6.1. Statement of the problem

Firstly, it must be observed that this problem has two natural parabolic variables, in the sense of partial differential equations theory: time $\tau$ and axial streamwise coordinate $\beta =z/R$. As the Reynolds number is constant in those experiments, we can establish a linear mapping between them as follows:

(6.1)\begin{equation} \beta=\tau \, Re / 2, \end{equation}

whereby the dynamic behaviour could either be described in terms of $\tau$ or $\beta$. Due to the character of the TULF, we shall be using $\tau$ in our reasoning. Moreover, we shall consider horizontal pipes and henceforth it will be assumed $\varGamma =0$ everywhere.

The statement of the physico-mathematical problem is as follows (see figure 3). We begin with a horizontal turbulent S-flow characterised by $\varPi _0$ and $Re=2\tilde {u}$, satisfying the zero theorem (3.6) with $Re_{L_0}>Re$ (and possibly $Re_{L_0} \gg Re$), which we wish to convert into a laminar Hagen–Poiseuille flow defined by $\varPi _f(1-\alpha ^2)/4=Re (1-\alpha ^2)$, with $\varPi _f<\varPi _0$ (and possibly $\varPi _f \ll \varPi _0$) and the same $Re=Re_{L_f}=2\tilde {u}$, fulfilling the zero theorem (2.8) with $\varPi _f=4Re$; that is, we must decrease considerably the MPG while maintaining constant the flow rate, since even for moderate $Re$ it will generally be $\varPi _f \ll \varPi _0$ (see figure 2). Otherwise put, we must decrease considerably the initial $Re_{L_0}$ and bring it down to $Re$, so that $Re_{L_f}=Re$ at the end of the process. Although $Re$ is constant throughout the process, and it would thus appear as a S-flow, there is an underlying laminar U-flow that must evolve from $Re_{L_0}=\varPi _0/4$ to $Re_{L_f}=\varPi _f/4$, and thereby we should write $Re_L(\tau )$. To reach this goal, beginning at $\tau =\beta =0$, we disturb the S-flow using an external supply of energy such that a lower MPG $\varPi _d\leq \varPi _f<\varPi _0$ is attained, and preferably $\varPi _d \ll \varPi _f$, if we wish to accomplish complete laminarisation in a short time (in case $\varPi _d=\varPi _f$, the U-flow would take infinite time to laminarise completely and, therefore, the laminar S-flow would never be observed). The particular sort of disturbance will be different for each experiment, but in all of them the common goal of decreasing the MPG must be achieved; this notion is of utmost importance to understand the laminarisation mechanism and we shall repeat it often. The only way of decreasing the ULF is to decrease its source of motion, which is the MPG. The external supply of energy decouples the MPG and the ULF (and also the mean-velocity field), and the U-ULF is forced to follow the changing MPG under its own dynamics. The disturbance does not bring the MPG down immediately; an interval ${\rm \Delta} \tau$ is needed for the U-flow to set $\varPi _d$. The MPG remains low, at approximately $\varPi _d$, during some time interval until at instant $\tau _d$ the effect of the disturbance is no longer felt. The time $\tau _d$ defines a pipe segment of dimensionless length $L_d= \tau _d \, Re/2$, which we shall call the disturbed domain. The disturbed domain is defined by the dimensionless streamwise coordinate interval $0\leq \beta \leq L_d$, outside of which the effect of the external energy supply is negligible. The MPG difference ${\rm \Delta} \varPi = \varPi _d-\varPi _0$, which is negative, drives the U-ULF to lower values, because ${\rm \Delta} \varPi$ is the source of motion for ${u}_L$, according to the TULF.

Figure 3. General pattern for complete laminarisation.

When the disturbance is over and the MPG has already experienced its main decrease, $\varPi _d$, which causes the major deceleration in the U-ULF, the U-flow is left with a residual MPG $\varPi _r$, such that $\varPi _d\leq \varPi _r\leq \varPi _f=4Re$, and a U-ULF that is not yet coupled to the MPG and is still decreasing. This residual MPG is all that remains to definitely bring the U-ULF down to $Re(1-\alpha ^2)$; the lower $\varPi _r$ the faster the laminarisation, which occurs at time $\tau =\tau _f$, when the ULF reaches $\tilde {u}_L(\tau _f)=\varPi _f/8=Re/2$ and its bulk evolution ceases because it recouples with the MPG: the resulting flow would be completely laminarised, $\mathfrak {I}(\alpha )=1,\alpha \in (0,1)$ (see figure 3). This justifies calling $\tau _f$ the laminarisation time. The limit case $\varPi _r=\varPi _f$ would formally imply $\tau _f \rightarrow \infty$ and thus in practical applications one would expect $\varPi _r<\varPi _f$. The difference, $\delta \varPi =\varPi _f-\varPi _r$, will be called the excess MPG. If $\varPi _r>\varPi _f$ the ULF will never reach $\tilde {u}_L=Re/2$, the PTC will never be zero, and the flow will never laminarise completely, $\mathfrak {I}(\alpha )<1$ for some interval within (0,1). In this last case, $\tau _f$ would represent the time at which the bulk ULF no longer changes, and the name ‘laminarisation time’ would be somewhat uncalled for. Either way, we can define a length for laminarisation or a length for constant bulk ULF, given by $L_f= \tau _f \, Re /2$.

The account we have just offered will be called the laminarisation pattern, and we shall refer to it frequently. The notion that the MPG must be greatly reduced to attain complete laminarisation is central to understanding this mechanism, and the reader is strongly recommended to ponder over this idea prior to carry on reading. Note that $Re_L(0)=Re_{L_0}=\varPi _0/4 \gg Re$ and $Re_L(\tau _f)=Re_{L_f}=\varPi _f/4=Re$. In figure 3 $\varPi _d$ has been depicted as roughly constant and generally would not be so; nevertheless, this fact should not alter the conclusions nor the mechanism for complete laminarisation described herein.

The reader would have noticed that no mention has been made of the PTC in the above explanation of the laminarisation pattern, despite the PTC being the component that encompasses the flow's turbulent motion. Otherwise put, we want $\tilde {u}_T=0$ but no direct action is made over the RSSRG, which is the source of the PTC; instead, we act directly upon the MPG, which is the source of the ULF, a laminar flow. This apparent contradiction is solved by the condition $\tilde {u}_L+\tilde {u}_T=Re/2={\rm const}.$, (i), which constrains the U-flow: one acts upon the ULF to provoke a change in the PTC. The lower MPG can be considered a seed for complete laminarisation. Once the seed is planted, there is no way back; the U-flow will laminarise completely, it is just a matter of time. Figure 4 shows an imaginary example of the evolution of $\tilde {u}_L$ and $\tilde {u}_T$ for the case $\tilde {u} = \tau _f=1$. Note $\tilde {u}_L$ cannot decrease below $\tilde {u}=Re/2$. In mathematical terms, the U-flow has non-denumerable infinite ways of fulfilling the condition $\tilde {u}_L + \tilde {u}_T = Re / 2$; for each $\tilde {u}_L$ there is a unique $\tilde {u}_T$ but $\tilde {u}_L$ only responds to the MPG; reaching $\varPi _f=4 Re$ yields necessarily $\tilde {u}_L=\tilde {u}=Re/2$ and to this ULF corresponds uniquely the PTC $\tilde {u}_T=0$, i.e. a laminar flow.

Figure 4. Evolution of $\tilde {u}_L$ and $\tilde {u}_T$ after a sufficiently low MPG (in arbitrary units, $\tilde {u}=\tau _f=1$). Each variation in $\tilde {u}_L$ must be matched by an equal and opposite variation in $\tilde {u}_T$.

Nevertheless, although the constraint $\tilde {u}_L + \tilde {u}_T = Re / 2$ relieves us of studying the behaviour of the U-PTC, it is obvious that a vanishing PTC must be caused by a vanishing RSSRG, which is its source. From a mathematical standpoint, $\varSigma (\tau,\alpha ) \rightarrow 0$ could be caused either by a decaying turbulence, $\sigma (\tau,\alpha ) \rightarrow 0$, or by a homogeneous turbulence, because the latter also implies a zero radial gradient of RSS. Eventually, a turbulence forced to be homogeneous within a pipe, surrounded by walls, has just one possible outcome: the vanishing of turbulence, and this is what is observed in the experiments of Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b).

