On turbulence decay of a shear-thinning fluidS. Rahgozar, and D. E. Rival

Citation: Physics of Fluids 29, 123101 (2017); doi: 10.1063/1.5012900View online: https://doi.org/10.1063/1.5012900View Table of Contents: http://aip.scitation.org/toc/phf/29/12Published by the American Institute of Physics

Articles you may be interested inThe physics of aerobreakup. IV. Strain-thickening liquidsPhysics of Fluids 29, 122101 (2017); 10.1063/1.4997009

Introduction to Focus Issue: Two-Dimensional TurbulencePhysics of Fluids 29, 110901 (2017); 10.1063/1.5012997

The culmination of an inverse cascade: Mean flow and fluctuationsPhysics of Fluids 29, 125102 (2017); 10.1063/1.4985998

Kolmogorov’s Lagrangian similarity law revisitedPhysics of Fluids 29, 105106 (2017); 10.1063/1.4993834

Flapping foil power generator performance enhanced with a spring-connected tailPhysics of Fluids 29, 123601 (2017); 10.1063/1.4998202

Effect of surface roughness on droplet splashingPhysics of Fluids 29, 122105 (2017); 10.1063/1.5005990

PHYSICS OF FLUIDS 29, 123101 (2017)

On turbulence decay of a shear-thinning fluidS. Rahgozar1,2,a) and D. E. Rival21Department of Energy, Materials and Energy Research Center (MERC), Tehran, Iran2Department of Mechanical and Materials Engineering, Queen’s University,Kingston, Ontario K7L 3N6, Canada

(Received 7 May 2017; accepted 8 November 2017; published online 4 December 2017)

An experimental investigation of turbulent flow in a shear-thinning fluid is presented. The experi-mental flow is a boundary-free, uniformly sheared flow at a relatively high Reynolds number (i.e.,Reλmax = 275), which decays in time. As just one example of decaying turbulence, the experiment canbe thought of as a simple model of bulk turbulence in large arteries. The dimensionless parametersused are Reynolds, Strouhal, and Womersley numbers, which have been adapted according to thecharacteristics of the present experiment. The working fluid is a solution of aqueous 35 ppm xanthangum, a well-known shear-thinning fluid. The velocity fields are acquired via time-resolved particleimage velocimetry in the streamwise/cross-stream and streamwise/spanwise planes. The results showthat the presence of xanthan gum not only modifies the turbulent kinetic energy and the dissipation ratebut also significantly alters the characteristics of the large-scale eddies. Published by AIP Publishing.https://doi.org/10.1063/1.5012900

I. INTRODUCTION

Turbulent flow in shear-thinning fluids is of fundamentalimportance in numerous industrial applications and biologicalsystems. These applications include flows in paper-making,1

food and paint industries, slurries and dilute polymer solu-tions in many industrial processes,2–5 and finally blood flowin large arteries.6,7 Despite these numerous applications, ourknowledge about the characteristics, scales, and structures ofnon-Newtonian turbulent flows is still far from complete. Thevast majority of research in turbulent flow for non-Newtonianfluids has concentrated on the understanding of wall-boundedflow of drag-reducing polymers.8–11 In addition, relativelyfew studies have been carried out in bulk flow, homogenousisotropic, and grid turbulence in order to isolate the effects ofpolymer on the flow from those of the wall.12–15 These theoret-ical, experimental, and numerical attempts have improved ourunderstanding of polymer-turbulence interactions. However,even for the case of turbulent flow of drag-reducing polymers,which has been studied intensively over the last decades, apersuasive physical understanding and a predictive model areboth still lacking. This patchy knowledge is not surprisingsince even turbulent flow with Newtonian fluids is in itselfa very challenging subject. Moreover, the direct numericalsimulation (DNS) has often been performed at low Reynoldsnumbers and has relied on limited polymer models.16,17 On theother hand, experimental studies have been mainly carried outin wall-bounded flows, where it is difficult to isolate the walland polymer roles from one another.11 Furthermore, accord-ing to several experimental studies, the interaction betweenpolymer additives and turbulence is highly sensitive to the con-centration and characteristics of polymers.16,18,19 Hence, theresults reported in these studies are often controversial,12,20

a)Electronic mail: s.rahgozar@merc.ac.ir

which means that more complete experiments are needed inorder to develop and test the physical models. As one exam-ple, Cai, Li, and Zhang15 carried out a numerical simulationof decaying, homogeneous, isotropic turbulence with polymeradditives. They reported a larger decay rate of the turbulentkinetic energy in the polymer solution case because of theinteraction between turbulence and polymer microstructures.However, by using laser anemometry and flow-visualizationtechniques, McComb, Allan, and Greated12 showed that thepolymer additive reduced both the turbulent intensity and therate of decay in their experiment with grid-generated turbu-lence. White and Mungal16 provided a review of polymer dragreduction in turbulent wall-bounded shear flows. They empha-size the complexities of these flows and the need for moredetailed experiments and simulations.

The theory of decaying homogeneous isotropic turbulenceand homogeneous shear flow dates back to the 1930s.21,22

Decaying turbulent flows have also been investigated rig-orously both numerically and experimentally over the pastdecades. The classic example is grid turbulence, which hasbeen studied extensively as the most fundamental case.23,24

The main motivation to study such a simple flow has beento characterize the dissipative nature of these flows. However,there is no turbulence-production mechanism in the absence ofa mean velocity gradient. On the other hand, the nearly homo-geneous turbulent shear flow has been achieved both numer-ically25 and experimentally in water and wind tunnels.26–28

Although this flow is theoretically simpler than traditionalwall-bounded flows such as channel and boundary-layer flows,it is more difficult to realize experimentally.27 In contrast togrid turbulence, a turbulence-production mechanism is presentin this kind of flow. Numerous attempts have been made overthe past decades to generate uniform mean shear in wind andwater tunnels by passing a flow through various condition-ing units at the test-section entrance.26–28 Alternatively, some

1070-6631/2017/29(12)/123101/10/$30.00 29, 123101-1 Published by AIP Publishing.

