Notes on Computational Fluid Dynamics: General Principles

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8.4 Flow around a cylinder

Flow around a cylinder was used as an example of boundary layer separation in Sec. 6.5 . It showed an image of vortices shedding from the cylinder in a periodic manner, known as the Kármán vortex street.

The image comes from a CFD simulation in two dimensions, representative of a cylinder of inﬁnite axial length, using the principal parameters: cylinder diameter $\text{[math]}$ ; freestream velocity $\text{[math]}$ in the $\text{[math]}$ -direction; and, ﬂuid kinematic viscosity $\text{[math]}$ . The corresponding Reynolds number $\text{[math]}$ , which falls within the laminar ﬂow regime.

The computational mesh used in the simulation is described in Sec. 8.3 . The centre height of the cells adjacent to the cylinder was $\text{[math]}$ , corresponding to a calculated $\text{[math]}$ .

The simulation used the transient solution algorithm in Sec. 5.19 , solving for momentum conservation for an incompressible ﬂuid, with $\text{[math]}$ and constant $\text{[math]}$ . No energy equation was solved and no turbulence modelling was required since the ﬂow was laminar.

The freestream boundary conditions from Sec. 4.16 were applied to $\text{[math]}$ and $\text{[math]}$ over the entire external boundary, with reference values $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ . The no-slip condition, $\text{[math]}$ and $\text{[math]}$ , was applied on the cylinder boundary.

The simulation ran for $\text{[math]}$ with a time step $\text{[math]}$ using recommended numerical schemes from Sec. 3.23 . Oscillations began in the wake of the cylinder at $\text{[math]}$ , soon leading to shedding of vortices which reached a stable pattern at $\text{[math]}$ .

The following ﬁgure, shaded by $\text{[math]}$ , shows the slower-moving vortices as the darker structures which propagate downstream of the cylinder.

$PICT\relax \special {t4ht=$

The repeated vortex shedding causes oscillations in the force $\text{[math]}$ of the ﬂuid on the cylinder. The force is calculated as the sum of viscous and pressure forces acting on the cylinder patch faces ( $\text{[math]}$ ), in kinematic units $\text{[math]}$ (from kinematic $\text{[math]}$ and $\text{[math]}$ ) by

X X X f = Sf ☐☐☐ = Sf (pI ☐☐☐) = Sfp Sf ☐☐☐: f f f \relax \special {t4ht=

(8.1)

The dimensionless lift coeﬃcient $\text{[math]}$ is shown below, calculated from: the $\text{[math]}$ -component of the force is $\text{[math]}$ , where $\text{[math]}$ is the unit vector in the $\text{[math]}$ -direction; and, the projected frontal area $\text{[math]}$ of the cylinder.

$PICT\relax \special {t4ht=$

The oscillation period $\text{[math]}$ corresponds to a Strouhal number $\text{[math]}$ , consistent with published data.¹

¹Christoﬀer Norberg, Flow around a circular cylinder: Aspects of ﬂuctuating lift, 2001.

Notes on CFD: General Principles - 8.4 Flow around a cylinder