# Drag enhancement and drag reduction in viscoelastic flow

###### Abstract

Creeping flow of polymeric fluid without inertia exhibits elastic instabilities and elastic turbulence accompanied by drag enhancement due to elastic stress produced by flow-stretched polymers. However, in inertia-dominated flow at high Re and low fluid elasticity , a reduction in turbulent frictional drag is caused by an intricate competition between inertial and elastic stresses. Here, we explore the effect of inertia on the stability of viscoelastic flow in a broad range of control parameters and . We present the stability diagram of observed flow regimes in coordinates and find that instabilities’ onsets show unexpectedly non-monotonic dependence on . Further, three distinct regions in the diagram are identified based on . Strikingly, for high elasticity fluids we discover a complete relaminarization of flow at Reynolds number of the order of unity, different from a well-known turbulent drag reduction. These counterintuitive effects may be explained by a finite polymer extensibility and a suppression of vorticity at high Wi. Our results call for further theoretical and numerical development to uncover the role of inertial effect on elastic turbulence in a viscoelastic flow.

###### pacs:

47.20.Gv, 47.50.Ef, 47.50.Gj## I Introduction

Long chain polymer molecules in Newtonian fluid alters the rheological properties of the fluid; the relation between stress and strain becomes nonlinear. Moreover, polymers being stretched by velocity gradient in shear flow engender elastic stress that modifies the flow via feedback mechanism. It results in pure elastic instabilities larson (); shaqfeh () and elastic turbulence (ET) groisman (), observed at and . Here Re is the ratio of inertial to viscous stresses, , and Wi defines the degree of polymer stretching ; where is the flow speed, is the characteristic length scale, is the longest polymer relaxation time, and and are the density and dynamic viscosity of the fluid, respectively bird ().

ET is a spatially smooth, random in time chaotic flow, which statistical, mean and spectral properties are characterized experimentally groisman (); groisman1 (); groisman2 (); teo1 (); jun (); jun1 (); arratia (); atul1 (), theoretically lebedev (); lebedev1 () and numerically boffetta (); berti (); berti1 (); gotoh (). The hallmark of ET is a steep power-law decay of velocity power spectrum with an exponent indicating that only a few modes are relevant to flow dynamics groisman (); groisman1 (); lebedev (); lebedev1 (). Further, an injection of polymers into a turbulent flow of Newtonian fluid at reduces the drag and also has a dramatic effect on the turbulent flow structures toms (). In recent investigations, a new state of small-scale turbulence associated with maximum drag reduction asymptote is observed in a pipe flow at and . This state is termed as elasto-inertial turbulence (EIT) and exhibits properties similar to ET despite the fact that it is driven by both inertial and elastic stresses, and their interplay defines EIT properties hof (); dubief (); dubief1 (). Thus, the fundamental question arises how inertial effect modifies ET in viscoelastic flow towards turbulent drag reduction.

Numerous studies were performed in various flow geometries to unravel the role of inertia on the stability of viscoelastic flow, albeit contradictory results were obtained. In Couette-Taylor flow between two cylinders, the instability sets in at which grows with elasticity number (=) saturating at sufficiently high , and reduces with increasing inertia groismanprl96 (); groismanepl (); groismanphfl (). Whereas the onset of the instability is almost constant at very low and reduces with increasing , in a rather limited range, in agreement with numerical simulations shaqfeh1 (); thomas (). Recent experiments in the Couette-Taylor flow with both inner and outer co- and counter-rotating cylinders at low show weak, smooth dependence on muller09 (); muller11 (); muller13 (). At the moderate , either stabilization or destabilization of the first bifurcation depending on co- or counter-rotation of cylinders is found muller09 (). However, the general tendency in the dependencies of the bifurcations on reported in Refs. groismanprl96 (); groismanepl (); groismanphfl () is confirmed later in Refs. muller09 (); muller11 (); muller13 (). On the other hand, a non-monotonic dependence of the first bifurcation in a wall-bounded, plane Poiseuille flow on in its narrow range of low values is revealed in numerical simulations using the Oldroyd-B constitutive equation. The reduced solvent viscosity strongly modifies this effect: the smaller polymer contribution to the viscosity, the less pronounced effect suresh (). In extensional viscoelastic flow mckinley (), e.g. planar flow with an abrupt contraction-expansion, and in flow past a cylinder kenney (), a role of both elasticity and inertia was investigated in a narrow range of Re and Wi, and for only three values. In extensional flow, the onset of the elastic instability turns out to be independent of Re in the range of 0.1 to 40 for three polymer solutions that correspond to , 8.4 and 89. However, in the case of flow past a cylinder kenney () decreases with increasing Re. Recent numerical studies kellay () on two-dimensional viscoelastic flow past a cylinder reveal the phase diagram in () coordinates and both drag enhancement and drag reduction (DR) were observed in the range of to 10 and to . Thus, despite extensive theoretical and experimental efforts, the influence of inertia on viscoelastic flow in a broad range of () and is still not understood and a stability diagram of different flow regimes is missing.