123101-2 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

research groups have developed active grids,29 and few studieshave also been carried out in water tanks to generate isotropicturbulence.30

The present study focuses on the modification of decay-ing turbulence in a shear-thinning fluid with specific viscousbehaviour rather than analyzing the effect of polymer addi-tives on turbulent drag reduction. The experimental flow isa boundary-free, uniformly sheared flow at relatively highReynolds numbers. The study is thus a combination of twoclassic flows, i.e., the decaying turbulence and a uniformlysheared flow. The idea is to keep the geometry as simpleas possible allowing the turbulent characteristics of the flowto be representative of turbulent motion in a broad range ofapplications using the relevant dimensionless parameters. Thesimple geometry also allows one to employ the particle imagevelocimetry (PIV) technique in order to accurately measurevelocity fields and consequently to investigate the details ofturbulence scales, parameters, and structures. As a referencefor the working fluid and the dimensionless parameters, flow inlarge arteries will be referred to throughout the paper. Althoughthe selection of the working fluid and the Reynolds number(tow speed) is made based on the order of magnitude of bloodflow in large arteries as a case study, the experimental approachand therefore results can be easily adapted to other appli-cations. The chosen working fluid is a solution of aqueousxanthan gum (XG), which among other cases replicates theviscous behaviour of blood as a well-known shear-thinningfluid.31,32 Due to the limitation in the spatial resolution, thefocus is on the large-scale structures. Like any study of sim-ple canonical flows, it is assumed that there exist importantuniversal mechanisms in these turbulent flows.33,34 As anexample, Sekimoto, Dong, and Jimenez34 reported important

common characteristics between the statistically stationaryhomogeneous shear turbulence and wall-bounded turbulence.A proper analysis of a simplified flow can thus significantlyimprove our understanding of more complex flows. More-over, as already demonstrated by several studies,12,15,19 theturbulence modification caused by polymer additives is notnecessarily dependent on the presence of a wall. Hence, adeeper understanding of wall-free mechanisms can be alsoof fundamental interest. It is certain that this simplified exper-iment does not represent the whole complexity of turbulentflow in a shear-thinning fluid, particularly in confined cases,but it can provide more insight into the way that turbulenceand polymer additives interact, especially farther from the wallregion. To the authors’ knowledge, there have been no previ-ous investigations of turbulent flow of a shear-thinning fluid ina boundary-free uniformly sheared flow, and as such, this firststudy could provide the basis for further experiments on thesubject.

It should be emphasized that the blood flow in large arter-ies is selected here as a case study mostly because it is avery typical example of decaying turbulence and blood is anarchetype of shear-thinning fluids. This does not mean that wecan easily ignore all complexities of such a complex flow suchas wall effects. However, assuming the smallest scale of turbu-lence to be of order of red blood cells, the relative length scalesof small to large-scale structures can be of order of 0.001–0.01in large arteries. This broad range of scales suggests that theflow in the center of large arteries may behave as bulk flow,where the surrounding wall does not necessarily play a dom-inant dynamic role. Nevertheless, this assumption needs tobe further evaluated based on rigorous high-resolution in vivoinformation, which to date is not available.

FIG. 1. (a) Optical towing tank with thetest section in the middle. (b) Side viewof the test section with the shear genera-tor. (c) The solidity distribution of thegrid. Y is the cross-stream coordinateand H is the tank height. (d) Three repre-sentative times in a cardiac cycle, wheret* is the dimensionless relative time.

123101-3 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

Throughout the paper, U, V, and W are instantaneousstreamwise, cross-stream, and spanwise velocities, respec-tively. 〈〉 denotes the mean value over independent realizations,and u, 3, and 4 are the fluctuating streamwise, cross-stream,and spanwise velocities, respectively. For viscosity, however,ν′

represents viscosity fluctuations, while ν stands for themean viscosity. The streamwise, cross-stream, and spanwisecoordinates are, respectively, x, y, and z (see Fig. 1).

II. DIMENSIONLESS PARAMETERS

In order to experimentally model the decaying turbulencein the case of flow in large arteries or other time varyingflows, it is vital to introduce the relevant dimensionless param-eters beforehand. These parameters allow us to design theexperiment with respect to the principal characteristics of, forinstance, an in vivo flow. As with canonical, rigid straight-pipe flow studies, it is common practice to use Reynolds andWomersley numbers based on global quantities (e.g., center-line velocity and diameter) for turbulent flow in large arter-ies. Hence, in such studies, the effects of wall deformabilityand curvature, arterial branching, valves, and heart pulsa-tion on turbulence are neglected. However, in the presentpaper, the dimensionless parameters are proposed in termsof the magnitude of velocity fluctuations, which is a directmeasure of turbulence, allowing us to take a more generalapproach.

A. Turbulent Reynolds number

Suppose that L is an integral length scale characterizingthe energy-containing turbulent eddies in vessels and U is acharacteristic velocity with U2 = 〈uiui〉 /3 = 2k/3 (Einsteinsummation applies), then the turbulent Reynolds number canbe defined as the ratio of the viscous decay time to the char-acteristic eddy decay time: ReT = UL

/ν = (L2/ν)

/(L/U).

Here, ν is the kinematic viscosity and k is the turbulent kineticenergy (TKE). The average L at high Re is generally ofthe order of the imposed scale by boundaries (e.g., pipe orcylinder diameter) or object producing the turbulence (e.g.,grid size). In the absence of boundaries, L can be obtainedfrom two-point correlation functions as will be shown inSec. IV C.35 Alternatively, from a different point of view, Lcan be roughly estimated using the well-known scaling law:21

ε ≈ CεU3/L, where L is the integral length scale obtainedfrom this equation, ε is the dissipation rate, and Cε is anundetermined constant close to unity.29 The equation ReT canbe rewritten by using L ≈ L = U3/ε and U2 = 2k/3 asReT ≈ U4/νε = 4k2/9νε , which is similar to the definitionin the literature.35 Note that k/ε ≈ L

/U is the characteris-

tic large-eddy decay time and ReT characterizes the ratio oflarge- to small-scale [τη = (ν

/ε)1/2 Kolmogorov] time scales.

