# Nonlinear hydrodynamic instability and turbulence in pulsatile flow

^{a}Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria;^{b}Center of Applied Space Technology and Microgravity, University of Bremen, 28359 Bremen, Germany;^{c}Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany;^{d}Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

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Edited by Michael D. Graham, University of Wisconsin–Madison, Madison, WI, and accepted by Editorial Board Member John D. Weeks March 31, 2020 (received for review August 14, 2019)

## Significance

The inner lining of blood vessels, the endothelium, is shear sensitive. Fluctuating shear stresses and disordered flow are responsible for cellular dysfunction, leading to the development of atherosclerotic lesions. We here identify a nonlinear hydrodynamic instability that gives rise to disordered motion in the parameter range of cardiovascular flow. During flow deceleration small geometrical imperfections trigger a helical vortex pattern that subsequently breaks down into bursts of turbulence. The resulting fluctuating shear stress level drops abruptly during the acceleration where flow relaminarization sets in. The observed instability occurs at considerably lower flowrates than either the linear instability or the classic route to turbulence. Our study shows that disordered motion is more common in pulsatile/cardiovascular flows than previous stability considerations suggest.

## Abstract

Pulsating flows through tubular geometries are laminar provided that velocities are moderate. This in particular is also believed to apply to cardiovascular flows where inertial forces are typically too low to sustain turbulence. On the other hand, flow instabilities and fluctuating shear stresses are held responsible for a variety of cardiovascular diseases. Here we report a nonlinear instability mechanism for pulsating pipe flow that gives rise to bursts of turbulence at low flow rates. Geometrical distortions of small, yet finite, amplitude are found to excite a state consisting of helical vortices during flow deceleration. The resulting flow pattern grows rapidly in magnitude, breaks down into turbulence, and eventually returns to laminar when the flow accelerates. This scenario causes shear stress fluctuations and flow reversal during each pulsation cycle. Such unsteady conditions can adversely affect blood vessels and have been shown to promote inflammation and dysfunction of the shear stress-sensitive endothelial cell layer.

Blood vessels react to hemodynamic forces and in particular the vessels’ inner layer, the endothelium, is highly shear sensitive. Fluctuating flow and low wall shear stress levels cause inflammation of the endothelium, which in turn can lead to the development of atherosclerosis lesions (1⇓–3). However, the hydrodynamic instabilities responsible for fluctuations and varying shear stress levels are often unknown. Already for the simpler case of steadily driven flow through a straight pipe it is nontrivial to predict whether the fluid motion will be smooth and laminar or highly fluctuating and turbulent. In that case the laminar state is linearly stable, yet turbulence arises as a result of finite-amplitude perturbations provided that the Reynolds number (

Pulsatile flows are more complex and governed by two additional control parameters, i.e., the pulsation amplitude and frequency (Womersley number). Depending on parameters the primary instability encountered differs qualitatively. For predominantly oscillatory flows, i.e., flows with small or no mean flow component, the flow becomes linearly unstable even though the cycle-averaged Reynolds number vanishes in this limit. This linear instability has been extensively investigated and is well understood (10⇓⇓⇓–14). In contrast, the flow in blood vessels is pulsatile; i.e., it is dominated by the mean flow and the oscillatory component is smaller. Although the aforementioned linear instability is also found in this case, it occurs only at very high flow speeds (15). The corresponding critical Reynolds number lies far above the values encountered in blood flows and hence this transition threshold is not relevant for cardiovascular flow. In addition, the aforementioned subcritical instability to turbulent puffs persists to pulsatile flow (16, 17). In cardiovascular flows, it is typically assumed that turbulence sets in at similar

In large arteries, Reynolds numbers can reach peak values considerably larger than this limit. However, mean values even in the aorta typically do not exceed

In the following we report a subcritical instability specific to pulsating flow. This nonlinear instability sets in during flow deceleration, downstream of small imperfections of the pipe, such as bends or protrusions. Initially, a helical wave arises which subsequently breaks down into turbulence, and fluctuation levels rise before they abruptly drop during the accelerating phase where flow relaminarization sets in. This helical instability is observed at

## Results

### Puff Turbulence (Experiment).

Initial experiments were carried out in a rigid straight pipe with an inner diameter of

### Helical Instability (Experiment).

When the pulsation amplitude surpasses 0.7, the above trend stops and the transition threshold begins to move to lower