Here we perform experiments, over a broad range of () and , in a channel flow of dilute polymer solution hindered by two-widely spaced obstacles (see Fig. 1 for experimental setup). Changing solvent viscosity by two orders of magnitude allows to vary elasticity number liu3 () by more than four orders of magnitude. Such approach enables us to investigate the role of inertia in viscoelastic flow in different flow regimes in a wide range of () and .

The main feature of viscoelastic flow at between two widely-spaced obstacles is an elastic wake instability in the form of a quasi-2D counter-rotating elongated vortices generated by a reversed flow atul (). The two vortices constitute two mixing layers with a non-uniform shear velocity profile filling the inter-obstacle space. Further increase of Wi leads to chaotic dynamics with properties similar to ET atul1 (). There are several reasons for the choice of the flow geometry: (i) Since blockage ratio , the flow between the cylinders is unbounded, like "an island in a sea" of otherwise laminar channel flow, contrary to all previous wall-dominated flow geometries that were used to study ET ( and w are the cylinders’ diameter and channel width, respectively) groisman (); groisman1 (); groisman2 (); teo1 (); jun (); jun1 (). Therefore, it is expected to observe mostly homogeneous, though anisotropic flow, closer to that considered in theory lebedev (); lebedev1 () and numerical simulations boffetta (); berti (); berti1 (). By employing unbounded flow we concentrate on variation in the bulk flow structures due to polymer additives, which results in a significant frictional loss. (ii) Large Wi and Re can be reached in the same system to scan the range from ET to DR. (iii) Several techniques can be simultaneously employed to quantitatively characterize the flow.

## Ii Experimental setup

### ii.1 The experimental setup and materials

The experiments are conducted in a linear channel of , shown schematically in Fig. 1. The fluid flow is hindered by two cylindrical obstacles of diameter mm made of stainless steel separated by a distance of mm and embedded at the center of the channel. Thus the geometrical parameters of the device are , and . The channel is made from transparent acrylic glass (PMMA). The fluid is driven by gas at a pressure up to and injected via the inlet into a rectangular channel. As a fluid, a dilute polymer solution of high molecular weight polyacrylamide (PAAm, homopolymer of molecular weight MDa; Polysciences) at a concentration ppm (, where ppm is the overlap concentration for the polymer used liu3 ()) is prepared using water-sucrose solvent with sucrose weight fraction varied from to (see Table 1 in sm ()). The solvent viscosity, , at is measured in a commercial rheometer (AR-1000; TA Instruments). An addition of polymer to the solvent increases the solution viscosity () of about . The stress-relaxation method liu3 () is employed to obtain ; for solution, is measured to be s. Linear dependence of on was shown in Ref. liu3 ().

### ii.2 Pressure measurements and Imaging system

High sensitivity differential pressure sensors (HSC series, Honeywell) of different ranges are used to measure the pressure drop across the obstacles and an additional absolute pressure sensor (ABP series, Honeywell) of different ranges are used to measure the pressure fluctuations after the downstream cylinder at a sampling rate of 200 Hz, as shown schematically in Fig. 1. The accuracy of the pressure sensors used is full scale. We measure pressure drop both for solvent and polymer solution as a function of flow speed, and the difference between these two measurements provides an information about the influence of polymers on the frictional drag.