Hence, ReT is a relevant dimensionless parameter quantify-ing the turbulent scales and strength. It can be shown, forisotropic turbulence, that the Taylor microscale Reynolds num-ber and this turbulent Reynolds number are not independentsince Reλ = Uλ

/ν =√

15Re1/2T , where the Taylor microscale

λ is estimated from this equation for isotropic turbulence,ε = 15νU2/λ2 = 10νk/λ2.21,36

B. Turbulent Strouhal number

Another important time scale of the flow is based on thedegree of unsteadiness (e.g., the cardiac cycle duration). Inthe present study, this time (cycle duration) is defined as thetime in which the flow loses a significant amount of its turbu-lent kinetic energy (≈80%). Based on in vivo data, the flowmay lose up to nearly 100% of its TKE in a single heartcycle. A new dimensionless parameter can be thus definedas StT = (L

/U)/(1/f ) = fL

/U, which is the time scale of

large-eddy decay to the unsteadiness time. Here, f = Uc/Td is

the frequency of oscillation (e.g., pulse), and T is the dimen-sionless unsteadiness or the cycle duration, normalized by thetowing speed (Uc) and the shear-generator rod diameters d (seeFig. 1 and Sec. III A). This parameter compares the lifetimeof a characteristic large eddy with the time between pulsesand indicates if large-scale eddies have sufficient time to begenerated and destroyed in a single cycle. A similar approachwas taken by Kiser et al.37 to define this parameter. By mul-tiplying ReT and StT , a new parameter can be defined asW2

T = fL2/ν, which is the ratio of the viscous decay time to theunsteadiness. WT can be thought of as a turbulent Womersleynumber.

III. EXPERIMENTA. Facility

The experiment is performed in a large optical towingtank at Queen’s University with an approximate dimension of8 × 1 × 1 m3. The test section is located in the middle of thetank to ensure negligible end wall effects. A high-speed tra-verse system tows a shear generator in the center of the towingtank at speeds up to 1 m/s over a maximum distance of 4 m[see Fig. 1(a)]. An approximately uniform mean shear is gen-erated in the tank by towing the shear generator consistingof a plane parallel-rod grid of uniform rod diameter (d) withnon-uniform spacing (s). Figure 1(b) shows the schematic ofthe shear generator passing through the field of view. In thepresent experiments, the towing speed (Uc) was 0.2 m/s andthe diameter of the rods was d = 42 mm. Figure 1(c) showsthe solidity distribution of the grid, which is based on the anal-ysis of Owen and Zienkiewicz.38 The blockage necessary forgenerating shear is kept minimal (i.e., about 30% of the areaof the cross section) in order to minimize potential large-scalesecondary flows. Since the measurement plane is located atthe midspan of the tank far from the side walls, the mean flowis expected to behave essentially in a bulk two-dimensionalsense [see also Figs. 5(b) and 5(c)]. However, an additionalmeasurement is performed in the streamwise/spanwise (x � z)plane in order to obtain all velocity components for our cal-culations and as a means to check this assumption of flowuniformity.

B. PIV measurements

Time-resolved 2D PIV measurements are performed inthe streamwise/cross-stream (x � y) and streamwise/spanwise(x � z) planes. The PIV images were acquired using a Fast-cam SA-4 camera (Photron, San Diego, CA, USA) at 125 Hz

123101-4 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

TABLE I. Relevant similarity and scale parameters for the present experiment and in vivo data of Les et al.43

at the peak TKE. Here, t1 denotes the duration of a cycle (cardiac cycle for the in vivo case), while L = U3 /εand U = (2k/3)1/2 is the characteristic velocity. L represents the integral length scale, which is assumed to be themean diameter of abdominal aortic aneurysm for the in vivo data and is obtained from two-point correlations inthe experimental data (L11).

Parameters ReT max Reλmax StT WT t1 k/ε L/U 1

/S τη λ η L L/L

s s s s s mm mm mmExpt. 1250 275 0.2 15 6 1.3 1.1 6.3 0.03 8.3 0.2 37.8 1.3In vivo data 1280 . . . 0.2 15 1 . . . 0.2 . . . . . . . . . . . . 30 . . .

with a full resolution of 1024 × 1024 pixels. Neutrally buoy-ant 100 µm hollow glass spheres (Potters Industries, Carlstadt,NJ, USA) were added to the water to serve as tracer particles.The particles were illuminated using a solid state, 532 nmcontinuous wave 1 W laser (Dragon Lasers, Changchun, Jilin,China) with a sheet thickness of about 1 mm in the middleof the field of view. Images of the illuminated tracer particleswere acquired within a field of view (FOV) of approximately160 × 160 mm2. Fifty independent measurement runs havebeen performed in each plane with 125 PIV image pairs every2 s. Each measurement run corresponds to a single cycle (seeSec. II). With respect to the experiment, a single cycle corre-sponds to each run of towing the shear generator. The timebetween each run is long enough (i.e., at least 15 min) tomake certain that the flow develops from a quiescent state(i.e., TKE <0.5% of TKEmax). The images are processed inDaVis 8.2.0 (LaVision GmbH, Goettingen, Germany); a multi-pass image iteration is performed with the final interrogationwindow size of 16 × 16 pixels. The spatial resolution is thusabout 2.5 mm, which is approximately equal to 10η (η is theKolmogorov length scale). Since a 50% overlap of the inter-rogation windows is used to compute the velocity vectors, thevector spacing is half that of the spatial resolution. Althoughthe sampling frequency (125 Hz) would in theory be appropri-ate to measure down to the Kolmogorov scale (see Table I), thespatial resolution is only sufficient to resolve large-scale struc-tures on the order of the integral length scale (see Sec. IV C).The peak validation and local neighbourhood validation are

