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Experiments in Fluids (2021) 62:185 https://doi.org/10.1007/s0034802103257w
RESEARCH ARTICLE
Ultrasonic velocity profiler applied to explore viscosity–pressure fields and their coupling in inelastic shear‑thinning vortex streets
· Yuichi Murai2
Received: 1 December 2020 / Revised: 2 June 2021 / Accepted: 24 June 2021 / Published online: 19 August 2021
© The Author(s) 2021
Abstract
A method for simultaneous estimation of viscosity and pressure fields in inelastic shearthinning fluids is developed by means of ultrasound velocity profiling technique (UVP). In the method, equation of continuity, rheological model and pressure Poisson equation are incorporated as data processing sequences for measured velocity distributions. The proposed method is applied to study the vortex street structure formed behind a circular cylinder, which shows viscosity–pressure coupling due to shearthinning property of fluid. For demonstration, aqueous solution of CMC (carboxy methyl cellulose) of weight concentration of 0.1% is chosen as the working fluid with shearthinning property. An alternating staggered pattern of lowpressure spots is successfully reconstructed for zeroshearbased Reynolds number, Re = 50–300. We have found that increasing Re resulted in decrease in vortex shedding Strouhal number because of vortex sustainability supported by shearthinning property.
Neetu Tiwari tiwari@aero.mech.tohoku.ac.jp
1 Faculty of Engineering, Tohoku University, 6601 Aza Aoba, Aramaki, Aobaku, Sendai 9808579, Japan
2 Faculty of Engineering, Hokkaido University, N13W8, Sapporo, Hokkaido 0608628, Japan
Graphical abstract
Introduction
Flow over cylinder is a widely studied topic for Newto nian fluid flows (Williamson 1996) due to its relevance in many engineering problems. Large amount of research exists investigating physics of phenomena and flow con trol (Zdravkovich 1997) by theoretical (Mittal et al. 2008), experimental (Norberg 1994) and numerical simulations (Rajani et al. 2009). Comparatively, fewer studies have been conducted for nonNewtonian fluids. Main concern in working with nonNewtonian fluids is that they do not fol low Newton’s law of viscosity and exhibit various relations between stress and strain rate. Accordingly, these fluids are classified as shear thinning, shear thickening, viscoelastic, thixotropic and rheopectic. Purely, shearthinning or thick ening fluids can be represented by a generalized Newto nian fluid model where local shear stress is a function of local shear rate, and viscosity can vary spatially in flow. To simulate flow over a cylinder in shearthinning fluids, such models like, power law, Casson, Cross and Carreau–Yasudamodel are generally used. For power law fluids, cylinder flow simulations were parametrically investigated in steady confined (Rao et al. 2011), unconfined (Sivakumar et al. 2007), circular (Bharti et al. 2007b, a; Bharti et al. 2008), square (Dhiman et al. 2006, 2007) and elliptical geometries (Koteswara Rao et al. 2010). However, power law model overpredicts viscosity for flows where shear rate is very low (e.g., free stream regions in external flow) and underpredicts viscosity when shear rate is very high. Carreau–Yasuda and Cross models incorporate zero and infinite shear rate viscos ity and thus are more appropriate to represent viscosity over a wide range of shear rate (Carreau 1972) (Cross 1968). For Carreau fluids, (Pantokratoras 2016) reported drag coeffi cient variations with model parameters. These studies were mostly limited to steady flows. Recently, (Bailoor et al. 2019) have performed numerical simulation for unsteady cylinder flow by immersed boundary method. Force coef ficients and vortex characterstics with Carreau number in shearthinning fluid were investigated at Re = 100. More recently, flow over impulsly moving cylinder has also been simulated using Carreau model (Yun et al. 2020) wherethe effect of Carreau number, power law index and Re on unsteady nonNewtonian fluid flow has been studied. There is a recent trend of simulations with models other than power law. On the other hand, experiments were performed for flow over cylinder of various shearthinning fluids by Coelho and Pinho (2003). They have measured velocity using laser Dop pler velocimeter (LDV) which provides point measurement of fluid velocity and reported point pressure measurements (Coelho and Pinho 2004). The details of unsteady wake flow field, viscosity and pressure fields have not been investigated experimentally for shearthinning fluids which are the moti vation for this work.Ultrasound velocity profiling (UVP) and particle image velocimetry (PIV) can provide the velocity information in 1D and 2D/3D, respectively, but their application to wake flows of shearthinning fluid flows is scarce. Note that PIV and UVP cannot directly provide the pressure field infor mation which is important in investigation of interaction of rheological property with dynamic pressure via velocity gradient tensor field. Some of the available pressure measurecylinder in opaque Newtonian fluid at Re = 1000. We have developed a pressure estimation algorithm based on UVP measurements targeting vortex streets formed behind a cyl inder. Equation of continuity and Navier–Stokes equations were incorporated into data processing of measured 1D1C velocity distribution to reconstruct vortex street structure as well as corresponding pressure distribution.The theory behind PIVbased pressure estimation is dis cussed below. From measured velocity field, pressure field can be estimated by substituting it into governing equations of fluid flows. For Newtonian fluids, PIVbased pressure field estimation methodology has been well established (van Oudheusden 2013). Its principle relies on substitution of measured velocity data into one of the equations describing the velocity–pressure relation is given below.Navier–Stokes equation: Relation between pressure and velocity is prescribed through the Navier–Stokes equation which can give local instantaneous pressure gradient vector as
