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A study of unsteady wake flows past a two-dimensional square cylinder with and without planar jet injection into the vortex formation region

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The present work describes the study of unsteady turbulent wake flows past two-dimensional square cylinders with and without planar jet injection into the vortex formation region. An experimental investigation provided LDV measurements of the mean and turbulent near and middle wake development at Reynolds numbers of 8520 and 14285 and for various jet to approach flow velocity ratios. In the computational work a Large Eddy Simulation employing the Smagorinsky subgrid scale model was initially performed for the higher Reynolds number plane wake configuration. Further, a hybrid phase-averaged Navier-Stokes model was formulated that encompasses aspects from both the LES formalism and the conventional eddy-viscosity procedures. Comparisons of the hybrid model calculations with the LES and measurements demonstrated the ability of the hybrid method to reproduce the range of complex unsteady flow phenomena studied here.

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C :

Courant number

C D :

drag coefficient

C S :

Smagorinsky's model coefficient

C P :

pressure coefficient

C μ :

turbulent viscosity coefficient

D :

square cylinder diameter

f :


H :

channel height

k :

turbulence kinetic energy

L,l :

turbulence length scales

u,v,w :


U 0 :

approach flow velocity

P :

static pressure

S t :

Strouhal number

t 0 :

characteristic time (D/U 0)

x i :

coordinate directions (i=1,2,3)

δ ij :

Kronecker delta (δ ij =0 forij;δ ij =1 fori=j)

ν t :

eddy-viscosity coefficient

λ :

non-dimensional diffusion coefficient

ρ :


τ ij :

subgrid-stress tensor



\(\bar \cdot\) :


\(\tilde \cdot\) :

grid-scale velocity (LES)


turbulent fluctuation


subgrid-scale fluctuation (LES)


periodic fluctuation

c :

explicitly (cumulatively) calculated

i :

1, 2, 3 Cartesian coordinates

i,j,k :

tensor notation

t :

turbulent flow, total


subgrid (e.g.k SGS)


characteristic filter length

ɛ :

turbulence energy dissipation rate

ν :

molecular viscosity


wall coordinates

n :

time level


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Koutmos, P., Mavridis, C. & Papailiou, D. A study of unsteady wake flows past a two-dimensional square cylinder with and without planar jet injection into the vortex formation region. Appl. Sci. Res. 55, 187–210 (1995). https://doi.org/10.1007/BF00867511

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Key words

  • bluff-body wake flow
  • LES of vortex shedding flow
  • hybrid eddy-viscosity models