used to detect and substitute false vectors at each iterationof image interrogation. A vector is considered valid if thedifference between the two highest correlation peaks is atleast 20%. The erroneous vectors were substituted but onlyif at least five valid vectors surrounded them. The maxi-mum particle displacement in pixels is about a quarter ofthe final interrogation window size, which ensures a mini-mal random error. With the typical noise level of 0.1 pixel,the maximum random errors of velocity fields are less than±1.5%. Based on the analysis reported in the studies ofSaarenrinne, Piirto, and Eloranta,39 and Sciacchitano andWieneke,40 the random uncertainty of other turbulent quan-tities, i.e., Reynolds stresses, turbulent kinetic energy, dis-sipation rate, two-point correlations, and also the integralscales, is estimated to be less than ±5%. In order to ver-ify the statistical convergence and also this level of uncer-tainty, turbulent statistics are compared between a randomsub-group of 20 realizations of the experiment and the wholedatabase (50 realizations). The results are shown in Fig. 2.The observed deviation is within 3% for the turbulent kineticenergy, the dissipation rate, and the integral length scale. ForReynolds shear stress, the average deviation is around 15%.However, by using 40 realizations, the results (red circles) tendto converge around those of the full database (black square)with an average deviation of about 5%. The details of calcu-lations are given in Sec. IV C. As will be shown in Sec. IV,this level of uncertainty is insignificant with respect to theresults.

FIG. 2. Comparison of (a) TKE, (b) dissipation rate, (c)Reynolds shear stress, and (d) integral length scale, calcu-lated with a number of different independent realizations.Black square symbols: N = 50 realizations, red diamondsymbols: N = 20 realizations, and red circle symbols:N = 40 realizations. All data are for the Newtonian case.

123101-5 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

FIG. 3. Viscous behaviour characterization of the shear-thinning fluid (red)compared with the literature (black)41 across a range of strain rates. The dashedline represents µ

/µhigh = 3.4γ�0.19.

C. Working fluids

The shear-thinning working fluid consisted of 275 g xan-than gum (XG) dissolved in 7.75 m3 of 0.4% by weightaqueous glycerol (i.e., about 35 ppm XG). This small amountof glycerol has little effect on the solution density (<0.1%)and the kinematic viscosity (<0.9%) compared to pure water.Nevertheless, the tiny amount of XG significantly alters theviscoelastic characters of the solution. Li, Walker, and Rival41

used the same concentration of XG but dissolved in 4% byweight aqueous glycerol to replicate blood viscosity. The glyc-erol only facilitates the dissolving of XG and has no importanteffect on the shear-thinning property of the solution itself. Inorder to determine viscosity at different shear rates, an AR2000rheometer with temperature control (TA Instruments Inc.) wasused. The measurement was performed under standard condi-tions, and the uncertainty is expected to be less than ±3%. Theresults were compared with those of Li, Walker, and Rival,41

and as shown in Fig. 3, a very good agreement was observed.A power expression of the measured data in the form of kγn�1,with k = 3.4 and n = 0.81, is also included in this figure(dashed line). This equation is in general agreement with otherreported models in the literature.42 It is worth mentioning thatLi, Walker, and Rival41 had already compared the viscositybehaviour of their blood analog fluid with that of real bloodat different hematocrits, demonstrating good agreement. Notethat for comparison, all measurements have been carried outtwice, i.e., with the above non-Newtonian fluid as well as withpure water as a Newtonian fluid.

IV. RESULTS AND DISCUSSIONA. Dimensionless parameters

Table I summarizes the main parameters described inSec. II for the Newtonian fluid at peak TKE. Kolmogorovtime τη and length η scales are also presented for comparison.

For all parameters in the experiment, the integral length scaleand the Taylor microscale are obtained from two-point corre-lations. The details of estimation are given in Sec. IV C. Thein vivo data are based on the work of Les et al.43 in which turbu-lent kinetic energy is estimated for abdominal aortic aneurysm(AAA). To estimate ReT , the integral length scale is assumed tobe equal to the mean diameter of the AAA, i.e., 3 cm. The fre-quency of oscillation is assumed to be 1/s for estimating the tur-bulent Strouhal number. The value of StT ranges from 0.2 to 5.3for the highest and lowest ReT (i.e., 1280 and 50), respectively.With the same assumptions, the turbulent Womersley numberis constant for all cases, i.e., WT ≈ 15. As shown in this table,excellent agreement between the experimental flow and thein vivo induced data is achieved. The fact that StT < 1 at a highReynolds number indicates that the flow cycle duration is longwith respect to the large-scale eddy lifetime.

B. Mean flow characteristics

Since the shear-thinning working fluid demonstrated noimportant effect on the mean flow characteristics, in the presentsection, only results for the Newtonian case are presented.However, the effects of the shear-thinning fluid on the turbulentstatistics and scales are provided in Sec. IV C.

In order to create the experimental flow, the shear gen-erator introduced in Sec. III passes through a fixed field ofview and the image recording starts right after its passage iscomplete. The considered time portion of the flow that can rep-resent the general turbulent flow in large arteries away fromwalls is selected based on two criteria: first, turbulent kineticenergy has to lose a substantial amount of its energy (e.g.,80%) as reported in the literature.43 Second, the dimensionlessparameters introduced in Sec. II have to be comparable to thoseestimated for the in vivo data. Moreover, in order to simplyestimate the turbulent statistics from the homogeneous turbu-lence approximation, the mean velocity gradient has to reacha nearly uniform state, while the turbulent Reynolds numberis comparable to that of the in vivo data. As will be shownin Sec. IV C by setting the in vitro cardiac cycle duration toT = t1Uc

/d = 28.6 (t1 is the cycle duration in physical unit s),

the values of StT , WT , and ReT become comparable to thoseobtained from in vivo data.

Rose26 studied the homogeneity of a turbulent shear flowgenerated with a shear generator, which was itself designedbased on the analysis of Owen and Zienkiewicz.38 As reportedby Rose,26 the flow approaches a state of uniform Reynoldsnormal stresses and nearly uniform mean shear at aboutτ = xS

/Uc = 0.4, where τ is a dimensionless time, and x, S,

and Uc denote the distance from the shear generator, the meanshear (∂〈U〉

/∂y), and the convection velocity, respectively.