The laminarisation mechanism that has just been described is only possible with a constant $Re$ during the U-flow evolution. Should changes in $Re$ be allowed, to achieve complete laminarisation it would also be necessary to act upon the PTC, i.e. to decrease its source, the RSSRG $\varSigma (\tau,\alpha )$. As was discussed in the previous paragraph, a convenient way to do it is to make the turbulence homogeneous throughout the pipe's cross-section, which irons out any gradient and vanishes the RSSRG. Curiously enough, usually it is necessary to very much increase the turbulence, so that it becomes uniformly high everywhere. Uniformly high or low, as long as it is homogeneous, in the end it will turn $\varSigma (\tau,\alpha )\approx 0$, leading to a negligible PTC, $\tilde {u}_T \approx 0$ (see § 6.2). Any turbulence that does not cause a radial gradient is irrelevant for the configuration of the mean-velocity field.

Whether $\tilde {u}_T=0$ or not would depend upon how effective the disturbance has been (see figure 3); if successful, the U-flow would become laminar and would remain laminar forever, unless the disturbance were removed. Only a further disturbance that raises the MPG, or a relaxation of condition $Re={\rm const}.$, would provide some margin for the PTC to increase again. The laminarised S-flow would only undergo a transition to turbulence if its $Re$ is allowed to decrease (Mullin Reference Mullin2011). As a summary of the laminarisation pattern, we propose the following schematic sequence: steady state $\rightarrow$ local disturbance $\rightarrow$ very low $\varPi _d$ $\rightarrow$ ULF decoupled from MPG $\rightarrow$ end of disturbed domain $\rightarrow$ residual MPG $\varPi _r<\varPi _f$ $\rightarrow$ $\tilde {u}_L$ approaches $Re/2$ and $\tilde {u}_T$ approaches zero $\rightarrow$ ULF recouples to MPG $\rightarrow$ laminar Hagen–Poiseuille S-flow with $\varPi _f = 8 \tilde {u}_L=8 \tilde {u} = 4Re_L=4 Re$. Of course, the disturbance must be strong enough, and the disturbed domain long enough, for this sequence to unfold, and that constitutes one of the extraordinary merits of the experimental research of Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b). The account we have just exposed is the general picture the TULF offers to explain the observed phenomena.

The physico-mathematical problem, as modelled in figure 3, is thereby characterised by five degrees of freedom, namely, $\varPi _d$ (or ${\rm \Delta} \varPi$), ${\rm \Delta} \tau$, $\tau _d$ (or $L_d$), $\varPi _r$ (or $\delta \varPi$) and $\tau _f$ (or $L_f$), which ought to be set in any experiment destabilising turbulence in pipe flow. Note $\varPi _0$ and $\varPi _f$ are determined by the initial conditions. The relationships among the energy input into the flow, $\varPi _d$, and $L_d$ will have to be found empirically in each case, as well as the residual MPG $\varPi _r$ beyond the perturbed domain. Arguably, the shorter $L_d$ the lower $\varPi _d$ to reach complete laminarisation, for a given energy-per-unit-length input. Most often, it would be necessary that $\varPi _d<0$ within the disturbed domain to attain complete laminarisation; otherwise put, the external energy supply would be converted into an adverse MPG. This surprising prediction will be explained in detail in § 6.3.

6.2. Four rotors experiment

We begin with the rotors experiment, which is the only one with a quasihomogeneous perturbation affecting the whole cross-section. The flow ($Re=3500$) is vigorously stirred with four rotors, in pairs of clockwise and anticlockwise rotation. Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b) claim that the enhanced turbulence created by the rotors deforms the velocity profile, flattening it, and the deformation causes the U-flow to completely laminarise. Although we shall discuss later the flattening of U-profiles, what actually occurs in the experiment is the laminarisation pattern we exposed above, see figure 5. As anyone would imagine, both sources of mean motion, MPG $\varPi (\tau )$ and RSSRG $\varSigma (\tau,\alpha )$, play their role in the observed laminarisation. Thus, the moving rotors have two main effects in the U-flow, coinciding with both sources: (1) the rotors become an obstacle for the motion of the fluid and, as all obstacles do, they create a pressure drop in the U-flow, leading to a MPG decrease after starting the rotors, $\varPi _d<\varPi _0$ (see figure 5). (2) The rotors increase the turbulence around them, meaning that the fluctuating components of velocity $u'_i$, or the turbulence intensity $u'_i/\tilde {u}$, increase noticeably about the rotors. However, the newly created turbulence is uniform, homogeneous around the rotors and so is the associated RSS field. But a homogeneous RSS has a very small radial gradient, that is, $\varSigma (\tau,\alpha ) \approx 0$ and, since the RSSRG is the source of the PTC, we must expect a vanishing PTC or, otherwise put, a laminarised U-flow. An everywhere-increased turbulence is a likely route to laminarisation, for only the RSSRG constitutes a source of mean momentum. Any turbulence that does not cause a radial gradient is irrelevant for the configuration of the mean-velocity U-field; this is a very important outcome from the TULF (see (5.3) and (5.12)). Note the turbulence index field $\mathfrak {I}(\tau,\alpha )$ is not fooled by the uniform increase of turbulence intensity: in any laminarisation process $\mathfrak {I}(\tau,\alpha ) \rightarrow 1$ regardless of the turbulence intensity.

Figure 5. Sketch of MPG $\varPi (\tau )$ evolution in the four rotors experiment (laminarisation pattern). Here $\varPi _L(\tau )$ is a virtual MPG, corresponding to the laminar Hagen–Poiseuille flow that at each instant has the same $Re_L$ as the U-ULF. (a) Successful complete laminarisation ($\varPi _d =\varPi _r< \varPi _f=4Re$); (b) unsuccessful complete laminarisation ($\varPi _d=\varPi _r > \varPi _f=4Re$).

No temporal evolution curves are offered for this experiment in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b), and thus not much can be said about its degrees of freedom, except that the U-flow needs some $130D$ to laminarise completely, $L_f=260$, according to Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, supplementary figure 1). The antagonistic thrusts of the rotor pairs cause a large pressure drop in the U-flow, which can be adjusted to reach $\varPi _d< \varPi _f= 4 Re=14\,000$, although the relationship between rotor's spin velocity and $\varPi _d$ would have to be found empirically. The residual MPG $\varPi _r$ would be roughly the same as that created in the disturbed domain $\varPi _d$, and the situation in the U-flow could be described by ${\rm \Delta} \tau = \tau _d$ and $\varPi _d=\varPi _r$. With such low $\varPi _d$, since the U-flow's Reynolds number is not allowed to decrease, all available bulk velocity would be occupied by the ULF, and the PTC would be forced to zero (see point (2) above). Therefore, the final S-flow we must expect is a new ULF, devoid of PTC, with a Reynolds number $Re_{L}=Re=3500$, in accordance with the experiment (see figure 5a). No other explanation is required, but the simple and easily checkable pressure drop associated with any obstacle inserted in a flow. Regrettably, no results for the MPG are reported in the rotors experiment.

If the MPG decrease were not enough, then the residual ULF would have $Re_{L_r}=2\tilde {u}_{L_r}>Re=2\tilde {u}$ and there would remain some residual bulk velocity available for a modest PTC, $\tilde {u}_{T_r}=\tilde {u}-\tilde {u}_{L_r}$: the laminarisation would not be complete and the final S-flow would be partially turbulent, with a greater turbulence index than that corresponding to the initial S-flow, $\mathfrak {I}_0(\alpha ) \leq \mathfrak {I}_r (\alpha ) \leq 1$ (see figure 5b).

Moreover, the laminarisation of a pipe flow would, in principle, be possible with any obstacle or device that could decrease sufficiently the MPG immediately downstream, like a honeycomb insert (Kuehnen, Scarselli & Hof Reference Kuehnen, Scarselli and Hof2019), an orifice plate (Kuehnen et al. Reference Kuehnen, Scarselli, Schaner and Hof2018a) or any other, while maintaining constant the U-flow's $Re$.

The remaining experiments do depend on a definite disturbed-domain length $L_d$, and the decrease of MPG is obtained through indirect methods (see §§ 6.3 and 6.4). We shall also see that the laminarisation pattern is only developed in the core flow, and an adverse MPG is generated therein. The external perturbation does not affect directly the whole cross-section, but rather the near-wall U-flow.

