# The Dependence of Flue Pipe Airflow Parameters on the Proximity of an Obstacle to the Pipe’s Mouth

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measurements

## 3. Data Analysis

_{0}) of the sound is located. To determine the precise frequency values, Discrete-time Fourier Transform (DTFT) [18] was calculated in the ranges found with the FFT. The DTFT X(e

^{jω}) for the discrete sequence of the signal x[n] is given by Equation (1):

_{M}of the obtained results. We used the standard deviation σ

_{M}for individual values of the fundamental frequency.

_{M}is the average of the set M = {M[1], …, M[n]}, M[k] is the k-th value of the M set, n is a number of measurements.

_{0}for each recording with an accuracy of 0.01 Hz, as we calculated DTFT for the frame length equal to the entire sound (about 5 s). The sound level for f

_{0}was also calculated in relation to the sound level generated by the calibrator and the measurement uncertainty σ

_{M}from Equation (2).

_{f0}were calculated, which are expressed in cents and are the musical distances between two sounds. Equation (3) was used to calculate each of the intervals, where x is the distance of the obstacle from the pipe lip and x

_{0}is the distance, at which the sound frequency does not depend on the obstacle. Figure 2 shows the relationship between f

_{0}and x.

- The fundamental frequency of the pipe decreases as the obstacle is closer to the lip. The changes of the pipe sound frequency depending on the distance of the obstacle to the mouth of the pipe are presented in Table 2;
- The spectrum of the sound recorded at the mouth of the pipe may differ significantly from the spectrum of the sound at the top of the resonator (i.a. significant difference in the number of harmonics may be observed);
- The fundamental frequency of the pipe, measured at the top of the pipe resonator, is sometimes different from the fundamental frequency measured at the lip of the pipe. It can be seen in the spectrum, especially on the logarithmic frequency scale, as the harmonics of the sound do not overlap, as shown in Figure 3;
- In the spectrum of some pipes, there are additional components that are not harmonics of the fundamental frequency of the pipe, see Figure 4.

## 4. Interval Calculus

**a**,

**b**are sets,

**min**is the minimum value of the set,

**max**is the maximum value of the set.

**g**of g. It is equivalent to the real-valued function g, satisfying the relationship (9).

**y**is the interval.

_{z}as the centered form of the inclusion function of h was used [22], satisfying Equation (10).

**y**is the interval, z is the middle point of

**y**interval, ∇h(

**y**) is a derivative of h(

**y**) function.

## 5. Fundamentals of a Sound Generation in a Labial Pipe

_{o}is the wavelength in an open pipe, and l

_{p}is the length of the resonator. In Formula (12), λ

_{s}is the wavelength for a stopped pipe. The length of the pipe is measured from the lower lip to the end of the sliding collar or the end of the pipe resonator (if there is no sliding collar).

_{pipe}is the fundamental frequency of the pipe’s sound [Hz], c is the speed of sound [m·s

^{−1}] and λ is the wavelength [m].

_{p}is the specific heat capacity of gas under constant pressure, C

_{v}is the specific heat capacity at constant volume, R is a molar gas constant 8.3144621 J·mol

^{−1}·K

^{−1}, T is temperature [K], μ is the molar mass of the gas. For dry air: κ = 1.401, μ = 0.029 kg·mol

^{−1}. For the measurement temperature of 19 °C, the speed of sound in the air is c ≈ 343 m·s

^{−1}.

_{D}, drag D increases, and velocity v decreases.

_{D}is the drag coefficient, S

_{D}is the cross-sectional area of the object exposed to the flow (i.e., the area of the orthographic projection on a plane perpendicular to the direction of the flow).

_{D}depends on the flow velocity, the shape of the streamlined body, and the Reynolds number. The Reynolds number value determines whether the fluid motion is laminar or turbulent. The transition between laminar and turbulent flow occurs at a critical Reynolds number. The critical Reynolds number, below which turbulent flow is not observed, is used in thermodynamics. There is no one, universal value of the critical Reynolds number that assures sound generation [31]. This value is determined empirically depending on the type of flow. The Reynolds number R

_{e}is calculated using Equation (16). For flows in the ranges of large Reynolds number values (over 2000), in which the boundary layer is turbulent, the aerodynamic drag D does not change because the drag coefficient C

_{D}is constant [30].