Note that for different flows and geometries very differentvalues for τ have been reported in the literature. As will beshown in the following, our results are consistent with thoseof Rose.26 Figure 4 shows the mean streamwise velocity pro-files in the x � y planes estimated by ensemble averaging of 50independent velocity vector fields. In each time, the profiles areshown in three streamwise locations, i.e., in the middle (blacksolid lines) and both sides of the field of view. All velocitiesare normalized by the speed of the shear generator (Uc), which

123101-6 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

FIG. 4. Mean streamwise velocity profiles in threestreamwise locations on the x � y plane, in the middle(black solid lines) and both sides of the field of view. (a)t* = �0.3 when the shear generator leaves the field ofview. (b) t* = 0. (c) t* = 0.5. (d) t* = 1.

is 0.2 m/s. In this figure, t* = t/T = �0.3 (t and T both nor-

malized by Uc and d) corresponds to the time that the sheargenerator leaves the field of view. As expected, Fig. 4(a) showsan important variation in 〈U〉 with sinusoidal-shaped profiles,which is related to the presence of the rods of the shear gen-erator. In Fig 4(b), t* = 0 (τ ≈ 0.4) corresponds to the time inwhich the mean streamwise velocity profiles become approx-imately linear, which is 2 s after t* = �0.3. Hence, t* = 0 isthe beginning of the cycle in which the flow has its highestvelocity fluctuations due to the passage of the shear generator.Based on the in vivo data for abdominal aortic aneurysm,43

t* = 0 can thus be considered as the mid-deceleration time,where the TKE is maximum. Figure 4(d) (t* = 1 with τ ≈1.3) corresponds to the end of the period where the minimumfluctuations exist and it corresponds to a time right before thenext mid-deceleration, i.e., near peak systole [see Fig. 1(d)].Figure 4(c) shows a time between peak systole and mid-deceleration in the diastolic phase [i.e., t* = 0.5 with τ ≈ 0.8,see Fig. 1(d)]. The mean shear (S) is about 0.15 1/s for allcases. It is worth mentioning that the local shear can be tens tohundreds of times higher. Figure 5 shows that the mean cross-stream and spanwise velocities are virtually zero at t* = 0.5 asexpected for this flow. It also shows that the mean streamwisevelocity is reasonably homogeneous in the spanwise direction.

FIG. 5. (a) Mean cross-stream velocity profiles in three streamwise locationsin the x � y plane. [(b) and (c)] Mean streamwise and spanwise velocitiesin three streamwise locations in the x � z plane, respectively. All data arecaptured at t* = 0.5.

Furthermore, the root-mean-square velocities shown in Fig. 6demonstrate a very good homogeneity in the spanwise andcross-stream directions. Note that a comparable homogeneityis also observed at other times and also for the non-Newtonianflow (not shown).

C. Turbulent statistics

Figure 7(a) shows the decay of the turbulent kinetic energyover a single cycle. TKE loses more than 80% of its energy witha decay exponent n of �1.3 (k(t) ∼ t−n

1 ) for the Newtonian case(black square symbols). This is consistent with the classicaldecay law reported in the literature.24 For the non-Newtoniancase (red nabla symbols), the decay rate is larger with com-parable TKE at t* = 0, the time in which the local shear rateis maximum. The difference in the turbulent kinetic energydecay can be attributed to the effective viscosity, which is muchhigher at a lower shear rate (see Fig. 3).9 Similar results havebeen reported in other studies with varying conditions.12,15,20

For a homogeneous turbulent shear flow, the turbulent kineticenergy evolves by dk

/dt = P � ε , where P is the rate of produc-

tion of turbulent kinetic energy: P = �S〈u3〉with S = ∂〈U〉/∂y,

which is the mean shear rate.35 It is important to note thatfor non-Newtonian flow, the turbulent kinetic energy equationhas two more dissipation terms because of viscosity fluctu-ations (ν′). Hence, in the case of the non-Newtonian flow,ε denotes the summation of all dissipation terms, i.e., meanviscous dissipation (2ν〈sijsij〉, with sij =

12 ( ∂ui∂xj

+∂uj

∂xi)), turbu-

lent viscous dissipation (2〈ν′

sijsij〉), and mean shear turbulent

viscous dissipation (2〈ν′sij〉Sij, with Sij =12 ( ∂Ui

∂xj+∂Uj

∂xi)).44,45

Figure 7(b) demonstrates that ε also decays with time but witha different rate. The reduction in the dissipation rate of thenon-Newtonian case up to t* = 0.2 and its increase afterwardmay be attributable to the fact that polymers absorb and storea portion of turbulent energy at high shear rates and thenrelease this energy at low shear rates, leading to a higheroverall dissipation rate. This assumption appears consistentwith the energy balance theory of Tabor and De Gennes,46

in which the Kolmogorov cascade is valid only down to a

123101-7 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

FIG. 6. [(a) and (b)] Root-mean-square of streamwiseand cross-stream velocities in three streamwise locationsin the x � y plane. [(c) and (d)] Root-mean-square ofstreamwise and spanwise velocities in three streamwiselocations in the x � z plane. All data are captured att* = 0.5.

certain limit where the polymer stresses balance the Reynoldsstresses.46,47 Hence, it can be inferred that at the end of acycle, when Reynolds stresses weaken, polymers have moreroom to modify turbulent structures. Moreover, the fact thatthe rate of dissipation decays more slowly for non-Newtonianthan for Newtonian fluids is also consistent with the analysis ofPinho.44