6.3. Annular streamwise jet experiment

The second experiment to be explained is the streamwise inject or annular jet ($Re=5000$). Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b) claim that the WSS increases in the U-flow upon injecting the annular jet. This is literally true but not effectively true, for we must distinguish between flow and core flow, the latter being what remains after removing from the former the thin annular near-wall portion, see figure 6. Note this wall-adjacent annulus contains an inordinately high portion of the whole cross-sectional area.

Figure 6. Mean-velocity U-profile and sketch of dual MPG, strong favourable and moderate adverse, within the disturbed domain of the streamwise inject experiment.

The initial S-flow enters the disturbed domain with MPG $\varPi _0$ and WSS $\sigma _{w_0}$, and the annular jet is input into the flow with moderately high velocity (we do not consider yet very high injection velocities, case $c4$ of Kuehnen et al. (Reference Kuehnen, Scarselli, Schaner and Hof2018a, figure 10b)), comparable to that of the core flow, and requires an external source of energy to do so. Therefore, the jet creates a sort of barrier between the core flow and the wall; the wall is shielded by the jet and loses contact with the core flow. The higher WSS becomes a private matter between wall and jet, not between wall and core flow. In the disturbed domain, figure 6, we see near the wall a strong favourable MPG that creates the jet itself, the accelerating wall flow, and also a lower (most often adverse) MPG in the core that decelerates the core flow, so that the constraint of fixed $Re$ could hold: if one portion of the U-flow accelerates then another must decelerate. Thus, the conditions to trigger the laminarisation pattern in the core flow are set, see figure 7. This local combination of favourable and adverse MPG in the same cross-sectional region is quite unstable, can only be sustained through an external energy input, cannot be maintained for long, and dissipates shortly downstream the disturbed domain, where the precarious cohabitation shown in figure 6 merges into a single MPG, the residual MPG $\varPi _r$ affecting the whole cross-section, which is the minimum found in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 2a), measured some 30D downstream of the disturbed domain. Thereby, the actual occurrence of a dual MPG in the disturbed domain cannot be reproduced downstream, and instead an evolution curve with a single residual MPG is recorded. The account of facts just exposed is also substantiated in Kuehnen et al. (Reference Kuehnen, Scarselli, Schaner and Hof2018a, figure 12), which illustrates how the mean-velocity U-profiles evolve, with the wall flow swelling and the core flow shrinking as the energy input increases.

Figure 7. Laminarisation pattern: sketch of MPG $\varPi (\tau )$ evolution in the core flow for the annular jet and radial jets experiments. Here $\varPi _L(\tau )$ is a virtual MPG, corresponding to the laminar Hagen–Poiseuille flow that at each instant has the same $Re_L$ as the U-ULF. (a) Successful complete laminarisation ($\varPi _r < \varPi _f=4Re$); (b) unsuccessful complete laminarisation ($\varPi _r > \varPi _f=4Re$).

Moreover, the velocity gradient between core flow and wall flow changes sign, see figure 6 and Kuehnen et al. (Reference Kuehnen, Scarselli, Schaner and Hof2018a, figure 12), and the disturbed domain can be defined as the region where this change of sign is present. The associated degree of freedom $L_d$ would thus be given by the streamwise distance at which the dual MPG situation illustrated in figure 6 no longer holds. It is not a sharp and clear-cut definition, but it ought to suffice for designing experiments, and the relationship between injection velocity (or injection MPG) and $L_d$ would have to be found empirically. In a situation like this, in which the core flow is rather detached from the wall flow, (5.20) would also be approximately valid for the core flow, with $\varPi (\tau )$ the MPG acting upon the core flow (red line in figure 7). The core U-ULF is thus driven by a lower (most likely adverse) MPG and must decrease under its own dynamics, trailing the new MPG $\varPi _d$ (figure 7a), but any change in U-ULF must be accompanied by a matching one from U-PTC, (i). It follows that if $L_d$ and $\varPi _d$ are duly adjusted, the laminarisation pattern is initiated causing a laminarised core flow. If the wall flow is thin enough, it will contain just a small U-PTC, (vii), insufficient to ruin the core's laminarisation process if the merged residual MPG $\varPi _r$ is still well below $\varPi _f=4Re$. In this case, the initial core-flow laminarisation extends to the whole U-flow farther downstream. When the whole U-flow laminarises completely, the MPG increases again and recouples with the ULF, see figure 7(a) and Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 2a about $tU/D \approx 50$), but it will only increase up to $\varPi _f=4Re=20\,000$, whereby it becomes bulk stationary. The agent causing this late MPG increase is the force set in action by the constraint $Re={\rm const}.$ If, on the other hand, the residual MPG $\varPi _r > \varPi _f=4 Re$ then the associated U-ULF would be $\tilde {u}_{L_r}>Re/2=\tilde {u}$ and there would remain some residual U-PTC, $\tilde {u}_{T_r}=\tilde {u}-\tilde {u}_{L_r}$: the laminarisation would not be complete and the final S-flow would be partially turbulent, with a greater turbulence index than that corresponding to the initial S-flow, $\mathfrak {I}_0(\alpha ) \leq \mathfrak {I}_r (\alpha ) \leq 1$ (see figure 7b).

It remains to suggest an explanation for the case $c4$ of Kuehnen et al. (Reference Kuehnen, Scarselli, Schaner and Hof2018a, figure 10b). Recall that, in figure 6, if $\delta \alpha = 1-1/\sqrt {2}\approx 0.2929$, then the area of the annulus containing the wall flow would be equal to that of the core flow. It is our belief that if the injection velocity is too high, the annular jet would not be so thin a few diameters downstream, and the core flow would not be dominant within the pipe's cross-section, see Kuehnen et al. (Reference Kuehnen, Scarselli, Schaner and Hof2018a, figure 12). Should this be the case, even if the core U-flow were to laminarise completely, there would remain a considerable U-PTC in the thicker annular jet itself that would prevent the complete laminarisation of the whole U-flow. Kuehnen et al. (Reference Kuehnen, Scarselli, Schaner and Hof2018a) even report that “$\ldots$ meaning that the controlled flow exhibits features of relaminarisation but finally returns to a turbulent state downstream $\ldots$”. As long as the annular jet remains close to the wall, it will not develop a significant PTC, for the near-wall region is the natural domain of the ULF (García García & Fariñas Alvariño Reference García García and Fariñas Alvariño2019c), (vii).

6.4. Radial jets experiment

The last experiment to be discussed is that with 25 radial jets ($Re=3100$). Again, in this flow we observe the same three performers on stage: jets; wall flow; and core flow. Of all quantities involved in the zero theorem for turbulent flow, (3.6), the one directly affected by the normal jets is the WSS $\sigma _w$. The jets change the WSS ‘felt’ by the core flow, see Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, supplementary figure 6), since they create a barrier between wall and core flow: an external energy input is again detaching the core flow from the wall, and a new effective WSS appears in the core flow, $|\sigma _{w_d}|<|\sigma _{w_0}|$, and a new MPG is set in the disturbed domain, $\varPi _d=-2\sigma _{w_d}<\varPi _0$, (iii); thereby MPG and U-ULF decouple and the U-ULF is forced to pursue the new MPG $\varPi _d$ under its own dynamics, see figure 7. Again, we have the conditions that trigger the laminarisation pattern in the core flow. As discussed in § 6.3, in this situation (5.20) is approximately applicable to the core flow, with $\varPi (\tau )$ the resulting MPG after the disturbance (red line in figure 7). If $\tau _d$ is sufficiently long to render the effective WSS sufficiently low, the U-PTC within the core would fade off and the core U-flow would laminarise completely; in turn, its laminarisation would extend to the whole U-flow downstream, because the PTC is already negligible near the wall, (vii). The degree of freedom $L_d= \tau _d Re / 2$ is clearly set in this experiment of Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b), ${L_d=50}$, as opposed to the fuzzily defined $L_d$ of the previous annular streamwise-jet experiment. Downstream the disturbed domain, $\tau >\tau _d$, a residual MPG $\varPi _r$ remains in the U-flow (figure 7): if $\varPi _r \leq \varPi _f=4 Re$ the U-flow will eventually laminarise completely, otherwise a residual PTC would remain and the final S-flow would be partially turbulent, with a greater turbulence index than that corresponding to the initial S-flow, $\mathfrak {I}_0(\alpha ) \leq \mathfrak {I}_r (\alpha ) \leq 1$. The relationship between the injection velocity of radial jets and $\varPi _d$ would have to be found empirically.

