their observations using DNS were confirmed experimentally (e.g. Hernon et al., 2007; Mandal et al., 2010) and complemented by computational studies (e.g. Brandt et al., 2004; Zaki and Durbin, 2005; Schlatter et al., 2008). In addition to the streak breakdown scenario that originates near the edge of the boundary layer, Nagarajan et al. (2007) identified another mechanism that originates near the wall and is initiated by the amplification of an instability wavepacket. To elucidate the numerical and experimental observations of breakdown, a number of studies evaluated the secondary instability of the streaky boundarylayer flow. Andersson et al. (2001) performed inviscid analysis of the linearly-optimal streaks, and could only predict the instability near the edge of the boundary layer. Vaughan and Zaki (2011) applied Floquet analysis to examine the secondary instability of unsteady streaks. They identified two types of instabilities: an outer mode similar to the observations in Jacobs and Durbin (2001) and an inner instability near the wall which bears a close resemblance to the wavepackets reported by Nagarajan et al. (2007). Nolan and Zaki (2013) showed that the onset of breakdown is due to highamplitude streaks, with streamwise perturbation exceeding 20% of the free-stream velocity. Recently, Hack and Zaki (2014) demonstrated that zero-pressure gradient (ZPG) boundary layers favor the amplification of outer instabilities, and the inner mode is promoted if an adverse pressure gradient (APG) is applied.

1.2. Bypass transition

Several parameters have been proposed to characterize the transient turbulent flows. He and Jackson (20 0 0) suggested that factors such as the initial Reynolds number, final Reynolds number, and various dimensionless ramp rate parameters define the transient flows inside pipes. The initial and final Reynolds numbers determine the initial and later states of the transient flows, respectively, while the flow increase rate is associated with the difference in the transient flow from the steady flow corresponding to the same Reynolds number. He and Seddighi (2015) examined the effects of the ratio of the final to initial Reynolds number on the bypass-like transition. They showed that in the case of rapid acceleration, a laminar flow similar to Stokes’ first problem appears at the early stage of transient flow and then develops into a final turbulent state. When the final Reynolds number is high, the development of the flow is similar to a bypass-like transition, which accompanies the presence of strong streak during pre-transition and the generation of turbulent spots. On the other hand, if the final Reynolds number is low, the transient process is qualitatively different, and the streak formation and breakdown are not dominant. Despite the significant difference in flow visualizations, He and Seddighi (2015) claimed that the transient flows with low Reynolds-number-ratio case can be characterized as the bypass-like transition because they exhibit strong similarity in various turbulence statistics (e.g., critical Reynolds number, energy growth, development of friction factor). Seddighi et al. (2014) have investigated effects of a mild acceleration and compared the results with those of rapid acceleration of He and Seddighi (2013). However, it should be noted that the test cases of Seddighi et al. (2014) are limited to the range of rapid acceleration rates that cause bypass-like transitions. In this work, we have performed a series of spectral DNS of the turbulent channel flow with temporal acceleration by the prescribed time-dependent mean pressure gradient to examine the flow parameters relevant to the transition behavior. Fig. 1(a) shows the schematic diagram of the accelerated flow in a channel. The initial Reynolds number based on the friction velocity was fixed to Reτ ,i = 180 and four different final Reynolds numbers (Reτ ,f =

Before proceeding further, it would be advantageous to review the bypass transition to turbulence in boundary layers. Transition to turbulence in boundary layers is often categorized by natural or bypass transition. When the background perturbation levels are inappreciable, the transition is governed by the amplification of twodimensional Tollmien–Schlichting (TS) waves (Kleiser and Zang, 1991). However, in the presence of moderate free-stream disturbances, a shorter path termed ‘bypass transition’ leads to boundary layer turbulence at lower Reynolds numbers [for a recent review, see Zaki (2013)]. The initial stage of bypass transition concerns the penetration of free-stream vortical disturbances into the boundary layer. In the inviscid limit, convected free-stream disturbances are not allowed to penetrate into the boundary layer due to the filtering effect of the mean shear, and this phenomenon is known as shear sheltering (Hunt and Durbin, 1999). At finite Reynolds number, however, the sheltering mechanism is less effective, and low-frequency disturbances from the free stream can cause a finite distortion within the boundary layer (Jacobs and Durbin, 1998; Zaki and Saha, 2009). Forcing due to the low-frequency perturbations generates an energetic response inside the boundary layer in the form of streamwise-elongated streaks. These distortions are termed Klebanoff modes (Klebanoff, 1971) and have been studied using linear theory (Butler and Farrell, 1992; Andersson et al., 1999; Luchini, 20 0 0), experiments (Westin et al., 1994; Matsubara and Alfredsson, 2001) and direct numerical simulations (Andersson et al., 20 01; Zaki and Durbin, 20 05; 20 06). The amplification of streaks is due to an inviscid lift up mechanism whereby quasi-streamwise vortices lift up low-velocity fluid away from the wall and sweep high-velocity fluid towards the wall (Landahl, 1980). This inviscid amplification mechanism is curtailed by viscous damping, and the outcome is only a transient growth followed by viscous decay (Butler and Farrell, 1992). The amplification of streaks can make the flow susceptible to secondary instability which precedes the breakdown to turbulence.