^{−1}], l is the characteristic dimension or length [m], and ν is the kinematic viscosity [m

^{2}·s

^{−1}].

_{D}can be assumed to be constant. For air at a temperature of about 20 °C, the kinematic viscosity is ν = 1.461·10

^{−5}m

^{2}·s

^{−1}[33].

_{e}= 2300 is the critical Reynolds number for flows in circular tubes. A flow is always turbulent above that value. In non-circular tube flow systems, the critical Reynolds numbers are different. Moreover, no constant values of critical Reynolds number can be used as they depend on the characteristic dimension or length of various measurable objects. In our case, the characteristic dimension l was the distance of the wedge cutting off the air jet in the form of the upper lip from the flue, from which the air is coming out. This selection was based on the papers [7,8,36].

_{0}generated by turbulent flow. This relationship is called the Strouhal number S

_{r}and is described by Equation (17):

_{0}, and the relationship between them is described by the Strouhal number [12], see Equation (17). In our case, approaching the obstacle to the pipe’s mouth causes the flow velocity changes, and we can use Equation (17) to solve for the fundamental frequency f

_{0}. Therefore, f

_{0}can be expressed as a function of the Strouhal number.

## 6. Results

#### 6.1. Determination of Strouhal and Reynolds Numbers for Labial Pipes

_{e}> 2300 [5]. Additionally, since f

_{0}>> 1, the Reynolds number is very large. Using Equation (19) and the interval calculus [22], it is possible to determine the range, in which the value of the Strouhal number S

_{r}for the fundamental frequency of the pipe f

_{0}will fall.

_{r}intervals, depending on the R

_{e}intervals, are presented in Table 4. For example, the S

_{r1}interval, depending on the interval R

_{e1}= [2300, 5000), is obtained as shown below. Using the interval arithmetic for Equation (19) and the values of the constants α and τ (Table 3) for a corresponding R

_{e1}range, the interval S

_{r1}can be calculated as in Equation (20):

_{r}is approximately constant and the mean for this range is S

_{r}≈ 0.2. Since the characteristic dimension l (the mouth height, called cut-up, see Figure 5) is also constant, then according to Equation (17), the higher the flow velocity u, the higher the frequency f

_{0}. As the value of u decreases, the frequency f

_{0}also decreases. Moreover, for the constants l and S

_{r}and the known fundamental frequency f

_{0}, the flow velocity u can be determined by Equation (17), and then the Reynolds number R

_{e}can be determined from Equation (19), as shown in Table 5.

#### 6.2. The Influence of an Obstacle on the Change of Flow Velocity

_{r}is constant. Let f

_{k}be the fundamental frequency of the pipe, calculated by DTFT for the k-th distance, and f

_{k+1}for k + 1 distance. We denote the airflow velocities in the pipe’s mouth for the k-th measurement as u

_{k}, and similarly u

_{k+1}for k + 1. Then we have the following dependencies, presented in Equations (21) and (22):

_{e}from Equation (18) will depend only on the fundamental frequency f

_{0}, which will depend on the flow velocity u (Equation (17)).

_{e}> 4000, the aerodynamic drag coefficient is very small, close to zero [38,39]. Therefore, in the area of influence of the obstacle on the flue pipe sound, the flow velocity should decrease linearly, assuming that the measurement is made every integer multiple of a distance value—in our case by k · 5 mm. Therefore, if for each pair (f

_{k}; f

_{k+1}) the value of the f

_{k+1}/f

_{k}ratio will be similar, then in the flue pipe the velocity of the airflow is directly proportional to the fundamental frequency f

_{0}of the generated sound.