The fluid exiting the channel outlet is weighed instantaneously as a function of time by a PC-interfaced balance (BA210S, Sartorius) with a sampling rate of and a resolution of . The time-averaged fluid discharge rate is estimated as . Thus the flow speed is calculated as . For flow visualization, the solution is seeded with fluorescent particles of diameter (Fluoresbrite YG, Polysciences). The region between the obstacles is imaged in the mid-plane via a microscope (Olympus IX70), illuminated uniformly with LED (Luxeon Rebel) at wavelength, and two CCD cameras attached to the microscope: (i) GX1920 Prosilica with a spatial resolution pixels at a rate of and (ii) a high resolution CCD camera XIMEA MC124CG with a spatial resolution pixels at a rate of , are used to record particles’ streak.

## Iii Results

Frictional drag () for each is calculated through the measurement of pressure drop across the obstacles () (see Fig. 1) as a function of and is defined as ; is the hydraulic radius and is the distance between locations of measurement atul (). Figure 2 shows variation of with Re for three values and a sequence of transitions can be identified for each . These transitions are further illustrated through high resolution plot of the normalized friction factor versus Re and Wi presented in top and bottom insets of Fig. 2, respectively.

Three flow regimes characterized by different scaling exponents are identified: (i) the first drag enhancement above the elastic instability follows for all values of explored; for high it is associated with a growth of two elongated vortices (or two mixing layers) atul (); (ii) further drag enhancement at high occurs due to ET atul1 () characterized by a steep algebraic decay in both the power spectra of velocity and pressure fluctuations with the exponents (see further), and intensive vorticity dynamics and a growth of average vorticity as and typical for ET atul1 (). For low , either a saturation or reduction of the friction factor with Re or Wi marks the DR regime. And (iii) for both high and intermediate , the DR regime with decreasing at increasing Re or Wi is observed and at low , the drag enhancement is noticed. Another striking finding is a complete relaminarization of flow, i.e. 100 drag reduction, that occurs for (also for and 3704; data not shown), where returns back to the laminar value at (). With decreasing , the transition points are shifted to a higher value of Re and Wi, and remarkably even at both drag enhancement and DR regimes can be recognized.

To elucidate further, the critical values of the respective transitions for each is mapped in (Fig. 3(a)), (Fig. 3(b)) and (Fig. 3(c)) coordinates. In the range explored for () three different transitions are observed, which are associated with elastic instability, drag enhancement and DR as shown in Fig. 3(a,b). These transitions persist for all elasticity values and the elastic instability occurs first followed by other two transitions. In addition, the complete flow relaminarization is observed only for , 2433 and 3704. Interestingly, the sequence of DR and drag enhancement changes as varies from low to high values; DR is followed by drag enhancement at low and this sequence reverses at high , as described above. This change in the sequence occurs in the intermediate range of elasticity at . Furthermore, three regions in Fig. 3(a-c) can be identified based on variation of the critical values () with . For low elasticity (), is independent of , while for high elasticity (), drops sharply with . For intermediate elasticity (), shows weak dependence on (see Fig. 3(a)). In Fig. 3(b), the dependence of on is non-monotonic: a strong growth with at low , a sharp decrease at high , and a gradual growth at intermediate . The transitions are further mapped in plane for different to emphasize the role of inertia on the stability of a viscoelastic fluid flow. Same three regions are identified: at high , grows with with a stabilizing effect of inertia, at low a steep drop of with , and in the intermediate region decreases with increasing with destabilizing effect of inertia (see Fig. 3(c)).

Long-exposure particle streak images in Fig. 4 illustrate the flow structures in three regions of elasticity and at different Re and Wi above the transitions’ values, see also corresponding S1-S9 Movies in sm (). In low and intermediate elasticity regions, a large scale vortical motion appears above the elastic instability, however, in DR and drag enhancement regimes small-scale turbulent structures dominate and the large scale vortical motion vanishes (top and middle panels in Fig. 4). In high elasticity region, e.g. , unsteady pair of vortices atul () spans the region between the obstacles (e.g. ) and at higher Re small scale vortices emerge with an intermittent and random dynamics (e.g. ) that constitute the ET regime atul1 (), whereas in DR regime (e.g. ) much smoother spatial scale and less vortical motion are found (bottom panel of Fig. 4). However, a quantitative analysis of velocity field at low and intermediate values of requires serious technical efforts and will be the subject of our further investigation.