ment devices such as piezoelectric transducer, manometer and pitot tube can provide pressure information only at a∇p = −p 6u + (u ⋅ ∇)u + µ∇2u6t(1)single point in fluid. These devices are applicable to non Newtonian fluid flows but are intrusive in nature. Pressure sensitive paint (PSP) (Bell et al. 2001) is nonintrusive and can be utilized to obtain pressure distribution over a solid surface but is not applicable in fluid flow domain. Table 1 classifies previous studies applied to cylinder wake flows. The studies described in Table 1 are purely experimental or numerical, except our previous paper (Tiwari et al. 2021) and (Tiwari et al. 2019). Tiwari et al. 2021 discussed PIV based pressure and viscosity investigation in pseudoplas tic flow for steady flows and focused on enlightening the change of vortex dynamics due to shearthinning property of fluid. A linear error propagation analysis was also dis cussed to compare the characteristics of error in a shear thinning fluid and a Newtonian fluid. In our previous work (Tiwari et al. 2019), UVP was applied to study flow overwhere u and μ are fluid flow velocity vector and viscos ity, respectively. Right hand side terms of the equation are provided by velocity measurements from PIV. By spatially integrating the local pressure gradient, scalar pressure field can be reconstructed. In literature, either Eulerian (Van Der Kindere et al. 2019) or Lagrangian (Charonko et al. 2010) methods are adopted depending on the sampling rate of velocity measurements. In our last work (Tiwari et al. 2021), we used a momentum equation with Eulerian method to reconstruct the pressure field albeit for a nonNewtonian fluid with variable viscosity.Pressure Poisson equation: For incompressible flows, Poisson equation for pressure is derived by taking diver gence of Eq. (1) and is written as
Table 1 Classification of research in literature on cylinder flow in shearthinning fluid
LDV: laser Doppler velocimetry, PIV: particle image velocimetry, UVP: ultrasound velocity profiler, RANS: Reynolds averaged Navier–Stokes equations, LES: large Eddy simulations, DNS: direct numerical simulations, IBM: immersed boundary method∇2p = −p∇ ⋅ (u ⋅ ∇)u → −2p 6u 6v − 6u 6vMethod
Experiment
Numerical
Type
LDV
PIV
UVP
RANS
LES
DNS
IBM
Published studies
Coelho and Pinho (2003)
Tiwari et al. (2021)
Nakashima
et al. (2016) Tiwari et al.
(2019)
Bharti et al. (2008) Pantokratoras (2016)

Lashgari et al. (2012)
Bailoor et al. (2019)
Data dimension
0D(Point)
2D/3D
1D
2D/3D
2D/3D
2D/3D
Nature
Real
Modelling
Application to opaque fluid
No
No
Yes
Yes
Steady/unsteady
Unsteady
Steady
–
Steady
–
Steady
Unsteady
(2)6x 6y 6y 6xif 2DThis equation does not contain the time derivative term. Viscous term in Eq. (1) also vanishes in Eq. (2) for Newto nian fluids. This avoids noise amplification due to second order derivatives of PIV data of low velocity resolution.For nonNewtonian fluids, these equations cannot be applied directly. In industry relevant opaque fluids, PIV is also not applicable. Thus, ultrasound velocity profiler (UVP) (Takeda 2012; Shiratori et al. 2015) is a promising and valid choice. UVP uses a single ultrasound transducer to provide onedimensional, onecomponent (1D1C) velocity distri butions. Therefore, majority of UVP measurement targets onedirectional dominant flow fields such as pipe flows or channel flows. Our team has also developed a novel rheom etry using UVP (Yoshida et al. 2017, 2019). It is based on substitution of UVPmeasured velocity distribution into rheological model to estimate complex rheological proper ties. Various physical relations between input and output quantities in flow structure analysis are illustrated in Fig. 1 for easy understanding of the present concept. The cases(a) to (d) of the figure had been established in literature as discussed above.In this study, we propose a novel algorithm for UVP to simultaneously estimate viscosity and pressure fields in vortex streets formed behind cylinder for unsteady flow of inelastic shearthinning fluids which may also be opaque. We present how UVP data can be best processed for evalu ating viscosity–pressure correlations. It is emphasized that our main objective is not to reveal rheological property of unknown fluid, but is to establish a pressure field estima tion method that couples with nonNewtonian fluid proper ties, using a known rheological constitutive equation that approximates the targeted fluid properties.The outline of paper is as follows: In Sect. 2, the experi mental arrangement and measurement systems are discussed. The algorithm is explained along with numerical governing equations and rheological model in Sect. 3. In Sect. 4, the results are discussed, and correlation plots are analyzed to understand the dependence of viscosity on enstrophy and pressure. Present technique is further tested at various Re in the range 50 < Re < 300.Experimental setup
