Lab Report: Inviscid Flow Experiments

Categories: Physics

Report, Pages 8 (1990 words)

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Abstract

In fluid dynamics, inviscid flow refers to a flow of a fluid with no viscosity. This report presents the results of two experiments aimed at studying the force of a jet impinging on a fixed plane and the development of a round air jet downstream of a nozzle. The theoretical foundations of these experiments are rooted in Newton's second law of motion and the momentum equation. Mass flow rate and theoretical force calculations were performed based on these principles. The experimental results, presented graphically, exhibited the expected behavior of the air jet.

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The experiments successfully demonstrated the characteristics of the air jet, but some errors in the results were observed, primarily related to measurements using a ruler for plate height and manometer readings. Despite these limitations, the tests provided valuable insights into the fundamental structure of an air jet and the concept of internal structures within it. The inaccuracies were attributed to both systematic and human factors.

Introduction

The objective of this laboratory report is to gain insight into a simple inviscid flow problem in fluid dynamics.

Simple inviscid flow, though not a realistic representation of real-world scenarios, serves as a valuable tool for studying fluid flow in conjunction with Bernoulli's and Euler's equations (R. Byron Bird, Stewart, and Lightfoot, 2007). The laboratory experiments focused on understanding the force exerted by a jet on a fixed plane and the development of a round air jet downstream of a nozzle. The theoretical basis for these experiments relies on Newton's second law, the momentum equation, and Bernoulli's principles.

Experimental Setup and Theory

Two experiments were conducted. The first experiment investigated the force acting on a vertical plate positioned above the jet axis at various heights. This experiment was repeated at different radial positions: 8 cm, 0 cm, and -8 cm. The second experiment examined the effects of a round air jet downstream of a nozzle. In this case, the pitot tube probe's radial distance above the jet varied from 8 cm to -8 cm, with different vertical heights for each radial distance.

The theoretical foundation for these experiments can be elucidated using Newton's Second Law of motion and the Momentum equation. Newton's second law of motion states that when fluid is ejected in the form of a jet from a nozzle outlet, and a plate is fixed above the jet path, the jet exerts a force on the plate. This law asserts that the external force applied to the fluid is equal to the rate of change of momentum, which can be expressed as:

F_vector = (dP_vector) / dt (1.1)

Where F_vector represents the force vector, and P_vector is the momentum vector. The momentum vector is derived using the momentum equation, defined as:

P_vector = mv_vector (2)

Here, 'm' denotes mass, and v_vector represents velocity. Substituting equation (2) into equation (1.1), we obtain:

F_vector = d(mv_vector) / dt (1.2)

Utilizing the chain rule of differentiation and assuming steady flow with no acceleration, equation (1.2) can be further simplified to:

F_vector = ṁv_vector (1.3)

Methodology

The apparatus used in both experiments included the following components: an air jet apparatus comprising a pressure chamber, a plate, an orifice plate, and a nozzle; manometers; a spring gauge; and a pitot tube. Prior to commencing the experiment, the fluid level in an inclined manometer was noted. In the first experiment, the fluid level in the manometer remained constant at 28.5 mm throughout the entire experiment.

Procedure for Experiment 1:

Place the plate at an appropriate height above the nozzle and record the initial reading on the spring gauge. This reading accounts for the weight of the plate. The initial height used in this laboratory was 95 cm.
Switch on the fan and measure the force on the spring gauge. Ensure that the reading is taken at eye level with the spring gauge.
Lower the plate to a new position and record the force again. In this report, the height decreases in 10 cm intervals until reaching 25 cm, followed by 5 cm intervals until 5 cm, and the last reading is taken at 3 cm.
Repeat step 3, moving the plate progressively closer to the jet axis, and record the observed phenomena.
Finally, repeat steps 1-4 at different radial positions: 8 cm, 0 cm, and -8 cm.

At the end of the experiment, a table with four columns, as depicted in Table 1, is generated. The first column represents the plate height (cm), while the other three columns display the force on the plate (N) for the three radial distances. To convert the force readings from kilograms (kg) to Newtons (N) and account for the initial plate weight, the following formula is applied:

F = (m₀ - m) * g (8)

Where F represents the force in Newtons, m₀ is the weight of the plate in kilograms (1.2 kg), m is the experimental reading in kilograms, and g is the acceleration due to gravity (9.81 m/s²).

This table is subsequently used to create a graph of 'force on the plate' against 'plate height' with three lines in Excel, as shown in Fig. 1.

Procedure for Experiment 2:

Remove the plate used in the previous experiment and replace it with the Pitot tube.
Position the Pitot tube at the nozzle exit and record the pressure on the manometer.
Move the Pitot tube to a new (higher) position and record the fluid level in the manometer.
Shift the Pitot tube radially and position it at different distances from the jet axis. Record the fluid level in the manometer.
Repeat step 3 for approximately 7 to 8 vertical positions.

At the conclusion of the experiment, a table with 14 columns, as depicted in Table 2, is generated. The first column represents the Pitot tube radial position, while the other 13 columns display the fluid level in the manometer for various heights. This table is used to create a graph of 'fluid level in the manometer' against 'Pitot tube radial location' with 13 lines in Excel, as illustrated in Fig. 2.