1.3. Motivations

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Fig. 1. (a) Schematic of temporal accelerating flow in a channel; (b) Temporal evolution of ensemble-averaged streamwise velocity for test case where Reτ ,f = 395 and T∗ =2.0. The circles and diamonds represent the DNS data of Moser et al. (1999) with the same Reynolds number as the initial and final state, respectively.

Table 1 Test cases and accelerating parameters. The cases accompanied by a distinct bypasslike transition are shown underlined and bold. Reτ ,i Reτ , f

Reτ ,f /Reτ ,i T ∗ (= T uτ ,i /h ) γ

δ

180 250

1.4

395

2.2

Seddighi et al. (2014)

178 414

2.3

0.05 2.0 5.0 1.0 2.0 5.0 1.0 2.0 5.0 0.1 1.0 3.0 5.0 10.0 0.0057

He and Seddighi (2013)

178 418

2.3

2.1 0.0053

1.51 613.84

0.00414 1.678

195 (S16) 216 (S15) 253 (S14) 179 310 (S13)

1.1 1.2 1.4 1.7

0.0 0 035 0.00102 0.00256 0.00271

608.73 481.09 384.87 650.45

1.673 1.3226 1.058 1.788

0.00509 0.005206 0.00512

636.59 1.75 975.05 2.680 1347.05 3.703

Present

He and Seddighi (2015)

300

1.7

350

1.9

414 (S01) 2.3 538 (S12) 3.0 658 (S11) 3.7

18.57 0.46 0.19 1.57 0.79 0.31 2.29 1.14 0.46 28.57 2.86 0.95 0.57 0.29 556.95

0.005105 0.00128 0.0 0 051 0.00432 0.00216 0.0 0 0864 0.00628 0.00314 0.00126 0.062836 0.0062836 0.00314 0.001257 0.0 0 0898 1.5225

250, 300, 350, and 395) were selected to show the effects of the Reynolds number ratio (Reτ ,f /Reτ ,i ) on the transient channel flows. The acceleration times for each final Reynolds number (T∗ ) are listed in Table 1. The acceleration time is defined as T ∗ = T uτ ,i /h, where uτ ,i is the initial friction velocity and h is the half-height of the channel. Note that the previous study (He and Seddighi, 2015) was limited to those cases wherein the temporal increase rate of the Reynolds number is very large, namely the acceleration time T∗ is very small. However, in the present study, by investigating a wide range of acceleration times, the effects of the flow acceleration rates on the near-wall turbulence were examined. The main emphasis of the present study is placed on the criteria on the occurrence of the bypass-like transition in temporally accelerating channel flows. Since the real bypass transition from laminar boundary layer flow accompanies the amplification of streaks (i.e., Klebanoff modes), the streak-dominated transient process (He and Seddighi, 2015) is referred to as the bypass-like transition in the present study. The remainder of this article is organized as follows. In Section 2, we present the governing equations, boundary conditions, and numerical algorithms. The DNS results are discussed in Section 3. Finally, conclusions are presented in Section 4.

2. Governing equations and numerical method The non-dimensional governing equations of an unsteady, incompressible flow are

Du 1 = −∇ p + ∇ 2u + Dt Reτ ,i



 ∂ p − (t ) ex ∂x

∇ ·u=0

(1)

(2)

where u is the velocity and p is the pressure. Here, all variables are non-dimensionalized by the channel half-height h, and the initial frictional velocity uτ ,i . The Reynolds number is defined as Reτ ,i = uτ ,i h/ν, where ν is the kinematic viscosity. The last term on the right-hand side of Eq. (1) denotes the mean pressure gradient related to the flow acceleration. The mean pressure gradient is 1.0 in the statistically steady flow. However, in the accelerated flow, the mean pressure gradient increases to a value greater than unity. To generate a linear increase of the flow rate during an acceleration period, a time-dependent mean pressure gradient was imposed in Eq. (1). By integrating the ensemble-averaged streamwise momentum equation over a cross-section of the channel, the time-dependent mean pressure gradient can be written as

 2 ∂ p 1 dRem Reτ (t ) − (t ) = + ∂x 2Reτ ,i dt Reτ ,i

(3)

