# Mathematical Description of the Aerodynamic Characteristics of Stationary Flows in a Vertical Conical Diffuser When Air Is Supplied through Various Tube Configurations

## Abstract

**:**

## 1. Introduction

- -
- Gas-dynamic improvement and the development of ways to control the gas dynamics of flows in a conical diffuser remain urgent tasks for fundamental and applied science;
- -
- Recent research into the gas-dynamic characteristics of flows in conical diffusers is carried out mainly through numerical simulation;
- -
- There is a lack of reliable experimental data (and their mathematical description) on the gas-dynamic characteristics of flows in a vertical conical diffuser for mathematical model verification.

- -
- To develop an experimental set-up for studying stationary flows in a vertical conical diffuser when air is supplied through tubes with various cross sections;
- -
- To choose measuring instruments and research methods taking into account the physical features of the processes under research;
- -
- To obtain data on the instantaneous values of the air flow velocity along the height and diameter of the diffuser’s cylindrical part for various initial conditions;
- -
- To establish the evolution of the velocity fields along the height of the diffuser’s cylindrical part for various configurations of the supply tubes;
- -
- To empirically determine the value of the drop in the average velocity along the height of the diffuser and provide a mathematical description of that process;
- -
- To establish and mathematically describe the patterns of changes in the intensity of turbulence along the height of the conical diffuser under various initial conditions.

## 2. Description of the Experimental Measurement Facility

_{o}= 0.1013 mPa. The typical measurement in this work was taken in the control section at a height of 100 mm and the centre of the cylindrical part of the diffuser (at a distance of 50 mm). During the experiments, the average air flow velocity in the typical section varied from 4 m/s to 8 m/s (26,500 < Re < 53,500). Therefore, the flow pattern in this study was developed and turbulent.

_{x}with a hot-wire anemometer at a constant temperature were made. A nichrome filament with a diameter of 5 μm and a length of 4 mm was used as the sensitive element of the hot-wire anemometer sensor. Five sensors with thread at distances of 10, 20, 30, 40, and 50 mm were made. The data from the hot-wire anemometer (output signal from 0 to 5 V) were received by an analogue-to-digital converter, and then they were transferred to a laptop for processing with custom-made software. The standard relative uncertainty of air flow measurement was 3.6%. The measuring system and the features of its functioning are described in more detail in articles [26,27].

_{x}= f (τ). One of the key aerodynamic characteristics of diffuser flows is the turbulence intensity TI. In this research, TI was calculated as the ratio of the root–mean–square pulsation velocity component to the average velocity of the flow under study [27].

## 3. Results and Analysis of Experimental Findings

_{x}= f (τ) for the sensor in the first section (H

_{1}= 100 mm) in the centre of the diffuser’s cylindrical part (l

_{5}= 50 mm) when air is supplied through various tube configurations. The data are selected so that the average speed in the first section is approximately the same for all the data presented and is about 4.5 m/s.

_{x}= f (τ) is white noise for all supply tube configurations. At the same time, visual observations show that the use of profiled tubes leads to the creation of small fluctuations in the w

_{x}= f (τ) function. This can also be confirmed by the calculated data: TI for round tubes is 0.231 and 0.285 for triangular tubes (almost a 20% difference). This is due to the fact that stable and vortex structures are created in the corners of the profiled tubes, which turbulise the flow in the diffuser. Similar gas-dynamic effects were discovered in [28,29]. It is inappropriate to describe the presented dependencies mathematically, as those data are special cases.

_{x}= f (τ) function, but for the sensor in the third section (H

_{3}= 300 mm) in the centre of the diffuser’s cylindrical part (l

_{5}= 50 mm) when air is supplied through tube configurations. Based on a comparison of the functions w

_{x}= f (τ) in Figure 3 and Figure 4, it is found that there is a significant decrease in small fluctuations in velocity upstream. This indicates a gradual relaxation of the flow along the height of the diffuser’s cylindrical part.

- -
- No clear patterns in the change in TI along the diameter of the diffuser’s cylindrical part were found;
- -
- The influence of the tubes’ cross-sectional shape on the function TI = f (D) has not been established (according to the author, the changes are random);
- -
- The amplitude of the fluctuations of turbulence intensity values relative to the average value is ±35%;
- -
- The mathematical formulation of the function TI = f (D) for the cases under study is inappropriate until the physical laws are established.

