Laminar Circular Cylinder at Re = 100: FW-H Against Literature

Overview

A circular cylinder in a laminar cross-flow sheds an alternating vortex street whose surface-pressure fluctuations radiate a tone at the shedding frequency. This case predicts the far-field sound of a Reynolds-number-100, Mach-0.15 cylinder with the Ffowcs Williams-Hawkings analogy, evaluated on the cylinder wall and on a closed permeable surface in the same flow solution, and compares the angular distribution of the shedding tone with a published reference.

Problem Setup

The flow is laminar with $M_{\infty }=0.15$ ($U_{\infty }=51.04$ m/s, $T_{\infty }=288.15$ K, $p_{\infty }=86985$ Pa) and a constant dynamic viscosity chosen so that $R{e}_D=100$ for a cylinder of diameter $D$. The O-grid is one diameter deep and periodic in the spanwise direction, and a sponge layer that absorbs the outgoing sound starts at $16D$ and is $18D$ thick. Time integration is second-order implicit with $\Delta t=2.5\times {10}^{-4}$ s.

The Ffowcs Williams-Hawkings analogy is evaluated on the cylinder wall and on a permeable box spanning $-8D\le x\le 15D$ and $-5D\le y\le 5D$, open on its spanwise faces. Both integrals are repeated over 50 copies of the span to represent a long cylinder. The acoustic pressure is recorded at 36 observers spaced ${10}^{\circ }$ apart on an arc of radius $12.9D$ about $x=1.86D$, at mid-span.

Quantities of Interest

The frequency of the shedding tone, and its level around the observer arc from both integration surfaces, normalized by its maximum between ${25}^{\circ }$ and ${175}^{\circ }$ and compared with the reference curve, with the RMS difference between the two over that arc.

Discussion

At this low Reynolds number the wake is a regular street of vortices that leave the top and the bottom of the cylinder in turn. Each one pushes the cylinder sideways, so the cylinder radiates a single clean tone, here at $8.2$ Hz from the cylinder-wall surface and $8.3$ Hz from the permeable surface. The sound has a dipole pattern: loud above and below the cylinder, quiet along the flow direction. In the directivity plot that shows up as two lobes, one up and one down.

The plot compares the shape of this pattern, with each curve divided by its own maximum between ${25}^{\circ }$ and ${175}^{\circ }$. Both Ffowcs Williams-Hawkings (FW-H) surfaces reproduce the reference: the two lobes, their width, and the quiet directions along the flow axis. Over the scored arc the root-mean-square difference from the reference is $0.93$ dB for the cylinder wall and $0.98$ dB for the permeable surface, about 11% in pressure amplitude.