3.1. The general picture

The flow structures at several instants during the transient period in selected cases are shown in figure 1 using three-dimensional isosurface plots of $u^{\prime }/U_{b0}$ and ${\it\lambda}_{2}/(U_{b0}/{\it\delta})^{2}$ . Here, $u^{\prime }(=u-\bar{u})$ is the instantaneous streamwise fluctuating velocity and ${\it\lambda}_{2}$ is the second largest eigenvalue of the symmetric tensor $\unicode[STIX]{x1D64E}^{2}+{\it\bf\Omega}^{2}$ , where $\unicode[STIX]{x1D64E}$ and ${\it\bf\Omega}$ are the symmetric and antisymmetric parts of the velocity gradient tensor $\boldsymbol{{\rm\nabla}}\boldsymbol{u}$ . This was initially proposed by Jeong & Hussain (Reference Jeong and Hussain1995) as an effective indicator for vortical structures and has since been used in the study of turbulence and transition. All the cases shown in the figure have the same initial Reynolds number ( $\mathit{Re}_{0}=2800$ ) but very different final Reynolds numbers, namely 12 600, 5000 and 3100 for S11, S13 and S16 respectively. These can be compared with the flow studied in HS2013 in which $\mathit{Re}$ was varied between 2800 and 7400.

Figure 1. Streaks and vortex structures in three-dimensional plots of isosurfaces in (a) S11, (b) S13 and (c) S16. Streaks are shown in green/blue with $u^{\prime }/U_{b0}=\pm 0.35$ and vortical structures are shown in red with ${\it\lambda}_{2}/(U_{bo}/{\it\delta})^{2}=-5$ .

We first consider the flow with the highest change of flow rate, i.e. case S11. In the initial flow ( $t^{\ast }=0$ ), patches of fluids with high or low fluctuating velocities are clearly in existence and some hairpin structures are also identifiable through the isosurfaces of ${\it\lambda}_{2}$ . These are rather uniformly distributed in the flow field, showing the picture of a typical turbulent shear flow. During the early period (appropriately, $t^{\ast }<29$ ), elongated streaks are formed, as evident by the long tubes of isosurfaces of positive and negative $u^{\prime }/U_{b0}$ which appear alternately to each other. Such flow structures are common and representative in the pre-transition and transition regions of boundary layers (e.g. Jacobs & Durbin Reference Jacobs and Durbin2001; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001). The number of hairpins appears to reduce during the early stage of this period, but new vortical structures start to appear in clusters at approximately $t^{\ast }=29$ , which indicates the generation of turbulent spots, signifying the onset of transition. During the period to follow (approximately, $29<t^{\ast }<55$ ), such turbulent spots grow to occupy more spaces, joining with each other, and eventually at approximately $t^{\ast }\sim 55$ , the entire surface is covered with newly generated turbulence. The isosurface tubes (streaks) break up along with the generation of turbulent spots as the transition progresses. The vortices often occur around the low-speed streaks accompanying their breakup in this case, which is again similar to those shown in boundary layer bypass transition (e.g. Jacobs & Durbin Reference Jacobs and Durbin2001; Schlatter et al. Reference Schlatter, Brandt, De Lange and Henningson2008). The basic features of the flow described above are similar to those found in HS2013, although the streaks during the pre-transition and transition periods are stronger and more striking in S11.

Figure 2. Development of the friction coefficient with equivalent Reynolds number ( $\mathit{Re}_{t}=tU_{b1}^{2}/{\it\nu}$ ): (a) effect of varying $\mathit{Re}_{0}$ (same $\mathit{Re}_{1}$ ); (b) effect of varying $\mathit{Re}_{1}$ (same $\mathit{Re}_{0}$ ).

Overall, case S13 (which has a ‘medium’ Reynolds number ratio) follows a similar trend (figure 1 b), but the strength of the streaks is much weaker and the generation of turbulent spots seems to occur at a lower rate and the changes are less striking. This trend is similar to the transition of a boundary layer subject to a higher level of FST. For example, in a boundary layer subject to a 7 % FST, Jacobs & Durbin (Reference Jacobs and Durbin2001) found that although streaks and jets are present in such flows, they are surrounded by elevated turbulence and distinct turbulent spots are much more difficult to discern. In the case of S16 where the Reynolds number was only increased by 10 %, the flow structures appear to be characteristically different (figure 1 c). It is clear that the turbulent activities at the end of the transient are stronger than those at the beginning, but the distinct transition is not seen. There are no clear signs of generation of elongated streaks at the pre-transition stage nor obvious enhancement of the generation of turbulent spots during the transition period.

The above results show that the phenomenological features of transition of the transient flow identified in HS2013 are strong and well pronounced when the difference between the final and initial Reynolds numbers is high, but less pronounced, though still clearly in existence, when it is reduced. When the Reynolds number difference is very low, however, the transitional process is difficult to identify from the visualization presented above.

In the rest of the paper, various statistics of the mean flow and turbulence are analysed to quantify the flow structures observed above to establish the effect of varying the initial and final Reynolds numbers. It will become clear that although the visualization of the flow structures shows that the transient process of lower-Reynolds-number-ratio cases appears to be qualitatively different, they in fact show as much a transition characteristic as the higher-Reynolds-number-ratio cases in the parameters used to identify such processes, and the critical equivalent Reynolds number (defined later) in all the flows studied can be correlated using a simple expression.