Finally, we characterize the observed flow regimes through frequency power spectrum of absolute pressure fluctuations for various Re and Wi in three regions of elasticity. The pressure spectra are presented as a function of normalized frequency to signify the time scales involved in flow with respect to . Figure 5 shows pressure power spectra in the drag enhancement regime for three values. For low elasticity, decay exponent evolves from -1.3 to -3 with increasing Re and Wi and in the range of to 1 (Fig. 5(a) for ). It is worth to notice that is reached at the highest Re and Wi. In the intermediate range of elasticity, exponent value is obtained in the range of to 1, same as for low (see Fig. 5(b) for ), whereas for high , exhibit steep decay with in a higher frequency range to 10 for all Re and Wi values for (Fig. 5(c)). This value of is one of the main characteristics of the ET regime jun (); atul1 (). The value at high is a relevant frequency to generate ET spectra with at higher frequencies jun (), as stretching-and-folding mechanism of elastic stresses due to velocity field redistributes energy across the scales lebedev (); lebedev1 (). Similar scaling of is observed in numerical simulations in the dissipation range of the turbulent drag reduction regime gotoh (). For low , decay at up to 1 is caused by the inertial effect. In the drag reduction regime, demonstrate completely different scaling behavior with , shown in Fig. 6. For low , one finds a steep decay of at high frequencies with a scaling exponent and a rather slow decay with an exponent between -0.5 and -1 at (Fig. 6(a) for ), in accord with numerical simulations gotoh (). For high , the spectra decay steeply at high frequency , and at low frequencies a slow decay with an exponent is observed (Fig. 6(c) for ). In the intermediate range of , the decay exponent varies between -1.8 and -2.5 (Fig. 6(b) for ) in the frequency range . To highlight the scaling dependencies of between different flow regimes in each region, we present the same data in Fig. S1 in sm () for various Re and Wi, and for three regions of .

For comparison we present as well as as a function of Re in the range between and for two Newtonian fluids, water () and a solution of sucrose in water (), see Fig. S2 in sm (). The dependencies of both and on Re are smooth and growing at that differs significantly from that found for polymer solutions at low . It is also rather different from the dependence of on in a channel flow past an obstacle, which is studied extensively. Thus, we can conclude that in viscoelastic flow, we observe inertia-modified elastic instabilities, contrary to inertial instabilities modified by elastic stress. This conclusion is further supported by the measurements of the power spectra of pressure fluctuations in a Newtonian fluid flow that exhibit power-law exponent at high Re (see Fig. S3 in sm ()), contrast to those presented in Figs. 5 and 6.

## Iv Discussion and Conclusion

Polymer degradation is often encountered under strong shear and in particular at high elongation rates due to velocity fluctuations at and Gyr (). As a result of degradation, the influence of polymers on the flow becomes ineffective. To ensure that drag enhancement and DR we observe in our experiments at is not the result of polymer degradation, we reuse the polymer solutions (after performing the experiment with two obstacles) in experiments on a channel flow with a single obstacle. Indeed we observe elastic instability, drag enhancement and DR with single obstacle for , e.g. see Fig. S4 in sm () as an example and all results will be published elsewhere. Moreover, the problem of polymer degradation was addressed in details in our paper on turbulent drag reduction in a large-scale swirling flow experiment conducted at burnishev (). It was pointed out that “the main technical achievement in the experiment was long term stability of polymers in turbulent flow that allowed us to take large data sets up to data points for up to 3.5 hours at the highest Re without a sign of polymer degradation.” Thus, we conclude that the observed flow regimes in our experiments are not caused by polymer degradation.