6.5. Flattening of mean-velocity profiles

Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b) suggest repeatedly that the flattening of mean-velocity U-profiles might be the cause of the observed laminarisation, and even the experiments’ design appears to have been influenced by a pursuit of the said flattening effect. Linking the progress of laminarisation to a flattening of U-profiles is essentially correct, and is a testimony of those authors’ perspicacity. However, it is important to understand that the flattening is a consequence of the laminarisation process, not its cause. The flattening means, among other things, a deformation (change) of the mean-velocity field, and since Newton it is known that changes in momentum are caused by forces. Therefore, the flattening is caused by a modification of the forces applied to the U-flow, and only two forces are available: the MPG that pushes the U-flow; and the RSSRG that pulls in opposite direction, since it encompasses the turbulent stress forces. Moreover, the constraint of constant $Re$ determines that both forces must be globally balanced at every instant, because the bulk acceleration is zero. However, they can differ locally, always provided the cross-section average of one be identical to the other, and the local differences tend to be greater near the wall, where the flattening manifests itself.

The defining characteristic of laminarisation is always a proportionally higher increase of the U-ULF respect to the U-PTC, hence its name. In any laminarisation process the U-ULF is proportionally greater than the U-PTC, and this occurs especially near the wall, where the U-PTC evolution is slower. Therefore, near the wall the subtracting term is lower in absolute value (the PTC is generally negative), and that yields a locally higher mean velocity, which is the observed phenomenon. The flattening of U-profiles is abundantly reported in our previous papers (García García & Fariñas Alvariño Reference García García and Fariñas Alvariño2019b,Reference García García and Fariñas Alvariñoc, Reference García García and Fariñas Alvariño2020, Reference García García and Fariñas Alvariño2021), and occurs every time the U-ULF takes precedence over the U-PTC, or otherwise put, any time the turbulence index increases. The mathematical details can be found in those references. All theoretical experiments we have performed in previous works show this effect: the route to laminarisation, either partial or total, implies the growth of the ratio $|u_L/u_T|$, that is, the flattening of mean-velocity U-profiles.

We shall perform now a simple exercise to demonstrate that the deformation of mean-velocity U-profiles is a mathematical necessity in any U-flow evolution, as it follows from (5.12). The reader should note that, in the present context, ‘deformation’ means that the family of mathematical curves describing a U-flow is qualitatively different from that of a S-flow. Because of its simplicity, the informal demonstration will only be applied to the PresGrad component, (5.14), with an analogous reasoning for all other components. Assume a transient flow evolution from steady-state $Re_1$ to steady-state $Re_2$. Since for horizontal S-flows the ULF and PresGrad are one and the same thing, the initial and final PresGrad components take the common form $\varPi _i (1-\alpha ^2)/4$ of a Hagen–Poiseuille flow. We begin by noting that the parabola function $1-\alpha ^2$ is expressed in Fourier–Bessel series as (see (7.16))

(6.2)\begin{equation} 1-\alpha^2 = 8 \sum _{n=1}^{\infty} \frac{J_0(\lambda_n \alpha )}{\lambda_n^3 J_1(\lambda_n)} = 4 \sqrt{2} \sum _{n=1}^{\infty} \frac{\phi_n(\alpha)}{\lambda_n^3}. \end{equation}

Let us imagine, just for a moment, that the PresGrad component were defined by

(6.3)\begin{equation} u_{P}(\tau,\alpha)=\sqrt{2} \left( \int _0^{\tau} \varPi(\tau') \, \mathrm{d} \tau' \right)\sum_{n=1}^{\infty} \frac{ \phi_n(\alpha)}{\lambda_n^3}, \end{equation}

with the integral over $\varPi (\tau )$ outside of the sum, instead of the correct (5.14) (the extra $\lambda _n^2$ in the denominator arises from a change of variables in the integral of (5.14), $\xi ' = \lambda _n^2 \tau '$, $\mathrm {d} \tau ' = \mathrm {d} \xi '/\lambda _n^2$ and $\mathrm {e} ^{\xi -\xi '} \approx 1$). The integral would thus act as a global multiplicative factor of the parabola $(1-\alpha ^2)$. In this imaginary scenario, for each $\tau$ the U-profile of $u_P$ would be a parabola and the evolution of PresGrad would be a continuous sequence of parabolas up to the final one, $\varPi _2(1-\alpha ^2)/4$. Otherwise put, the temporal evolution of this component would be self-similar with respect to the variables $(\tau,\alpha )$. However, the correct mathematical result has the integral inside the sum, and the evolution U-profiles cannot be parabolas, but other curves that would appear as deformed when compared with a parabola. The particular form of such curves would depend on the disturbing factor within the sum, i.e. the integral over $\varPi$. The temporal evolution of PresGrad is not, and cannot be, self-similar with respect to $(\tau,\alpha )$, which is a pity because it would make life much easier. Arguably, the evolution of PresGrad could still be self-similar with respect to other variables, but that is a story to be told elsewhere. In summary, the deformation of mean-velocity U-profiles must occur in any temporal evolution of a U-flow; it is a prediction of the TULF and it is what actually happens in experiments.

7.1. A mathematical model for steady-state pipe flow

The following dimensionless polynomials (Pai Reference Pai1953), which we shall call a Pai flow, are a solution of the RANSE (3.2) and, therefore, $u(\alpha )$ would correspond to the turbulent S-flow generated by the sources $\varPi$ and $\sigma (\alpha )$ in the natural normalisation,

(7.1)$$\begin{gather} u(\alpha)= \frac{\varPi}{4 \chi} \left( 1+\frac{\chi-q}{q-1} \alpha^2 - \frac{\chi-1}{q-1} \alpha^{2q} \right)= \frac{\varPi}{4}(1-\alpha^2)+ \frac{\hat{\varPi}}{4q} [ 1 - q(1-\alpha^2)-\alpha^{2q} ], \end{gather}$$

(7.2)$$\begin{gather}\sigma(\alpha)=\frac{\varPi}{2} \frac{q(\chi-1)}{\chi(q-1)}\alpha (1-\alpha^{2q-2})=\frac{\hat{\varPi}}{2} \alpha (1-\alpha^{2q-2} ), \end{gather}$$

with $q\mathop {\geq }2$ a best-fitting integer and $\chi \in \mathbb {R}$, $1\mathop {\leq } \chi \mathop {<}q$, the so-called centreline turbulent dissipation (CTD), which is equal to the ratio

(7.3)\begin{equation} \chi = \frac{u_L(0)}{u(0)}=\frac{Re_L}{u(0)}=\frac{1}{\mathfrak{I}(0)}, \end{equation}

where $u_L$ is the ULF generated directly by $\varPi$, i.e. a laminar Hagen–Poiseuille flow. According to its definition, $\chi =1$ for laminar flow, and in such case the value of $q$ is irrelevant. The functions $u(\alpha )$ and $\sigma (\alpha )$ thus defined belong to the mean space $\mathfrak {M}$ introduced in § 3. The first expression in (7.1) is called the compact form of a Pai flow, whereas the second is the decomposed form, for it shows explicitly the ULF ($\varPi (1-\alpha ^2)/4$) and the PTC (the remainder). Note how $\hat {\varPi }=\varPi q (\chi -1)/\chi (q-1)$ plays in the PTC an analogous role to $\varPi$ in the ULF, and it is named the weighted MPG, being $\hat {\varPi } \lesssim \varPi$. The parameters $\varPi$, $q$ and $\chi$, which collectively characterise any Pai flow, are known as the spatial degrees of freedom (SDoF). These polynomials seem to have been largely ignored in the literature, despite representing quite accurately any turbulent S-flow of moderate $Re$. To assess the suitability of Pai polynomials, figure 8 compares the mean velocity of (7.1) with the experimental results of the Princeton Superpipe (see Zagarola Reference Zagarola1998), for the five lowest $Re$ of the set, which we name SP1 to SP5. The best-fitting SDoF parameters are given in table 1. We can see a small deviation in SP4, and a more noticeable one in SP5, albeit from SP1 to SP3 the coincidence is remarkable. In figure 9 is shown a Pai flow for $Re=5300$, together with the corresponding DNS data. This figure will be explained below.