3.2. Effect of initial and final Reynolds numbers

The variations of the friction coefficient ( $C_{f}={\it\tau}_{w}/({\it\rho}U_{b}^{2}/2)$ , where ${\it\tau}_{w}$ is the wall shear stress) in the various cases investigated in this study are inspected to establish the overall behaviour of the flow response (figure 2). Here, we use an alternative non-dimensional time to $t^{\ast }$ , namely

which is referred to as the equivalent Reynolds number herein. Considering the bulk velocity $U_{b1}$ as a characteristic convective velocity, together with the time $t$ , it defines a length $x=U_{b1}t$ , representing the distance that a fluid particle has travelled after the commencement of the transient. As a result, the equivalent Reynolds number can be written as $\mathit{Re}_{t}=(xU_{b1})/{\it\nu}$ , mirroring the Reynolds number $\mathit{Re}_{x}$ used in the boundary layer based on the free-stream velocity and the distance from the leading edge. It will be demonstrated later that $\mathit{Re}_{t}$ indeed has the same significance in a transient flow transition as $\mathit{Re}_{x}$ in the boundary layer transition, although direct quantitative comparison between the two quantities is not appropriate. It is useful to note that $\mathit{Re}_{t}=t^{\ast }\mathit{Re}_{1}$ , where $t^{\ast }=tU_{b1}/{\it\delta}$ and $\mathit{Re}_{1}=U_{b1}{\it\delta}/{\it\nu}$ .

Focusing on case S11 first, it can be seen that the $C_{f}$ reaches a very high value (off scale) immediately after the commencement of the excursion due to the inertia resulting from the rapid flow acceleration in a very short period of time. From there, it decreases monotonically until approximately $\mathit{Re}_{t}=3.6\times 10^{5}$ (or, $t^{\ast }=29$ ) where it reaches a minimum, after which it increases again to approximately the final steady value at $\mathit{Re}_{t}=7.0\times 10^{5}$ , or $t^{\ast }=55$ . Comparing with the flow visualization of the corresponding case (figure 1 a), it is clear that the timing of the minimum $C_{f}$ roughly coincides with the initial stage of the generation of turbulence spots. Following HS2013, we refer to the time when $C_{f}$ reaches the minimum as the point of the ‘onset of transition’ and the corresponding non-dimensional time as the critical equivalent Reynolds number $\mathit{Re}_{t,cr}$ or, equivalently, the critical time $t_{cr}^{\ast }$ .

The effect of varying the initial and final Reynolds numbers ( $\mathit{Re}_{0}$ and $\mathit{Re}_{1}$ ) on the overall flow behaviour is now studied by comparing the responses of $C_{f}$ in the various cases. Figure 2(a) shows that $\mathit{Re}_{t,cr}$ reduces monotonically with increase of $\mathit{Re}_{0}$ . For a fixed $\mathit{Re}_{1}$ of 7400, $\mathit{Re}_{t,cr}$ reduces from $1.47\times 10^{5}$ to $5.50\times 10^{4}$ when $\mathit{Re}_{0}$ is increased from 2800 to 5300. In addition, alongside the reduction of $\mathit{Re}_{t,cr}$ , the minimum friction coefficient increases significantly, showing a progressively smaller ‘undershooting’ of the final $C_{f}$ . Figure 2(b) shows that increasing $\mathit{Re}_{1}$ results in an increase in $\mathit{Re}_{t,cr}$ . For a fixed $\mathit{Re}_{0}$ of 2800, as $\mathit{Re}_{1}$ is increased from 3100 to 12 600, $\mathit{Re}_{t,cr}$ increases from $3.30\times 10^{4}$ to $3.65\times 10^{5}$ . In addition, the minimum $C_{f}$ varies from a very small ‘undershooting’ at $\mathit{Re}_{1}=3100$ to a strong one at $\mathit{Re}_{1}=12\,600$ . The final $C_{f}$ reduces with increase of $\mathit{Re}_{1}$ . It is of most interest that the friction factor in all cases, including those with a very small $\mathit{Re}$ ratio, shows the same characteristic behaviour even though $\mathit{Re}_{t,cr}$ and the level of undershooting of $C_{f}$ can be very different in the different cases. It will be shown later that the flow before the critical point always behaves like a laminar flow despite the significant differences in the initial Reynolds number and the level of flow perturbation. Overall, the critical time $t_{cr}^{\ast }$ shows a similar trend to that of the equivalent Reynolds number $\mathit{Re}_{t,cr}$ described above, reducing with increase of the initial Reynolds number or decrease of the final Reynolds number.

The mechanisms by which the initial and final Reynolds numbers affect the transition process and the critical equivalent Reynolds number are no doubt very complex. It has been well established in boundary layer research that the transition is strongly influenced by the level of FST, referring to Andersson et al. (Reference Andersson, Berggren and Henningson1999), Luchini (Reference Luchini2000), Brandt et al. (Reference Brandt, Schlatter and Henningson2004), Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005), Nagarajan et al. (Reference Nagarajan, Lele and Ferziger2007) and Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008). Moreover, Brandt et al. (Reference Brandt, Schlatter and Henningson2004) and Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008) also showed that both the critical Reynolds number and also possibly the mechanisms of transition are affected by the length scales of the FST. In light of such understanding, the following factors are candidates for consideration, which can potentially influence the behaviours of the transient process when the Reynolds numbers are varied.

1. (i) $\mathit{Re}_{0}$ ( $=U_{b0}{\it\delta}/{\it\nu}$ ), which defines the initial turbulence in terms of the amplitude and time/length scales. The higher $\mathit{Re}_{0}$ is, the lower the initial turbulence intensity is but also the smaller the time/length scales are. It also defines the initial mean velocity profile.

2. (ii) $\mathit{Re}_{1}$ , which defines the ‘free-stream’ velocity. Arguably this is the most important velocity of the transient flow.

3. (iii) $(\mathit{Re}_{1}-\mathit{Re}_{0})$ , which defines $(U_{b1}-U_{b0})$ , is the cause of the change. Indeed, the time-developing boundary layer is characterized by this velocity (see § 3.3).

4. (iv) The acceleration rate, $(U_{b1}-U_{b0})/{\rm\Delta}t$ . This could potentially be a factor. However, in all cases considered herein, the acceleration is very rapid and the flow increase can be viewed as a step change. Tests with the acceleration rate increased by an order of magnitude show no effect on the transition behaviour. For a transient with a much slower rate, the acceleration rates does affect the $\mathit{Re}_{t,cr}$ but the general transition process remains similar (Seddighi et al. Reference Seddighi, He, Vardy and Orlandi2014).