A schematic of the experimental arrangement for velocity measurement in wake of a cylinder is shown in Fig. 2. A towing tank of dimensions 5.00 m in length, 0.54 m in width and 0.45 m in depth (h) was used. The diameter D and the length of cylinder are 0.04 m and 0.40 m, respectively. TheFig. 1 Relations between input and output quantities depend on measurement information. u and v denote streamwise and perpen dicular velocity components in twodimensional flow, respectively. p and μ are pressure and viscosity distributions. Case a is a standard CFD simulation for Newtonian fluids, where fluid property is input to compute velocity and pressure. b PIVbased pressure field estimation for Newtonian flows. c for nonNewtonian flow (Tiwari et al, 2021). d UVP for Newtonian flow (Tiwari et al. 2019). e UVP for nonNewto nian flow which is the present topiccylinder is made of acrylic resin and fixed on a mobile car riage with vertical shafts. The position of the cylinder is set horizontally at h/2 (> 5 D) from free surface of fluid as well as the bottom floor. The effect of free surface is negligible if the cylinder is placed 3–4 diameters below the free surface (Abdul Khader 1979). The influence of bottom wall is also minimal because the effective blockage ratio is more than 10 (Bharti et al. 2007a). Hence, effect of free surface or bottom plane on the wake of cylinder is also neglected in this study.Test fluid and experimental conditions
The test fluid is an aqueous CMC (carboxy methyl cellulose) solution of 0.1% by weight. Structural formula of CMC is shown in Fig. 3, in which R depends on degree of etherifi cation (DoE). Manufacturer of CMC is Daicel FineChem Ltd, Japan.Table 2 shows nominal viscosity of the CMC solutions for 19 types of product codes, measured by the manufac turer using a conetype rheometer at 25 °C in temperature.
Fig. 2 A schematic of experi mental setup with UVP meas urement system. Ultrasound transducer is set vertically to measure y component of veloc ity in the wake of a circular cyl inder, which is carried together with the cylinder along a linear slider attached on the container
Fig. 3 Structural formula of carboxy methyl celluloseTable 2 Viscosity of aqueous CMC solution manufactured by Daicel Co. LtdWe chose the CMC of the code 1390 (indicated in bold) in Table 2, that has molecular weight of 2 × 106 kg/kmol, because our group also collected rheological data for this type (Yoshida et al. 2019) and confirmed that the measured viscosity was the same as these nominal values. We also note that there is a little elasticity taking place above 0.5% in weight percentage for the type chosen. However, this becomes ignorable at 0.1%, and treated as inelastic shear thinning fluid in the later analysis of UVP data. For 0.1 wt% solution, the nominal viscosity is 10.8 mPas at 25 °C.Uniform mixing of CMC in water is ascertained prior to experiments. The measurements are performed at Reynolds number in the range 50 < Re < 300, where Re = ρDU/μ0; here, μ0is zeroshear viscosity and U is the towing speed. UDegree ofCode Viscosity [mPas] at 25 °Cranges from 20 to 100 mm/s in the present parametric study.etherification The carriage is accelerated for initial 1–3 s after which the1.0 wt.% 0.5 wt.% 0.1 wt.%
*Viscosity is representative value in each classified range measured by Etype rheometerspeed is constant and is decelerated again before stopping. The presented measurement is taken from few seconds after the carriage reaches uniform velocity, and thus, the flow at location measurement is not affected by initial acceleration. Towing experiment is done in stationary fluid; thus, uniform flow inlet velocity profile ahead of the cylinder is same as the towing speed of the cylinder. Influence of the upper and lower boundaries is negligible in present experimental con dition because transducer is set at 3D (three times diameter) downstream. However, some influence might be felt in the wake sufficiently far from the cylinder, where momentum diffusion reaches the boundary. At 3D downstream, the cyl inder symmetry is confirmed as is evident by the symme try in measured velocity profile (shown later in Fig. 5) and resultant viscosity distribution that had a value close to unity outside wake until both boundaries (shown later in Fig. 7). Spherical tracer particles (HPSS20, Mitsubishi Chemi cal), made of highporous polymer with hydrophilic sur face property to aqueous solution, are mixed as ultrasonic reflectors for UVP measurement. Density and diameter of the particles are 990 kg/m3 and 50–120 µm, respectively. The particles are dispersed homogeneously in the container.0.7
1110
20
3
1.1
1120
35
5
1.1
1130
75
12
1.2
1140
150
25
1.4
1150
250
42
1.7
1160
400
68
2.1
1170
650
110
2.8
1180
1150
195
4.2
1190
1650
280
5.6
0.9
1220
15
2
1.0
1240
35
5
1.1
1250
55
9
1.2
1260
105
17
1.3
1.2
1330
75
12
1.2
1350
250
42
1.7
1380
1500
255
5.2
1390
3500
595
10.8
1.0
2220
2250
382
7.3
2260
5000
850
15.0
UVP measurement system
UVP monitor model Duo (Metflow S.A., Switzerland) was used for signal processing of ultrasound pulse echo waves. An ultrasonic transducer of basic frequency 4 MHz was used in vertical position for measuring velocity profiles in y direction. The transducer was fixed at 3D (3 times the diameter) downstream of the cylinder which is not close to cylinder so the threedimensionality during vortex formation phenomenon can be avoided (Azmi and Zhou 2018). Dur ing each run, cylinder and transducer were towed together by the carriage at a constant controlled speed. The detailed specification of UVP system is given as below: maximum measurable range: 121.84 mm; sampling period: 123 ms; cycle per pulse: 4; spatial resolution: 0.74 mm; velocityresolution: 0.477 mm/s; number of channels (velocity data points): 450; and number of velocity profiles stored: 450. The UVP spatial resolution did not change with the depth, and a constant sampling rate was adopted.
