## Abstract

Turbulence is the major cause of friction losses in transport processes and it is responsible for a drastic drag increase in flows over bounding surfaces. While much effort is invested into developing ways to control and reduce turbulence intensities^{1,2,3}, so far no methods exist to altogether eliminate turbulence if velocities are sufficiently large. We demonstrate for pipe flow that appropriate distortions to the velocity profile lead to a complete collapse of turbulence and subsequently friction losses are reduced by as much as 90%. Counterintuitively, the return to laminar motion is accomplished by initially increasing turbulence intensities or by transiently amplifying wall shear. Since neither the Reynolds number nor the shear stresses decrease (the latter often increase), these measures are not indicative of turbulence collapse. Instead, an amplification mechanism^{4,5} measuring the interaction between eddies and the mean shear is found to set a threshold below which turbulence is suppressed beyond recovery.

## Main

Flows through pipes and hydraulic networks are generally turbulent and the friction losses encountered in these flows are responsible for approximately 10% of the global electric energy consumption. Here turbulence causes a severe drag increase and consequently much larger forces are needed to maintain desired flow rates. In pipes, both laminar and turbulent states are stable (the former is believed to be linearly stable for all Reynolds number (Re) values; the latter is stable if Re > 2,040 (ref. ^{6})), but with increasing speed the laminar state becomes more and more susceptible to small disturbances. Hence, in practice most flows are turbulent at sufficiently large Re. While the stability of laminar flow has been studied in great detail, little attention has been paid to the susceptibility of turbulence, the general assumption being that once turbulence is established it is stable.

Many turbulence control strategies have been put forward to reduce the drag encountered in shear flows^{7,8,9,10,11,12,13,14,15,16,17}. Recent strategies employ feedback mechanisms to actively counter selected velocity components or vortices. Such methods usually require knowledge of the full turbulent velocity field. In computer simulations^{7,8}, it could be demonstrated that under these ideal conditions, flows at a low Re number can even be relaminarized. In experiments, the required detailed manipulation of the time-dependent velocity field is, however, currently impossible to achieve. Other studies employ passive (for example, riblets) or active (oscillations or excitation of travelling waves) methods to interfere with the near-wall turbulence creation. Typically here drag reduction of 10 to 40% has been reported, but often the control cost is substantially higher than the gain, or a net gain can be achieved only in a narrow Re number regime.

Instead of attempting to control or counter certain components of the complex fluctuating flow fields, we will show in the following that by appropriately disturbing the mean profile, turbulence can be pushed outside its limit of existence and as a consequence the entire flow relaminarizes. Disturbance schemes are developed with the aid of direct numerical simulations (DNSs) of pipe flow and subsequently implemented and tested in experiments. In the DNS, a flow is simulated in a five-diameter (*D*)-long pipe and periodic boundary conditions are applied in the axial direction. Initially we perturb laminar pipe flow by adding fluctuation levels of a fully turbulent velocity field rescaled by a factor (*k*) to a laminar flow field. As shown in Fig. 1a (dark blue curve), for small initial perturbations (that is, small *k*), the disturbance eventually decays and the flow remains laminar. For sufficiently large amplitudes (*k* of order unity), turbulence is triggered (purple, red and cyan curves). So far, this is the familiar picture of the transition to turbulence in shear flows, where turbulence is triggered only if perturbation amplitudes surpass a certain threshold. However, when increasing the turbulent fluctuations well beyond their usual levels (*k* > 2.5), surprisingly the highly turbulent flow almost immediately collapses and returns to laminar (light and dark green curves). Here the initially strong vortical motion leads to a redistribution of shear resulting in an unusually flat velocity profile (black profile in Fig. 1c).

To achieve a similar effect in experiments, we increase the turbulence level by vigorously stirring a fully turbulent pipe flow (Re = 3,500), employing four rotors located inside the pipe 50*D* downstream of the pipe inlet (see Supplementary Movie 1 and Supplementary Fig. 1). As the highly turbulent flow proceeds further downstream, it surprisingly does not return to the normal turbulence level but instead it quickly reduces in intensity until the entire flow is laminar (Fig. 1b top to bottom and Supplementary Movie 1). Being linearly stable, the laminar flow persists for the entire downstream pipe. In a second experiment, turbulent flow (Re = 3,100) is disturbed by injecting fluid through 25 small holes (0.5 mm diameter) in the pipe wall (holes are distributed across a pipe segment with a length of 25*D*, see Supplementary Fig. 3). Each injected jet creates a pair of counter-rotating vortices, intensifying the eddying motion beyond the levels of ordinary turbulence at this Re. The additional vortices redistribute the flow and as a consequence the velocity profile is flattened (Fig. 1c, purple dotted line). When the perturbation is actuated downstream, fluctuation levels drop and the centre line velocity returns to its laminar value (Fig. 1d). Laminar motion persists for the remainder of the pipe. In this case, the frictional drag is reduced by a factor of 2. Overall the injected fluid amounts to only ~1.5 % of the total flow rate in the pipe. With the present actuation device, we achieved a net power-saving (taking all actuation losses into account) of 31% over the remainder of the pipe. On the other hand, the minimum actuation cost required to create the necessary flow disturbance is substantially lower (~1%), so that the net saving potential at this Re is 45% (see Methods).

