# zbMATH — the first resource for mathematics

A fully implicit combined field scheme for freely vibrating square cylinders with sharp and rounded corners. (English) Zbl 1390.76324
Summary: We present a fully implicit combined field scheme based on Petrov-Galerkin formulation for fluid-body interaction problems. The motion of the fluid domain is accounted by an arbitrary Lagrangian-Eulerian (ALE) strategy. The combined field scheme is more efficient than conventional monolithic schemes as it decouples the computation of ALE mesh position from the fluid-body variables. The effect of corner rounding is studied in two-dimensions for stationary as well as freely vibrating square cylinders. The cylinder shapes considered are: square with sharp corners, circle and four intermediate rounded squares generated by varying a single rounding parameter. Rounding of the corners delays the primary separation originating from the cylinder base. The secondary separation, seen solely for the basic square along its lateral edges, initiates at a Reynolds number, $$Re$$ between 95 and 100. Imposition of blockage lowers the critical $$Re$$ marking the onset of secondary separation. For free vibrations without damping, $$Re$$ range is 100–200 and mass ratio, $$m^{\ast}$$ of each cylinder is 10. The rounded cylinders undergo vortex-induced motion alone whereas motion of the basic square is vortex-induced at low $$Re$$ and galloping at high $$Re$$. The flow is periodic for vortex-induced motion and quasi-periodic for galloping. The lower branch and desynchronization characterize the response of rounded cylinders. For the square cylinder, the components of response are the lower branch, desynchronization and galloping. Removal of the sharp corners of square cylinder drastically alters the flow and vibration characteristics.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74H45 Vibrations in dynamical problems in solid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
##### References:
 [1] Lighthill, M. J., Boundary layer theory, (Rosenhead, L., Laminar boundary layer, (1963), Oxford University Press Oxford) [2] Sohankar, A.; Norberg, C.; Davidson, L., Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition, Int J Num Methods Fluids, 26, 39-56, (1998) · Zbl 0910.76067 [3] Robichaux, J.; Balachandar, S.; Vanka, S. P., Three-dimensional Floquet instability of the wake of square cylinder, Phys Fluids, 11, 560-578, (1999) · Zbl 1147.76482 [4] Sharma, A.; Eswaran, V., Heat and fluid flow across a square cylinder in the two-dimensional laminar flow regime, Num Heat Trans, 45, 247-269, (2004) [5] Dalton, C.; Zheng, W., Numerical simulations of a viscous uniform approach flow past square and diamond cylinders, J Fluids Struct, 18, 455-465, (2003) [6] Hu, J. C.; Zhou, Y.; Dalton, C., Effects of the corner radius on the near wake of a square prism, Exp Fluids, 40, 106-118, (2006) [7] Kumar, M. B.S.; Vengadesan, S., Influence of rounded corners on flow interference due to square cylinders using immersed boundary method, ASME J Fl Eng, 134, 091203-1-091203-3, (2012) [8] Amandolese, X.; Heḿon, P., Vortex-induced vibration of square cylinder in wind tunnel, Comptes Rendus Mećanique, 338, 12-17, (2010) [9] Sen, S.; Mittal, S.; Biswas, G., Flow past a square cylinder at low Reynolds numbers, Int J Num Methods Fluids, 67, 1160-1174, (2011) · Zbl 1426.76303 [10] He, T.; Zhou, D.; Bao, Y., Combined interface boundary condition method for fluid-rigid body interaction, Comput Methods Appl Mechan Eng, 81-102, (2012) · Zbl 1253.74034 [11] Jaiman, R.; Geubelle, P.; Loth, E.; Jiao, X., Combined interface boundary conditions method for unsteady fluid-structure interaction, Comput Meth Appl Mech Eng, 200, 27-39, (2011) · Zbl 1225.74091 [12] Jaiman, R.; Geubelle, P.; Loth, E.; Jiao, X., Transient fluid-structure interaction with non-matching spatial and temporal discretizations, Comput Fluids, 50, 120-135, (2011) · Zbl 1271.76242 [13] S Leontini, J.; Thompson, M. C.; Hourigan, K., The begining of branching behaviour of vortex-induced vibration during two-dimensional flow, J Fluids Struct, 73, 