Figure 8. Princeton Superpipe experimental velocity data and Pai polynomials for cases of table 1 (from García García & Fariñas Alvariño Reference García García and Fariñas Alvariño2019b).

Figure 9. The DNS velocity data for $Re=5300$ in Wu & Moin (Reference Wu and Moin2008) and best-fitting Pai polynomial.

According to (3.3a,b), $u(\alpha )$ and $\sigma (\alpha )$ determine each other uniquely; it follows that if (7.1) is a good approximation for the actual mean velocity, then (7.2) is also as good an approximation to the actual RSS. The plots of figure 2 have been calculated directly from the Pai polynomial (7.1) and the SDoF of table 1. It must be stressed, again, that Pai polynomials are a solution of the RANSE, (3.2), belonging to the mean space $\mathfrak {M}$, and not simply some functions presumed ad hoc (an example of an ad hoc mean-velocity profile would be the famous power law $u(\alpha )=u_{max}(1-\alpha )^{1/n}$, which does not fulfil the governing equation).

If we are just given a turbulent S-flow of Reynolds number $Re$, and no further data, the SDoF can be obtained from the following expressions, which are as accurate as the correlations they emanate from, namely the Colebrook correlation and $u(0)= (1+1.3 \sqrt {f})\, \tilde {u}$, (see White Reference White2016, (6.43)). The demonstration is given in Appendix B:

(7.4)$$\begin{gather} \varPi=\frac{f\, Re^2}{16}, \end{gather}$$

(7.5)$$\begin{gather}\chi_1=\frac{f\, Re}{32(1+1.3 \sqrt{f})}, \end{gather}$$

(7.6)$$\begin{gather}q_1=\frac{\chi_1 (f \, Re-64)}{64 \chi_1 - f \, Re}, \end{gather}$$

((7.4) was also deduced in Marusic et al. (Reference Marusic, Joseph and Mahesh2007, (4.1))). Equation (7.6) will not generally yield an integer value for $q_1$. In that case, we round $q_1$ to the nearest integer $q\in \mathbb {N}$, always provided $q>\chi _1$, and we reckon a new $\chi$ from the following equation:

(7.7)\begin{equation} \chi=\frac{q\ f\, Re}{64(q+1)-f\, Re}. \end{equation}

Moreover, we have the interesting relationship,

(7.8)\begin{equation} f\, Re = 64\frac{\chi (q+1)}{\chi+q}, \end{equation}

which is exactly $f\, Re = 64$ for laminar flow ($\chi =1$), regardless of the value of $q$.

The bulk velocity corresponding to a Pai flow, (7.1), is

(7.9)\begin{equation} \tilde{u}= \tilde{u}_L + \tilde{u}_T = \frac{\varPi}{8} - \frac{\hat{\varPi}}{8} \frac{q-1}{q+1}= \frac{\varPi}{8} \frac{\chi+q}{\chi(q+1)} = \frac{Re}{2}. \end{equation}

Note that if $q> \chi \gg 1$, then $\hat {\varPi } \approx \varPi$, $\tilde {u}_L \approx |\tilde {u}_T|$ and $\tilde {u} \ll \tilde {u}_L$, which is the result depicted in figure 2. The bulk velocity $\tilde {u}$ is usually a dwarf compared with those two giants, $\tilde {u}_L$ and $\tilde {u}_T$.

Given a turbulent S-flow with bulk velocity $\tilde {u}$, the laminar Hagen–Poiseuille flow with the same bulk velocity satisfies $\tilde {u}=\tilde {u}_L=\varPi _f/8$ (see (2.8)), whereby the MPG of the laminar flow must fulfil (see (7.9))

(7.10)\begin{equation} \varPi_f=\varPi \frac{\chi+q}{\chi(q+1)} , \quad \varPi_f<\varPi. \end{equation}

To assess how accurate the Pai polynomial resulting from the above procedure is, we shall apply it now to the DNS flow of $Re=5300$ obtained in Wu & Moin (Reference Wu and Moin2008), which has a Reynolds number within the order of magnitude of those reported in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b). Applying (7.4)–(7.7) to $Re=5300$, we obtain $f=0.03677$, $\varPi =64560.46$, $\chi _1=4.87524$ and $q_1=5.44901$. Rounding $q_1$ to $q=6$, we get $\chi =4.62029$. The DNS data of Wu & Moin (Reference Wu and Moin2008) is furnished in wall units, $u^+=U/U_{\tau }$. To convert wall units into the natural normalisation, suffices to apply $u=u^+ \, Re_{\tau }$, being $Re_{\tau }=U_{\tau } R /\nu$ the friction Reynolds number, also called the von Kármán number. For the chosen DNS flow, it is $Re_{\tau }=181.37$. The resulting Pai polynomial is plotted in figure 9, together with the DNS velocity data. The DNS data appear to have been time-averaged rather than ensemble-averaged (see Wu & Moin Reference Wu and Moin2008, § 2.3), whereas the Pai flow is always a mathematically ensemble-averaged field. A slightly better fit would have been obtained if $q$ were allowed to have non-integer values. However, in such case the solution would not be a polynomial, and much of the analytical advantage would be lost.

The case of a turbulent Hagen–Poiseuille flow with MPG $\varPi$ and moving-wall velocity $V_w$ is somewhat more involved. Recall we mentioned in § 4 that given a turbulent Hagen–Poiseuille–Couette flow $u(\alpha )$, the field $u_w(\alpha ) \equiv u(\alpha )-V_w$ corresponds mathematically to a turbulent Hagen–Poiseuille flow with the same sources, $\varPi$ and $\sigma (\alpha )$. We shall now use this result to characterise a turbulent Hagen–Poiseuille–Couette flow through polynomials. The mean-velocity field is (see (4.2a,b))

(7.11)\begin{align} u(\alpha)&=V_w+ \frac{\varPi}{4 \chi_w} \left( 1+\frac{\chi_w-q}{q-1} \alpha^2 - \frac{\chi_w-1}{q-1} \, \alpha^{2q} \right) \nonumber\\ &= V_w+\frac{\varPi}{4}(1-\alpha^2)+ \frac{\check{\varPi}}{4q} [ 1 - q(1-\alpha^2)-\alpha^{2q} ], \end{align}

in which $q$ is, again, a best-fitting integer, but $\chi _w$ now responds to

(7.12)\begin{equation} \chi_w=\frac{u_{w_L}(0)}{u_w(0)}=\frac{u_L(0)-V_w}{u(0)-V_w}, \end{equation}

according to the definition of CTD $\chi$ offered above for turbulent Hagen–Poiseuille flow. Likewise, the Couette-weighted MPG $\check {\varPi }$ is defined by

(7.13)\begin{equation} \check{\varPi}=\varPi \frac{q(\chi_w-1)}{\chi_w(q-1)}. \end{equation}

Thereby, if we are only given a turbulent Hagen–Poiseuille–Couette flow characterised by $Re$ and $V_w$, then we calculate $Re_w=Re-2V_w$ (see (3.12) and (4.5)), and with $Re_w$ we reckon $f_w$ through the Colebrook correlation, and now we can apply the set of (7.4)–(7.7) to determine the SDoF parameters defining $u_w(\alpha )$ and (7.11): $\varPi$, $\chi _w$ and $q$. The bulk velocity of a turbulent Hagen–Poiseuille–Couette flow is

(7.14)\begin{equation} \tilde{u}= \tilde{u}_L + \tilde{u}_T = V_w +\frac{\varPi}{8} - \frac{\check{\varPi}}{8} \frac{q-1}{q+1}= V_w+\frac{\varPi}{8} \frac{\chi_w+q}{\chi_w(q+1)} =\frac{Re}{2}. \end{equation}