^{–b·H}. At the same time, the shape of the tube cross section does not actually affect the intensity of the drop in the air flow velocity along the set-up’s height.

_{1}≈ 6.5 m/s, is described by the following equation (approximation reliability 0.85):

_{1}≈ 6.5 m/s, approximation reliability 0.92):

_{1}≈ 8.25 m/s, is described by the following equation (approximation reliability 0.83):

_{1}≈ 8.25 m/s, approximation reliability 0.86):

_{1}= 100 mm) when air is supplied through round (5), square (6), and triangular (7) tubes has the following form (approximation reliability not less than 0.89):

_{4}= 800 mm) when air is supplied through round (8), square (9), and triangular (10) tubes have the forms (approximation reliability not less than 0.92):

- -
- The tubes’ cross-sectional shape has a significant effect on turbulence intensity along the height of the vertical diffuser’s cylindrical part;
- -
- The dependency TI = f (H) has a pronounced maximum in the region of H = 300 mm, which is typical of all initial average flow rates and all configurations of the supply tubes;
- -
- The turbulence intensity has significantly higher values (up to 50%) when air is supplied through profiled tubes compared to round tubes, which is especially typical of a low initial w.

## 4. Conclusions

- Experimental data on the instantaneous values of the stationary flow velocity along the height and diameter of the diffuser’s cylindrical part for various initial conditions and when air is supplied through tubes with different configurations are obtained.
- The velocity fields along the height of the diffuser’s cylindrical part for various initial conditions when air is supplied through tubes with cross sections in the form of a circle, a square, and a triangle are determined. The evolution of the velocity fields in the upward flow direction is shown.
- The flow’s turbulence intensity TI along the height and diameter of the diffuser is calculated for various initial conditions and when air is supplied through tubes with various configurations.
- A mathematical description (exponential equations) of the change in the average flow velocity along the height of the diffuser’s cylindrical part for various initial conditions and configurations of the supply tubes is presented.
- The regularities of changes in the intensity of turbulence along the height of the diffuser for various initial conditions and tube configurations are established.
- The obtained data on the aerodynamic characteristics of flows in a conical diffuser can be useful for refining and verifying mathematical models and improving engineering calculations.
- Further research could be conducted to obtain dimensionless equations for describing the aerodynamic characteristics of flows in vertical diffusers of various designs, as well as to refine mathematical models for modelling aerodynamics.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

w_{x} | local air flow velocity, m/s |

w | average flow velocity, m/s |

τ | time, s |

p_{o} | barometric pressure, kPa |

t | temperature, °C |

(d) | diameter, mm |

l | linear dimension, mm |

H | height, mm |

Re | Reynolds number |

TI | turbulence intensity |

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**Figure 1.**Size dimensions (

**a**) and photograph (

**b**) of the experimental set-up for studying the aerodynamic characteristics of flows in a vertical diffuser: 1—cylindrical part of the set-up; 2—hot-wire anemometer sensor; 3—diffuser; 4—base with supply tubes; 5—hot-wire anemometer for measuring instantaneous values of flow velocity; 6—analogue-to-digital converter; 7—laptop for collecting and processing experimental findings; H, H

_{1}…H

_{4}—height of the cylindrical part in the diffuser; l

_{1}…l

_{4}—linear dimension along the diameter of the cylindrical part of the diffuser.

**Figure 2.**Plan view and key dimensions of the tubes for supplying air to the vertical diffuser: (

**a**)—circular tube; (

**b**)—tube with square cross section; (

**c**)—tube with triangular cross section.

**Figure 3.**Dependences of local values of air flow velocity w

_{x}on time τ in the first control section (H

_{1}= 100 mm) for a sensor in the centre of the installation (l

_{5}= 50 mm) for supply tube configurations: (

**a**)—round tube (average parameters in cross section: w

_{1}= 4.2 m/s, TI = 0.231); (

**b**)—square tube (w

_{1}= 4.45 m/s, TI = 0.18); (

**c**)—triangular tube (w

_{1}= 4.48 m/s, TI = 0.285).