Here, Rem is the bulk mean Reynolds number (Rem = Um 2h/ν ). In Eq. (3), the first term on the right-hand side contributes to the increase in the flow rate, while the second term contributes to the increase in the skin friction when the flow is accelerated by the time-dependent mean pressure gradient. The second term on the right-hand side of Eq. (3) requires the variations in the friction velocity with time, which is unknown. In the present study, Reτ (t) is .25 computed using the empirical relationship (c f = 0.073Re−0 ) by m Dean (1978) with an assumption of quasi-steady state conditions during the acceleration period. Time integration of the governing equations is achieved by a semi-implicit method where the implicit Crank–Nicolson scheme is used for the viscous terms and the explicit Adams–Bashforth scheme is employed for the nonlinear convective terms. The spatial derivatives are obtained by using a spectral method with Fourier representations in the streamwise and spanwise directions, and Chebyshev expansion in the wall-normal direction. Periodic boundary conditions are applied in the streamwise and spanwise directions, and the no-slip boundary condition is imposed on the velocity at the solid walls. More details on the numerical method are given in Kim et al. (2008). The domain size is (Lx , Ly , Lz ) = (4π h, 2h, π h) in the streamwise, wall normal, and spanwise directions, respectively, which is sufficiently large in comparison with other DNS studies. The grid used is (Nx , Ny , Nz ) = (257, 129, 129). The computational domain size and grid

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Fig. 2. (a) Prescribed mean pressure gradient given by Eq. 3; (b) Development of bulk mean Reynolds number for Reτ ,f = 395; (c) Comparison of skin friction coefficient development for different flow accelerating methods. The prescribed mean pressure gradient increase (present, T∗ = 0.1) and the prescribed mass flow rate increase (He and Seddighi, 2013, T∗ = 0.0053) are shown. + + + resolution are (L+ x , Lz ) = (2262, 565), (x , z ) = (8.84, 4.42), + + + + and (Lx , Lz ) = (4964, 1241), (x , z ) = (19.39, 9.69) in wall units corresponding to the initial Reynolds number Reτ ,i = 180, and final Reynolds number Reτ ,f = 395, respectively. For every test case, four independent simulations are performed to calculate the ensemble-averaged quantities. Fig. 1(b) shows that the mean velocity profiles obtained by the present DNS of the transient channel flow are in good agreement with the statistically steady channel flow results for a given Reynolds number (Moser et al., 1999). These results suggest that the grid resolution is sufficiently fine and the number of repeated runs is enough to get the converged ensemble averages. More information on the computational details is given in the Appendix.

3. Results and discussion Fig. 2(a) and (b) show the imposed mean pressure gradient and temporal variation of the bulk mean Reynolds number for Reτ ,f = 395. Simulations were conducted for five different values of acceleration time (T∗ = 0.1, 1.0, 3.0, 5.0 and 10.0). In Fig. 2(a), significant changes in the mean pressure gradient are clearly seen as the duration of acceleration becomes shorter. This is due to the rapid increase of the flow rate as shown in Fig. 2(b), which results in a significant increase in the first term on the right-hand side of Eq. (3). After the acceleration period, the mean pressure gradient is imposed as a constant value corresponding to the steady flow at Reτ ,f . In Fig. 2(b), for T∗ = 0.1, a nonlinear increase in the bulk mean Reynolds number is observed, which is due to the use of Dean’s correlation for calculating the friction velocity in Eq. (3). In the case of small T∗ , the relationship between Rem and cf during acceleration is significantly different from Dean’s correlation. However, when T∗ is small, dRem /dt on the right-hand side of Eq. (3) becomes dominant whereas the influence of (Reτ /Reτ ,i )2 on the total mean pressure gradient is meager. He and Jackson (20 0 0) also reported that dRem /dt provides a dominant contribution to the mean pressure gradient in the pipe flow with a strong acceleration. This can be confirmed from Fig. 2(c) which shows that the temporal variation in cf of the present study, imposing the mean pressure gradient with Dean’s correlation, is very similar to that of He and Seddighi (2013) where the exact linear increase in the mean flow rate was imposed. Fig. 3(a) and (b) show the evolution of the vortical structures for T∗ = 0.1 and T∗ = 5.0, respectively. The vortices are visualized using iso-surfaces of the swirling strength (Chakraborty et al., 2005). For T∗ = 0.1 (Fig. 3a), it was found that the turbulence structures are sparsely distributed at t∗ = 0.4 after the acceleration. At t∗ = 0.6, vortex packets occur, wherein the turbulent structures cluster locally. These structures are further developed with increasing time until the entire wall is covered with vortical structures. This is con-

