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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
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 [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
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