**Figure 4.**Dependences of local values of air flow velocity w

_{x}on time τ in the third control section (H

_{3}= 500 mm) for a sensor in the centre of the set-up (l

_{5}= 50 mm) for various supply tube configurations: (

**a**)—round tube (average parameters in the section: w

_{3}= 4.81 m/s, TI = 0.124); (

**b**)—square tube (w

_{3}= 4.85 m/s, TI = 0.313); (

**c**)—triangular tube (w

_{3}= 4.79 m/s, TI = 0.199).

**Figure 5.**Change in local values of the flow velocity w

_{x}along the cylinder diameter D when air is supplied to the diffuser through round tubes for the initial average flow velocity in the first section w

_{1}≈ 6.31 m/s along the set-up height: (

**a**)—H

_{4}= 800 mm; (

**b**)—H

_{3}= 500 mm; (

**c**)—H

_{2}= 300 mm; (

**d**)—H

_{1}= 100 mm.

**Figure 6.**Change in local values of the flow velocity w

_{x}along the cylinder diameter D when air is supplied to the diffuser through square tubes for the initial average flow velocity in the first section w

_{1}≈ 6.45 m/s along the set-up height: (

**a**)—H

_{4}= 800 mm; (

**b**)—H

_{3}= 500 mm; (

**c**)—H

_{2}= 300 mm; (

**d**)—H

_{1}= 100 mm.

**Figure 7.**Change in local values of the flow velocity w

_{x}along the cylinder diameter D when air is supplied to the diffuser through triangular tubes for the initial average flow velocity in the first section w

_{1}≈ 6.65 m/s along the set-up height: (

**a**)—H

_{4}= 800 mm; (

**b**)—H

_{3}= 500 mm; (

**c**)—H

_{2}= 300 mm; (

**d**)—H

_{1}= 100 mm.

**Figure 8.**Change in turbulence intensity TI (in the first section H

_{1}= 100 mm) along the cylinder diameter D when air is supplied to the diffuser through tubes of various configurations: (

**a**)—round tube (w

_{1}≈ 6.65 m/s, TI

_{1}= 0.213); (

**b**)—square tube (w

_{1}≈ 6.45 m/s, TI

_{1}= 0.180); (

**c**)—triangular tube (w

_{1}≈ 6.65 m/s, TI

_{1}= 0.245).

**Figure 9.**Dependences of the average air flow rate w along the installation height H when air is supplied through round (1), square (2), and triangular (3) tubes for the initial average flow rate in the first section: (

**a**)—w

_{1}≈ 6.5 m/s; (

**b**)—w

_{1}≈ 8.25 m/s.

**Figure 10.**Dependences of the turbulence intensity TI on the average air flow velocity w when air is supplied through round (1), square (2), and triangular (3) tubes for various control sections: (

**a**)—control section at the height of H

_{1}= 100 mm; (

**b**)—H

_{4}= 800 mm.

**Figure 11.**Dependences of the intensity of turbulence TI on the height of the diffuser H when air is supplied through round (1), square (2), and triangular (3) tubes for the initial average flow velocity in the first section: (

**a**)—w

_{1}≈ 6.5 m/s; (

**b**)—w

_{1}≈ 8.25 m/s.

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**MDPI and ACS Style**

Plotnikov, L.
Mathematical Description of the Aerodynamic Characteristics of Stationary Flows in a Vertical Conical Diffuser When Air Is Supplied through Various Tube Configurations. *Axioms* **2023**, *12*, 244.
https://doi.org/10.3390/axioms12030244

**AMA Style**

Plotnikov L.
Mathematical Description of the Aerodynamic Characteristics of Stationary Flows in a Vertical Conical Diffuser When Air Is Supplied through Various Tube Configurations. *Axioms*. 2023; 12(3):244.
https://doi.org/10.3390/axioms12030244

**Chicago/Turabian Style**

Plotnikov, Leonid.
2023. "Mathematical Description of the Aerodynamic Characteristics of Stationary Flows in a Vertical Conical Diffuser When Air Is Supplied through Various Tube Configurations" *Axioms* 12, no. 3: 244.
https://doi.org/10.3390/axioms12030244