sistent with the results obtained by He and Seddighi (2013). They reported that, following a rapid increase in the flow rate, the turbulent channel flow undergoes a transition that is strikingly similar to the bypass transition in a boundary layer subject to free-stream turbulence. On the other hand, for T∗ = 5.0 (Fig. 3b), the development of the transient turbulent flow is progressive from the initial state to the final one. The instantaneous pictures represent that, even if the Reynolds number ratio (Reτ ,f /Reτ ,i ) is high, the transition does not occur in the case of the long duration of acceleration. It has been reported that an outer instability appears near the edge of the pre-transitional boundary layer with favorable pressure gradient (e.g. Nolan and Zaki, 2013; Hack and Zaki, 2014). To observe the outer instability in our flow configuration, we show the three-dimensional plots of streaks visualized using iso-surfaces of the fluctuating component of streamwise velocity at a pretransitional period (t ∗ = 0.3 ∼ 0.5) and an early stage of transition (t ∗ = 0.6) in Fig. 4. During the period of pre-transition, elongated high- and low-speed streaks are alternatively generated. As marked by the red dashed line in Fig. 4, a low-speed streak exhibits a sinuous pattern as observed in the DNS study of the transitional boundary layer (Hack and Zaki, 2014). This feature is closely related to the outer instability introduced by Vaughan and Zaki (2011). In the transient channel flow, the pre-existing turbulence acts as initial disturbances like the free-stream turbulence in bypass transition. Although the intensity of the initial turbulence is reduced by the increase of flow rate, the initial turbulent structure still resides near the edge of the wall layer during the pretransitional period. Once a low-speed streak is lifted near the wall layer, it undergoes a sinuous instability which subsequently develops into the formation of vortex packets. The outer instability phenomenon is also indicated by the DNS studies of turbulent transient channel flows (He and Seddighi, 2013, 2015). Fig. 5(a) displays the temporal changes in the skin friction coefficient for various acceleration times in the case of Reτ ,f = 395. When the acceleration time is short (T∗ = 0.1 and 1.0), a sudden increase in cf is observed in the initial stage of acceleration. This happens because a large pressure gradient is applied abruptly. After the abrupt increase of cf , a rapid decrease of the coefficient can be observed. This is because the turbulence structure is not yet fully developed, in comparison with the increased flow rate, which results in a significant difference in cf , relative to the value in the steady state. After attaining the minimum value, cf increases and then eventually reaches the steady state. On the other hand, when the acceleration time is long (i.e., T∗ = 5.0 and 10.0), the value of cf does not depart significantly from that of the steady state. In Fig. 5(b), cf is plotted against the instant bulk Reynolds number. If the acceleration time is short (i.e., T∗ = 0.1 and 1.0), cf is proportional to the inverse of Rem during the acceleration, which indicates a feature of laminar flow. This is consistent with the quasi-

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Fig. 3. Instantaneous vortical structures in transient channel flows for case of Reτ ,f = 395: (a) T∗ = 0.1 and (b) T∗ = 5.0. The vortices are shown as iso-surfaces of the swirling strength for λci h/uτ ,i = 30.

laminar flow, observed prior to the transition in the acceleration flow (He and Seddighi, 2013). When the acceleration time is long (i.e., T∗ = 5.0 and 10.0), the variation in cf with Rem is similar to .25 Dean’s correlation (c f ∼ Re−0 ), which implies that the turbulence m structure adapts to the altered mean flow. It has also been reported that the flow evolves progressively and the bypass-like transition process does not occur during the transient flow when the Reynolds number ratio between the final and initial stages is low (He and Seddighi, 2015). To examine the effects of the Reynolds number ratio (Reτ ,f /Reτ ,i ) on the transient channel flows, four different final Reynolds numbers (Reτ ,f = 250, 300, 350, and 395) were tested for a fixed initial Reynolds number of Reτ ,i = 180. Fig. 6(a), and (b) show the changes in the vortical structures for T∗ = 0.05 and T∗ = 2.0 in the case of Reτ ,f = 250, respectively. Unlike the case of Reτ ,f = 395, even in the case

where the acceleration time is short (T∗ = 0.05), the locally clustered vortical structures are not clearly shown. However, as will be shown in Fig. 8(c), the rapid acceleration gives rise to a laminarlike flow similar to Stokes’ solution at the early stage of the transient flow, which leads to a laminar-turbulent transition without involving bypass-like transition. This transient flow behavior due to a rapid acceleration for low final Reynolds number is consistent with He and Seddighi (2015). Fig. 7(a) plots the temporal changes in the bulk Reynolds number for a range of acceleration times when Reτ ,f = 250. A linear increase in the bulk Reynolds number during the acceleration period can be confirmed. Fig. 7(b) shows the temporal changes in the skin friction coefficient. In the case of T∗ = 0.05, cf rapidly increases and then decreases to the local minimum and increases again to reach the value for the steady state. This trend is similar

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Fig. 4. Time sequence of streaks for the case of Reτ , f = 395 and T ∗ = 0.1; top view with iso-surfaces of high-speed (u /uτ ,i = 4, yellow) and low-speed streaks (u /uτ ,i = −4, blue). Example of an outer instability is marked by the red dashed lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Development of friction coefficient with (a) non-dimensional time t∗ and (b) bulk mean Reynolds number Rem .

to those cases in which Reτ ,f = 395, and T∗ ≤ 1.0 (Fig. 5a), in which the bypass-like transition occurs. However, although the local minimum in the variation of cf can be clearly seen, the bypass-like transition was not distinct when Reτ ,f = 250 and T∗ = 0.05. Therefore, for the accelerated turbulent channel flow, the occurrence of a local minimum in the cf time history does not necessarily indicate a bypass-like transition in the flow. Fig. 7(c) shows cf plotted against the bulk Reynolds number.