7.2. A mathematical model for transient pipe flow

As already explained in § 5, defining a model within the framework of the TULF means assigning explicit mathematical functions to the abstract quantities $\varPi (\tau )$, $\varGamma (\tau )$, $\varSigma (\tau,\alpha )$, $V_w(\tau )$ and $u_0(\alpha )$ appearing in (5.12) and (5.20). When applied to the experiments of Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b), (5.12) and (5.20) become greatly simplified, for ${\varGamma (\tau )=V_w(\tau )=0}$. Moreover, we are not concerned about the particular details of mean-velocity U-profiles, and thus we can work only with bulk velocities and (5.20). Last, but by no means least, the constraint $\tilde {u}_L(\tau )+\tilde {u}_T(\tau )=Re/2$, (i), allows us to study exclusively the evolution of the U-ULF $\tilde {u}_L(\tau )$, and we are thereby spared the burden of finding suitable functions $\varSigma (\tau,\alpha )$ for these flows. Therefore, the bulk U-ULF applicable in this research is limited to (see (5.20))

(7.15)\begin{equation} \tilde{u}_L(\tau)=4 \sum _{n=1}^{\infty} \left\{ \frac{u_{L_n}^{(0)}}{\sqrt{2}} \frac{\mathrm{e}^{-\lambda_n^2 \tau}}{\lambda_n} + \frac{1}{\lambda_n^2} \int _0^{\tau} \varPi(\tau')\, \mathrm{e}^{-\lambda_n^2 (\tau-\tau')} \, \mathrm{d}\tau' \right\}. \end{equation}

Since the initial flow in all experiments is a turbulent S-flow as in (3.3a,b), its corresponding S-ULF is the Hagen–Poiseuille flow $u_{0_L}(\alpha )=\varPi _0 (1-\alpha ^2)/4$.The components of $u_{0_L}(\alpha )$ in the basis $\{\phi _n(\alpha ) \}$ of the Hilbert space $L^2_{\alpha }(0,1)$ are given by the inner product

(7.16)\begin{equation} u^{(0)}_{L_n}=\left\langle \frac{\varPi_0}{4}(1-\alpha^2),\phi_n(\alpha) \right\rangle= \frac{\varPi_0}{4}\int _0^1 \alpha (1-\alpha^2) \frac{\sqrt{2} \ J_0(\lambda_n \alpha)}{J_1(\lambda_n)} \, \mathrm{d} \alpha= \frac{\sqrt{2}\ \varPi_0}{\lambda_n^3}, \end{equation}

and (7.15) is written as

(7.17)\begin{equation} \tilde{u}_L(\tau)=4 \sum _{n=1}^{\infty} \frac{ \mathrm{e}^{-\lambda_n^2 \tau}}{\lambda_n^2} \left\{ \frac{\varPi_0 }{\lambda_n^2} + \int _0^{\tau} \varPi(\tau')\, \mathrm{e}^{\lambda_n^2 \tau'} \, \mathrm{d}\tau' \right\}. \end{equation}

It remains to find an expression for the MPG $\varPi (\tau )$. The actual MPG might be a rather complicated function, but in figure 3 we offered a likely model and in § 6.1 it was reduced to defining five degrees of freedom ($\varPi _d$, ${\rm \Delta} \tau$, $\tau _d$, $\varPi _r$ and $\tau _f$), because $\varPi _0$ and $\varPi _f$, the initial and final MPG, are uniquely determined by $Re$. Unfortunately, the data reported in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b) for any of the experiments are not sufficient to uniquely fix those five parameters, and there are infinitely many possibilities for them. Therefore, we shall devise a much-simplified model for $\varPi (\tau )$ and assume that the MPG drops instantaneously to $\varPi _d$, and stays there until complete laminarisation is attained, i.e. $\varPi _r=\varPi _d$ and ${\rm \Delta} \tau = \tau _d=0$. The model is purposely chosen to be very simple, so that no doubt remains that the final outcome (a laminar S-flow) is predicted even with a low-accuracy model. We are left with only two degrees of freedom, $\varPi _d=\varPi _r$ and $\tau _f$. This simplifying assumption is based upon (iv) and mathematically can be formulated as the MPG following a Heaviside step ${\rm \Delta} \varPi$ at dimensionless time $\tau =0$,

(7.18)\begin{equation} \varPi(\tau)= \varPi_0 +{\rm \Delta} \varPi \mathcal{H}(\tau) = \varPi_0 + (\varPi_d-\varPi_0)\mathcal{H}(\tau), \end{equation}

with $\mathcal {H}(\tau )$ the Heaviside unit distribution. Under such an assumption and inserting this $\varPi (\tau )$ into (7.17), the TULF provides the following continuous function for a fully developed $\tilde {u}_L(\tau )$ that attains complete laminarisation at $\tau _f$ ($\varPi _r<\varPi _f=4 Re$):

(7.19)\begin{equation} \tilde{u}_L(\tau)=\begin{cases} \displaystyle 4 \sum _{n=1}^{\infty}\frac{\varPi_0 \, \mathrm{e}^{-\lambda_n^2 \tau} + \varPi_d(1-\mathrm{e}^{-\lambda_n^2 \tau})}{\lambda_n^4}\\ \qquad =\frac{\varPi_d}{8}-4 {\rm \Delta} \varPi \sum _{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_n^2\tau}}{\lambda_n^4} =\frac{\varPi_L(\tau)}{8} & 0\leq \tau \leq \tau_f ,\\ \displaystyle \frac{Re}{2}=\frac{\varPi_f}{8} & \tau > \tau_f. \\ \end{cases} \end{equation}

If, on the other hand, complete laminarisation is not reached ($\varPi _r>\varPi _f=4 Re$), the bulk U-ULF takes the form

(7.20)\begin{equation} \tilde{u}_L(\tau)= 4 \sum _{n=1}^{\infty}\frac{\varPi_0 \, \mathrm{e}^{-\lambda_n^2 \tau} + \varPi_d(1-\mathrm{e}^{-\lambda_n^2 \tau})}{\lambda_n^4}=\frac{\varPi_d}{8}-4 {\rm \Delta} \varPi \sum _{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_n^2\tau}}{\lambda_n^4} =\frac{\varPi_L(\tau)}{8}. \end{equation}

Note that at $\tau =0$ only the terms with $\varPi _0$ survive, whereas for $\tau \rightarrow \tau _f$, (7.19) (respectively for $\tau \rightarrow \infty$, (7.20)) only the terms with $\varPi _f/8$ (respectively, $\varPi _d/8$) remain. In the meantime, $\tilde {u}_L(\tau )$ would be equal to the bulk velocity of a laminar Hagen–Poiseuille flow driven by an intermediate virtual MPG $\varPi _L(\tau )$, $u^S(\alpha ;\tau )=\varPi _L(\tau ) (1-\alpha ^2 )/4$, see (viii), although $\varPi _d$ be already enforced downstream of the disturbed domain. The MPG corresponding to this virtual SB-flow, $\varPi _L(\tau )$, is represented in green colour in figures 5 and 7. The excess MPG, $\delta \varPi = \varPi _f-\varPi _r=\varPi _f-\varPi _d$, needed by the U-flow to reach $\varPi _f$ within a finite time interval, will be recovered once the ULF has attained $\tilde {u}_L=Re/2$ (see red lines in figures 5a and 7a). If, on the contrary, $\varPi _r>\varPi _f$, then the ULF will be limited by $\tilde {u}_L \rightarrow \varPi _d/8=\varPi _r/8>Re/2$, and the difference $\tilde {u}_T=Re/2 - \tilde {u}_L$ would correspond to the S-PTC, which would not be zero and the flow would not laminarise completely. This last situation is illustrated in figures 5b and 7b. In any case, $\tau _f$ denotes the time at which the ULF does not measurably change, either because it has reached $Re/2$ with an excess MPG $\delta \varPi >0$, or because it is asymptotically arriving to $\varPi _r/8=\varPi _d/8$ and there is no significant MPG to drive the ULF any lower ($\delta \varPi <0$). This account of facts, based upon (7.19) and (7.20), constitutes, again, the laminarisation pattern introduced above.