Inspection of the pipe revealed that the pipe segment directly upstream of the location where the helical (wave) instability occurred was slightly bent (with an axial misalignment of approximately

It should be noted that the misalignment considered above is only a fraction of a pipe diameter, and in the cardiovascular context virtually all blood vessels show deviations from the idealized straight pipe case, which are of that order or larger. To trigger the helical instability in a more controlled manner, we inserted a short pipe segment with a chosen moderate curvature (sketched in Fig. 1*B*; see *Materials and Methods* for details), while keeping the rest of the pipe straight and well aligned. With a more strongly curved pipe segment, the instability occurs at considerably lower

### Helical Instability in Simulation.

To elucidate the origin of the instability, we carried out numerical simulations of the Navier–Stokes equations. Albeit the laminar flow is linearly stable over the parameter range studied in the experiments, this does not preclude the possibility that perturbations can grow over part of the pulsation cycle, as long as they experience a net decay over the full cycle (13, 23). We determined the optimal perturbations of pulsating pipe flow by performing a linear nonmodal transient growth analysis with an adjoint-based method (see *Materials and Methods* for technical details). As shown in Fig. 4*A*, the energy of infinitesimal perturbations can be amplified by more than four orders of magnitude during part of the cycle. Interestingly, the optimal perturbation has a helical shape and yields its maximum energy amplification toward the end of the deceleration phase. Overall, this helical perturbation dominates during the deceleration phase, and it has an optimal azimuthal wavenumber *A* there are several families of highly amplified (suboptimal) helical perturbations parameterized by the axial wavelength.

To compare to experiments, we carried out direct numerical simulations initialized with a helical suboptimal perturbation of wavelength *B*, in close agreement with experiments. The strong fluctuations and abrupt changes in shear stress that occur during this period are shown in Fig. 4*C*. Again like in experiments, the fluctuations decayed during the acceleration phase and the flow returned to laminar. The helical vortex pattern in the radial–azimuthal plane and the waviness in the radial–streamwise cross-section resemble those in experiments, as shown in Fig. 5 (Movies S2 and S3). It can hence be concluded that the large transient amplification of disturbances during flow deceleration provides a generic mechanism for the generation of helical vortices and a subsequent breakdown into turbulence.

In a recent investigation, Pier and Schmid (25) studied in detail how pulsation modifies the classic linear instability of channel flow (two-dimensional [2D] Tollmien–Schlichting waves). In agreement with von Kerczek (13) they found that pulsation leads to a modulation of the growth rate of Tollmien–Schlichting waves. More specifically, they noted strong modal transient growth during deceleration and decay during acceleration. While this phase relationship is in very good agreement with the one observed here, pipe flow is linearly stable and hence does not support Tollmien–Schlichting waves. On the other hand, the linear instability of pulsatile pipe flow identified by Thomas et al. (15) occurs only when the oscillatory component is predominant, i.e., for parameters far from cardiovascular conditions. It stems from the thin Stokes layer near the pipe wall and it occurs at much larger pulsation amplitudes (and Reynolds numbers) and is 2D (axisymmetric,

### Lumen Constriction.

The cross-sections of blood vessels frequently deviate from the idealized circular case; for example, protrusions may arise during wound healing or stenosis formation. To test whether the helical instability may also arise under such conditions, we replaced the curved pipe segment by a straight section that includes a local constriction in the form of a spherical cap (up to *C*). For increasing Reynolds number at

In an earlier study Blackburn et al. (26) investigated linear nonmodal transient growth after a severe axisymmetric stenosis for steady and pulsatile flows. They found that nonaxisymmetric disturbances with

To further test the robustness of the helical instability, we changed the waveform of the pulsatile driving. The idealized sinusoidal flow rate modulation was replaced by the waveform typically observed in the aorta (27). Experiments were carried out in the 20-mm pipe and the flow parameters were

### Blood Flow Experiments.

While the experiments reported so far were carried out in water, we next used blood as the working fluid. Blood has non-Newtonian properties and is a dense suspension of blood cells (e.g., red blood cells take up approximately *D*). Flows were deemed unsteady if deviations in pressure were larger than twice the background noise level of the sensor. Like in the Newtonian flow also the pulsatile blood flow became unstable during flow deceleration, and a considerable drag increase was detected approximately

## Discussion and Conclusion

In summary, we report a generic instability for pulsatile pipe flow that occurs for large pulsation amplitudes and precedes the normal turbulence transition. The helical vortex pattern characteristic for this instability sets in at unusually low Reynolds numbers. As shown, weak curvature and modest pipe constrictions are sufficient to destabilize the laminar flow. It is interesting to note that the geometrical perturbations that appear to be most efficient in pulsatile flow are inefficient in the context of steady pipe flow. Curvature in fact has a stabilizing effect under steady conditions (28) and can even lead to relaminarization (29) at not too large *C* and *SI Appendix*, Fig. S1) encountered during flow deceleration offer a possible cause for endothelial activation.