For T∗ = 0.05, i.e., for a significant flow acceleration, cf curve shows a laminar-like behavior (c f ∼ Re−1 m ), which is consistent with the result of the rapid acceleration condition for higher final Reynolds number (Reτ ,f = 395) as shown in Fig. 5(b). Meanwhile, for the cases of T∗ = 2.0 or 5.0, the changes in cf with time are similar to those for the steady state, as can be seen in Fig. 7(c). As shown in Figs. 5(b), and 7(c), the occurrence of the transition in accelerated channel flow can be determined by whether

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Fig. 6. Instantaneous vortical structures in transient channel flows for case of Reτ ,f = 250: (a) T∗ = 0.05 and (b) T∗ = 2.0. The vortices are shown as iso-surfaces of swirling strength for λci h/uτ ,i = 25.

the slope in the log Rem - log cf graph equals -1 in the acceleration period (0 < t∗ < T∗ ), which indicates the occurrence of a quasi-laminar state. However, this determination, being based on the slope in the log-log plot between Rem and cf , utilizes information on the flow only during the acceleration period. When the acceleration time is very short, the determination of the transition using only the information of the acceleration period can give (a)

rise to misleading results since the flow can undergo significant changes after the acceleration. As a criterion for determining the transition which includes flow information after the acceleration period, the existence of the local minimum in cf temporal variation can be provided (He and Seddighi, 2015). If the local minimum of cf does not occur clearly, the transition will not occur. However,

(b)

(c) 0.02

10000

0.018 0.01

0.016

9000

0.014

8000

cf

cf

Rem

0.01 0.006

0.008

7000 *

T = 0.05 T* = 2.0 T* = 5.0

6000

5000

-1

cf ~ Rem

0.012

0.008

0

1

2

t*

3

4

*

T = 0.05 T* = 2.0 * T = 5.0

0.004

5

0.006

0.002 0

1

2

*

t

3

4

5

0.004 5000

*

T = 0.05 * T = 2.0 * T = 5.0 cf = 0.073 Re-0.25 m 6000

7000

Rem

8000

9000

10000

Fig. 7. Flow development for case of Reτ ,f = 250: (a) bulk mean Reynolds number; (b) temporal variation of the friction coefficient; (c) skin friction coefficient versus bulk mean Reynolds number Rem .

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√ Fig. 8. Development of the mean velocity profiles in the very early stage for Reτ ,f = 395 (a, b) and Reτ ,f = 250 (c, d). Perturbation velocity versus y/2 ν t (a, c) and mean + velocity versus y (b, d). Solid lines: rapid acceleration; dashed lines: slow acceleration; circles: Stokes’ solution; triangles: Moser et al. (1999) (denoted by MKM).

even if the distinct local minimum of cf does exhibit as shown in Fig. 7(b) (T∗ =0.05), the bypass-like transition may not appear. Fig. 8 shows the change in velocity distribution at the beginning of the acceleration in terms of the perturbing velocity (Fig. 8(a) and (c)) and mean velocity normalized by the friction velocity (Fig. 8(b) and (d)). The perturbation velocity is defined as (He and Seddighi, 2013; 2015):

u∧ (y, t ) =

u(y, t ) − u(y, t = 0 ) u(y = h, t ) − u(y = h, t = 0 )

(4)

where the bracket denotes the ensemble-averaged quantity, y is the distance from the wall and t is the elapsed time after the commencement of the acceleration. He and Seddighi (2015) reported that in the case of rapid acceleration, a laminar flow similar to Stokes’ first problem appears at the beginning of acceleration and then develops into a final turbulent state and this laminarturbulent transition occurs even if the final Reynolds number is low. The present results also show that even for the small difference between final and initial Reynolds numbers (Fig. 8(c)) as well as a large difference (Fig. 8(a)), if the acceleration rate is high (T ∗ = 0.05 or 0.1), the change in mean velocity is very similar to the Stokes’ solution. As time progresses, it develops into a final turbulent flow as shown in Figs. 3(a) and 6(a). When the final Reynolds number is high, the development of the flow is similar to a bypass-like transition. On the other hand, if the final Reynolds number is low, the flow progressively evolves to the final turbulent state, and the formation and instability of the streaks is not dominant mechanism in the transient process. Fig. 9 shows the development of mean velocity and streamwise turbulent intensity profiles during transitional and turbulent periods for the cases of rapid acceleration. The mean velocities are non-dimensionalized by the friction velocity of the respective instants. The red line indicates the mean velocity, where cf is a local minimum after the acceleration period. As shown in Fig. 9(a),