Algorithm development
Figure 4 shows the flowcharts for pressure estimation using velocity measurement data from PIV/UVP for Newtonian/ nonNewtonian fluids. The pressure estimation process from PIV data is rather straightforward and has been established in literature. Here, it is shown with UVPbased algorithms for a clear comparison. Planar PIV measurement can provide velocity information of two components in two dimensions
Fig. 4 Flowchart for pressure estimation from velocity measurement in the case of a transparent Newtonian fluids with PIV data b opaque/trans parent Newtonian fluid with UVP data c opaque/transparent nonNewtonian fluid with UVP datawhich is sufficient to estimate twodimensional pressure field from Navier–Stokes or pressure Poisson equation. On the other hand, a UVP transducer can provide one component of velocity along the measurement line. The steps of pressure estimation in Newtonian fluid by PIV data and UVP data are shown in algorithms in Fig. 4a, b. In nonNewtonian fluids, the number of steps increases even more due to inclusion of rheological model as shown in Fig. 4c. These steps are further explained in following sections.Velocity measurement
Measurement by UVP provides fluid velocity distribution in a spatiotemporal domain. Measured y component of velocity in the wake of cylinder is shown in Fig. 5a. Here, wake region can be clearly identified by observing alternat ing positive and negative regions of velocity. This shows the presence of regular vortices being shed in the wake as expected. This highlights the capability of present measure ment in capturing flow structures. Before using this data in CFD equations, adequate filtering is required. Y and t are nondimensionalized, respectively, by the diameter of cylinder D and D/U.Velocity data filtering
Like PIV, velocity measurements by UVP also suffer from various noises. If proper attention in not rendered, thesedata are further analyzed with POD which is most efficient physicsbased filter in removing nonorganized errors (Cha ronko et al. 2010). POD (Sirovich 1987a, b) decomposes the experimental data in orthonormal basis functions and resolves the kinetic energy into POD modes. In present data, first mode captured the highest kinetic energy, 76% of total energy, which includes dominant structures of flow and modes 1 to 10 contain 99.5% of energy. In Fig. 5c, the reconstructed data after POD filtering are shown considering first ten POD modes. We applied FFT and POD combined filtering to all the results shown in this work. Importantly, using only FFT or only POD filtering for velocity data does not yield expected results for pressure profiles.In the further sections, all governing equations are given in nondimensional form. Following scaling variables, ren dered dimensionless terms: d for length variables, U for velocity terms, ρU2 for pressure. These equations are solved by finite difference method, and all the equations are discre tized by central difference scheme.Reconstruction of 2D velocity field
Velocity information in ty plane is converted to x–y plane based on Taylor frozen hypothesis. The hypothesis com bined with velocity measurement was found to be sufficient for pressure estimation with known convection velocity of dominant vortex structures (de Kat and Ganapathisubramani 2013) asnoises can magnify in calculation of second component of velocity and subsequently in viscosity and pressure. Combiv(x, y)= v(x0− Us⋅ t, y),(3)nation of temporal (FFT, fast Fourier transform) and spatial (POD, proper orthogonal decomposition) filtering in UVP data had proved very efficient for noise reduction with con serving coherent flow structures as explained in our pre vious work (Tiwari et al. 2019). So, here we adopted the same filtering technique. After application of FFT low pass filtering, the velocity data are reconstructed as shown in Fig. 5b. Here, high frequency noises are filtered, and thesewhere Us stands for the vortex migration velocity, which ranges from 0.55U to 0.65U relatively to the free stream velocity U (Lin and Hsieh 2003). x0 represents the refer ence position of streamwise coordinates which is position of measurement. The 2D velocity is reconstructed by sub stituting the filtered v component of velocity in equation of continuity as
Fig. 5 Distribution of y component velocity measured by UVP: a original velocity data, b velocity after FFT filtering for noise reduction, cvelocity after FFT and POD filtering considering the first ten POD modes6u + 6v = 0 → u(x, y) =−6v(x, y) dx.(4)at Re ≤ 293, we have charaterized it as shearthinning fluid6x 6yJ 6yhaving purely viscous characteristics.Based on this, we adopted Carreau–Yasuda model, toZero velocity is given in upper and lower boundaries,and the integration in multiple directions is applied to mini mize the cumulative residual. The method is the same as our previous paper (Tiwari et al. 2019). The reconstructed 2D velocity vector field is shown in Fig. 6. Here, dominant vortices can be seen when only mode 1 of POD is considrepresent shearthinning property of test fluid. It has benefit over power law model in representing accurate shearthin ning property for zero to infinite range of strain rate (Tiwari et al. 2021). For Carreau–Yasuda fluids, nondimensional viscosity is formulated as:ered for velocity reconstruction. When modes 1–3 are con µ∞ µ∞II2D1 a m−1∕a2(5)sidered, very clear presence of staggered arrangement of vortices can be identified. Similarly, as more higher modesη = + 1 −µ0 µ01 + ß 2are considered cumulatively, smaller vortices appear along with staggered pattern.Rheological model