To calculate the mass flow rate, the following equations are employed:

Area A = πr² (9)

Pressure (p_c - p_a) = ρ_{liquid in manometer} * g * L * sin(15) (10)

Velocity at the orifice (v_d) = √(2(p_c - p_a) * ρ_air) (11)

Exit velocity (v_vector) = A_d * v_d / A_e (12)

Mass flow rate (ṁ) = C_D * ρ_air * v_vector * A_e (13)

Where:

ρ_{liquid in manometer} is the density of the liquid in the manometer (1000 kg/m³).
g is the acceleration due to gravity (9.81 m/s²).
L is the inclined manometer reading (28.5 mm).
A_d and A_e are the areas of the orifice plate and the nozzle, respectively (3.85e-3 m² and 4.91e-4 m²).
C_D is the discharge coefficient (0.88).
ρ_air is the density of air (1.225 kg/m³).
v_vector is the exit velocity (75.45 m/s).
v_d is the velocity at the orifice (9.62 m/s).

Once the mass flow rate is calculated (0.0399 kg/s), the theoretical force of the jet on the plate can be determined using equation 1.3, resulting in a force value of 3.013 N.

Results and Discussion

Experiment 1 Results:

The results of the first experiment are graphically presented below:

Table 1. Summary of Experiment 1 Results
Plate Height (cm)	Force at Radial Position 8cm (N)	Force at Radial Position 0cm (N)	Force at Radial Position -8cm (N)
Plate Close to Nozzle	Force Increases	Maximum Force	Force Increases
10 cm - Above	Negative Force Readings	Negative Force Readings	Negative Force Readings

The table illustrates an expected behavior. As the plate moves closer to the air jet nozzle, the force exerted on the plate increases due to the distribution of the air. Notably, the force values for radial positions of 8 cm and -8 cm are nearly identical, with overlapping lines. Conversely, when the plate is positioned at radial position 0 cm, it exerts the maximum force when directly above the air jet nozzle.

At heights above 10 cm, an interesting phenomenon is observed. The plate begins to be pulled toward the nozzle rather than being pushed upward, resulting in negative force readings. This behavior is attributed to the formation of vortex rings. Vortex rings form due to the mutual transfer of momentum between the air jet and the surrounding air. The surrounding air entrained in the air jet acquires momentum and flows forward with the jet, creating a velocity gradient due to the momentum difference between the two (Chang et al., 2020) (Muppidi and Mahesh, 2005).

In theory, the air jet should maintain a uniform shape and velocity along its path, resulting in a straight-line graph at a force of 3.013 N. However, in reality, the air jet leaves the nozzle, travels in a straight line for a short distance, and then transitions into a free jet region where the air disperses outward from the main path, forming a cone-like shape.

Experiment 2 Results:

The results of the second experiment are presented below:

Table 2. Summary of Experiment 2 Results
Pitot Tube Radial Position (cm)	Fluid Level Change at Far Radial Position (mm)	Fluid Level Change at Radial Position 0cm (mm)	Fluid Level Change at Near Radial Position (mm)
Far from Jet	No Significant Change	Fluid Level Increases	Fluid Level Increases
Near 0 cm Radial Position	Fluid Level Increases	Fluid Level Increases (Symmetrical)	Fluid Level Increases
Lower Pitot Tube Height	Fluid Level Increases	Fluid Level Increases	Fluid Level Increases
Higher Pitot Tube Height	Weak Force	Weak Force	Weak Force
Outliers	Point (0, 22) at 35 cm Height	Point (0, 22) at 35 cm Height (Repeat)	Repeat Readings at 45 cm Height

The table illustrates an expected behavior. When the Pitot tube is farther away from the air jet, there is no significant change in the fluid level in the manometer. However, as the radial location approaches 0 cm, the fluid level increases. Additionally, the table exhibits symmetry when considering the axis of the 0 cm radial position. The height of the Pitot tube also influences the results; as the height decreases, the fluid level increases. At higher heights, between 95 cm and around 65 cm, the fluid level does not increase significantly due to the outward distribution of air, resulting in a weaker force.

Some outliers are observed in the results, notably the point (0, 22) for a height of 35 cm, which is lower than expected. This reading should be repeated to achieve a more accurate curve. Similarly, the readings for a height of 45 cm should be repeated to improve the curve.

Error Analysis and Uncertainties:

These experiments successfully demonstrated the characteristics of an air jet, but several errors influenced the results. The primary source of error was the use of a ruler to measure the plate's height and the fluid level along the manometer. This introduced both human error, due to estimating the 'zero position' and the end location of the fluid meniscus, and systematic error, caused by the ruler's ±1 mm uncertainty, resulting in inaccurate measurements. Additionally, human error affected the placement of the pitot tube's radial positions.

Prior to the experiment, the presence of air bubbles in the manometer fluid introduced systematic inaccuracy, affecting every manometer distance reading. These air bubbles may have caused the findings to consistently read higher than expected. Random error was associated with the shifting location of the static tube.

Conclusions

In conclusion, the experiments conducted in this laboratory report effectively showcased the behavior of an air jet when interacting with a plate and a Pitot tube. They also demonstrated how an air jet interacts with its surrounding fluids. While the results provided valuable insights into the fundamental structure of an air jet, minor variations between the measurements may be attributed to inaccuracies in measuring the fluid meniscus location in the manometer using a ruler, potentially resulting in incorrect results.

Despite the presence of inaccuracies stemming from both equipment limitations and human factors, the experiments were still valuable in highlighting the interactions of an air jet with its surroundings and introducing the concept of internal structures within an air jet.