In another experiment, we attempted to disrupt turbulence (Re = 5,000) by injecting fluid parallel to the wall in the streamwise direction (see Supplementary Figs. 2 and 4). Unlike for the previous case, this disturbance does not result in a magnification of cross-stream fluctuations, but instead it directly increases the wall shear stress and hence also the friction Re number, Re_{τ}. Directly downstream of the injection point, the latter is increased by about 15%. The acceleration of the near-wall flow automatically causes deceleration of the flow in the pipe centre (the overall mass flux is held constant), hence again resulting in a flatter velocity profile (blue in Fig. 1c). Despite the local increase in Re_{τ}, further downstream the fluctuation levels begin to drop and the turbulent flow has been sufficiently destabilized that eventually (30*D* downstream) it decays and the flow returns to laminar. As a result, friction losses drop by a factor of 2.9 (see Fig. 2a) and the potential net power saving (not including actuation losses) is 55% (see Methods). For this type of perturbation, we find that relaminarization occurs for an intermediate injection range (~15% of the flow rate in the pipe), while for smaller and larger rates, the flow remains turbulent. A property common to all above relaminarization mechanisms is their effect on the average turbulent velocity profile.

To test a possible connection between the initial flat velocity profile and the subsequent turbulence collapse, we carried out further computer simulations where, this time, a forcing term was added to the full Navier–Stokes equations. The force was formulated such that it decelerates the flow in the central part of the pipe cross-section while it accelerates the flow in the near-wall region. The mass flux through the pipe and hence Re remain unaffected (see Supplementary equation (17) and Supplementary Fig. 7). Unlike in the experiments where the disturbance is applied locally and persists in time, here the forcing is applied globally. As shown in Fig. 2b, when turning on the forcing with sufficient amplitude the initially fully turbulent flow completely relaminarizes. Hence, a profile modification alone suffices to destabilize turbulence. Interestingly, the energy required for the forcing is smaller than the energy gained due to drag reduction (even for intermediate forcing amplitudes, see Supplementary Fig. 8). In this case, we therefore obtain a net energy-saving already in the presence of the forcing (in experiments, the saving is achieved downstream of the perturbation location). After removal of the forcing (see Supplementary Fig. 12), turbulence fluctuation levels continue to drop exponentially and the flow remains laminar for all times. This effect has been tested for fully turbulent flow for Re numbers between 3,000 and 100,000, and in all cases a sufficiently strong force was found to lead to a collapse of turbulence resulting in drag decrease and hence energy-saving in the numerical simulations of up to 95% (in practical situations, finite-amplitude perturbations may limit the persistence of laminar flow at such high Re).

We next investigate whether a profile modification on its own also relaminarizes turbulence in experiments. While body forces such as that used in the simulations are not available (at least not for electrically non-conducting fluids), profiles can nevertheless be flattened by a local change in the boundary conditions. For this purpose, one pipe segment is replaced by a pipe of slightly (4%) larger diameter that is pushed over the ends of the original pipe and can be impulsively moved with respect to the rest of the pipe (see Supplementary Movie 2 and Supplementary Fig. 5). The pipe segment is then impulsively accelerated in the streamwise direction and abruptly stopped, the peak velocity of the 300*D*-long movable pipe segment is equal to or larger than (up to three times) the bulk flow speed in the pipe. The impulsive acceleration of the near-wall fluid leads to a flattened velocity profile (red profile in Fig. 1c). Despite the fact that overall the fluid is accelerated and additional shear is introduced (Re_{τ} is increased), after the wall motion is stopped (abruptly, over the course of 0.2 s) turbulence also in this case decays (see Fig. 2c and Supplementary Movie 2). If, on the other hand, the wall acceleration is reduced, with wall velocities lower than 0.8*U*, turbulence survives. The impulsive wall motion is found to relaminarize turbulence very efficiently up to the highest Re number (Re = 40,000) that could be tested in the experiment (here the wall was moved at the bulk flow speed).