5. (v) The initial FST intensity ( $\mathit{Tu}_{0}$ ). This is dependent on both $\mathit{Re}_{0}$ and $\mathit{Re}_{1}$ , which can be represented by $(u_{rms,0}^{\prime })_{max}/U_{b1}$ , as explained later, where $(u_{rms,0}^{\prime })_{max}$ is the peak value of the wall-normal profile of the r.m.s. of the streamwise turbulent fluctuating velocity at $t=0$ .

We have investigated the various mechanisms discussed above and correlated the data against alternative parameters. It has become evident that, as far as the critical Reynolds number is concerned, the dominant effect of varying $\mathit{Re}_{0}$ and $\mathit{Re}_{1}$ is through changing the initial FST intensity, as demonstrated below.

First, let us derive a way of describing the initial turbulent intensity, that is, the equivalent FST. We consider a very early instant of the transient flow following the step increase of the flow rate. At this stage, the turbulence remains unchanged from that of the initial flow and the mean velocity is that of the final flow. The turbulence in a fully developed channel is very different from the FST of the boundary layer, being highly anisotropic and non-uniform normal to the wall. For simplicity and unambiguity, we choose the peak value of the wall-normal profile to represent the turbulence level. Consequently the initial free-stream turbulence intensity can be written as

The ratio $(u_{rms,0}^{\prime })_{max}/U_{b0}$ is the peak turbulence intensity of the initial flow before the commencement of the transient. Recently, there has been considerable interest in the effect of the Reynolds number on the peak turbulence intensity in wall units, i.e.  $(u_{rms}^{\prime +})_{max}=(u_{rms}^{\prime })_{max}/u_{{\it\tau}}$ (Hultmark, Bailey & Smits Reference Hultmark, Bailey and Smits2010; Ng et al. Reference Ng, Monty, Hutchins, Chong and Marusic2011; Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013). It has been established that $(u_{rms}^{\prime +})_{max}$ varies with the Reynolds number in a boundary layer and a channel flow, whereas there are still some dispute on whether $(u_{rms}^{\prime +})_{max}$ is also dependent on the Reynolds number for pipe flows. In any case, it is well established that the turbulence intensity expressed in the outer scaling, $(u_{rms}^{\prime })_{max}/U_{b}$ , is a function of the Reynolds number. The present DNS data for $2800<\mathit{Re}<12\,600$ show that $(u_{rms}^{\prime })_{max}/U_{b}\sim \mathit{Re}^{-0.1}$ , and hence the free-stream turbulence ( $\mathit{Tu}_{0}$ ) defined in (3.2) is proportional to $(U_{b0}/U_{b1})\mathit{Re}_{0}^{-0.1}$ . In fact, the following expression represents the DNS $\mathit{Tu}_{0}$ extremely closely:

Figure 3. Dependence of the critical equivalent Reynolds number on the velocity ratio.

Figure 4. Dependence of the critical equivalent Reynolds number on the initial FST intensity.

Figure 3 shows the critical equivalent Reynolds number ( $\mathit{Re}_{t,cr}$ ) plotted against the velocity ratio (equivalent to the Reynolds number ratio) in double logarithmic scale. The data correlate reasonably well, which suggests a strong dependence of $\mathit{Re}_{t,cr}$ on the velocity ratio. In fact, all the data of the $\text{S}1X$ series (with the same $\mathit{Re}_{0}$ but different $\mathit{Re}_{1}$ ) lie nearly perfectly on a straight line, implying that $\mathit{Re}_{t,cr}$ and $U_{b0}/U_{b1}$ are related in a power-law form. On the other hand, all of the data of series $\text{S}0X$ (with fixed $\mathit{Re}_{1}$ but varying $\mathit{Re}_{0}$ ) appear also to lie on a straight line, which suggests a systematic $\mathit{Re}_{0}$ effect. By trial and error, it has been established that the effect of varying the initial Reynolds number is closely represented by $\mathit{Re}_{0}^{-0.1}$ , and the two series of data are brought closely together when $\mathit{Re}_{t,cr}$ is shown as a function of $(U_{b0}/U_{b1})(1/\mathit{Re}_{0}^{0.1})$ . Now, comparing this knowledge with (3.3), it is clear that the critical equivalent Reynolds number is likely to be a function of the free-stream turbulence. It can indeed be seen from figure 4, where $\mathit{Re}_{t,cr}$ is shown with respect to $\mathit{Tu}_{0}$ , that the data from both series now lie strikingly closely along a straight line, which can be well represented by

where $\mathit{Re}_{t,cr}=t_{cr}U_{b1}^{2}/{\it\nu}$ and $\mathit{Tu}_{0}$ is defined by (3.2), which can be estimated using (3.3).

This result shows that the effect of varying $\mathit{Re}_{0}$ or $\mathit{Re}_{1}$ on $\mathit{Re}_{t,cr}$ simply comes down to the variation of the initial turbulence intensity. All other factors discussed above in the list are insignificant as far as the critical Reynolds number is concerned. For example, the change of the Reynolds number of the initial flow leads to some change in the length scale of the initial FST, comparing for example cases S01 and S04, but this has no direct effect on the critical Reynolds number except for that through changing $\mathit{Tu}_{0}$ . This result is in contrast to the conclusions reached by Brandt et al. (Reference Brandt, Schlatter and Henningson2004) and Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008), who showed that the length scales of the FST have a major effect on $\mathit{Re}_{cr}$ . It should be pointed out, however, that the insensitivity of the results to the length scale observed in the present study should be treated carefully since the change of $\mathit{Re}_{0}$ is limited.

