Notes on Computational Fluid Dynamics: General Principles

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6.5 Boundary layer separation

Boundary layers provide the main source of vorticity for turbulence as discussed in Sec. 6.4 . A boundary layer breaks away from the boundary when it reaches the end of the surface or by separation before reaching the end. Vorticity and turbulence are thereby swept into regions of ﬂuid away from solid boundaries.

Flow around a cylinder illustrates boundary layer separation, vorticity and turbulence. The ﬂuid ﬂows at uniform velocity $\text{[math]}$ upstream of the cylinder. It decelerates to stagnation, $\text{[math]}$ , at point A on the surface (in the $\text{[math]}$ -normal direction).

High pressure at A introduces a favourable pressure gradient $\text{[math]}$ which increases the ﬂow speed $\text{[math]}$ around the cylinder towards B, developing a boundary layer in the process.

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The ﬂow reaches a peak speed at B, then decelerates over the downstream side of the cylinder. The adverse pressure gradient $\text{[math]}$ causes $\text{[math]}$ to decrease. At some point C, the velocity gradient can reach $\text{[math]}$ . Beyond C, the boundary layer can separate such that $\text{[math]}$ along its proﬁle, see point D.

Boundary layer separation in a cylinder depends on Reynolds number $\text{[math]}$ , Eq. (2.68 ), using $\text{[math]}$ and $\text{[math]}$ . For $\text{[math]}$ , there is no separation, with the ﬂow exhibiting a pattern downstream that mirrors the upstream ﬂow.⁶

For $\text{[math]}$ , the boundary layer separates with its vorticity sustaining a pair of vortices attached to the rear of the cylinder.

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At $\text{[math]}$ , vorticity is released downstream as vortices break oﬀ from the cylinder in a periodic manner known as the Kármán vortex street, shown above.

At $\text{[math]}$ , the vorticity starts to become become chaotic, with turbulence beginning to appear in the vortices. At $\text{[math]}$ , the entire wake region becomes turbulent.

The frequency of vortex shedding is characterised by another dimensionless number from Eq. (2.68 ), the Strouhal number $\text{[math]}$ . For $\text{[math]}$ , experiments show⁷ $\text{[math]}$ , where $\text{[math]}$ is the period at which the vortex pattern repeats.

⁶Sadatoshi Taneda, Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers, 1956.

⁷Vincenc Strouhal, Über eine besondere Art der Tonerregung, 1878 and Anatol Roshko, On the development of turbulent wakes from vortex streets, NACA Report 1191, 1954.

Notes on CFD: General Principles - 6.5 Boundary layer separation