in the case of high final Reynolds number (Reτ ,f = 395), the mean velocity profile at the time of cf minimum is significantly shifted upward as compared with those at other instants since the friction velocity at that time is much less than that corresponding to the final steady flow. However, as shown in Fig. 9(b), when the difference between the initial and final Reynolds numbers is not significant, the mean velocity profile corresponding to the local minimum of cf does not exhibit a significant difference relative to the mean velocity at other instants and is similar to that of the quasi-steady state. Fig. 9(c) and (d) show the development of the streamwise turbulent intensity through the transition. Here, the profiles are plotted in outer-region scaling. For a high Reynoldsnumber-ratio case (Reτ ,f = 395), the peak value of urms decreases first due to the abrupt increase in mass flow rate, and it overshoots and then decreases to the final value corresponding to the final Reynolds number. The peak location moves closer to the wall during the transitional period. In the outer region (y/h > 0.3), urms monotonically increases in time after the acceleration. The significant variation in the urms profile indicates that the streak plays a dominant role in the transitional period as shown in Fig. 4. On the other hand, a low Renolds-number-ratio case shows that variation in urms profile is not significant in Fig. 9(d), although it is similar to that observed in the high Reynolds-number-ratio case. Consistently with He and Seddighi (2015), this suggests that streaks do not play a significant role in the transient period when the Reynolds-number-ratio is low. Fig. 10 shows the development of mean velocity and streamwise turbulent intensity profiles during transient period under mild acceleration (T ∗ = 5.0) for Reτ , f = 395 and 250, respectively. During the transient period, the mean velocities are almost the same as those of the non-accelerating flow. The profile of r.m.s. of fluctuating streamwise velocity changes progressively during the accel+ eration period. Even for high Reτ ,f /Reτ ,i case (Fig. 10(c)), the urms profiles at the beginning and end of the acceleration are in good

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Fig. 9. Development of the profiles of mean velocity (a, b) and streamwise turbulent intensities (c, d) during transitional and turbulent periods under rapid acceleration: (a, c) Reτ ,f = 395, T∗ = 0.1; (b, d) Reτ ,f = 250, T∗ = 0.05. Points in inset cf (t) diagram correspond to respective instants in velocity profiles. Red lines correspond to time when skin friction is a local minimum.

Fig. 10. Development of the profiles of mean velocity (a, b) and streamwise turbulent intensities (c, d) during transient period under very mild acceleration (T ∗ = 5.0) for Reτ , f = 395 (a, c) and Reτ , f = 250 (b, d). Points in inset cf (t) diagram correspond to respective instants in the velocity and urms profiles.

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Fig. 11. Development of maximum value of rms of fluctuating velocities normalized by bulk mean velocity: (a) Reτ ,f = 395; (b) Reτ ,f = 250.

agreement with the profiles of the steady case corresponding to the initial and final Reynolds numbers, respectively. Furthermore, + the maximum value of urms remains unchanged and occurs at almost the same wall-normal location during the transient period. Fig. 10 clearly indicates the quasi-steady characteristics of the transient flows and thus suggests that the laminar-turbulent transition does not occur for the mild flow acceleration. Fig. 11 shows the development of the maximum value of the root mean square (rms) of the fluctuating velocities normalized by the ensemble-averaged bulk mean velocity. When examining the case in which Reτ ,f = 395 and T∗ = 0.1 (Fig. 11(a)), the maximum streamwise turbulent intensity, urms,max decreases abruptly in its initial phase because the turbulence cannot react to a rapid change in the mean flow. Thereafter, it increases and reaches the final value after the overshooting (as indicated by the vertical arrow). It has also been reported that, in experiments examining pipe flow accompanied by a strong acceleration, the streamwise turbulent intensity decreases after an overshoot occurs (He and Jackson, 20 0 0). The developments of the vrms,max and wrms,max are similar, but significantly different from that of urms,max . The streamwise component is increased first, while the other two components denote a long delay before they increase. Such behaviors of the turbulent intensities reflect the flow characteristics of the bypass-like transition in the accelerated channel flow. The occurrence of a strong acceleration leads to a streaky structure and this structure is exposed to a secondary instability to form turbulent spots, which in turn develop into turbulent structures before reaching the final state. The generation of turbulent vortical structures promotes energy redistribution (Jeong et al., 1997). Thereby, the cross-stream component turbulent intensities increase, while urms,max overshoots and then decreases slightly. Since the localized peak in urms,max is associated with the enhanced streaky structure, followed by the generation of turbulent spots, the overshoot in the evolution of urms,max can be used as an indicator of the transition in the present transient channel flow. Meanwhile, for Reτ ,f = 250 and T∗ = 0.05 (when the acceleration rate is significant), urms,max increases after

decreasing during the initial phase. However, no clear overshoot is observed. This confirms to the fact that no bypass-like transition has occurred in this case. Fig. 12(a) is a map of the occurrence of the bypass-like transition for every acceleration condition tested in the present study. The closed symbol corresponds to the bypass-like transition. If all three of the conditions below are satisfied, it is determined that a bypass-like transition has occurred. 1) The laminar-like behaviors (e.g., c f ∼ Re−1 and u∧ ≈ m UStokes ) are observed at the early stage of the transient flow. 2) Distinct turbulent spots are generated which then develop into vortical structures in clusters. 3) The overshoot in the evolution of urms,max is clearly observed. In Fig. 12(a), when the acceleration time is long (i.e., T∗ ≥ 3.0), corresponding to a low acceleration rate, the bypass-like transition is not observed. When the ratio of the final to initial Reynolds number is small (Reτ ,f = 250), the bypass-like transition is not clear regardless of the acceleration time. Such changes in the flow field are expected to be closely related to the amount of the impulse applied to the flow during the temporal acceleration. By integrating the pressure gradient (Eq. (3)) over the acceleration period, the total impulse exerted on the flow during the acceleration can be calculated as