It has been well established through both theoretical and experimental investigations that $\mathit{Re}_{cr}\sim \mathit{Tu}_{0}^{-2}$ for a spatially developing boundary layer (Andersson et al. Reference Andersson, Berggren and Henningson1999; Brandt et al. Reference Brandt, Schlatter and Henningson2004; Fransson et al. Reference Fransson, Matsubara and Alfredsson2005; Ovchinnikov et al. Reference Ovchinnikov, Choudhari and Piomelli2008). In particular, Andersson et al. (Reference Andersson, Berggren and Henningson1999) proposed

It is interesting to see that (3.4) and (3.5) are similar in form, even though both the multiplier and the exponent are different. As mentioned before, the value of $\mathit{Re}_{t,cr}$ in the transient channel flow and that of $\mathit{Re}_{x,cr}$ of the boundary layer are not directly comparable; indeed, the two flows are not equivalent and hence the differences in the multipliers are trivial. On the other hand, the fact that the exponent of (3.4) for the transient flow is different from the theoretical value of ‘ $-2$ ’ for the boundary layer may be of interest, but the implications of this observation are not explored here. It is noted that, previously, Blumer & Van Driest (Reference Blumer and Van Driest1963) established an empirical correlation for boundary layer transition based on experimental data as

where $a=10^{-4}$ and $b=62.5\times 10^{-8}$ . The value of $\mathit{Re}_{x,cr}$ calculated from this expression is quite similar to that of (3.5), but $\mathit{Re}_{x,cr}$ is not strictly related to $\mathit{Tu}_{0}$ through a $-2$ power law.

Another interesting feature of the transition is the period of the transition phase, that is, the time between the onset of the transition and the completion of it. As for the onset of transition ( $t_{cr}$ ), we again use $C_{f}$ to define the completion of transition, and assume that the transition is completed ( $t_{turb}$ ) when $C_{f}$ reaches its first peak. The period of the transition phase is the difference between these two times. We can again express it in terms of the equivalent Reynolds number as

Figure 5 shows ${\rm\Delta}\mathit{Re}_{t,cr}$ versus $\mathit{Re}_{t,cr}$ . The trend of the data can be reasonably well represented by the straight line shown in the figure, but there are some scattered points. Whether such scattered points may be related to potentially different transition mechanisms needs further investigation. Various researchers have previously investigated the transitional length for boundary layers. Dhawan & Narasimha (Reference Dhawan and Narasimha1958) and Narasimha, Narayanan & Subramanian (Reference Narasimha, Narayanan and Subramanian1984) suggested a power-law relation between ${\rm\Delta}\mathit{Re}_{x,cr}$ and $\mathit{Re}_{x,cr}$ , and the latter proposed

Figure 5. Relationship between the period of the transition phase and the critical Reynolds number. Here, $\mathit{Re}_{cr}$ refers to $\mathit{Re}_{x,cr}$ and $\mathit{Re}_{t,cr}$ respectively for boundary layer and transient channel flow.

The onset and the completion of transition are defined based on the level of intermittency at 0.25 and 0.75 respectively. In addition, in contrast to the $\mathit{Re}_{t,cr}$ used herein, which is based on the onset of the transition, $\mathit{Re}_{x,cr}^{\prime }$ in the above equation is defined as the location at an intermittency level of 0.5, that is, halfway between the onset and the completion of the transition. More recently, Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005) proposed an alternative formulation based on a large number of experimental data:

The onset and completion of transition are defined based on intermittency levels of ${\it\gamma}=0.1$ and 0.9 respectively and ${\mathit{Re}^{\prime }}_{x,cr}$ is again for ${\it\gamma}=0.5$ . Equations (3.8) and (3.9) are shown in figure 5 so that the general trends can be compared with the data from the present transient channel flow. In order for such a comparison to be made, ${\mathit{Re}^{\prime }}_{x,cr}$ in (3.8) and (3.9) is replaced by $\mathit{Re}_{x,cr}+0.5{\rm\Delta}\mathit{Re}_{x,cr}$ , to ensure that the definition of the critical Reynolds number is consistent. It is noted that, incidentally, the present data can be reasonably well represented by the correlations of Narasimha et al. (Reference Narasimha, Narayanan and Subramanian1984) and Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005) if a factor of 0.5 is applied to the ${\rm\Delta}\mathit{Re}_{cr}$ of the present data. In particular, the present data appear to support Fransson et al.’s idea that ${\rm\Delta}\mathit{Re}_{t,cr}$ and $\mathit{Re}_{t,cr}$ are linearly related, and that there is a minimum ${\rm\Delta}\mathit{Re}_{t,cr}$ for small values of  $\mathit{Re}_{t,cr}$ .

3.3. The time-developing boundary layer

The flow response in the initial (pre-transition) stage of the transient channel flow was described as a time-developing boundary layer in HS2013. Following the rapid increase in the flow rate, a very thin boundary layer of high strain rate is formed adjacent to the wall, but the bulk of the flow increases as a ‘solid body’ with no change in its velocity profile. This boundary layer, which progressively develops into the flow with time, exhibits a character that is similar to the laminar boundary layer before the onset of transition. Here, we study the behaviour of the time-developing boundary layer in the various cases by examining the perturbing velocity:

where $\bar{u}_{c}$ is the centreline velocity, $y^{+0}=yu_{{\it\tau}0}/{\it\nu}$ and $t^{+0}=tu_{{\it\tau}0}^{2}/{\it\nu}$ , where $y$ is the wall-normal distance from the wall, $t$ is the time that has elapsed after the commencement of the transient flow and $u_{{\it\tau}0}$ is the friction velocity at $t=0$ . The benefit of using the wall units based on the initial flow is that the onset of transition occurs at roughly the same time ( $t^{+0}=80{-}110$ , see below), and hence the flow at the same $t^{+0}$ is roughly at the same stage of the pre-transition development.

