We analyse the incompressible flow past a square cylinder immersed in the wake of an upstream splitter plate which separates two streams of different velocities, UT (top) and UB (bottom). The Reynolds number associated to the flow below the plate is kept constant at ReB=D UB/ν=56, based on the square cylinder side D as characteristic length. The top-to-bottom flow dissymmetry is measured by the ratio R≡ ReT/ReB∈[1,5.3]. The equivalent bulk Reynolds, taken as the mean between top and bottom changes with R in the range Re∈( ReT+ReB)/2∈[56,178]. A Hopf bifurcation occurs at R=2.1{plus minus}0.1 ( Re=86.8{plus minus}2.8), which results in an asymmetric Kármán vortex street with vortices only showing on the high-velocity side of the wake. A spanwise modulational instability is responsible for the three-dimensionalisation of the flow at R≈3.1 ( Re≈115) with associated wavelength λ z≈2.4.For velocity ratios R{greater than or equal to}4, the flow becomes spatio-temporally chaotic. The migration of the mean stagnation and base pressure points on the front and rear surfaces of the cylinder as R is increased determine the boundary layer properties on the top and bottom surfaces and, with them, the shear layers that roll up into the formation of Kármán vortices, which in turn help clarify the evolution of the lift and drag coefficients. The symmetries of the different solutions across the flow transition regime are imprinted on the top and bottom boundary layers and can therefore be analysed from the time evolution and spanwise distribution of trailing edge boundary layer displacement thickness at the top and bottom rear corners.
We investigate the incompressible flow past a square cylinder immersed in the wake of an upstream nearby splitter plate separating two streams of different velocity. The bottom stream Reynolds number, based on the square side,
$Re_B=56$
is kept constant while the top-to-bottom Reynolds numbers ratio
$R\equiv Re_T/Re_B$
is increased in the range
$R\in [1,6.5]$
, corresponding to a coupled variation of the bulk Reynolds number
$Re\equiv (Re_T+Re_B)/2\in [56,210]$
and an equivalent non-dimensional shear parameter
$K\equiv 2(R-1)/(R+1)\in [0,1.4667]$
. The onset of vortex shedding is pushed to higher
$Re$
as compared with the square cylinder in the classic configuration. The advent of three dimensionality is triggered by a mode-C-type instability, much as reported for open circular rings and square cylinders placed at an incidence. The domain of minimal spanwise-periodic extension that is capable of sustaining spatiotemporally chaotic dynamics, namely one accommodating about twice the wavelength of the dominant eigenmode, has been chosen for the analysis of the wake transition regime. The path towards spatiotemporal chaos is, in this minimal domain, initiated with a modulational period-doubling tertiary bifurcation that also doubles the spanwise periodicity. At slightly higher values of
$R$
, the flow has become spatiotemporally chaotic, but the main features of mode C are still clearly distinguishable. Although some of the nonlinear solutions found along the wake transition regime employing the minimal domain might indeed be unstable to long wavelength disturbances, they still are solutions of the infinite cylinder problem and are apt to play a relevant role in the inception of spatiotemporally chaotic dynamics.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.