_{k+1}/f

_{k}calculated for a single pipe is almost constant and is approximately 1.003. The measurement uncertainty, i.e., the divergence between the calculated and the measured frequency, does not exceed 1% for five-fold measurements of the fundamental frequency of the pipe sound. The smallest dispersion of these proportions is 0.1%. Moreover, calculations from the interval calculus prove to be a good representation of this phenomenon, since the Strouhal number is approximately constant.

#### 6.3. The Dependence of the Fundamental Frequency on the Change in the Distance of the Obstacle from the Pipe’s Mouth

_{0}of the pipe sound as the obstacle is located closer to the mouth of the pipe looks similar for all analyzed types of mouth. The characteristics of this phenomenon are very close to the logarithmic curve (see e.g., Figure 6). Since the occurring phenomena are similar, even though they take place in different geometric conditions, logarithmic regression coefficients were determined according to Equation (23), describing the dependence of the fundamental frequency of the pipe sound on the distance of the obstacle. The data necessary to calculate the sound frequency using Equation (23) for specific pipes are shown in Table 6, i.e., the logarithmic regression coefficients a and b, the distance x

_{0}at which the fundamental frequency of the pipe sound is not changed by the obstacle, and the coefficient of determination r

^{2}, that is a measure of the extent to which the model fits into the sample.

_{0}is the distance, at which the sound frequency does not depend on the obstacle.

_{2}coefficient, shown in Equation (24), connecting the b coefficient with the fundamental frequency of the pipe f

_{0}. The calculated values of b

_{2}are presented in Table 6.

_{2}coefficient does not change much. The measurement uncertainty σ

_{M}for different mouth types is 0.001, which is 0.1% of the mean. Thus, it can be assumed that b

_{2}is constant for all pipes and equals 0.998. This generalization allows using Equation (25) for all pipes in stops, within a specific type of mouth.

## 7. Discussion

_{0}and the distance from the upper lip to the flue l is constant. Although the authors did not explicitly specify that, this value is the Strouhal number. Hruška and Dlask [7] proved that there is a strong correlation between the Principal Component Analysis component and the reciprocal of the Strouhal number in the initial transient of the pipe sound. Cheong et al. [9], as well as Selfridge et al. [12], confirm the thesis that for Aeolian tone the Strouhal number is constant. In the case of the cylinder, for the fundamental frequency, which dominates the fluctuating lift force, the value of S

_{r}≈ 0.2 was determined. The stability of the Strouhal number is also confirmed by other researchers [15], analyzing turbulent flow generated by airflow through a flue. They prove that for Reynolds numbers R

_{e}> 2000, the value of the Strouhal number is approximately constant. The authors examining the whistling effect in the tubular system came to similar conclusions [41].

^{2}for proposed Equation (25) is 91% on average. It is not an ideal match but this indicates that the function model is sufficiently fitted to the data obtained from measurements and DTFT, which is confirmed by the plots of DTFT values and the values calculated from Equation (25), as shown in Figure 7. Future work may address the issue of developing an improved model of the function f

_{0}(x), presented in Equation (25).

## 8. Conclusions

- The fundamental frequency of the sound is directly proportional to the speed of the airflow in the pipe’s mouth;
- The speed of the airflow in the pipe’s mouth increases with the distance of the obstacle from the pipe’s mouth;
- The value of the Reynolds number in the pipe increases with the distance of the obstacle from the pipe’s mouth;
- The value of the Strouhal number for a labial pipe does not change significantly and can be approximated by a constant value.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The interval of the fundamental frequency changes in relation to the distance between the obstacle and the pipe lip of Flute 4-foot D sharp.

**Figure 3.**The spectrum of the Bass Principal 16-foot D pipe at the lip (red dotted plot) and the top (grey solid line plot) after normalization to the maximum sound level value for the immediate proximity of the obstacle (0 mm) at the pipe’s lip. As we can see, the harmonics do not overlap. The fundamental frequency, calculated by DTFT with 0.01 Hz frequency resolution, is 69.91 Hz at the lip and 69.69 Hz at the top (a difference of about 5 cents, which can be heard by a trained musician, especially when such a sound is accompanied by another playing pipe).