Let us forget for the moment the unsuccessful case. If the experiment is successful and a complete laminarisation is attained, then from (7.19) it follows that

(7.21)\begin{align} \sum _{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_n^2\tau_f}}{\lambda_n^4} &=\frac{\varPi_d-4Re}{32(\varPi_d-\varPi_0)} =\frac{-\delta \varPi}{32 {\rm \Delta} \varPi}\nonumber\\ &\Leftrightarrow \varPi_d=\left( 32 \varPi_0 \sum _{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_n^2\tau_f}}{\lambda_n^4}-4Re \right) \left( 32 \sum _{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_n^2 \tau_f}}{\lambda_n^4} - 1 \right)^{{-}1}, \end{align}

which relates $\varPi _d$ and $\tau _f$ for the initial S-flow defined by $Re$ and $\varPi _0$. If $\tau _f$ is known, $\varPi _d$ can be reckoned rather easily with the equation on the right-hand side, for the series converges rapidly and is quite accurate with some 50 terms or fewer. If, instead, $\varPi _d$ is known, the equation on the left-hand side can be solved recursively, starting with an estimated value for $\tau _f$. With the first eigenvalue $\lambda _1 \approx 2.404826$ being the most significant of the series, it is a smart idea to begin the recursion with $\tau _{f_1}$ given by

(7.22)\begin{equation} \frac{\mathrm{e}^{-\lambda_1^2\tau_{f_1}}}{\lambda_1^4} = \frac{\varPi_d-4Re}{32(\varPi_d-\varPi_0)} \Rightarrow \tau_{f_1} = \frac{-1}{\lambda_1^2} \ln \left[\frac{\lambda_1^4 (\varPi_d-4Re)}{32(\varPi_d-\varPi_0)} \right]= \frac{-1}{\lambda_1^2} \ln \left[ \frac{-\lambda_1^4 \delta \varPi}{32 {\rm \Delta} \varPi} \right], \end{equation}

knowing that, typically, $\tau _{f_1}$ yielded by (7.22) is rather similar to the actual $\tau _f$, i.e. $\tau _{f_1} \sim \tau _f$ ($\delta \varPi \geq 0$, ${\rm \Delta} \varPi <0$, and $\delta \varPi <|{\rm \Delta} \varPi |$). On the other hand, if $\tau _{f_1}$ defined by (7.22) were good enough to design an experiment, the researcher would not need to write the software that calculates (7.21). Finally, should higher accuracy be needed, the reader can always devise a more sophisticated model for $\varPi (\tau )$ than (7.18) (figure 3 might serve as guide).

To end this section, and as a concomitant of the solution just found, we shall reckon the time constant of the dynamical system constituted by a laminar pipe U-flow. To our knowledge, the consideration of a laminar flow as a dynamical system, and thus liable of having time constant, phase space, orbits, Lyapunov exponents and like notions, is not frequently reported in the literature, and might be of interest to the reader. For an energy-increasing system, such a time constant $\tau _c$ is defined as the time needed for the dynamical system to yield the value $\tilde {u}_L(\tau _c) \equiv (1-\mathrm {e}^{-1}) \tilde {u}_L(\infty ) \approx 0.6321 \tilde {u}_L(\infty )$, where $\tilde {u}_L(\infty )$ is the limit steady-state response obtained after applying a unit Heaviside step at $\tau =0$ in the system's source. For an energy-decreasing system, it would be $\tilde {u}_L(\tau _c) \equiv \mathrm {e}^{-1} \tilde {u}_L(\infty ) \approx 0.3679 \tilde {u}_L(\infty )$. Applied to our case, we assume that the MPG (the source) undergoes a unit Heaviside step from $\varPi _0$ to $\varPi _0+1$ at $\tau =0$, i.e. (7.18) would become $\varPi (\tau )=\varPi _0+\mathcal {H}(\tau )$, since $\varPi _d=\varPi _0+1$ and ${\rm \Delta} \varPi =1$. The equation defining the time constant $\tau _c$ can either be derived from (7.19) or (7.20), as follows:

(7.23)\begin{equation} \tilde{u}_L(\tau_c)=\frac{\varPi_0}{8}+(1-\mathrm{e}^{{-}1})\frac{1}{8}= \frac{\varPi_0+1}{8}- 4 \sum_{n=1}^{\infty} \frac{\mathrm{e}^{\lambda_n^2 \tau_c}}{\lambda_n^4} \Rightarrow \sum _{n=1}^{\infty} \frac{\mathrm{e}^{(1-\lambda_n^2\tau_c)}}{\lambda_n^4}=\frac{1}{32}. \end{equation}

This is an extremely interesting result: the time constant $\tau _c$ does not depend upon the Reynolds number of the laminar flow (or the MPG $\varPi _0$, which is the same thing). Moreover, the same result is obtained if, instead of a unit Heaviside step (${\rm \Delta} \varPi =1$), a general Heaviside step ${\rm \Delta} \varPi$ is applied to the flow. Here $\tau _c$ seems to be a universal constant, valid for any laminar pipe flow. Since $\tau _c=t_c \nu /R^2$, it should be stated that the dimensional time constant $t_c$ depends exclusively upon the fluid's viscosity and the pipe's radius. The response of a laminar flow to any step increase in MPG is the same for all initial bulk velocities of such a flow or, otherwise put, for given pipe and fluid the time constant of a laminar flow is the same for all $Re$. Moreover, the reader should be aware of the following identity involving the zeros $\lambda _n$ of $J_0(x)$:

(7.24)\begin{equation} \sum _{n=1}^{\infty} \frac{1}{\lambda_n^4}=\frac{1}{32}, \end{equation}

which helps one understand the relevance of (7.23). Reckoning the series (7.23) with 500 terms, we get for the universal time constant the value

(7.25)\begin{equation} \tau_c= \frac{t_c \nu}{R^2} = 0.165381775. \end{equation}

Future experimental research should take into account this universal time constant; to get a swift laminarisation (or any other unsteady effect), one should use small-diameter pipes and high-viscosity fluids. Be warned that it is not a matter of Reynolds number, since the bulk velocity plays no role in how fast the flow reacts to any perturbation. This advice also serves to minimise the necessary disturbed-domain size to attain complete laminarisation. The dimensionless length corresponding to $\tau _c$ is $L_c=\tau _c Re/2$ (see (6.1)). As $\tau _c$ is constant, we have the result $L_c \propto Re$ (dimensionless) or L$_c \propto R\ Re$ (dimensional). In both cases, we arrive at the conclusion that the “development length required to reach a fully parabolic profile increases linearly with $Re$”, as expressed literally in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, caption of figure 3b).

7.3. Four rotors experiment

We begin with the four rotors experiment, $Re=3500$. With such a $Re$, (7.4)–(7.10) yield $f=0.041528$, $\varPi _0=31795$, $q=4$ and $\chi =3.328906$. The final MPG to attain complete laminarisation is $\varPi _f=14\,000$, according to (7.10). At 18 $^\circ$C, the kinematic viscosity of water is $\nu =1.0533 \times 10^{-6} \ {\rm m}^2\ {\rm s}^{-1}$, and as $D=2R=54 \ {\rm mm}$ for this experiment, we have $\tilde {U}=0.06827 \ {\rm m}\ {\rm s}^{-1}$ ($\tilde {u}=1750$). Since the flow is reported to laminarise completely at some $130D$ downstream ($L_f \approx 260$), we can estimate $\tau _f= L_f/\tilde {u} \approx 0.14857$, i.e. $t_f \approx 102.83\ {\rm s}$. Thereby, according to (7.21) with 50 terms in the series, we have $\varPi _d=\varPi _r=1858.7652$, a low favourable MPG. No adverse MPG is necessary in this case to attain complete laminarisation. Additionally, the MPG drop in the disturbed domain is ${\rm \Delta} \varPi = \varPi _d-\varPi _0=-29937.24$, and the excess MPG recovered at complete laminarisation ($\tilde {u}_T(\tau _f)=0$) is $\delta \varPi = \varPi _f-\varPi _r=12141.24$ (see laminarisation pattern in figure 5). Note $\tau _f$ is slightly below the universal time constant $\tau _c$ of (7.25), $\tau _f \lesssim \tau _c$, which means that the final laminar Hagen–Poiseuille flow is attained later, as already reported in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, caption in figure 3b); that is, the mean flow is laminar for $\tau > \tau _f$, but it is still a U-flow with a not-yet parabolic profile shortly after $\tau _f$ (see § 6.5). We shall use the ratio $|{\rm \Delta} \varPi | / \varPi _0$ as a figure of merit to assess how significant the MPG decrement is to achieve complete laminarisation within the requested time interval. This ratio gives information about how effective the laminarisation method is to rapidly accomplish the desired result. In the rotors experiment it is $|{\rm \Delta} \varPi | / \varPi _0=0.9416$.