Figure 6. Development of the perturbation velocity profiles ( $\bar{u}^{\wedge }$ ) in the pre-transition period in all cases studied and comparison with the Stokes solution. Solid lines: present data; dash-dotted: Stokes solution. (a) Perturbation velocity $\bar{u}^{\wedge }$ versus $y^{+0}$ at various non-dimensional times ( $t^{+0}$ ); the profiles are shifted by an amount proportional to the time. Lines (1)–(3) show the thickness of the boundary layer of $\bar{u}^{\wedge }$ , ${\it\delta}_{99}$ ; the displacement boundary layer of $\bar{u}$ , ${\it\delta}_{u}^{\ast }$ ; and that of $\bar{u}^{\wedge }$ , ${\it\delta}^{\ast }$ . (b) Perturbation velocity $\bar{u}^{\wedge }$ versus $y/2\sqrt{{\it\nu}t}$ for all cases at various instants during the pre-transition phase ( $t^{+0}\leqslant 80$ ).

The profiles of $\bar{u}^{\wedge }$ at various $t^{+0}$ of all the cases are plotted in figure 6(a) against $y^{+0}$ . The profiles at any time $t^{+0}$ are shifted by a distance proportional to $t^{+0}$ . Also shown in the figure for comparison is the solution of Stokes’ first problem (Schlichting & Gersten Reference Schlichting and Gersten2000):

where ${\it\eta}=y/2\sqrt{t{\it\nu}}$ and $\text{erf}({\it\eta})=2/\sqrt{{\rm\pi}}\int _{0}^{{\it\eta}}\text{e}^{-{\it\xi}^{2}}\text{d}{\it\xi}$ . The equation is recast in terms of $\bar{u}^{\wedge }$ , $y^{+0}$ and $t^{+0}$ for plotting. It is clear that, visually, the time-developing boundary layer shows strong similarities to the spatially developing boundary layer. Most interestingly, the profiles in different cases collapse closely on top of each other for $t^{+0}<100$ (that is, in the pre-transition stage). At earlier stages (say, $t^{+0}<50$ ), these velocity profiles are closely represented by the Stokes solution. Later, $50<t^{+0}<100$ , the profiles of the various cases are still in close agreement with each other, but they deviate from the Stokes solution. Further proceeding with time (say $t^{+0}>110$ ), the profiles of the various cases start diverge from each other. The close accordance between the velocity developments of the various flow cases during the pre-transition phase is extraordinary, noting the wide range of Reynolds number ratios examined. The response of the mean velocity in the transient flow is the same in a 10 % increase flow as in a 400 % increase one, irrespective of their very different flow patterns shown in figure 1, for example. Further, the profiles of the velocity $\bar{u}^{\wedge }$ with respect to the non-dimensional parameter ${\it\eta}$ at various times of the pre-transitional phase ( $t^{+0}<80$ ) in all the cases are shown in figure 6(b) together with the Stokes solution. It can be seen that the profiles collapse very well onto a single curve, close to the Stokes solution, with only small spreads, hence showing the similarity of the velocity profiles at different times. Figure 7 shows that the values of the displacement thickness $({\it\delta}^{\ast })$ of the perturbation velocity in the various cases correlates reasonably well when they are normalized by the initial flow wall units, that is, ${\it\delta}^{\ast +0}={\it\delta}^{\ast }u_{{\it\tau}0}/{\it\nu}$ .

Figure 7. Displacement thickness of the perturbation velocity ( $\bar{u}^{\wedge }$ ).

Figure 8. Development of the modified friction coefficient $C_{f,du}^{\prime }$ defined by (3.14) with non-dimensional time $t^{+0}$ in all cases.

A friction coefficient for the perturbation flow can be defined as

where ${\it\tau}_{w,du}=\{{\it\mu}\partial [\bar{u}(y,t^{\ast })-\bar{u}(y,0)]/\partial y\}|_{y=0}={\it\tau}_{w}-{\it\tau}_{w,0}$ is the wall shear stress of the perturbation flow, and ${\it\tau}_{w}$ and ${\it\tau}_{w,0}$ are the values of the wall shear stress of the full flow at times $t$ and 0 respectively. Noting that the wall shear stress of the theoretical solution of Stokes’ first problem is ${\it\tau}_{w}={\it\mu}(U_{b1}-U_{b0})/\sqrt{{\rm\pi}{\it\nu}t}$ (Schlichting & Gersten Reference Schlichting and Gersten2000), the friction coefficient for a transient laminar flow is given by

Now, returning to the data shown in figure 6(a), the fact that the velocity profiles of the various cases overlap each other at any $t^{+0}$ implies that the values of the wall shear stress of $\bar{u}^{\wedge }$ normalized in the manner defined below should collapse on top of each other when expressed in terms of $t^{+0}$ :

which can be reorganized as

The present data in the form of the modified friction coefficient $C_{f,du}^{\prime }$ defined above plotted against $t^{+0}$ are shown in figure 8, together with the Stokes solution for comparison. It can be seen that all the data indeed collapse well onto a single curve during the pre-transitional stage. This curve is elevated from the Stokes solution, but can be well represented by the following expression:

which is shown in the figure. The expression can be recast in terms of $C_{f,du}$ as

Equation (3.16) together with figure 8 demonstrates that the detailed variation of the wall shear stress in the pre-transient phase of all the transient flow cases can be closely correlated in terms of the modified friction coefficient ( $C_{f,du}^{\prime }$ ) against the wall-unit time of the initial flow ( $t^{+0}$ ). This correlation deviates somewhat from the solution of Stokes’ first problem for laminar flow, but only needs a relatively small correction based on $t^{+0}$ . In practice, either (3.16) or (3.17) can be used to predict the wall shear stress in a transient turbulent flow. Another useful piece of information that can be derived from figure 8 is that the timing of the onset of transition in terms of $t^{+0}$ does not change significantly as the flow conditions are varied, being in the range of $t^{+0}=80{-}110$ for the conditions examined in this study. As a result, $t^{+0}$ provides a good measure of the stage of the flow development, and hence the reason for using it for comparison for the flow developments under the various conditions.

Figure 9. Development of the profiles of $u_{rms}^{\prime }$ , $v_{rms}^{\prime }$ , $w_{rms}^{\prime }$ at various times. The profiles are shifted to the right by a distance proportional to the time. (a) S11; (b) S15. Thin lines: initial profiles; thick lines: profiles at time $t^{+0}$ ; lower horizontal lines: displacement boundary layer ( ${\it\delta}^{\ast }$ ); higher horizontal lines: velocity boundary layer ( ${\it\delta}_{99}$ ).

