Vortex deformation and turbulent energy of polymer solution in a two-dimensional turbulent flow

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Highlights

  • The vortex shedding and deformation in turbulent flow were categorized based on relaxation time of a solution.

  • Turbulent energy of polymer solution was observed without specific turbulent production.

  • Turbulent energy transportation of polymer solution was affected by relaxation time of a solution.

Abstract

An experimental study was performed to investigate the effects of the extensional rheological properties of polymer solutions on vortex deformation in turbulent flow and turbulent statistics. To focus on the extensional properties, a self-standing two-dimensional (2D) turbulent flow was used as an experimental setup, and the flow was observed through interference patterns and particle image velocimetry (PIV). Vortex shedding and the resulting deformation in the 2D flow were categorized into three types. The vortex flow regime was defined by the shedding frequency and relaxation time of the polymer solution. Turbulent energy was suppressed by polymer additives; however, a characteristic peak reappeared with increasing concentration. The results imply that the characteristic turbulent energy peak is influenced by the relaxation process of the extended polymers in the flow.

Introduction

The addition of a small amount of polymer to a Newtonian fluid reduces the frictional drag of the flow and delays the transition to turbulent flow. To understand the drag reduction phenomenon, many experimental and numerical studies have been conducted in the past decades [1], [2], [3], [4], [5], [6]. An important feature of this phenomenon is the anisotropic effect, which is due to the polymer extension in the flow [7], [8], [9], [10]. The extension of polymers, caused by the polymer coil-stretch transition, influences the flow, that is, it modifies energy transportation in the flow. To describe the subsequent onset of drag reduction to the coil-stretch transition, Min et al. [11] adopted the elastic theory on the transport equations for kinetic and elastic energy. Kinetic energy is transported through the elastic energy created by the polymer extension at the wall in a turbulent flow. Thus, the polymer extension estimated from the Weissenberg number and the time criterion, such as the relaxation time, is an important parameter that influences energy transportation. Min et al. [11] suggested that the relaxation time of a polymer should be sufficiently long to transport the elastic energy from the near-wall region to the buffer or log layer. The influence of the polymer extension and relaxation time of fluids on the turbulent drag reduction has also been demonstrated in other experimental studies [12], [13], [14]. An important consensus among researches in these previous studies has been that drag reduction occurs as a consequence of the dynamic interaction between polymer molecules and turbulence only when the polymer relaxation time becomes comparable to the characteristic time-scale of the flow [12]. It became possible to conduct quantitative experimental analysis, especially after the development of the capillary break-up extensional rheometer (CaBER). Owolabi et al. [12] proposed an equation to formulate the drag reduction efficiency in terms of polymer extension based on the Weissenberg number, which is calculated using the relaxation time measured by a CaBER.

Polymer extension and spatiotemporal non-uniform turbulent drag reduction have become popular areas of research in recent numerical studies. Turbulent flows are extensional, and the straining motion of the fluids, which is caused by an extensional flow, overcomes the rotation of the fluids, thereby stretching the polymers to their fully extended length. Xi and Graham, and the following related works, revealed that a highly extended polymer induces the intermittent dynamics of Newtonian and viscoelastic turbulence near the wall [15], [16], [17]. A minimal flow unit at the wall is laminarized by extended polymers and modified larger domains. Indeed, the instantaneous degree of polymer stretching and drag reduction are temporally anti-correlated. The polymers stretch in active turbulence and induce a subsequent hibernation period, that is, a weak turbulent flow. During the hibernation period, the drag is low, and the polymers relax [18], [19], [20], [21].

Another approach to comprehend the turbulent flow dynamics influenced by the rheological properties of the fluids is to study vortex shedding from a cylinder in viscoelastic fluids. Cadot [22] suggested that anisotropic vortex deformation at a cylinder in viscoelastic fluids appeared to be similar to what is observed in turbulence, and thus, could lead to a better comprehension of vortex inhibition and drag reduction. Therefore, numerous experimental and numerical studies on visualizing vortex shedding at a cylinder have been conducted [22], [23], [24], [25], [26], [27], [28], [29], [30]. Usui et al. first proposed that the Strouhal number of vortex shedding in a viscoelastic solution was correlated to the Weissenberg number [24]. Cadot et al. mentioned that the shear viscosity of a polymer solution only marginally influences vortex shedding; conversely, the elastic properties of the solution influence the vortex shedding frequency and the formation length [22]. Here, the formation length is the distance required to form vortices from the cylinder surface. It has been suggested that when the relaxation time related to the elasticity approaches the diffusive time scale in the fluids, the stabilization mechanism of vortex shedding occurs. Here, again, the time criterion is important for the formation and stability of the vortices. Coelho and Pinho studied the effects of elasticity and shear thinning on the Strouhal number of vortex shedding [27]. They suggested that the Strouhal number decreases by elasticity and increases by shear thinning. In these studies, the Reynolds numbers of the flows were relatively low, that is, approximately 50.