In turbulent wall-bounded shear flows, energy has to be transferred continuously from the mean shear into eddying motion, and a key factor here is the interplay between streamwise vortices (that is, vortices aligned with the mean flow direction) and streaks. The latter are essentially dents in the flow profile that have either markedly higher or lower velocities than their surroundings. Streamwise vortices ‘lift up’ low-velocity fluid from the wall and transport it towards the centre (see Supplementary Fig. 8). The low-velocity streaks created in the process give rise to (nonlinear) instabilities and the creation of further vortices. Key to the efficiency of this ‘lift-up mechanism’ is that weak vortices suffice to create large-amplitude streaks. This amplification process is rooted in the non-normality^{4} of the linear Navier–Stokes operator and its magnitude is measured by the so-called transient growth (see also Supplementary Figs. 10 and 11).

Computing transient growth for the forced flow profiles in the DNS, we indeed observe that transient growth monotonically decreases with forcing amplitude (see Supplementary Fig. 11) and it assumes its minimum value directly before turbulence collapses. Generally, the flatter the velocity profile the more the streak vortex interaction is suppressed, and in the limiting case of a uniformly flat profile the lift-up mechanism breaks down entirely.

Revisiting the experiments, the velocity profiles of all the disturbed flows considered exhibit a substantially reduced transient growth (Fig. 1c). For the streamwise injection, amplitudes relaminarizing the flow also show the minimum amplification (Fig. 2d) while at lower and higher injection rates where turbulence survives the amplification factors are higher and above the threshold found in the simulations. Similarly for the moving wall at sufficiently large wall acceleration where relaminarization is achieved, the lift-up efficiency is reduced below threshold, while at lower wall speeds it remains above.

Some parallels between the present study and injection and suction control in channels and boundary layers^{18,19,20} can be drawn. While for boundary layers during the injection phase the drag downstream increases, during the suction it decreases. Suction applied to a laminar Blasius boundary layer leads to a reduction of the boundary layer thickness and this is well known to delay transition^{21} and push the transition location downstream.

The drag reduction achieved for the different methods used to destabilize turbulence is summarized in Fig. 3. In each case, the friction value before the profile modification corresponds to the characteristic Blasius law for turbulence (upper line) and after the disturbance it drops directly to the laminar Hagen–Poiseuille law. Hence, the maximum drag reduction feasible in practice is reached (Fig. 3b), and at the highest Re numbers studied in experiments, 90% reduction is obtained. Although the numerical and experimental relaminarization methods affect the flow in different ways, the common feature is that the velocity profile is flattened.

The presented control schemes require manipulation of only a single velocity component and moreover they do not require any information about the instantaneous turbulent velocity field. The overall control strategy is far simpler compared with recently proposed active and feedback control schemes, while at the same time it offers the maximum possible drag reduction. The future challenge is to develop and optimize methods that lead to the desired profile modifications in high-Re-number turbulent flows.

## Methods

### Experimental set-up for the rotors

The facility consists of a glass pipe (poly(methyl methacrylate); PMMA) with inner diameter *D* = 54 ± 0.2 mm and a total length of 12 m (222*D*) made of 2 m sections (see Supplementary Fig. 1). The flow in the pipe is gravity driven and the working fluid is water that enters the pipe from a reservoir. The flow rate and hence the Re number (Re = *UD*/*ν*, where *U* is the mean velocity, *D* is the diameter of the tube and *ν* is the kinematic viscosity of the fluid) can be adjusted by means of a valve in the supply pipe. The temperature of the water is continuously monitored at the pipe exit. The flow rate is measured with an electromagnetic flowmeter (ProcessMaster FXE4000, ABB). The accuracy of Re is within ±1%. To ensure fully turbulent flow, the flow is perturbed by a small static obstacle (a 1-mm-thick, 20-mm-long needle located 10*D* after the inlet). Two metres downstream from the inlet the turbulent flow is perturbed by four small rotors that are mounted on a support structure within the pipe as indicated in Supplementary Fig. 1. The wiring of the motors is incorporated in the support structure of the motors. The rotors are small rectangular bars with even smaller rectangular bars at the tips. Their only purpose is to induce perturbations to the flow but no propelling motion or thrust. The rotors are turned at a rate of seven rotations per second. For the purpose of visual observations and video recordings of the flow field, the flow is seeded with neutrally buoyant anisotropic particles^{22}. The three locations where Supplementary Movie 1 was recorded are indicated in Supplementary Fig. 1.

