×

zbMATH — the first resource for mathematics

Floquet stability analysis of viscoelastic flow over a cylinder. (English) Zbl 1359.76029
Summary: A Floquet linear stability analysis has been performed on a viscoelastic cylinder wake. The FENE-P model is used to represent the non-Newtonian fluid, and the analysis is done using a modified version of an existing nonlinear code to compute the linearized initial value problem governing the growth of small perturbations in the wake. By measuring instability growth rates over a wide range of disturbance spanwise wavenumbers \(\alpha \), the effects of viscoelasticity were identified and compared directly to Newtonian results.
At a Reynolds number of 300, two unstable bands exist over the range \(0 \leq \alpha \leq 10\) for Newtonian flow. For the low \(\alpha \) band, associated with the “mode A” wake instability, a monotonic reduction in growth rates is found for increasing polymer extensibility \(L\). For the high \(\alpha \) band, associated with the “mode B” instability, first a rise, then a significant decrease to a stable state is found for the instability growth rates as \(L\) is increased from \(L = 10\) to \(L = 30\). The mechanism behind this stabilization of both mode A and mode B instabilities is due to the change of the base flow, rather than a direct effect of viscoelasticity on the perturbation.

MSC:
76A10 Viscoelastic fluids
76E17 Interfacial stability and instability in hydrodynamic stability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Richter, D.; Iaccarino, G.; Shaqfeh, E. S. G.: Simulations of three-dimensional viscoelastic flows past a circular cylinder at moderate Reynolds numbers, Journal of fluid mechanics 651, 415-442 (2010) · Zbl 1189.76057
[2] Williamson, C. H. K.: The existence of two stages in the transition to three-dimensionality of a cylinder wake, Physics of fluids 31, 3165-3168 (1988)
[3] Williamson, C. H. K.: Three-dimensional wake transition, Journal of fluid mechanics 328, 345-407 (1996) · Zbl 0899.76129
[4] Barkley, D.; Henderson, R.: Three-dimensional Floquet stability analysis of the wake of a circular cylinder, Journal of fluid mechanics 322, 215-241 (1996) · Zbl 0882.76028
[5] Henderson, R.: Nonlinear dynamics and pattern formation in turbulent wake transition, Journal of fluid mechanics 352, 65-112 (1997) · Zbl 0903.76070
[6] Cadot, O.; Kumar, S.: Experimental characterization of viscoelastic effects on two- and three-dimensional shear instabilities, Journal of fluid mechanics 416, 151-172 (2000) · Zbl 0948.76521
[7] Noack, B.; Eckelmann, H.: A global stability analysis of the steady and periodic cylinder wake, Journal of fluid mechanics 270, 297-330 (1994) · Zbl 0813.76025
[8] Robichaux, J.; Balachandar, S.; Vanka, S. P.: Three-dimensional Floquet instability of the wake of square cylinder, Physics of fluids 11, 560-578 (1999) · Zbl 1147.76482
[9] Camarri, S.; Giannetti, F.: Effect of confinement on three-dimensional stability in the wake of a circular cylinder, Journal of fluid mechanics 642, 477-487 (2010) · Zbl 1183.76719
[10] Schmid, P.; Henningson, D.: Stability and transition in shear flows, (2001) · Zbl 0966.76003
[11] Blackburn, H. M.; Lopez, J. M.: On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes, Physics of fluids 15, L57-L60 (2003) · Zbl 1186.76064
[12] Williamson, C. H. K.: Vortex dynamics in the cylinder wake, Annual review of fluid mechanics 28, 477-539 (1996)
[13] Thompson, M. C.; Leweke, T.; Williamson, C. H. K.: The physical mechanism of transition in bluff body wakes, Journal of fluids and structures 15, 607-616 (2001)
[14] Leweke, T.; Williamson, C. H. K.: Three-dimensional instabilities in wake transition, European journal of mechanics B/fluids 17, 571-586 (1998) · Zbl 0948.76505
[15] Pierrehumbert, R. T.: Universal short-wave instability of two dimensional eddies in an inviscid fluid, Physical review letters 57, 2157-2159 (1986)
[16] Bayly, B. J.: Three-dimensional instability of elliptical flow, Physical review letters 57, 2160-2163 (1986)
[17] Landman, M. J.; Saffman, P. G.: The three-dimensional instability of strained vortices in a viscous fluid, Physics of fluids 38, 2339-2342 (1987)
[18] Waleffe, F.: On the three-dimensional instability of strained vortices, Physics of fluids A 2, 76-80 (1990) · Zbl 0696.76052
[19] Giannetti, F.; Camarri, S.; Luchini, P.: Structural sensitivity of the secondary instability in the wake of a circular cylinder, Journal of fluid mechanics 651, 319-337 (2010) · Zbl 1189.76219
[20] Barkley, D.: Confined three-dimensional stability analysis of the cylinder wake, Physical review E 71, 1-3 (2005)
[21] Kumar, S.; Homsy, G.: Direct numerical simulation of hydrodynamic instabilities in two- and three-dimensional viscoelastic free shear layers, Journal of non-Newtonian fluid mechanics 83, 249-276 (1999) · Zbl 0946.76020
[22] Lagnado, R. R.; Simmen, J. A.: The three-dimensional instability of elliptical vortices in a viscoelastic fluid, Journal of non-Newtonian fluid mechanics 50, 29-44 (1993) · Zbl 0812.76036
[23] Haj-Hariri, H.; Homsy, G. M.: Three-dimensional instability of viscoelastic elliptic vortices, Journal of fluid mechanics 353, 357-381 (1997) · Zbl 0905.76037
[24] Lagnado, R. R.; Phan-Thien, N.; Leal, L. G.: The stability of two-dimensional linear flows, Physics of fluids 27, 1094-1101 (1984) · Zbl 0585.76045
[25] Lagnado, R. R.; Phan-Thien, N.; Leal, L. G.: The stability of two-dimensional linear flows of an Oldroyd-type fluid, Journal of non-Newtonian fluid mechanics 18, 25-59 (1985) · Zbl 0625.76006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.