# Calculation of the Acoustic Spectrum of a Cylindrical Vortex in Viscous Heat-Conducting Gas Based on the Navier–Stokes Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### 2.1. The Initial Value Problem

_{0}and of a height z

_{0}. The cylinder is situated on a flat wall. The axis of the cylinder is normal to the wall (4).

#### 2.2. The Solution to the Problem

## 3. Results

#### 3.1. Low-Frequency Oscillations

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}/2. As seen, there are two temporal intervals, in which the amplitude of the oscillations is large enough but the maximum amplitude is reached in the second area of oscillations. The oscillation spectrum (Figure 4) has two natural frequencies (about 115 Hz and 280 Hz), but there is no continuous spectrum.

_{0}/2, r

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}/2, r

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}, r = 1.709 cm, r

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}, r = 3.418 cm, r

_{0}= 0.376 cm, z

_{0}= 5.818 cm, z = 3.4 cm.

_{0}, r = 5.127 cm, r

_{0}= 0.564 cm, z

_{0}= 8.727 cm, z = 5.1 cm.

_{0}, r = 6.836 cm, r

_{0}= 0.752 cm, z

_{0}= 11.636 cm, z = 6.8 cm.

_{0}, r = 1.709 cm, r

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

#### 3.2. High-Frequency Oscillations

_{0}= 0.188 cm, Figure 12), 790 Hz and 1150 Hz (r

_{0}= 2 cm, Figure 13).

_{0}, r = 1.709 cm, r

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}, r = 18.12 cm, r

_{0}= 2 cm, z

_{0}= 30.84 cm, z = 18.02 cm.

_{0}, r = 1.709 cm, r

_{0}= 0.188 cm, z

_{0}= 2.909 cm, z = 1.7 cm.

_{0}, r = 18.12 cm, r

_{0}= 2 cm, z

_{0}= 30.84 cm, z = 18.02 cm.

## 4. Discussion

^{−}

^{1}(corresponding to Reynolds number Re = 6.8 × 10

^{4}). The ring noise was determined from the averaged spectrum in a series of 12 selected time samples of length 31.2 ms starting after 220 ms from the initiation of the ring (this corresponds to the part of the path at a distance from 200 to 230 cm from the nozzle orifice), manifesting itself as strong peaking of the spectrum in a narrow frequency band (∆ω = 300 Hz) with the maximum near the frequency ω

_{0}= 1200 Hz.

## 5. Conclusions

- (i)
- It was found that there are high and low frequencies corresponding to the frequencies experimentally observed for the vortex ring and atmospheric frequencies, respectively.
- (ii)
- As seen, the pattern of oscillations is different inside the initial cylinder and outside it. This fact may be explained as follows. There are multiple reflections of acoustic waves inside the initial cylinder. The reflected waves must be weaker in the domain outside.
- (iii)
- There are high-frequency oscillations modulated by a low-frequency signal. The value of high frequency remains constant during a long time. Thus it is possible to consider the high frequency as the natural frequency of the vortex. The value of the natural frequency depends on the initial radius of the vortical cylinder and does not depend on the intensity of the initial vorticity. Namely, it diminishes if the radius of the cylinder increases, as expected from physical considerations. The natural frequency has different values inside the initial cylinder and in the outer domain.

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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**Figure 14.**High-frequency oscillations. Dependence of the oscillation frequency on the scaling factor.

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Petrova, T.; Shugaev, F.
Calculation of the Acoustic Spectrum of a Cylindrical Vortex in Viscous Heat-Conducting Gas Based on the Navier–Stokes Equations. *Computation* **2016**, *4*, 32.
https://doi.org/10.3390/computation4030032

**AMA Style**

Petrova T, Shugaev F.
Calculation of the Acoustic Spectrum of a Cylindrical Vortex in Viscous Heat-Conducting Gas Based on the Navier–Stokes Equations. *Computation*. 2016; 4(3):32.
https://doi.org/10.3390/computation4030032

**Chicago/Turabian Style**

Petrova, Tatiana, and Fedor Shugaev.
2016. "Calculation of the Acoustic Spectrum of a Cylindrical Vortex in Viscous Heat-Conducting Gas Based on the Navier–Stokes Equations" *Computation* 4, no. 3: 32.
https://doi.org/10.3390/computation4030032