In recent studies on vortex shedding at a cylinder in viscoelastic fluids, numerical studies have achieved higher Reynolds numbers, and it has been revealed that viscoelastic fluids induce drag enhancement and drag reduction [31,32]. These two regimes, drag enhancement and drag reduction, are influenced by the Weissenberg number and Reynolds number. The addition of polymers varied the velocity fields, vortex shedding, and local elongation of the fluid elements [32], which implies the extension of the polymers around the cylinder. Relatively recently, drag enhancement and the extension of the polymers around the cylinder have been experimentally investigated by visualizing fluorescently stained deoxyribonucleic acid (DNA) [33]. Moreover, polymer elongation around a cylinder was observed in a recent numerical study with molecular dynamics simulation [34]. These results again confirmed that the extended polymer effects on drag reduction are not linear.

To focus on the extensional stress and polymer extension in a flow, a two-dimensional (2D) flow is useful [35,36]. A self-standing flowing soap film is an example of a 2D flow, where the flow surface is a free surface [37]. Thus, the flowing soap film is relatively free from shear stress at the wall. When a comb of equally spaced cylinders is inserted into the flow, vortices are generated at each cylinder, and the vortices merge with each other to develop a turbulent flow downstream. The extensional flow appears around each cylinder, and thus, the vortices in the flow are influenced by the extensional rheological properties of the fluids [38], [39], [40].

In our previous studies, we focused on the effects of the extensional rheological properties of polymer solutions on the modification of turbulent flow in a 2D flow [41]. In one of these studies, we investigated whether vortex shedding at a comb is influenced by the relaxation time of the fluids, as measured by a CaBER. Vortex shedding was categorized into three types, and the vortex deformation altered the 2D turbulent flow. In this study, extensional rates at the comb are varied by changing the spacing between the cylinders, and how that influences vortex shedding of the polymer-doped 2D flow is tested. Here, the extensional rate is represented by a mean value, as mentioned in Section 2.2, which induces the succeeding changes in the 2D turbulent flow. The 2D turbulent flow statistics are measured and analyzed by the particle image velocimetry (PIV) method. The energy transfer in the 2D flow of the polymer solution is precisely analyzed by the turbulent flow statistics, and we discuss the characteristic energy peak appearing behind a cylinder in the polymer-doped 2D flow.

Section snippets

Measurement of material and rheological properties

Sodium dodecylbenzenesulfonate (SDBS) was dissolved in ultrapure water at a concentration of 2 wt%. Polyethyleneoxide (PEO, molecular weight: 3.5 × 106) was used as a drag-reducing flexible polymer, and the solution concentrations were varied from 0.25 to 1.5 × 10−3 wt%. The overlap concentration of PEO was approximately 1.2 × 10−2 wt%.

The viscosity of the sample solutions was measured using a rheometer (MCR301: Anton Paar) with a cone-plate device at shear rates from 1 to 1000 s−1. The

Rheological properties of sample solutions

The zero shear viscosities, η0 [mPa·s], and relaxation times, λ [ms], of the sample solutions are listed in Table 1. Specifically, the η0 of the sample solutions containing PEO was virtually identical to that of the polymer-free solution. Whereas the λ of the polymer-free solution was low, the λ of the PEO solution increased with its concentration. The relaxation time measured with a CaBER represents the extensional rheological properties of the solutions.

Vortex shedding at comb influenced by polymers and extensional rates

In the experiments, vortices were shed

Conclusion

An experimental study was performed to investigate the effects of the extensional rheological properties of polymer solutions on vortex shedding in 2D turbulent flow and on turbulent statistics. The mean extensional rate was varied by changing the spacing of the cylinders of the combs that induced turbulent flow. The main conclusions of this study are as follows.

  • 1

    Vortex shedding at the cylinder of each comb was visualized through interference patterns. The vortices shed at the cylinder were

Declaration of Competing Interest

There are no conflicts to declare.

Acknowledgements

The present study was supported in part by a Grant-in-Aid for Scientific Research (B) (Project No.: 19H02497) and a Challenging Research (Exploratory) (Project No.: 19K22083) from the Japan Society for the Promotion of Science (JSPS KAKENHI).

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