**Figure 4.**Dolce Flute 4-foot B spectrum for the immediate proximity to the obstacle (0 mm) at the pipe lip. A linear frequency scale has been used to help locate harmonics that are equally spaced on this scale.

**Figure 6.**The dependence of the distance of the obstacle from the lips for the Bass Principal 16ft open D.

**Figure 7.**The comparison of the plots representing the dependence of the fundamental frequency f

_{0}on the distance of the obstacle x from the pipe’s mouth for Flute 4ft open D sharp. The grey squares represent the measured data (using DTFT). The grey solid line plot shows the interpolated fundamental frequency for the measured data. The red dotted plot shows data calculated from Equation (25).

Labium Photo | ||||||||
---|---|---|---|---|---|---|---|---|

Labium type | English bay leaf | Beard | Beard | Beard | Beard | Plate | Roller | Roller |

Stops | Principal 4ft | Bourdon 8ft | Bourdon 16ft | Dolce Flute 8ft | Flute 4ft | Gamba 8ft | Bass Principal 16ft | Geigen Principal 8ft |

Construction | Open, pewter (75% tin, 25% lead) | Open, oakwood | Stopped, oakwood | Open, pine | Open, spruce | Open, metal (55% lead, 45% tin) | Open, pine | Open, pine |

Lip width [mm] | 29.4 | 41.1 | 41.8 | 18.5 | 35.4 | 29 | 108 | 80.5 |

Internal pipe dimensions [mm] | 37.9 | 41.9 × 51.9 | 42.4 × 54.5 | 19.4 × 29.2 | 35 × 51 | 35.3 | 106 × 140 | 79 × 96 |

Wavelength [mm] | 530 | 580 | 590 | 290 | 975 | 920 | 2096 | 2471 |

**Table 2.**The dependence of the pipe sound frequency on the distance of the obstacle to the mouth of the pipe (the symbol “-” denotes that the pipe generated no sound in this case).

Distance from the Obstacle | Bourdon 16ft Stopped C Sharp | Bourdon 8ft Open B | Principal 4ft E1 | Gamba 8ft f Sharp | Geigen Principal 8ft Open C | Bass Principal 16ft Open D | Dolce Flute 8ft Open B | Flute 4ft Open D Sharp |
---|---|---|---|---|---|---|---|---|

x [mm] | f_{0} [Hz] | |||||||

0 | 131.44 | 238.14 | - | 185.68 | 64.45 | 69.69 | 458.15 | 149.15 |

5 | 135.12 | 246.02 | 320.59 | 186.26 | 64.80 | 71.15 | 464.61 | 153.18 |

10 | 136.95 | 249.22 | 324.14 | 186.46 | 64.92 | 71.85 | 466.35 | 154.70 |

15 | 137.82 | 250.61 | 325.28 | 186.55 | 65.06 | 72.41 | 467.53 | 155.16 |

20 | 138.30 | 251.30 | 325.81 | 186.60 | 65.16 | 72.61 | 468.13 | 155.46 |

25 | 138.60 | 251.76 | 325.90 | 65.17 | 72.79 | 155.58 | ||

30 | 251.99 | 326.16 | 65.22 | 72.98 | 155.68 | |||

35 | 252.19 | 326.28 | 65.28 | 73.10 | 155.74 | |||

40 | 252.30 | 326.37 | 65.31 | 73.19 | ||||

45 | 252.38 | 326.44 | 65.33 | 73.28 | ||||

50 | 326.49 | 65.35 | 73.32 | |||||

55 | 326.52 | 65.39 | 73.38 | |||||

60 | 65.39 | 73.41 | ||||||

65 | 65.40 | 73.41 | ||||||

70 | 65.45 | 73.26 | ||||||

75 | 73.31 | |||||||

80 | 73.32 | |||||||

85 | 73.34 | |||||||

x_{0} (no obstacle) | 139.24 | 252.76 | 327.19 | 186.91 | 65.49 | 73.49 | 469.52 | 155.98 |

**Table 3.**The values of the constants α and τ, depending on the intervals of the Reynolds number, used in Equation (19) [13].