## Materials and Methods

### Experimental Methods.

Experiments were carried out in straight, rigid pipes of circular cross-section: 1) A 12-m-long acrylic pipe (inner diameter *A*). The rear end of the pipe is connected to a piston system. The volume of the piston can provide approximately 3,000 to 20,000 advective time units for observation of approximately 15 to 450 pulsation cycles for the Reynolds number investigated. The plunger of the piston is driven by a motor through a gearbox. The speed of the motor is precisely controlled by a PC with a National Instruments card. The piston bore and the plunger speed set the cross-section–averaged flow speed in the pipe,

The perturbation method applied to generate turbulent puffs was as follows: A small amount of fluid, corresponding to approximately

To trigger the helical instability, two perturbation methods as sketched in Fig. 1 *B* and *C* were used and they were produced using a 3D printer. The ends of the perturbation sections were further finished in a milling machine to ensure a smooth connecting with the adjacent pipe segment. The curved pipe segment (Fig. 1*B*) is of cosinusoidal shape and has the same inner diameter as the pipe. The constriction perturbation (Fig. 1*C*) is straight and has a protrusion in the form of a spherical cap which is extended in the streamwise direction by

For this set of experiments, a V10 Phantom high-speed camera (in resolution of

The velocity fields recorded during the occurrence of the helical instability were obtained by PIV measurements. The data were recorded in the 20-mm glass pipe. The 2D planar PIV measurements were carried out in the mid–cross-section (radial–streamwise) of the pipe. To obtain all three velocity components, stereo-PIV measurements were carried out in the cross-section perpendicular to the pipe axis (radial–radial). The measurements were performed approximately

### Numerical Methods.

We numerically computed the motion of an incompressible Newtonian fluid driven through a circular straight pipe at a pulsatile flow rate. In the axial direction, periodic boundary conditions were considered. The Navier–Stokes equations were rendered dimensionless by scaling lengths and velocities with the pipe diameter D and the mean velocity

For the linear analysis, we employed the adjoint-based method of Barkley et al. (31) to calculate the optimal growth for our system. Note, however, that in our problem the base flow is time dependent,

In pulsatile flow, the laminar base flow is time dependent and hence the transient growth depends on the time **1** and **2**, respectively. Operationally, this method integrates Eq. **1** forward from **2** backward from

We solved the linearized equations using a Chebyshev–Fourier–Fourier spectral method, in which velocity and pressure are represented as

A multiparameter optimization process was carried out using the adjoint analysis. We computed the optimal growth at time t, *C*). Helical perturbations

In addition, we carried out direct numerical simulations of the nonlinear Navier–Stokes equations in cylindrical coordinates

### Data Availability.

The data can be found in Datasets S1–S8.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft and the Austrian Science Fund in the framework of the research unit FOR 2688 “Instabilities, Bifurcations and Migration in Pulsatile Flows,” Grants AV 120/6-1 and I4188-N30. D.X. gratefully acknowledges the support from the Alexander von Humboldt Foundation (3.5-CHN/1154663STP). A.V. acknowledges support from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant 754411. B.S. acknowledges the support from the National Natural Science Foundation of China under Grant 91852105. We thank Davide Scarselli for his help with the PIV measurements.

## Footnotes

↵

^{1}D.X. and A.V. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: bhof{at}ist.ac.at or duo.xu{at}zarm.uni-bremen.de.

Author contributions: M.A. and B.H. designed research; D.X., A.V., X.M., B.S., and M.R. performed research; D.X., A.V., and M.R. analyzed data; and D.X., A.V., M.A., and B.H. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission. M.D.G. is a guest editor invited by the Editorial Board.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1913716117/-/DCSupplemental.

Published under the PNAS license.

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