3.4. Energy growth

Figure 9 shows the development of the profiles of $u_{rms}^{\prime }$ , $v_{rms}^{\prime }$ and $w_{rms}^{\prime }$ in S11 and S15 for Reynolds number ratios of 4.5 and 1.25 respectively, to represent a low- and a high-FST case. The profiles at any time $t^{+0}$ are shifted by a distance proportional to $t^{+0}$ . Despite the large differences in the level of the changes in the two cases, some features of the response of turbulence can be identified in both scenarios, which are also common in all other cases. (i) The value of $u_{rms}^{\prime }$ starts to increase from the start of the transient, reaching a level that is close to the maximum value at the time of the onset of transition, only making slow adjustments afterwards. The pre-transition rapid increase is an important feature of transition, which reflects the formation and enhancement of the streaky structures, and is comparable to the pre-transitional energy growth in a boundary layer. (ii) By contrast, $v_{rms}^{\prime }$ and $w_{rms}^{\prime }$ remain largely unchanged (in fact reduce very slightly, as shown below) during the pre-transition period. They only start to increase when transition occurs.

To facilitate a direct comparison between the energy growths in the various cases, the development of the profile of $(u_{rms}^{\prime }-u_{rms,0}^{\prime })$ normalized by the increase of the bulk velocity ( $U_{b1}{-}U_{b0}$ ) for all cases is shown in figure 10(a). It is clear that the profiles for all cases at any $t^{+0}$ agree very well with each other. In fact, the cases with the same final Reynolds number (series $\text{S}0X$ ) fall rather closely on top of each other, whereas the scattered points shown in the figure have largely originated from series $\text{S}1X$ with a variation in the final Reynolds number, reflecting some albeit small unaccounted effect of $\mathit{Re}_{1}$ . Nevertheless, the close correlation is remarkable considering the fact that in some cases the flow is increased by only a very small amount, and that the flow structures appear to be so different between the cases, as shown in figure 1. Whether the flow shows strong or weak streaks, or even no streaks, the energy growth is apparently described by the same non-dimensional parameter. Also remarkable are the responses of $v_{rms}^{\prime }$ shown in figure 10(b). During the most part of the pre-transition stage, $v_{rms}^{\prime }$ decreases slightly in all cases although only slightly, and then increases rapidly after approximately $t^{+0}=80$ . The $w_{rms}^{\prime }$ component shows a similar trend to $v_{rms}^{\prime }$ (not shown). These results demonstrate that, even though the transition is very difficult to observe through visualizations in cases such as S16 due to the very small increase in flow rate, the process of a distinct quiescent pre-transition period followed by a ‘sudden’ and rapid generation of turbulence exhibits itself unambiguously in all cases in the turbulence statistics as well as in the velocity profiles discussed earlier.

Figure 10. Development of the profiles of (a)  $u_{rms}^{\prime \wedge }\,[=(u_{rms}^{\prime }-u_{rms,0}^{\prime })/(U_{b1}-U_{b0})]$ and (b)  $v_{rms}^{\prime \wedge }\,[=(v_{rms}^{\prime }-v_{rms,0}^{\prime })/(U_{b1}-U_{b0})]$ at various times in the pre-transition period for all cases.

The maximum streamwise energy growth, $u_{rms,max}^{\prime 2}\,\{=\max _{y}(u_{rms}^{\prime 2})\}$ , with respect to $\mathit{Re}_{t}$ , is shown in figure 11(a). It is clear that the growth rate increases with increase of $\mathit{Tu}$ , which is consistent with the trend exhibited in the experiments by Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005). Inspired by figure 10(a), the energy growth in terms of ${\rm\Delta}E=\max _{y}\{(u_{rms}^{\prime }-u_{rms,0}^{\prime })^{2}/(U_{b1}-U_{b0})^{2}\}$ is plotted against $t^{+0}$ (figure 11 b). Again, the data from the series $\text{S}0X$ practically collapse on top of each other whereas the data from $\text{S}1X$ have some deviations from the overall trend but still follow it rather closely. At the very early stage, there is a ‘delay’ in the increase of the energy, which was also noted by Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005), who attributed it to an early receptivity stage when the flow adjusts itself to the boundary layer. Following this stage, the energy growth is linear, which in many cases extends much beyond the point of the onset of transition. Noting that $u_{{\it\tau}0}/U_{b1}=((u_{rms,0}^{\prime })_{max}/U_{b1})(u_{{\it\tau}0}/(u_{rms,0}^{\prime })_{max})=\mathit{Tu}_{0}/(u_{rms,0}^{\prime +})_{max}$ and that $(u_{rms,0}^{\prime +})_{max}$ is only a very weak function of the Reynolds number, $t^{+0}=(tu_{{\it\tau}0}^{2}/{\it\nu})=(tU_{b1}^{2}/{\it\nu})(u_{{\it\tau}0}^{2}/U_{b1}^{2})\propto \mathit{Re}_{t}\mathit{Tu}_{0}^{2}$ . Consequently, figure 11(b) implies that

That is, the energy growth is linearly proportional to the initial FST intensity squared when $\mathit{Re}_{t}$ is fixed, or vice versa. The same expression has been reached by Fransson et al. (Reference Fransson, Matsubara and Alfredsson2005) based on their boundary layer experimental data. It should be noted that, strictly speaking, ${\rm\Delta}E$ is not the energy growth. Either $u_{rms}^{\prime 2}$ or $u_{rms}^{\prime 2}-u_{rms,0}^{\prime 2}$ instead of $(u_{rms}^{\prime }-u_{rms,0}^{\prime })^{2}$ would be a better representation of the energy in defining ${\rm\Delta}E$ , but no correlation can be found in these forms. This suggests that there may exist some nonlinear interactions between the existing flow structure and the growth of the ‘disturbances’.