Here, ß, a and m are the Carreau–Yasuda parameters. II2D is the second invariant of strain rate tensor which can be written in case of twodimensional incompressible flow as:II2D = , , . = 2 + 2 + 2 + 2As the fluid considered is nonNewtonian, proper selec tion of rheological model and its parameters are needed.ij jii jxx xy yx yy(6)In present study, low concentration of CMC (0.1% by wt) is used for which CMC chains are in their most extended form (Bayarri et al. 2009). The viscoelastic property of CMC was evaluated by creep test in study of (Benchabane and Bekkour 2008) and they observed that elastic behavior of CMC appeared only above concentration of 2.5% by weight. However, under certain flow conditions elastic properties has been observed at much lower wt% of CMC. Even our group found a hint of elasticity in 0.5 wt% aqueous CMC solution using USR (ultrasonic spinning rheometry), but in 0.1wt%, such elasticity could not be measured. Experiments with 0.1% CMC solution were also reported by (Pinho and Whitelaw 1990) to measure drag reduction in pipe flow which is a manifestation of elastic effects. At Re of 5100, they found drag reduction of 2.4%, but at Re of 1480, the drag reduction was found to be negligible. In contrast at higher Re (> 8000), significant drag reduction (> 20%) was observed. Hence, the elastic effects are significant only for turbulent flows at higher Re. Since CMC solution used in present study is quite low in concentration and our flow isHere, i,j is strain rate tensor. The parameters for Carreau–Yasuda model were obtained from literature where rheometer experiments were performed using same solution as the one in present work at = 10–4000 s−1 (Coelho and Pinho 2003). The parameters are reproduced in Table 3. The representative viscosity of present 0.1 wt% aqueous CMC solution is 10.8 mPas (as noted in Sect. 2.1), and the repre sentative viscosity for the solution used by Coelho and Pinho 2003 is 10.835 mPas (average of zero and infinite shear rate viscosity). These values are almost same. Moreover, litera ture shows that the flow curves obtained by two very differ ent group of researchers for 0.1wt% aqueous CMC solution are almost identical (varying within 5%). These data are shown in appendix (Fig. 16). This gives us the confidenceTable 3 Parameters of Carreau–Yasuda model Solution μ0 (Pa s) μ∞ (Pa s) ß(s)0.1%CMC 0.01367 0.0008 0.01317 0.6671 0.5864
Fig. 6 Reconstructed velocity vector field considering POD a mode 1, b modes 1 to 3 and c modes 1to 10to believe that the rheological parameters for our test fluid will be very close to the one proposed by (Coelho and Pinho 2003) and is justified in current work.taking divergence of the pressure gradient vector field, the pressure Poisson equation is obtained asIn typical rheometer tests, the strain rate is varied and cor 2 62p 62p 6 6p 6 6presponding viscosity is measured. By repeating such experi∇ p = 6x2 + 6y2 = 6x 6x+6y 6y(13)ment for a range of strain rate, a relationship can be found between strain rate and viscosity. In actual experiments, e.g., flow behind cylinder, a range of strain rate would appear depending on location and flow velocity. Still the local vis cosity follows the same relationship with local strain rate as measured by controlled rheometer measurements.Pressure estimation method
Unlike Newtonian fluid flows, the source term of pressure Poisson eq. in nonNewtonian fluid flows is a function of viscosity, so even after application of equation of continuity, various terms do not cancel out. Substitution of Eqs. (11) and (12) into Eq. (13) leads to a complicated source term of the Poisson equation, which produces higher order and mixed derivatives as below:6x2 + 6y2 = −26x 6y − 6y 6x62p 62p 6u 6v 6u 6vThe dimensionless momentum equation can be written as= −6x 6t–6x6yRe6x6y6p 6u6u 6u1 6rxx6ryx+ 1 6η 6 6v + 6u + 1 6η 6 6v + 6uu+ v++,(7)2 62η 6u+6η 62u+(14)6p 6v 6v 6v1 6ryx6ryyRe 6y 6x6xRe 6x2 6x6yRe 6x 6y6x6y6x 6x2= −
−6yu + v
6x 6y+ Re+ ,6x 6y(8)2 62η 6v6η 62v2 62η 6v6u6twhere represents stress tensor excluding pressure. Relation+ Re6y2 6y + 6y 6y2+ Re 6x6y6x + 6y .between the stress and strain rate in twodimensional flow is given byWith limited spatial resolution of experimental data,it is impossible to directly compute all the higher order derivative terms in Eq. (14) accurately. So, for presentr = η→ rxx rxy = η xx xy ,(9)pressure calculation, we adopt a twostep approach: righti,ji,jryx ryyyx yyhand side of Eqs. (11) and (12) is computed from UVP data at all grid points with secondorder finite differencwhere i,j is computed from measured velocity field as:ing scheme, and then, these results are substituted in thei,j =xx xy =2 6u 6x6v 6u6v + 6u 6x 6y.6v(10)right hand side of Eq. (13) to obtain source term of Pois son equation. Equation (13) is then solved for pressureyx yy+ 26x 6y 6ydistribution. This process provides two advantages: one is unnecessity of finite differencing for the secondorderAll