R_{e} | [360, 1300) | [1300, 5000) | [5000, 2 × 10^{5}) | [2 × 10^{5}, 10^{6}) |

α | 0.2257 | 0.2040 | 0.1776 | 0.5760 |

τ | −0.4402 | 0.3364 | 2.2023 | −175.956 |

**Table 4.**Interval values of the Strouhal number S

_{r}depending on the intervals of Reynolds number R

_{e}for labial pipes.

R_{e} | [2300, 5000) | [5000, 10^{4}) | [10^{4}, 2 × 10^{5}) |

S_{r} | (0.2088, 0.211] | (0.1996, 0.2088] | (0.1825, 0.1996] |

**Table 5.**Values of Reynolds number R

_{e}and flow velocity u for analyzed flue pipes at S

_{r}= 0.2.

Pipe | Bourdon 16ft Stopped c Sharp | Bourdon 8ft Open b | Principal 4ft e1 | Gamba 8ft F Sharp | Geigen Principal 8ft Open C | Bass Principal 16ft Open D | Dolce Flute 8ft Open B | Flute 4ft Open D Sharp |
---|---|---|---|---|---|---|---|---|

f_{0} [Hz] | 139.240 | 252.76 | 327.2 | 186.91 | 65.49 | 73.49 | 469.52 | 155.98 |

l [mm] | 16.53 | 15.65 | 7.51 | 8.1 | 15 | 33 | 8 | 12.4 |

u [m·s^{−1}] | 11.51 | 19.78 | 12.29 | 7.57 | 4.91 | 12.13 | 18.78 | 9.67 |

R_{e} | 13021 | 21187 | 6315 | 4197 | 5043 | 27389 | 10284 | 8208 |

**Table 6.**Logarithmic regression coefficients and the coefficient of determination for the dependence of the fundamental frequency on the distance between the obstacle and the pipe’s mouth.

Pipe | Bourdon 16ft Stopped c Sharp | Bourdon 8ft Open b | Principal 4ft e1 | Gamba 8ft F Sharp | Geigen Principal 8ft Open C | Bass Principal 16ft Open D | Dolce Flute 8ft Open B | Flute 4ft Open D Sharp |
---|---|---|---|---|---|---|---|---|

x_{0} [mm] | 30 | 50 | 60 | 25 | 75 | 90 | 25 | 40 |

a [Hz] | 0.91 | 1.76 | 2.14 | 0.13 | 0.12 | 0.45 | 1.32 | 0.83 |

b [Hz] | 138.37 | 252.50 | 327.35 | 186.65 | 65.36 | 73.39 | 468.16 | 155.83 |

b_{2} | 0.994 | 0.999 | 1.001 | 0.999 | 0.998 | 0.999 | 0.997 | 0.999 |

r^{2} | 91% | 96% | 85% | 88% | 84% | 90% | 95% | 97% |

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**MDPI and ACS Style**

Węgrzyn, D.; Wrzeciono, P.; Wieczorkowska, A.
The Dependence of Flue Pipe Airflow Parameters on the Proximity of an Obstacle to the Pipe’s Mouth. *Sensors* **2022**, *22*, 10.
https://doi.org/10.3390/s22010010

**AMA Style**

Węgrzyn D, Wrzeciono P, Wieczorkowska A.
The Dependence of Flue Pipe Airflow Parameters on the Proximity of an Obstacle to the Pipe’s Mouth. *Sensors*. 2022; 22(1):10.
https://doi.org/10.3390/s22010010

**Chicago/Turabian Style**

Węgrzyn, Damian, Piotr Wrzeciono, and Alicja Wieczorkowska.
2022. "The Dependence of Flue Pipe Airflow Parameters on the Proximity of an Obstacle to the Pipe’s Mouth" *Sensors* 22, no. 1: 10.
https://doi.org/10.3390/s22010010