Figure 11. Energy growth: (a)  $u_{rms,max}^{\prime 2}/U_{b1}^{2}$ versus $\mathit{Re}_{t}$ , (b)  ${\rm\Delta}E=\max _{y}\{(u_{rms}^{\prime }-u_{rms,0}^{\prime })^{2}/(U_{b1}-U_{b0})^{2}\}$ versus $t^{+0}$ .

Figure 12 shows the profiles of $u_{rms}^{\prime }$ normalized by its maximum value plotted against $y/{\it\delta}_{u}^{\ast }$ for a number of cases, where ${\it\delta}_{u}^{\ast }$ is the displacement thickness of the full velocity profile (i.e.  $\bar{u}$ not $\bar{u}^{\wedge }$ ). For the cases shown in figure 12(a), it is clear that the peak location remains fixed during the pre-transition period, where the profiles show close similarity. During the transition period, the peak location moves towards the wall. For the cases with a small Reynolds number ratio (figure 12 b), the change of the peak location is small throughout the transient period. Although the peak location remains fixed during the pre-transition for each flow, it is different for different flows, varying from 0.85 for S11 to 0.57 for S16 as the final Reynolds number reduces. This is in contrast to the boundary layer transition, in which the peak location is always at $y/{\it\delta}^{\ast }=1.3$ , which has been derived from optimal growth theory (Andersson et al. Reference Andersson, Berggren and Henningson1999) and demonstrated by DNS and experiments (e.g. Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Brandt et al. Reference Brandt, Schlatter and Henningson2004). A major reason for the transient channel to be different is likely to be associated with the fact that the free-stream velocity is not a flat profile but varies with the distance from the wall.

Figure 12. Profiles of streamwise r.m.s. velocity normalized by its peak value at several instants. Solid lines with markers: initial profile; solid lines: pre-transitional period ( $t^{+0}<100$ ); dashed lines: transitional and turbulent periods ( $t^{+0}>100$ ).

Figure 13. Time variation of the spanwise correlation of the streamwise velocity at a plane $y^{+0}=0.5$ (the magnitude of negative correlations is shown; positive correlations are set to zero).

3.5. Flow structures

Due to the presence of high- and low-speed streaks in turbulent and transitional flows, the spanwise correlation of the streamwise velocity ( $R_{11}$ ) shows a negative peak, a phenomenon that is commonly used to illustrate the behaviour of the streaks. A higher value of the negative peak of the correlation, $|R_{11,min}|$ , can be associated with the existence of stronger and/or more regularly spaced streaks relative to the background disturbances. Furthermore, the ‘ $y$ ’ location of the peak negative correlation ( $y_{min}$ ) indicates the averaged central location of the streaks above the wall. Twice the $z$ value at which this occurs ( $z_{min}$ ) can be seen as a measure of the averaged streak spacing. Figure 13 shows the time variation of the spanwise correlation on a horizontal plane near the wall ( $y^{+0}=0.5$ ) for various cases with the steady flow values shown in $t^{+0}<0$ . The absolute values of the negative correlations are shown while the positive correlations are set to zero for clarity. The overall behaviour of the streaks during the transient flows is well displayed. The streaks are enhanced relative to the background turbulence (that is, $|R_{11,min}|$ is greater) during the pre-transition period in all cases, but much more strongly in some than in others. The streak spacing appears to reduce initially but then mostly stay unchanged during the pre-transition period. The ‘strength’ and location/spacing of the streaks are quantified next: the variations of the minimum value ( $|R_{11,min}|$ ) with time for all cases are shown in figure 14, and the time variations of the wall-normal distance ( $y_{min}$ ) as well as the spanwise location ( $z_{min}$ ) are shown in figure 15.

Figure 14. Development of the minimum spanwise correlation of the streamwise velocity for streaks close to the wall ( $y<0.2$ ). Lines only: $\text{S}1X$ series; lines and markers: $\text{S}0X$ series.

Figure 15. Development of the wall-normal distance ( $y_{min}$ ) and spanwise location ( $z_{min}$ ) of the strongest near-wall streaks ( $y<0.2$ ) normalized with inner and outer length scales. Lines only: $\text{S}1X$ series; lines and markers: $\text{S}0X$ series.

The general trend of the variation of $|R_{11,min}|$ is similar in the various cases. It first increases, reaching a peak at approximately $t^{+0}=50{-}70$ , and reduces thereafter. The increase of $|R_{11,min}|$ is greater with increasing $\mathit{Tu}_{0}$ in both series, but the final value in most cases in $\text{S}0X$ is much lower due to their starting point (i.e. the streaks are weaker and/or fewer at higher Reynolds numbers). On the other hand, there appears to be an absolute maximum in $|R_{11,min}|$ , just below 0.3, which has been reached in S01, S11 and S12. For these three cases, the streak strength reduces sharply at the time of the onset of transition, reaching a minimum value towards the end of the transition. This is consistent with the visualization of the flow field in figure 1(a), namely strong streaks are formed during the pre-transition phase, but destroyed during transition as turbulence spots are generated and spread. By contrast, in the cases of a low Reynolds number ratio, such as in S15 and S16, the strengthening of the streaks is minor, and, probably more significantly, the streaks are only slightly weakened during the period of transition. It is hence likely that streak formation and instability are not the dominant mechanisms under such flow conditions. The change from streak-dominated to streak-‘inactive’ cases appears to be progressive as the Reynolds number ratio is decreased and hence no sudden transition from one mode to the other is displayed.