J∗ =



T∗ 0



−

 ∂ p Rem (t = T ∗ ) − Rem (t = 0 ) dt = ∂x 2Reτ ,i    = Increase o f f low rate (A )

 +

T∗ 0





ReDean (t ) τ Reτ ,i



2

dt

(5)



= Increase o f viscous f riction (B )

Here, J∗ refers to the dimensionless total impulse which consists of the increased flow rate (A) and increased viscous friction

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Fig. 12. Conditions for transient flows similar to bypass transition due to temporal accelerations based on the (a) the final Reynolds number, Reτ ,f and acceleration duration, T∗ and (b) the dimensionless total impulse exerted on the flow during the acceleration, J∗ and its relative contribution to the increase in the flow rate, A/J∗ . Red symbols denote the test cases of the present study. Closed symbols indicate the occurrence of the bypass-like transition. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(B). Fig. 12(b) shows the relative contribution of the total impulse to the increase in the flow rate (A/J∗ ), relative to the total impulse applied during the acceleration. Fig. 12(b) reveals that the transition in the accelerated channel flow occurs when the following conditions are satisfied: 1) The total impulse exerted on the flow exceeds a critical value: J∗ > Jcr . 2) The relative increase in the flow rate (A) should be significant in comparison with the increased viscous friction (B). In other words, most of the total amount of impulse contributes to the increase in the flow rate: A/J∗ > c. The condition 2) can be interpreted as a condition for the acceleration duration T∗ . When T∗ is short, the turbulence in the initial flow cannot immediately adapt to the change in the mean flow, i.e., it can be regarded as a frozen turbulence. The initial turbulence acts to the accelerated flow as a disturbance and it gives rise to a transition. In the frozen turbulence stage, Reynolds shear stress is much less than that in the non-transient flow because the turbulence does not grow yet. Because a contribution of the Reynolds shear stress to the skin friction is significant in wall-bounded flows (Fukagata et al., 2002), this results in the lower increase in the wall skin friction during the flow acceleration as compared with the increase in the flow rate. On the other hand, when T∗ is long, the bypass-like transition does not occur even if the Reynolds number ratio between initial and final states is high enough to cause the bypass-like transition by very rapid acceleration. In this case, the initial turbulence progressively develops according to the increase in the mean flow, which causes a significant increase in the wall skin friction relative to the increase in the flow rate. Meanwhile, He and Jackson (20 0 0) suggested the following dimensionless ramp rate parameters to characterize the transient flow inside a pipe:

γ=

dUm Um,i / dt D/uτ ,i

(6)

δ=

dUm Um,i / dt ν /u2τ ,i

(7)

where γ refers to the ratio of the initial flow time scale (D/uτ ,i ) to the accelerating time scale. If this value is significantly greater than unity, the difference from the steady flow becomes large. When it is lower than unity, the transient flow exhibits a similar pattern to that of the steady flow. The variable δ refers to the ratio of the time scale of the turbulence production (ν /u2τ ,i ) to the accelerating time scale. This ramp rate parameter can be applied to the fast

transient flow and is relevant to the time delay of turbulence production in the acceleration flow. In the same way as γ , if its value is large during acceleration, the flow pattern becomes very different from that of the steady flow. The above ramp rate parameters in the present study and existing studies are listed in the two right-hand columns of Table 1. Those underlined and bold correspond to the cases where the bypass-like transition is clearly observed. When the bypass-like transition occurs clearly in the present study, γ was calculated as being greater than 1. Also, in previous studies, when γ > 1, the bypass-like transition was clear in most cases. Additionally, δ was relatively large when the bypass-like transition was distinct. However, there are some cases when the bypass-like transition is not clear even when the ramp rate parameter is large. When the difference between the initial and final Reynolds numbers is small (for instance, when Reτ increases from 180 to 250 in the present study, and when Reτ increases from 179 to 195, as observed by He and Seddighi (2015)), a bypass-like transition does not occur although the ramp rate parameter was significant. Therefore, the ramp rate parameters suggested in the previous study were shown not to correctly characterize the transient flow in every case. The existing ramp rate parameters (γ and δ ) include only the effects of the initial Reynolds number (Rei ), and acceleration rate (dRe/dt) but the information regarding the final Reynolds number (Ref ) is ignored. Therefore, a new parameter including all of the factors (Rei , Ref , dRe/dt) is required to characterize the accelerated flow correctly. The parameter suggested in the present study, the dimensionless impulse J∗ , incorporates all such factors. In Eq. (5), J∗ is the sum of the increased flow rate (A) and the increased viscous friction (B). The increased flow rate (A) becomes large when the difference between the initial and final Reynolds number increases. The increased viscous friction (B) becomes more important as the acceleration time (T∗ ) increases. In addition, the finding of the present study, namely, that the bypass-like transition can occur when J∗ exceeds a critical value, supports the results of the previous studies which state that the localized vortex packet is observed when the ratio of the final and initial Reynolds number is relatively high (He and Seddighi, 2015). This cannot be explained by using only the existing ramp rate parameters (γ and δ ). Moreover, since J∗ takes the impulse consumed by the increased viscous friction into account, it can be explained that, in the ramp-up case, the bypass-like transition is not clear even when the Reynolds number ratio is large. Therefore, the dimensionless impulse suggested in the present study can be utilized as a general criterion for the bypass-like transition in the accelerated flow, instead of the exist-