the components of the tensor, Eq. (10), are obtainable on applying the continuity equation, Eq. (4), under the assumption of twodimensional flow. Substitution of Eqs.u+ v+2 ++xx+ yx,(9) and (10) into Eqs. (7) and (8), we get= −6x 6t–6p 6u6u 6u6x6yη 62uRe6x62u6y22 6ηRe6x6η6y(11)= −6y 6t–6p 6vu+ v+6v 6v6x6yη 62vRe6x62v6y22 6ηRe6x6η6y(12)2 ++xy+ yy.The last four terms in these equations capture the dynamic effects due to nonNewtonian nature of fluid flow. Aftermixed derivatives of velocity components that appears in the last five terms in Eq. (14). The other is elimination ofrotation in the pressure gradient vector field. Note that if spatial continuity of measured velocity data is lacking due to lower resolution, even Eq. (13) cannot provide a converged solution of pressure which can be the case with PIV data (Tiwari et al. 2021).After applying Taylor frozen hypothesis and equation of continuity, the measurement domain was treated as compu tational domain for pressure estimation. The spatial resolu tions in x and y directions are 1.13 and 0.74 mm, respec tively. The boundary conditions of pressure are as follows: at left (upstream) and right (downstream) boundaries, Neu mann boundary condition of pressure is applied (Eqs. (11, 12)). Zero pressure gradient conditions are applied at top and bottom boundaries. At four corners, Dirichlet condition of pressure is applied to control the absolute pressure.
Results and discussion
A demonstration of reconstruction of viscosity and pres sure fields from UVP measurements and their relationship is discussed for Re = 88 in Sects. 4.1, 4.2 and 4.3. Then, a parametric study for Re in the range 58 to 293 is presented in Sect. 4.4.Viscosity distribution
In Fig. 7, viscosity distribution in the wake of cylinder esti mated with Carreau–Yasuda model is shown for three dif ferent cases corresponding to UVP data considering, POD mode 1, modes 1 to 3 and modes 1 to 10. POD mode 1 shows lowering of viscosity inside the wake region due to the shearthinning property of aqueous solution of CMC. As higher POD modes are considered, additional low viscosity regions is observed in wake due to appearance of smaller eddies. Nearly 20% reduction of viscosity is observed at vor tex centers. As the strain rates increase in vortex centers, the viscosity should decrease.Pressure distribution
Once viscosity profiles are obtained, pressure distribution can be obtained after solving Eq. (13). Computed pressure distribution is shown in Fig. 8. Similar to the viscosity field, pressure distributions are obtained by considering POD mode 1, modes 1 to 3 and modes 1 to 10 of the velocity. The alternating low and highpressure regions are observed in the wake region due to staggered arrangement of vorti ces. The presence of vortices can be easily distinguished by the local minimum of pressure. Here, the range of pressure
Fig. 7 Viscosity variation in computational domain for data corresponding to POD a mode 1 b modes 1 to 3 and c modes 1 to 10, respectively (Re = 88)
Fig. 8 Pressure distribution in computational domain considering velocity data of POD a mode 1, b modes 1 to 3 and c modes 1 to 10coefficient, Cp = p/ρU2, between −1 to + 1 is reasonable in cylinder flows which confirms the quantitative accuracy of the algorithm. To understand the influence of rheological model on pressure distribution, rheological parameters a and m are varied by 5% and pressure distribution is recomputed in Appendix Fig. 17. There is corresponding change in the Cp by 5%. This indicates that viscosity plays a nonnegli gible role in determination of pressure distribution. This is different than the case of usual Newtonian fluids where viscous term is often neglected during pressure estima tion procedure. However, (McClure and Yarusevych 2017) noted in their work that viscous terms should be retained if Re is below 100. For the case of nonNewtonian fluids, the viscous term calculation is even more indispensable as it directly affects the flow dynamics. More work needs to be done to completely understand the role of viscous termsintensity of vortex structures is enstrophy. In this section, an attempt is made to understand the correlation between these scalars viscosity, pressure and enstrophy. First, phase averaging was performed in the range 7.5 ≤ x/D ≤ 17.5 and−2 ≤ y/D ≥ 2 for pressure, and viscosity in the wake. In Fig. 9, distribution of phase averaged pressure and viscosity distribution are plotted excluding parts of the free stream region. The lowpressure regions corresponding to the high viscosity regions are indicated in squares in Fig. 9a, b which represents the inverse relation between pressure and viscos ity. These relations can also be confirmed by performing a quantitative assessment with crosscorrelation analysis. The correlations between pressure, enstrophy and viscosity were investigated for wake region by quantifying them using the following parameter. 1 ,A − B − and specifically the parameters of rheological parametersfor different nonNewtonian fluids in pressure distribution estimation.Cc =i A i BNN − 1 i=1 σA σB(15)Correlation of viscosity to enstrophy and pressure