It can be seen from figure 15(a) that the $z$ -location of the minimum correlation, or the half-streak spacing, normalized with the initial-flow wall units ( $z_{min}^{+0}$ ), initially reduces rapidly after the commencement of the transient flow, by an amount that is roughly proportional to the Reynolds number ratio of the flow. It then remains as a constant until the onset of transition, implying that the streak spacing in absolute terms remains unchanged during the most part of the pre-transitional phase, noting that the normalization is based on the initial flow and remains unchanged during the transition. The initial and final streak spacings ( $2z_{min}^{+}$ ) based on their respective wall units of the flow are both approximately 100, as expected for an equilibrium turbulent flow (the latter is not shown). When expressed with respect to the displacement thickness of the full flow field, $z_{min}/{\it\delta}_{u}^{\ast }$ (figure 15 b), the distance of the minimum location in different cases reduces from various initial values during the pre-transition period, converging to approximately 2.5 at the point of onset of transition for the $\text{S}1X$ series where $\mathit{Re}_{0}$ is maintained the same, but to different values for $\text{S}0X$ where $\mathit{Re}_{1}$ is the same. These results compare favourably with those of Ovchinnikov et al. (Reference Ovchinnikov, Choudhari and Piomelli2008) for bypass transition; they have shown that the absolute streak spacing remains unchanged for a period during the pre-transition stage, but the value is different for FSTs with different amplitudes or length scales. They showed that $z_{min}/{\it\delta}_{u}^{\ast }$ in their cases reduces to converge to approximately 2 in most cases, whereas the experimental data of Matsubara & Alfredsson (Reference Matsubara and Alfredsson2001) found that it tends to 3.

The wall-normal distance of the location of the minimum correlation coefficient normalized using the initial-flow wall units ( $y_{min}^{+0}$ ) behaves in a very similar way in all the cases (figure 15 c). Following a brief reduction from its initial value of approximately 10, it increases roughly linearly throughout the pre-transition phase and approaches a value between 12 and 14 just before the onset of transition. The location in relation to the momentum displacement thickness ( ${\it\delta}_{u}^{\ast }$ ) in each case also increases, but only slightly (figure 15 d), and more significantly it assumes very different values for different cases, spreading from approximately 0.3 to nearly 1. The implication is that the streaks lift up in both absolute terms and relative to the boundary layer, but the former is more significant.

The peak values of the streaks are naturally much greater than the r.m.s. values of the corresponding fluctuating velocities and Wundrow & Goldstein (Reference Wundrow and Goldstein2001) suggested that it is likely that the strong events that are associated with these peaks are relevant to the onset of instability rather than the mean values. Hence the fact that experiments often report that r.m.s. values may be a major contributing factor to the differences often observed between the theory and measurements in levels of disturbance energy at the onset of transition. Brandt et al. (Reference Brandt, Schlatter and Henningson2004) and more recently Nolan & Zaki (Reference Nolan and Zaki2013) have further evaluated the idea and provided additional explanation and evidence. We represent the peak amplitudes of the streaks with the maximum and minimum fluctuating velocities defined as follows:

where $\langle u(y,t)\rangle$ denotes the span and streamwise average over a $y$ -plane. The $u_{max}^{\prime }$ and $u_{min}^{\prime }$ for the $\text{S}0X$ and $\text{S}1X$ series are shown in figure 16. The initial value of $u_{max}^{\prime }$ varies in a wide range, from 0.1 to 0.5, among the various cases with a higher-Reynolds-number-ratio flow resulting in a lower $u_{max}^{\prime }$ as expected. After the commencement of the transient, the cases with a lower initial value, however, increase faster than those with a higher initial value and, by the time of the onset of transition, all cases converge to a value of approximately 0.55. The $u_{min}^{\prime }$ tends to follow a similar trend but converges to a slightly smaller value of approximately $-0.5$ . Incidentally, these values are comparable to the values 0.5 and $-0.45$ of Brandt et al. (Reference Brandt, Schlatter and Henningson2004) for boundary layers, which are taken from figure 7 of that paper. It is interesting to note that the $u_{max}^{\prime }$ increases sharply at the time of the onset of transition in S11 and S12, and to a smaller extent in S01, but not in the remaining cases. This observation seems again to suggest that these three cases show different characteristics from the other cases. In fact, it is rather surprising that the streak peaks appear to reach approximately the same values in all flow cases around the time of the onset of transition, considering that streaks are probably not the dominant transition mechanism in some low-Reynolds-number-ratio cases. Figure 17 shows that, in the pre-transition stage, the wall-normal location at which the $u_{min}^{\prime }$ occurs is approximately 3 times higher than that of $u_{max}^{\prime }$ , indicating that high-speed streaks are located closer to the wall whereas low-speed streaks are pushed away from the wall, which is consistent with the results of Brandt et al. (Reference Brandt, Schlatter and Henningson2004).

Figure 16. Development of the peak positive and negative streak amplitudes $u_{max}^{\prime }$ and $u_{min}^{\prime }$ : (a) and (c)  $\text{S}0X$ series; (b) and (d)  $\text{S}1X$ series.

Figure 17. Evolution of the locations of the peak positive and negative streaks, $(y/{\it\delta})_{+peaks}$ and $(y/{\it\delta})_{-peaks}$ : (a) and (c)  $\text{S}0X$ series; (b) and (d)  $\text{S}1X$ series.

Finally, it is of interest to inspect the flatness of $v^{\prime }$ , which responds strongly to the intermittence and hence provides a good measure of the strength of turbulent spots in comparison with the background turbulence. A large value of the flatness occurs when there is a small number of burst events, but each one, when it happens, has a high amplitude. Consequently, the peak of the flatness of $v^{\prime }$ can be associated with the onset of transition. Figure 18 shows the flatness of $v^{\prime }$ at $y^{+0}=5$ in the various cases studied herein. In both the $\text{S}0X$ and $\text{S}1X$ series, the peak value is the highest when the Reynolds number ratio is maximum, and reduces with decrease of the ratio. In consistency with the observations in figure 16, the peak values of S11 and S12 are much greater than those of the rest, being 1800 and 600 respectively, followed by 200 for S01. In addition, the period during which an increased value is observed is narrower, that is, the process is more abrupt, when the Reynolds number ratio is higher.

Figure 18. Development of flatness of the wall-normal velocity with non-dimensional time at $y^{+0}=5$ .