Figures 7(c) and 7(d) present Reynolds shear stresses.As expected for the present flow, 〈u4〉 deviates around zerowith an average deviation of less than 1% and a maximumdeviation of about 2.5% at t* = 0.2, all with respect to thecorresponding TKE. Except for t* = 0, a significant reduc-tion in 〈u3〉 is demonstrated for the non-Newtonian case. Thisis consistent with several previous studies; see the study ofPinho44 for a review on this matter. This finding has beenattributed to the appearance of an extra elastic shear stressbecause of polymer additives.44 To explain the change intrend between t* = 0 � 0.2, the values at t* = 0.1 have beenalso added to Fig. 7(c). It is clearly confirmed that the jumpbetween t* = 0 and 0.2, in the non-Newtonian case, is anactual physical trend, which can be attributed to the fact thatat a high shear rate the shear-thinning fluid behaves virtu-ally like the Newtonian one. Furthermore, although 〈u2〉

/k,

〈32〉/k, and 〈42〉

/k are approximately self-similar during a

cycle,27�〈u3〉

/k shows an increasing trend. It is also found

that the assumption of P = ε is not at all acceptable in this flowsince the rate of change of turbulent kinetic energy (dk

/dt)

is very high compared to the rate of production of turbulentkinetic energy, although the absolute value of P is compara-ble to similar but non-decaying flows at comparable Reynoldsnumbers; see, for example, the work of Vanderwel andTavoularis.28

As mentioned earlier, the focus of the present work is onthe large-scale structures due to the limitation of spatial res-olution. This part of the analysis is based on the two-pointcorrelations and the characteristic eddy decay time, which is ameasure of the lifetime of the dominant turbulent eddies.Figures 8(a)–8(c) show two-point correlations of the stream-wise velocity fluctuation Ruu in the x � y plane at t* = 0.5for the Newtonian fluid. This figure gives us an idea of thespatial extent of large-scale eddies in this plane. The integraland microscale length scales can be estimated from two-pointcorrelation functions. The longitudinal integral length scaleL11 can be defined as the area under the curve of Ruu in thestreamwise direction [Fig. 8(c)], i.e., L11(t) = ∫

∞0 Ruu(r, t)dr,

r = ∆x.35 Note that for calculating the area under this curvea double-term exponential function is fitted to the tail of thecurve. Furthermore, the Taylor microscale is estimated fromthe second derivative of R33 at the origin, ∆x = 0,35 using asecond-order backward differentiation. Figure 8(d) comparesthe integral length scales estimated from two-point correla-tions and that obtained from the energy cascade hypothesis.The fact that these two length scales have virtually similartrends suggests that a fully developed cascade exists in which

FIG. 7. Variation of (a) TKE, (b) dissipation rate, (c)〈u3〉, and (d) 〈u4〉 in a single cycle. Square black symbols:Newtonian and nabla red symbols: non-Newtonian. Thedifference between the Newtonian and non-Newtoniancases at, for instance, t* ≈ 0.8 for both the TKE anddissipation rate is about 25%. Note that the discrepancyobserved here is significant with respect to the level ofuncertainty, as introduced in Sec. III B. Error bars arealso shown for 〈u3〉 and 〈u4〉 to illustrate the significanceof the discrepancies.

123101-8 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

FIG. 8. (a) Contours of two-point correlation coeffi-cients of the streamwise velocity fluctuations Ruu in thestreamwise/cross-stream planes at t* = 0.5. Contour lev-els are from 0.2 to 0.9 by increments of 0.1. [(b) and(c)] Cross-stream variation and streamwise variation ofRuu at the origin, respectively. Only one out of three datapoints is shown. (d) Integral length scale obtained fromtwo-point correlations (black nablas) and from the scal-ing theory (red triangles) in one cycle. All figures are forthe Newtonian fluid.

FIG. 9. (a) Integral length scale obtained from two-pointcorrelations (solid red line) and from the scaling theory(dashed red line) in one cycle for the non-Newtonian fluidcompared with that obtained from two-point correlationsfor the Newtonian fluid (black squares). (b) The charac-teristic eddy decay time, for Newtonian (black squares)and non-Newtonian (red nablas).

the dissipation can be inferred from large-scale quantities.29

This can also be deduced from the rough similarity of decaytimes L

/U and k

/ε presented in Table I.

Figure 9(a) contains the same information as Fig. 8(d)but for the non-Newtonian case. The evolution of the integrallength scale for the Newtonian fluid is also shown for compar-ison. Similar to the Newtonian case, the integral length scalesestimated from two-point correlations and the energy cas-cade hypothesis follow virtually the same trend. However, thisfigure clearly shows that the addition of XG significantly mod-ifies the integral length scale. This occurs despite a consensusthat the direct action of polymer is on the small scales of tur-bulent flows.48 The characteristic eddy decay time is shownin Fig. 9(b) for both the Newtonian and non-Newtonian cases.Again, a large difference is observed here, particularly at theend of the cycle. For both the integral length scale and thecharacteristic eddy decay time, the difference betweenthe Newtonian and non-Newtonian cases is not significant atthe beginning of the cycle, i.e., the time in which the localshear rate is maximum and consequently the viscosity is closeto the Newtonian fluid. As the local shear rate decreases,the difference between the Newtonian and non-Newtoniancases gradually increases for both the integral length scaleand the characteristic eddy decay time. Although the mecha-nism behind such an effect cannot be explored in the presentstudy, considering Fig. 7 and previous theories,9,46 the fol-lowing can be conjectured. As mentioned earlier, higher vis-cosity and released elastic energy result in a decrease in theTKE and an increase in the dissipation rate. Consequently,the combination of these two mechanisms leads to relatively

high dissipation of TKE (i.e., lower k/ε). This means that

the dominant eddies have a shorter lifetime and a shorterlength scale, but further experiments are needed to confirmthis hypothesis. Furthermore, Fig. 9(b) shows that at leastfor the first generated eddies, the analog cardiac cycle dura-tion is long with respect to the large-scale eddy lifetime. Inother words, large-scale eddies can be generated and destroyedwithin a single cycle, and thus large-scale turbulent structuresmay have an important role in the dynamics of the presentflow. This behaviour is even more pronounced for the non-Newtonian case in which the dominant eddies have a shorterlifetime.