In last sections, viscosity and pressure estimation was pre sented. Another scalar function roughly representing theHere, Cc represents the linear dependence of two vari ables A and B, having N observations. and are mean and standard deviation, respectively.In Fig. 10, the viscosity variations with enstrophy and pressure are presented with correlation coefficient (Cc). The
Fig. 9 Phase averaged a pres sure and b viscosity distribu tions in wake region
Fig. 10 Viscosity variation with a enstrophy and b pressure in computational domainscatter plots of η vs ω2 (Fig. 10a) and η vs Cp (Fig. 10b) show that they have inverse relationship which is confirmed by negative sign of Cc. This is also in the support of visual inspection of inverse relation of pressure and viscosity as shown in Fig. 9a, b. Quantitatively, viscosity is more cor related with enstrophy than with pressure which means that the shearthinning characteristic of fluid is more dependent on kinematics than the dynamics of fluid. This is expected because viscosity is estimated from velocity and its gradients via Carreau Yasuda model and enstrophy is also calculated from velocity gradients. However, the velocity gradients0.3St0.20.10Williamson 1996Present 0.1% CMC Unal & Rockwell 1988 Newtonian presentUsui et al. (1980), C=200ppm, D=3mm Usui et al. (1980), C=200ppm, D=1.6mm Usui et al. (1980), C=200ppm, D=6mm
40 80 160 320 640 1280Reemployed in their calculation are not identical. In Fig. 10b, the negative pressure region represents the interior of indi vidual vortices, while positive pressure region represents the shear layer between vortices.Parametric study for variable Re
A parametric study is conducted for flows at Re = 58, 88, 146 and 293. This range of Re is targeted because stable two dimensional vortex streets were formed in the wake in this range. The reconstructed 2D velocity vectors after consider ing POD mode 1 to 5 are shown for several Re in Fig. 11.In Fig. 11d, the measurement time was almost half of previous cases. In Newtonian fluids, Strouhal number (St) of vortex shedding increases with Re (Williamson 1996)Fig. 12 Variation of vortex shedding frequency by Strouhal number with Re for Newtonian and nonNewtonian fluid flowsfor the range of Re considered here (Fig. 12). However, in shearthinning fluid flows we found that Strouhal number decreases with Re as shown in Fig. 12. For Newtonian fluid’s von Karman wake at 30 < Re < 1000, Strouhal number from literature is also plotted in Fig. 12. We have also included the St measured in present rig for water flow at Re = 1000 to show the accuracy of UVP system with equation of continu ity in capturing wake dynamics. Interestingly for Re < 100, polymer additives are found useful to reduce the vortex shedding frequency (Kalashnikov and Kudin 1970). But as Re reaches ∼104, the shedding frequency is not much
Fig. 11 Velocity vector plots obtained by UVP measurements filtered by FFT and POD for Re a 58, b 88, c 146, d 293different from the Newtonian fluid flows (Sarpkaya et al. 1973). Decrement of Strouhal number with polymer concen tration, diameter of cylinder and Re was observed in experi mental study on flow over cylinder in viscoelastic fluids by (Usui et al. 1980). In Fig. 12, we have included their results obtained for polyethylene oxide (PEO) of concentration of 200 ppm and three cases of varying diameters of 1.6 mm, 3 mm and 6 mm. In case of diameter 1.6 mm, they have found increasing trend of St and as diameter increases fur ther to 3 mm decrement trend of St observed. Hence, the decrement in St for nonNewtonian fluids is not unexpected and can be observed under various flow configurations. The PEO solutions are now known to be elastic so that decrease of St measured by Usui et al. (1980) can be attributed to combination of shearthinning with elasticity. However in our CMC solution, we think that shearthinning effect is dominant over elastic effects and as such the decrement in St can be mostly due to shearthinning effect.Figure 13 shows viscosity distribution in the wake for various Re. It is observed that the reduction in local viscos ity is greater inside the vortex street as Re increases. This means that effective Re inside the street further increases with nominal Re because Re is inversely proportional to viscosity. Hence, decrease of St in the shearthinning fluid, which we found in present UVP measurement, cannot be explained only by the concept of effective Re. (This is because St increases with Re in Newtonian fluid flows.) Itshould be discussed based on sustainability of vortex struc tures that is a matter of coupling between viscosity (which decides viscous force) and pressure around individual vor tices. Due to lower viscosity in the wake, the sustainability of vortex structures increases, i.e., the vortices stay for the longer period by the virtue of increased inertial forces. So, the vortex shedding frequency may not increase as much it increases in flow of a corresponding Newtonian fluid.Velocity and viscosity data obtained and shown in Figs. 11 and 13 are substituted in Eq. (13) to determine the pressure distribution. Pressure distributions for various Re are shown in Fig. 14. The alternating positive and negative pressure spots are accurately reconstructed in wake region for all Re which indicates the applicability of the new pro posed method using pressure Poisson equation in unsteady nonNewtonian fluid flows.