7.4. Annular streamwise jet experiment

Let us now examine the annular jet experiment, whose results are reported in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 2a), which correspond to measurements made far downstream the disturbed domain, where the U-flow is fully developed. With $Re=5000$, (7.4)–(7.10) yield $f=0.037393$, $\varPi _0=58426$, $q=5$, $\chi =4.7444$, $\tilde {u}=Re/2=2500$ and $\varPi _f=4Re=20\,000$. Note that $\varPi _0/\varPi _f=2.9213$, which is approximately the result reported by Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b), “$\ldots$ by a factor of 2.9 (see Fig. 2a) $\ldots$”. Inside the disturbed domain the situation is analogous to figure 6, with a dual MPG; however, at the measurement station the dual MPG cannot be recorded, as they merge into a single MPG. The instant at which the MPG begins to increase can be identified with the laminarisation time, $\tau _f$, see figure 7(a), and $\tau _f$ can be measured directly on Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 2a), yielding $t_f^* = t_f \tilde {U}/D \approx 35$ dimensionless time units. With $\tilde {u}=2500$, $D=2R=0.03 \ {\rm m}$ and $\nu =1.0533 \times 10^{-6} \ {\rm m}^2\ {\rm s}^{-1}$, we have a dimensional bulk velocity $\tilde {U}\approx 0.17555 \ {\rm m}\ {\rm s}^{-1}$. Therefore, a time unit in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 2a) is equivalent to $D/\tilde {U}=0.171 \ {\rm s}$, and the laminarisation time would be $t_f \approx 6 \ {\rm s}$, which in the natural normalisation, (2.3a–c), corresponds to dimensionless time $\tau _f = t_f \nu /R^2 = 0.0281$. The corresponding laminarisation length would be $L_f= \tilde {u} \tau _f\approx 70$, or $35D$, which is quite close to the $30D$ distance reported in the actual experiment (Kuehnen et al. Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b). With these values, (7.21) applied to the core flow yields $\varPi _d=-165825.9642$ using 50 terms in the series, and $\varPi _d=-165825.9229$ with only five terms; the error is negligible, for the series converges very fast. Using, instead, (7.22) with $\tau _{f_1}=0.0281$, we obtain $\varPi _d=-147367.64$, with an error of $11.13\,\%$. Either way, the effect of the annular jet upon the core flow is an adverse MPG: the jet itself causes a brusque acceleration in the wall flow, but since the flow's $Re$ is not allowed to change, the core flow must decelerate (through an adverse MPG) to compensate for. The result is not so surprising, if we consider how much energy the turbulence dissipates from the flow, see figure 2; it is only natural that a powerful disturbance be needed to eliminate the turbulence. Obviously, the adverse MPG is only active within the disturbed domain; farther downstream of the disturbed domain, where the U-flow is fully developed, the dual MPG merges into a residual MPG $\varPi _r$ that is transmitted and recorded, and this is what Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 2a) shows. In this experiment $\tau _f \ll \tau _c$, with $\tau _c$ given by (7.25), which implies the final laminar Hagen–Poiseuille flow is reached much later, as reported in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, caption in figure 3b). Shortly after $\tau _f$ the flow is laminar, but it has a mean-velocity U-profile that is far from being a parabola (see § 6.5). The figure of merit for this experiment, in the core flow, is $|{\rm \Delta} \varPi | / \varPi _0=3.8382$, and complete laminarisation is accomplished much sooner than in the rotors experiment.

7.5. The 25 radial jets experiment

Finally, we study the 25 radial jets experiment, $Re=3100$. Equations (7.4)–(7.10) yield $f=0.0430845$, $\varPi _0=25878$, $q=4$, $\chi =2.8656$, $\tilde {u}=1550$ and $\varPi _f=12\ 400$. According to (7.1), the centreline mean velocity is $u(0)=\varPi _0/4\chi =2257.64$, and the ratio between $u(0)$ and the bulk velocity $\tilde {u}=Re/2$ is $2u(0)/Re \approx 1.46$, which is coincident with the average value shown in Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 1d, bottom panel). The disturbed domain is characterised by $L_d=50$ and $\tau _d = 2 L_d/ Re =0.03226$, whereas $\tilde {U}=0.10884\ {\rm m}\ {\rm s}^{-1}$ as calculated from $\nu =1.0533 \times 10^{-6} \ {\rm m}^2\ {\rm s}^{-1}$ and $D=2R=0.03\ {\rm m}$. Measuring directly over Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 1d, top panel), we see that the transient flow begins at approximately $t_1^*=t_1 \tilde {U}/D \approx 28.1$ (corresponding to $\tau =0$), and ends at $t_2^*=t_2 \tilde {U}/D \approx 35.7$ (corresponding to $\tau _f$), thus yielding $t^*_f=t_2^*-t_1^* \approx 7.6$, which in the natural normalisation is $\tau _f \approx 0.009807$. With such data, (7.21) applied to the core flow yields $\varPi _d=-174814.42$ with 50 terms in the series. No wonder that $\varPi _d$ is a strong adverse MPG, if we have into account that $\tau _f = 0.009807$ is over three times lower than $\tau _d = 0.03226$ (or $L_d \approx 3.3 L_f$), that is, the U-flow is completely laminarised before exiting the disturbed domain, see figure 10. Apparently, the radial jets set-up is a very compelling technique for laminarisation. Therefore, $\tau _f \ll \tau _c$ for this experiment, $\tau _c$ defined in (7.25), from which follows that the final parabolic laminar Hagen–Poiseuille flow is achieved much later, in accordance with Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, caption in figure 3b). The figure of merit for this experiment is $|{\rm \Delta} \varPi | / \varPi _0=7.7553$, the highest of them all.

Figure 10. Sketch of MPG $\varPi (\tau )$ evolution in the core flow for the 25 radial jets experiment as reported by Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b, figure 1d).

7.6. Epilogue to the experiments

If the figure of merit $|{\rm \Delta} \varPi | / \varPi _0$ were a guide to ascertain how proportionally difficult it is to achieve complete laminarisation, because it takes longer, we might conclude that the preferred methods would be, in order of decreasing efficiency: radial jets (7.76); annular jet (3.84); and, finally, four rotors (0.94), which takes the longest to reach the goal. Apparently, methods involving an adverse MPG in the core flow are more effective than those decreasing MPG in the whole flow. However, only limited attention should be paid to $|{\rm \Delta} \varPi | / \varPi _0$, since we have not yet proved its significance.

Regardless of the actual intermediate processes occurring during laminarisation, one outstanding fact remains: all experiments begin with a turbulent S-flow and end with a laminar S-flow of identical $Re$. The initial and final states of the evolution are given, and the TULF matches them with mathematical certainty. In-between, the process, the stages, the details, the timing or other aspects might be different in our approach with respect to the actual ones occurring in the experiments, but such differences cannot obscure the fact that the general picture, the qualitative mechanism, is like we have described herein. The laminarisation process must be as we say, give or take some degrees of freedom, since it is supported upon the general solution for mean pipe flow. Regrettably, the reported data are insufficient to devise better models that would adjust to whatever intermediate steps might have occurred in the U-flows.

Even a very simple model, with almost no features in-between the initial and final states, yields the desired result. With more data, more sophisticated models can be forged that would adjust to any observed minutia of the transient mean flow, including number of stages, duration of each, limit values, degrees of freedom, ... until complete laminarisation is attained. There are, in principle, infinite ways to arrive at complete laminarisation, but all of them imply the reduction of MPG beyond the limit of $\varPi _f=4Re$, because the Hagen–Poiseuille flow $Re(1-\alpha ^2)$ is the only one compatible with the given $Re$. This is a mathematical necessity and cannot be doubted.

We would like to end this epilogue with a plea to Kuehnen et al. (Reference Kuehnen, Song, Scarselli, Budanur, Riedl, Willis, Avila and Hof2018b), so that they repeat their brilliant experiments measuring thoroughly the MPG and RSSRG (or RSS) field. Such data are not simply important because we say so; they are important because they appear at the right-hand side in the governing equation for mean pipe flow, (5.3). They are important because they, and only they, constitute the sources of mean motion (forces). We take the compromise of inserting the measured data into our equations and reproduce the observed mean-velocity field.