### Experimental set-up for the wall-normal jet injection and the streamwise injection through an annular gap

The facility consists of a glass pipe with inner diameter *D* = 30 ± 0.01 mm and a total length of 12 m (400*D*) made of 1 m sections (see Supplementary Fig. 2). The flow in the pipe is gravity driven and the working fluid is water that enters the pipe from a reservoir. The flow rate and hence the Re number (Re = *UD*/*ν*, where *U* is the mean velocity, *D* is the diameter of the tube and *ν* is the kinematic viscosity of the fluid) can be adjusted by means of a valve in the supply pipe. The temperature of the water is continuously monitored at the pipe exit. The flow rate is measured with an electromagnetic flowmeter (ProcessMaster FXE4000, ABB). The accuracy of Re is within ±1%. To ensure fully turbulent flow, the flow is perturbed by a small static obstacle (a 1-mm-thick, 20-mm-long needle located 10*D* after the inlet). Two metres downstream from the inlet, the turbulent flow can be perturbed in a controlled way by two different devices that are mounted within the pipe (see Supplementary Figs. 3 and 4). The velocity field is measured ~330*D* downstream from the disturbance (control) at the position of the light sheet. The measurement plane is perpendicular to the streamwise flow direction (pipe *z* axis). All three velocity components within the plane are recorded using a high-speed stereo particle image velocimetry system (Lavision GmbH) consisting of a laser and two Phantom V10 high-speed cameras with a full resolution of 2,400 × 1,900 px. The resulting spatial resolution is 77 vectors per *D*. The data rate is 100 hertz. Hollow glass spheres (mean diameter 13 μm, *ϱ* = 1.1 g cm^{–1}) are used as seeding particles. Around the measurement plane, the pipe is encased by a water-filled prism such that the optical axes of the cameras are perpendicular to the air/water interface to reduce refraction and distortion of the images. Downstream of the perturbation, the pressure drop Δ*p* is measured between two pressure tabs with a differential pressure sensor (DP 45, Validyne, full range of 220 Pa with an accuracy of ±0.5%) separated by 39.5*D* in the axial direction. As the difference in pressure drop between laminar and turbulent flows is very distinct even at moderate Re numbers, the signal is utilized to observe whether the flow is laminar or turbulent.

### Experimental set-up for the moving pipe

A movable Perspex pipe with inner diameter *D* = 26 ± 0.1 mm and a total length of 12 m (461*D*) is fitted to very thin-walled stainless-steel pipes (MicroGroup) with outer diameter *d*_{st,o} = 25.4 ± 0.13 mm and a wall thickness of 0.4 ± 0.04 mm such that the Perspex pipe overlaps the steel pipes at the upstream and downstream end (see Supplementary Fig. 5). The steel pipes are stationary (mounted on fixed bearings). With respect to the Perspex pipe, they act as support and slide bearing, allowing the Perspex pipe to be moved back and forth in the axial direction. To prevent sagging, the Perspex pipe is supported by six additional bushings (polymer sleeve bearings, Igus). To avoid leakage, a radial shaft seal is mounted at both ends of the Perspex pipe. The length of the control section between the stationary upstream and downstream stainless-steel pipes (that is, the actual length where the wall of the Perspex pipe is in contact with the fluid and can be moved relative to the mean flow by moving the Perspex pipe) is *L*_{control} = 385*D*. The Perspex pipe is connected to a linear actuator (toothed belt axis with a roller guide driven by a servomotor, ELGA-TB-RF-70-1500-100H-P0, Festo; not shown in the figure). The linear actuator can move the Perspex pipe for an adjustable distance (traverse path) *s ≤ s*_{max} = 1.5 m at an adjustable velocity *U*_{pipe}* ≤ U*_{pipe,max} = 5.5 m s^{–1}. The maximum acceleration is *a* = 50 m s^{-2}. The resulting wall velocity of the Perspex pipe is specified as a ratio to the mean flow velocity *U*, such that *u*_{wall}* = U*_{pipe}*/U*. The flow rate and hence the Re number (Re = *UD*/*ν*, where *U* is the mean velocity, *D* is the diameter of the tube and *ν* is the kinematic viscosity of the fluid) can be adjusted by means of a valve in the supply pipe. The temperature of the water is continuously monitored at the pipe exit. The flow rate is measured with an electromagnetic flowmeter (ProcessMaster FXE4000, ABB). The accuracy of Re is within ±1%. The flow is always turbulent when entering the control area. The velocity field is measured ~50*D* upstream from the downstream steel pipe. The measurement plane is parallel to the streamwise flow direction (pipe *z* axis) and located in the centreline of the pipe. The two velocity components within a plane of ~3.5*D* length are measured using a high-speed 2D particle image velocimetry system (LaVision) with a full resolution of 2,400 × 1,900 px. The resulting spatial resolution is 56 vectors per *D*. The data rate is 100 hertz. Hollow glass spheres with a mean diameter of 13 μm are used for seeding. Around the measurement plane, the pipe is encased by a small rectangular Perspex box (50 × 50 × 350 mm) filled with water, such that the optical axis of the camera is perpendicular to the air/water interface to reduce refraction and distortion of the images. A differential pressure sensor (DP 45, Validyne, full range of 550 Pa with an accuracy of ±0.5%) is mounted onto the movable Perspex pipe. Here, the pressure drop Δ*p* in the Perspex pipe is measured between two pressure taps (axial spacing 260 mm).

