Two-dimensional unsteady inertial flows of a yield stress fluid around a cylinder
Introduction
It may be stated without exaggeration that hundreds of articles concerning experimental and numerical studies of instability behind obstacles have been published since the pioneering work of Von Karman and Rubach in 1912 [1]. For example, Williamson [2] summarised the regimes observed for the unconfined flow of a Newtonian fluid behind a cylinder. In the case of Reynolds numbers ranging up to about 260, Williamson identified four main regimes in the inertial domain. First of all, for a Reynolds number of about 49, the flow morphology changes from that of a stationary regime with two recirculating areas behind the cylinder to a periodic 2D unsteady regime with vortex shedding. Then, with a Reynolds number of around 190, there is a transition between the two-dimensional regime and three-dimensional regimes. The works of Zdravkovich [3], [4] should be consulted for further details on the various regimes.
The available documentation concerning experimental studies of the flow of yield stress fluids around obstacles has concentrated mainly on stationary regimes with negligible inertia [5], [6], [7], [8], [9].
Mossaz et al. [10], [11], [12] studied the occurrence of the various inertial regimes as well as the non-recirculating and recirculating morphology of yield stress fluids flowing around a cylinder, both numerically and experimentally. To the best of our knowledge, no experimental study has been conducted on the unsteady flow of viscoplastic fluids around obstacles and around a cylinder in particular.
In this mainly experimental study, the two-dimensional unsteady regime with vortex shedding will be studied using polymer gels with a shear-thinning elastoviscoplastic behaviour. The morphology behind the flow and the Strouhal number of the flow will be paid particular attention. It is well documented that yield stress fluids are likely to slide to the solid wall in flows [17, 19, 32, 33]. That's why, the influence of surface states that are particularly favourable to wall slip or adherence of the fluid will be established.
This article is organised as follows: i) the first part validates the experimental set-up in the unsteady regime, using a Newtonian glucose solution; ii) the second part discusses yield stress fluids, considering first of all the influence of cylinder/fluid interface conditions on the wake behind the cylinder and the Strouhal number. It then examines the criterion for the occurrence of the unsteady regime. Lastly, the experimental results are compared with those obtained numerically using a Herschel-Bulkley viscoplastic model.
In an appendix are further numerical results concerning the influence of the Reynolds and Oldroyd numbers and the power-law index on the value of the Strouhal number. Special attention is paid to the definition of relations for quickly determining the Strouhal number.
Section snippets
Theory
The study has been performed with aqueous gels of Carbopol 940. These gels are widely used as model fluids of viscoplastic materials in many experimental studies because they are nonthixotropic and transparent [13]. They behave as an elastic solid below the yield stress and as a shear-thinning viscoelastic fluid above it. The model presented by Piau [13] and Piau and Debiane [14] will be used to obtain the rheological characteristics of the gel and thus take more effectively into account its
Experimental device, methods and rheometry
This study focuses on displaying the wake behind the cylinder and measuring the Strouhal number. First, the experimental set-up was validated with a Newtonian glucose solution. Second, we studied the flow of two physical Carbopol gels.
The experimental set-up used in this study consisted of a Plexiglas® flume with a width Dinf =200mm and a cylinder of diameter D=19mm, situated along the axis of symmetry of the tank. The length of the flume is 3 m. The aspect ratio here was D/h≈0.095 with h: the
Newtonian validation
The results were validated with a Newtonian fluid in order to determine first of all the extent of two-dimensional flow. To do so, the position of the sheet was studied so that flow in the visualisation area would not be disturbed by the end effects of the cylindrical obstacle. Flow of the glucose solution is certainly 2D up to Reynolds numbers of 80.
The Newtonian validation also involved a comparative study. This was performed for non-recirculating and recirculating stationary inertial
Case of yield stress fluids
A few preliminary remarks should be made with regard to yield stress fluids. First of all, owing to the stabilising effect of plasticity, the flows remain two-dimensional with Reynolds values higher than those found for the Newtonian fluid. The flows are definitely 2D with Reynolds numbers Re≈90 for gel 1 and Re≈85 for gel 2.
The results given in this section have an uncertainty of 10% for the Reynolds numbers and 15% for the Oldroyd numbers.
Conclusion
Unsteady inertial flow of viscoplastic shear-thinning fluids behind a cylinder was examined both experimentally and numerically The experiments were performed with Carbopol® gel, a fluid that displays a shear-thinning elastoviscoplastic behaviour. The results were compared with numerical simulations performed by regularising the Herschel-Bulkley law with Papanastasiou's law. The results were thus validated via a study performed with a Newtonian glucose solution and a bibliographical study.
This
Declaration of Competing Interest
None.
Acknowledgements
We should like to thank the technical staff working at the laboratory for their assistance and advice. LRP is part of the LabEx Tec 21 (Investissements d'Avenir– grant agreement #ANR-11-LABX-0030). LRP is part of Institut Carnot PolyNat (Investissements d'Avenir – grant agreement #ANR-11-CARN-030-01).
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