857-864, (2006) [14] Hughes, T. J.R.; Brooks, A. N., A multi-dimensional upwind scheme with no crosswind diffusion, Finite Element Methods Convect, 19-35, (1979) · Zbl 0423.76067 [15] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 32, 199-259, (1982) · Zbl 0497.76041 [16] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: viii. the Galerkin/least-squares method for advective-diffusive equations, Comput Methods Appl Mech Eng, 73, 173-189, (1989) · Zbl 0697.76100 [17] Liu, J.; Jaiman, R. K.; Gurugubelli, P. S., A stable second-order scheme for fluid-structure interaction with strong added-mass effects, J Comput Phy, 270, 687-710, (2014) · Zbl 1349.76236 [18] van Brummelen, E. H.; van der Zee, K. G.; Garg, V. V.; Prudhomme, S., Flux evaluation in primal and dual boundary-coupled problems, J Appl Mech, 79, (2011) [19] Hughes, T. J.R.; Liu, W. K.; Zimmerman, T. K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput Meth Appl Mech Eng, 29, 329-349, (1981) · Zbl 0482.76039 [20] Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-$$\alpha$$ method, J Appl Mech, 60, 370-375, (1993) · Zbl 0775.73337 [21] Dettmer, W.; Perić, A computational framework for fluid-rigid body interaction: finite element formulation and applications, Comput Methods Appl Mech Eng, 195, 1633-1666, (2006) · Zbl 1123.76029 [22] Bazilevs, Y.; Takizawa, K.; Tezduar, T. E., Computational fluid-structure interaction: methods and applications, (2013), Wiley [23] Johnson, C., Numerical solutions of partial differential equations by the finite element method, (1987), Cambridge University Press [24] Tezduyar, T.; Mittal, S.; Ray, S. E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order- interpolation velocity-pressure elements, Comput Meth Appl Mech Eng, 95, 221-242, (1992) · Zbl 0756.76048 [25] Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput Methods Appl Mech Eng, 59, 85-99, (1986) · Zbl 0622.76077 [26] Tezduyar, T. E., Finite elements in fluids: stabilized formulations and moving boundaries and interfaces, Comput Fluids, 36, 191-206, (2007) · Zbl 1177.76202 [27] Hron, J.; Turek, S., A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics, (2006), Springer · Zbl 1323.74086 [28] Wick, T., Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal. ii library, Arch Numer Software, 1, 1, 1-19, (2013) [29] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J Sci Stat Comput, 7, (1986) [30] Sen, S.; Mittal, S.; Biswas, G., Steady separated flow past a circular cylinder at low Reynolds numbers, J Fluid Mech, 620, 89-119, (2009) · Zbl 1156.76381 [31] Lima E Silva, A. L.F.; Silveira-Neto, A.; Damasceno, J. J.R., Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J Comput Phy, 189, 351-370, (2003) · Zbl 1061.76046 [32] Posdziech, O.; Grundmann, R., A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder, J Fluids Struct, 23, 479-499, (2007) [33] Vorobieff, P.; Georgiev, D.; Ingber, M. S., Onset of the second wake: dependence on the Reynolds number, Phys Fluids, 14, L53-L56, (2002) [34] Farhoud, R. K.; Amiralaie, S.; Jabbari, G.; Amiralaie, S., Numerical study of unsteady laminar flow around a circular cylinder, J Civil Eng Urban, 2, 63-67, (2012) [35] Lin, P. T.; Baker, T. J.; Martinelli, L.; Jameson, A., Two-dimensional implicit time-dependent calculations on adaptive unstructured meshes with time evolving boundaries, Int J Num Methods Fluids, 50, 199-218, (2006) · Zbl 1161.76378 [36] Sarpkaya, T., A critical review of the intrinsic nature of vortex-induced vibrations, J Fluids Struct, 19, 389-447, (2004) [37] S Leontini, J.; Thompson, M. C.; Hourigan, K., Three-dimensional transition in the wake of a transversely oscillating cylinder, J Fluid Mech, 577, 79-104, (2007) · Zbl 1178.76127 [38] Williamson, C. H.K.; Govardhan, R., Vortex induced vibration, Ann Rev Fluid Mechan, 36, 413-455, (2004) · Zbl 1125.74323 [39] Sen, S.; Mittal, S., Free vibration of a square cylinder at low Reynolds numbers, J Fluids Struct, 27, 875-884, (2011) [40] Williamson, C. H.K.; Roshko, A., Vortex formation in the wake of an oscillating cylinder, J Fluids Struct, 2, 355-381, (1988)
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.