### Numerical method

We solve the incompressible Navier–Stokes equations

in a straight circular pipe in cylindrical coordinates (*r*, *θ*, *z*), with *r*, *θ* and *z* being the radial, azimuthal and axial coordinate respectively. Throughout this study, the flow is driven by a constant mass flux. The velocity u is normalized by the mean velocity *U* and length by pipe diameter *D*. *F* is the external body force and *p* is pressure. A Fourier–Fourier-finite difference code is used for the integration of the governing equations, with periodic boundary conditions in the axial and azimuthal directions. In the radial direction, a central finite-difference scheme with a nine-point stencil is adopted. In this formulation, velocity can be expressed as

where α*k* and *m* give wavenumbers of the modes in axial and azimuthal direction respectively, 2π/*α* gives the pipe length and ${\widehat{\mathbf{u}}}_{k,m}$ is the complex Fourier coefficient of mode (*k*,*m*). The governing equations are integrated with a second-order semi-implicit time-stepping scheme (for details, see^{23}). The code has been verified and extensively used in many studies (for example,^{24,25,26}).

In Supplementary Table 1, we list the Re numbers, pipe lengths and resolutions we considered in our simulations. To avoid significant domain size effect, the pipe lengths are selected to contain a few low-speed streaks, whose streamwise length is typically around 500 wall units in our normalization (see^{27}). The pipe length was doubled for Re = 4,000 and 5,000 to verify that the pipe lengths here in the table are sufficient. The resolutions are set to be able to sufficiently resolve the near-wall structures (see reference grid sizes shown in Table 1 of^{24}). Note that there is a difference of a factor of two in length scales between our normalization and theirs (double ours to compare with theirs).

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## Additional Information

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## Acknowledgements

We acknowledge the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement 306589, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 737549) and the Deutsche Forschungsgemeinschaft (Project No. FOR 1182) for financial support. We thank our technician P. Maier for providing highly valuable ideas and greatly supporting us in all technical aspects. We thank M. Schaner for technical drawings, construction and design. We thank M. Schwegel for a Matlab code to post-process experimental data.

## Author information

### Author notes

Jakob Kühnen and Baofang Song contributed equally to this work.

### Affiliations

#### Nonlinear Dynamics and Turbulence Group, IST Austria, Klosterneuburg, Austria

- Jakob Kühnen
- , Baofang Song
- , Davide Scarselli
- , Nazmi Burak Budanur
- , Michael Riedl
- & Björn Hof

#### Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, Bremen, Germany

- Baofang Song
- & Marc Avila

#### Center for Applied Mathematics, Tianjin University, Tianjin, China

- Baofang Song

#### School of Mathematics and Statistics, University of Sheffield, Sheffield, UK

- Ashley P. Willis

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### Contributions

J.K. and B.H. designed the experiments. J.K. and D.S. carried out the experiments and post-processed the data. M.R. carried out the rotor experiments. J.K., D.S. and B.H. analysed the experimental results. J.K. and B.H. supervised the experimental work. B.S. and N.B.B. performed the computer simulations of the Navier–Stokes equations. B.S., M.A. and N.B.B. analysed the numerical results. M.A., A.P.W. and B.H. supervised the computer simulations. D.S., A.P.W., M.A. and B.S. performed the theoretical analysis. J.K., B.S., D.S., M.A. and B.H. wrote the paper.

### Competing interests

The authors declare no competing financial interests.

### Corresponding authors

Correspondence to Jakob Kühnen or Björn Hof.

## Supplementary information

### Supplementary Information

Destabilizing turbulence in pipe flow.

## Videos

### Supplementary Movie

Relaminarization by vigorously stirring a turbulent pipe flow with four rotors.

### Supplementary Movie

Relaminarization by impulsive movement of a pipe segment.