Conclusion
Pressure estimation algorithm was proposed for UVPbased measurement in shearthinning fluid flow in cylinder wake. Velocity data obtained by UVP were substituted in equation of continuity, rheological model and pressure Poisson equa tion to explore local distributions of viscosity and pressure. This sequence successfully estimated the pressure distribu tion that couples with viscosity fluctuation inside the vortex
Fig. 13 Distribution of viscos ity obtained by substitution of velocity data in Carreau–Yasuda model for Re a 58, b 88, c 146,
d 293
Fig. 14 Distribution of pressure for Re a 58, b 88, c 146, d 293
street. Parametric study was conducted for Re in the range 55–293. In our analysis, we have found that the Strouhal number decreases with Re up to 293 due to increment of vor tex sustainability which causes the longer stay of vortices in wake. Capability of UVP measurement to capture the crucial flow structures and dynamical parameters was successfully demonstrated. It provides a tool to study the flow dynamics of more complex nonNewtonian fluids in conjunction with usual rheometer experiments.
Appendix
CFD simulation of the same flow
Since there is no alternative measurement technique avail able for validation, we performed a numerical simulation on the same conditions as the experiment using finite volume solver. We applied the Carreau–Yasuda model with same parameters as used in present work. A good agreement can be confirmed for the pressure field inside the wake. In this simulation, secondorder finite difference spatial discretiza tion scheme and a secondorder backward Euler implicit time marching scheme were used. The equations were
discretized and solved on a twodimensional unstructured mesh for Re = 146 (U = 50 mm/s). Total number of nodes were 30,000 in computational domain of size 40DX40D. Velocity and pressure fields are shown in Fig. 15.
Here, v component of velocity varies between −0.6 to
0.6 (v/U) which is similar to our measurement shown in Fig. 4. Alternating lowpressure regions represent the stag gered vortices in the wake which were successfully recon structed similarly to Fig. 10c. Cp data extracted at a point at 3D downstream of cylinder at center line of computational domain and shown in Fig. 15d. In the pressure distribution plot, pressure coefficient varies between (−1 to 1), which is similar in our case as shown in Fig. 14c.
Uncertainty propagation of rheometry into dynamics
In present study, rheological parameters are referred from rheometer of (Coelho and Pinho, 2003). Here, we attempt to evaluate the error propagation in estimated pressure due to uncertainty present in rheological parameter. In literature, we have found that the rheological parameters of CMC (0.1% concentration) change only slightly (less than 1%) among published results of Coelho and Pinho (2003), Benslimane et al. (2016, 2018) due to different
Fig. 15 Result of CFD simulation at U = 50 mm/s using Carreau– Yasuda model with same parameters as used in present measurement. a Streamwise velocity, b perpendicular velocity, c pressure distribu
tion and d pressure fluctuation due to vortex shedding at 3D down stream the cylinder on the central axis
Fig. 16 Comparison of viscosity and strain rate flow curve obtained in three different CMC samples
Fig. 17 Pressure variation at the center of domain for different cases of increment/decrement of rheological parameters
To account for some variation, we altered the rheological parameters by 5% and recalculated the pressure profiles. In Fig. 17, pressure profiles are obtained at the center line of measurement domain. The reconstructed pressure profiles changed under 5% in both increment/decrement cases.
Acknowledgements This work was financially supported by Japan Society for Promotion of Science (JSPS KAKENHI, 17H01245 and 18KK0105, and 20J10276). The authors express their thanks to Dr. Yuji Tasaka and Dr. Taiki Yoshida, Hokkaido University, for their technical support and to Dr. Ajit Kumar Dubey, Tohoku University, for various discussions and computational support. The authors also thank representatives from Daicel FineChem Ltd, Japan, for providing rheological data.
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