V. CONCLUSIONS

A boundary-free, uniformly sheared, decaying turbulentflow has been produced by towing a shear generator in alarge-scale optical towing tank in order to study the modifi-cation of turbulence in a shear-thinning fluid. The experimentis designed to model the bulk turbulent flow in, for instance,large arteries as well as with other industrial environments.The working fluid is a non-Newtonian analog of 275 g xan-than gum dissolved in 7.75 m3 of 0.4% by weight aqueousglycerol, which replicates the viscous behaviour of blood as awell-known shear-thinning fluid. Although the turbulent flowin large arteries is used here as a reference, the experiment canbe adapted for a broad range of applications using the rele-vant dimensionless parameters. Time-resolved particle imagevelocimetry has been used to quantify this experimental flow inboth the streamwise

/cross-stream and streamwise

/spanwise

123101-9 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

planes, allowing us to have three components of velocityfluctuations.

It is shown that this small amount of xanthan gum modifiesboth the turbulent kinetic energy and dissipation rate, albeitat a different rate. More interestingly, both characteristic timeand length scales of large-scale structures are also significantlyaltered for the non-Newtonian case. A number of explanationsfor these observations are also proposed. The concentrationof xanthan gum in the present study is lower than that in themajority of studies,2,49 which may allow us to generalize theseresults, at least qualitatively, to a wider range of applications.Furthermore, due to the shorter lifetime of large-scale turbulenteddies, it is expected that these eddies play a more importantrole in the non-Newtonian case compared to the Newtonianone in this specific unsteady flow.

This current study could be further extended by perform-ing higher spatial resolution measurements and investigatingthe interactions of large- and small-scale vortical and momen-tum structures. Moreover, by characterizing polymer scalesand comparing them with those of turbulent flow, eventuallydeeper understanding can be achieved.

ACKNOWLEDGMENTS

The authors acknowledge the funding provided by NaturalSciences and Engineering Research Council of Canada. Theyalso wish to thank J. N. Fernando for his help with the setup,R. Mendez for his assistance in the measurements and dataprocessing, and A. M. Walker for his comments regarding thexanthan gum experiment.

1R. J. Kerekes, “Rheology of fibre suspensions in papermaking: An overviewof recent research,” Nord. Pulp Pap. Res. J. 21, 598 (2006).

2A. S. Pereira and F. T. Pinho, “Turbulent characteristics of shear-thinningfluids in recirculating flows,” Exp. Fluids 28, 266–278 (2000).

3R. J. Poole and M. P. Escudier, “Turbulent flow of non-Newtonian liq-uids over a backward-facing step: Part 1. A thixotropic and shear-thinningliquid,” J. Non-Newtonian Fluid Mech. 109, 177–191 (2003).

4M. Rudman, H. M. Blackburn, L. J. W. Graham, and L. Pullum, “Turbulentpipe flow of shear-thinning fluids,” J. Non-Newtonian Fluid Mech. 118,33–48 (2004).

5V. Presser, C. R. Dennison, J. Campos, K. W. Knehr, E. C. Kumbur, andY. Gogotsi, “The electrochemical flow capacitor: A new concept for rapidenergy storage and recovery,” Adv. Energy Mater. 2, 895–902 (2012).

6A. M. Walker, C. R. Johnston, and D. E. Rival, “On the characterization of anon-Newtonian blood analog and its response to pulsatile flow downstreamof a simplified stenosis,” Ann. Biomed. Eng. 42, 97–109 (2014).

7D. Biswas, D. M. Casey, D. C. Crowder, D. A. Steinman, Y. H. Yun, andF. Loth, “Characterization of transition to turbulence for blood in a straightpipe under steady flow conditions,” J. Biomech. Eng. 138, 071001 (2016).

8B. A. Toms, “Some observations on the flow of linear polymer solutionsthrough straight tubes at large Reynolds numbers,” in Proceedings 1st Inter-national Congress on Rheology (North-Holland, Amsterdam, 1949), Vol. 2,pp. 135–141.

9J. L. Lumley, “Drag reduction in turbulent flow by polymer additives,”J. Polym. Sci. Macromol. Rev. 7, 263–290 (1973).

10P. K. Ptasinski, B. J. Boersma, F. T. M. Nieuwstadt, M. A. Hulsen, B. H. A. A.Van den Brule, and J. C. R. Hunt, “Turbulent channel flow near maximumdrag reduction: Simulations, experiments and mechanisms,” J. Fluid Mech.490, 251–291 (2003).

11F. C. Li, Y. Kawaguchi, T. Segawa, and K. Hishida, “Reynolds-numberdependence of turbulence structures in a drag-reducing surfactant solutionchannel flow investigated by particle image velocimetry,” Phys. Fluids 17,075104 (2005).

12W. D. McComb, J. Allan, and C. A. Greated, “Effect of polymer additiveson the small-scale structure of grid-generated turbulence,” Phys. Fluids 20,873–879 (1977).

13E. van Doorn, C. M. White, and K. R. Sreenivasan, “The decay of gridturbulence in polymer and surfactant solutions,” Phys. Fluids 11, 2387–2393(1999).

14N. T. Ouellette, H. Xu, and E. Bodenschatz, “Bulk turbulence in dilutepolymer solutions,” J. Fluid Mech. 629, 375–385 (2009).

15W. H. Cai, F. C. Li, and H. N. Zhang, “DNS study of decaying homogeneousisotropic turbulence with polymer additives,” J. Fluid Mech. 665, 334–356(2010).

16C. M. White and M. G. Mungal, “Mechanics and prediction of turbulent dragreduction with polymer additives,” Annu. Rev. Fluid Mech. 40, 235–256(2008).

17P. C. Valente, C. B. da Silva, and F. T. Pinho, “Energy spectra in elasto-inertial turbulence,” Phys. Fluids 28, 075108 (2016).

18R. Vonlanthen and P. A. Monkewitz, “Grid turbulence in dilute polymersolutions: PEO in water,” J. Fluid Mech. 730, 76–98 (2013).