The presented results on the friction coefficient and the pressure power spectra obtained in a wide range of the controlled parameters exhibit two remarkable features: (i) presence of three flow regimes with distinctive and different scaling behavior in both and ; and (ii) three regions on the stability diagrams in the planes of Re, Wi and parameters depending on the value of fluid elasticity. In spite of the fact that rather high values of Re are reached, inertial turbulence is not attained in the region between the obstacles and channel flow outside this region. As known from literature, turbulence in a flow past obstacle is attained at much higher Re kundu ().

The different scaling dependencies of in three flow regimes and in three regions of elasticity indicate the intricate interaction between elastic and inertial stresses. A two-way of energy transfer between turbulent kinetic energy and elastic energy of polymers also results in a modification of the velocity spectra scaling exponents at dubief (); dubief1 (). Effect of inertia at on scaling behavior of velocity power spectra with the exponent instead of in pure ET was first observed experimentally in the Couette-Taylor viscoelastic flow groisman2 () and later confirmed numerically khomami (). What is remarkable that in the drag enhancement regime about the same scaling exponent in is found for low and high at close values of Wi, and three orders of magnitude difference in Re values. It indicates the elastic nature of drag enhancement regimes both at low and high . Indeed, the scaling exponents of the pressure power spectra decay for (Fig. 5(a)) show at and . The observations of scaling , exponent of pressure spectra decay and exponent of velocity spectra decay are characteristics of ET flow atul1 (). Thus, the drag enhancement regime in low regions is typical of ET.

The striking and unanticipated observation, in high elasticity region, is significant DR and a complete flow relaminarization at and (see Figs. 2 and 3). The obtained result is different from turbulent DR observed at , where Reynolds stress exceeds the elastic one prior to the onset of turbulent DR and becomes comparable to elastic stress at the onset. Similar effect of the saturation and even weak reduction of was observed and discussed in the planar geometry with an abrupt contraction-expansion of a micro-fluidic channel flow, where the saturation of at higher polymer concentrations and even its reduction at lower were revealed in the range for three polymer solutions of different polymer concentrations mckinley (). For the highest , reached a value of at high in agreement with the early measurements in a pipe flow with an axisymmetric contraction-expansion at much lower mckinley1 ().

To find a possible explanation of DR in a wide range of and , we discuss the effect in details. At low between 1.4 and 31, either drag saturation or weak DR occurs just before the drag enhancement regime associated with ET and discussed above. It is worth to mention that due to the intricate interplay between elastic and inertial stresses the strength of DR is a non-monotonic function of and depends on the relation between Wi and Re. The higher and the lower , the more pronounced DR regime at low . The range of observed in DR regime corresponds to the vorticity suppression by elastic stress generated by polymer additives injected into a Newtonian fluid flow cadot2 (); cadot3 (); goldburg (), which is indeed confirmed by the snapshots at and and 153, shown in Fig. 4.

At high in the range and , reduces significantly. However, the complete flow relaminarization is observed only at , 2433 and 3704, where and . It means that high values of Wi and Re stabilize DR prior to the relaminarization due to finite polymer extensibility. The snapshot at and in Fig. 4 shows a vorticity-free flow, contrast to the snapshots at the same and low Re. Thus, at and high Wi, the inertial effects are negligible to suppress the growth of , whereas at and intermediate values of Wi, the drag can saturate, as seen, for example, at . As suggested in Ref. mckinley (), the saturation of observed for high is probably a consequence of polymer chains attaining their finite extensibility limit at very high Wi values. Here, we emphasize again that DR observed, at low and high Re, is not related to turbulent drag reduction realised in a turbulent flow at higher values of Re than achieved in our experiment.

The theory of ET and the corresponding numerical simulations do not consider the inertial effects and their role in ET, and therefore they are unable to explain the DR and flow relaminarization. Thus, the results reported call for further theoretical and numerical development to uncover inertial effects on viscoelastic flow in a broad range of () and .

## V Acknowledgements

We thank Guy Han and Yuri Burnishev for technical support. We are grateful to Dr. Dongyang Li for his help in the measurements of the friction factor and the pressure spectra of Newtonian fluid flow. A.V. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411. This work was partially supported by the grants from Israel Science Foundation (ISF; grant #882/15) and Binational USA-Israel Foundation (BSF; grant #2016145).

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