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ing parameters. The quantitative criterion for a clear bypass-like transition in the accelerated flow, (J∗ > 10 and A/J∗ > 0.8), as suggested in the present study, should be identified by further studies. 4. Conclusions In the present study, a detailed investigation of accelerated turbulent flows in a channel was undertaken by performing direct numerical simulations of transient turbulent flows. The simulations were started with a fully developed turbulent channel flow to which temporal accelerations were then applied. During the acceleration, nearly linear increasing excursions in the flow rate were imposed between the constant initial and final values. The initial Reynolds number based on the friction velocity was fixed at Reτ ,i = 180 and four different final Reynolds numbers (Reτ ,f = 250, 300, 350, and 395) were selected to determine the effects of the Reynolds number ratio (Reτ ,f /Reτ ,i ) on the transient channel flows. To elucidate the effects of the increase in the rate of the flow rates on the near-wall turbulence, we also examined a wide range of durations of acceleration. The turbulent statistics and instantaneous flow fields revealed that a rapid increase in the flow rate invokes similar phenomena to the bypass transition in the laminar boundary layer flow, which consist of distinct turbulent spots, followed by new vortical structures, starting to appear in clusters. It has also been observed that the bypass-like transition is accompanied by the laminar-like behaviors at the early stages and the overshoot in the time evolution

of the maximum streamwise turbulent intensity. In contrast, the flow evolves progressively and the transition does not occur clearly during mild flow acceleration. When the difference in the Reynolds number between the initial and final states is small during the acceleration, the bypass-like transition phenomena becomes obscure even for a strong acceleration. The present study proposed new criteria based on the impulse applied to the flow by the temporal acceleration in order to explain the bypass-like transition to the new turbulence. This transition is mainly due to the larger contribution of the impulse to the flow rate increase compared with that to the viscous friction. Acknowledgments This work was supported under the framework of international cooperation program managed by National Research Foundation of Korea (NRF-2014K2A1A2048497) and the National Institute of Supercomputing and Network/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2016-C3-0 0 08). KK would like to acknowledge the support for the code optimization from National Institute of Supercomputing and Networks (NISN). Appendix. Computational details To ascertain the reliability and accuracy of the present simulation, comparison of the turbulent statistics with DNS data of

Fig. A-1. Comparison of the simulations of steady channel flows with Moser et al. (1999) (denoted by MKM). (a) Reτ = 180; (b) Reτ = 395.

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that the present grid resolution is sufficient to resolve the turbulent statistics. It has been shown that for the rapid acceleration, a larger streamwise domain is required since the domain should be long enough to encounter significantly elongated streaks which are formed during the early transient stage (He and Seddighi, 2013). In Fig. A-2, the two-point correlations of the streamwise velocity in the x- and z-directions are displayed for a rapid acceleration case (Reτ , f = 250 and T ∗ = 0.05). Fig. A-2 demonstrates that the computational domain is sufficiently large since the two-point correlations fall off to zero values for large separations. Since the present flow configuration is not statistically stationary, the flow statistics have been computed by averaging in the homogeneous spatial directions and across the several independent simulations. We have computed the ensemble-averaged Reynolds stresses for a rapid acceleration case according to the number of realizations. As seen in Fig. A-3, four realizations are enough to obtain converged turbulent statistics. References

Fig. A-2. Development with time of profiles of (a) streamwise and (b) spanwise correlations of the streamwise velocity at y/h = 0.11 for the case of Reτ , f = 250 and T ∗ = 0.05.

Fig. A-3. Profiles of turbulent intensities and Reynolds shear stress at t ∗ = 0.5 for the case of Reτ , f = 395 and T ∗ = 1.0. M is the number of realizations used for averaging the statistics.

Moser et al. (1999) are made and presented in Fig. A-1. The mean velocity profiles at the initial and final Reynolds numbers are shown. Comparisons are extended to the turbulent intensities and r.m.s. of vorticity fluctuations. The present results are in excellent agreement with the DNS data both Reynolds numbers, indicating

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