19A. de Chaumont Quitry and N. T. Ouellette, “Concentration effects on tur-bulence in dilute polymer solutions far from walls,” Phys. Rev. E 93, 063116(2016).

20P. Perlekar, D. Mitra, and R. Pandit, “Manifestations of drag reduction bypolymer additives in decaying, homogeneous, isotropic turbulence,” Phys.Rev. Lett. 97, 264501 (2006).

21G. I. Taylor, “Statistical theory of turbulence,” Proc. R. Soc. A 151, 421–444(1935).

22T. V. Karman, “The fundamentals of the statistical theory of turbulence,”J. Aeronaut. Sci. 4, 131 (1937).

23A. Thormann and C. Meneveau, “Decay of homogeneous, nearly isotropicturbulence behind active fractal grids,” Phys. Fluids 26, 025112 (2014).

24P.-Å. Krogstad and P. A. Davidson, “Freely decaying, homogeneous tur-bulence generated by multi-scale grids,” J. Fluid Mech. 680, 417–434(2011).

25R. S. Rogallo, “Numerical experiments in homogeneous turbulence,”Technical Representative 81315 (NASA Technical Memorandum, 1981).

26W. G. Rose, “Results of an attempt to generate a homogeneous turbulentshear flow,” J. Fluid Mech. 25, 97–120 (1966).

27S. Tavoularis and S. Corrsin, “Experiments in nearly homogenous turbulentshear flow with a uniform mean temperature gradient. Part 1,” J. Fluid Mech.104, 311–347 (1981).

28C. Vanderwel and S. Tavoularis, “Coherent structures in uniformly shearedturbulent flow,” J. Fluid Mech. 689, 434–464 (2011).

29L. Mydlarski and Z. Warhaft, “On the onset of high-Reynolds-numbergrid-generated wind tunnel turbulence,” J. Fluid Mech. 320, 331–368(1996).

30I. P. D. De Silva and H. J. S. Fernando, “Oscillating grids as a source ofnearly isotropic turbulence,” Phys. Fluids 6, 2455–2464 (1994).

31K. A. Brookshier and J. M. Tarbell, “Evaluation of a transparent bloodanalog fluid: Aqueous xanthan gum/glycerin,” Biorheology 30, 107–116(1993).

32A. M. Walker, C. R. Johnston, and D. E. Rival, “The quantification ofhemodynamic parameters downstream of a Gianturco zenith stent wireusing Newtonian and non-Newtonian analog fluids in a pulsatile flowenvironment,” J. Biomech. Eng. 134, 111001 (2012).

33J. Jimenez, “Near-wall turbulence,” Phys. Fluids 25, 101302 (2013).34A. Sekimoto, S. Dong, and J. Jimenez, “Direct numerical simulation of

statistically stationary and homogeneous shear turbulence and its relationto other shear flows,” Phys. Fluids 28, 035101 (2016).

35S. B. Pope, Turbulent Flows (Cambridge University Press, 2000).36S. Corrsin, “Local isotropy in turbulent shear flow,” Technical Representa-

tive 58B11 (NASA Technical Memorandum, 1958).37K. M. Kiser, H. L. Falsetti, K. H. Yu, M. R. Resitarits, G. P. Francis, and

R. J. Carroll, “Measurements of velocity wave forms in the dog aorta,”J. Fluids Eng. 98, 297–304 (1976).

38P. R. Owen and H. K. Zienkiewicz, “The production of uniform shear flowin a wind tunnel,” J. Fluid Mech. 2, 521–531 (1957).

39P. Saarenrinne, M. Piirto, and H. Eloranta, “Experiences of turbulencemeasurement with PIV,” Meas. Sci. Technol. 12, 1904–1910 (2001).

40A. Sciacchitano and B. Wieneke, “PIV uncertainty propagation,” Meas. Sci.Technol. 27, 084006 (2016).

41L. Li, A. M. Walker, and D. E. Rival, “The characterization of a non-Newtonian blood analog in natural- and shear-layer-induced transitionalflow,” Biorheology 51, 275–291 (2014).

42A. M. Robertson, A. Sequeira, and R. G. Owens, “Rheological models forblood,” in Cardiovascular Mathematics: Modeling and Simulation of theCirculatory System (MS & A), edited by L. Formaggia, A. Quarteroni, andA. Veneziani (Springer, 2009), Vol. 1, pp. 211–241.

123101-10 S. Rahgozar and D. E. Rival Phys. Fluids 29, 123101 (2017)

43A. S. Les, S. C. Shadden, C. A. Figueroa, J. M. Park, M. M. Tedesco,R. J. Herfkens, R. L. Dalman, and C. A. Taylor, “Quantification of hemody-namics in abdominal aortic aneurysms during rest and exercise using mag-netic resonance imaging and computational fluid dynamics,” Ann. Biomed.Eng. 38, 1288–1313 (2010).

44F. T. Pinho, “A GNF framework for turbulent flow models of drag reducingfluids and proposal for a k–ε type closure,” J. Non-Newtonian Fluid Mech.114, 149–184 (2003).

45J. Singh, M. Rudman, and H. M. Blackburn, “The influence of shear-dependent rheology on turbulent pipe flow,” J. Fluid Mech. 822, 848–879(2017).

46M. Tabor and P. G. De Gennes, “A cascade theory of drag reduction,”Europhys. Lett. 2, 519 (1986).

47P. G. De Gennes, “Towards a scaling theory of drag reduction,” Phys. A140, 9–25 (1986).

48A. Liberzon, M. Guala, W. Kinzelbach, and A. Tsinober, “On turbulentkinetic energy production and dissipation in dilute polymer solutions,” Phys.Fluids 18, 125101 (2006).

49M. P. Escudier, A. K. Nickson, and R. J. Poole, “Turbulent flow of vis-coelastic shear-thinning liquids through a rectangular duct: Quantifica-tion of turbulence anisotropy,” J. Non-Newtonian Fluid Mech. 160, 2–10(2009).