# Two-dimensional compressible viscous flow around a circular cylinder

## TL;DR

This study uses direct numerical simulation to analyze how compressibility and viscosity influence flow around a circular cylinder, revealing effects on wake elongation, instability growth, and vortex shedding frequencies.

## Contribution

It provides new semi-empirical relationships for growth rate and frequency of flow instability considering compressibility and viscosity effects.

## Key findings

- Compressibility elongates the wake near the critical Reynolds number.
- Frequency of vortex shedding increases with Reynolds number above critical.
- Compressibility reduces growth rate and dominant frequency in linear instability.

## Abstract

Direct numerical simulation is performed to study compressible, viscous flow around a circular cylinder. The present study considers two-dimensional, shock-free continuum flow by varying the Reynolds number between 20 and 100 and the freestream Mach number between 0 and 0.5. The results indicate that compressibility effects elongate the near wake for cases above and below the critical Reynolds number for two-dimensional flow under shedding. The wake elongation becomes more pronounced as the Reynolds number approaches this critical value. Moreover, we determine the growth rate and frequency of linear instability for cases above the critical Reynolds number. From the analysis, it is observed that the frequency of the B\'enard-von K\'arm\'an vortex street in the time-periodic, fully-saturated flow increases from the dominant unstable frequency found from the linear stability analysis as the Reynolds number increases from its critical value, even for the low range of Reynolds numbers considered. We also notice that the compressibility effects reduce the growth rate and dominant frequency in the linear growth stage. Semi-empirical functional relationships for the growth rate and the dominant frequency in linearized flow around the cylinder in terms of the Reynolds number and freestream Mach number are presented.