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Today, we'll discuss the Froude number and its relationship with solitary waves. Can anyone explain what the Froude number is?
Is it a ratio of the flow's inertia to its gravitational forces?
Exactly! The Froude number helps us understand flow behavior in open channels. It’s defined as Fr = V/c, where V is the velocity of the fluid, and c is the wave speed.
What happens if the Froude number is less than one?
Great question! If Fr is less than one, we have subcritical flow, which means the wave can travel upstream.
Let’s calculate wave speed using our earlier derived equation, c = √(gy). If the depth y is 2 meters, what will the speed c be?
Using g = 9.81 m/s², c will equal √(9.81 * 2). That's about 4.43 m/s.
Correct! Now, if the fluid velocity V is 3 m/s, how do we find the Froude number?
Fr = V/c = 3 / 4.43, which is roughly 0.678.
Right, and since Fr < 1, we can confirm it’s subcritical flow. Well done!
What happens when the fluid speed V exceeds the wave speed c?
The flow would be supercritical, and waves wouldn’t travel upstream.
Exactly! If V = c, what do we observe?
The waves become stationary.
Correct! These relationships are crucial for predicting fluid behavior in engineering applications.
Now, let’s talk about how finite amplitude waves differ from small amplitude waves. How does this impact wave speed?
For larger amplitude waves, the speed increases compared to small amplitude waves.
Exactly! For finite amplitude waves, the wave speed formula updates to incorporate amplitude. Remember that!
So, a higher amplitude means a faster wave speed?
Yes, that’s correct! This change indicates different dynamics at play in larger wave scenarios.
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In this section, we delve into the interaction between wave speed, fluid velocity, and water depth through the concept of the Froude number. We also investigate how these relationships determine whether a flow is subcritical or supercritical and the implications for wave propagation in open channels.
The Froude number (Fr) is a dimensionless parameter that plays a crucial role in understanding fluid flow and wave behavior in open channels. It is defined as the ratio of the inertia of the flow to the gravitational forces acting on it.
In solitary waves, the speed of a small amplitude wave (c) can be expressed as:
Where g is the acceleration due to gravity and y is the depth of water. This indicates that the wave speed is dependent on water depth but is independent of wave amplitude.
By relating flow velocity (V) to wave speed (c), we can define the Froude number for solitary waves as:
Depending on the relationship between V and c, we can categorize flows into:
- Subcritical Flow (Fr < 1): The wave’s speed is greater than the fluid's velocity, allowing waves to travel upstream.
- Supercritical Flow (Fr > 1): The fluid's velocity exceeds the wave speed, preventing waves from traveling upstream.
If V equals c, stationary waves are formed. The section also touches on the behavior of finite amplitude waves, indicating that larger amplitudes lead to increased wave speeds compared to small amplitude waves.
Understanding these dynamics helps civil engineers design effective hydraulic structures and manage water resources more efficiently.
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So, actually, I have written down the reason, it was not supposed to, but it is okay. We will I will tell you; the reason is the wave motion is a balance between the inertial effects, which is proportional to rho and also the hydrostatic pressure effect, which is proportional to rho g.
The Froude number is a dimensionless number that is used to compare inertial forces to gravitational forces in fluid dynamics. It is defined as V/sqrt(g*y), where V is fluid velocity, g is gravity, and y is the flow depth. It helps us understand whether the flow is subcritical (Froude number < 1) or supercritical (Froude number > 1). In terms of wave motion, the relationship indicates that inertial forces (which depend on density) and hydrostatic pressure forces are in balance.
You can think of the Froude number like a seesaw at a playground. One side represents the weight of gravitational forces (like the kids sitting on one end), while the other side represents inertial forces (like the kids on the other end). For balance, the see-saw needs to have the right ratio of weight on each side, similar to how fluid motion is balanced between these forces.
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So, the Froude number for solitary wave is V / c, where c is the speed of the wave and V is the velocity of the fluid with the water with the velocity of the fluid, which will be definitely different from the waves speed.
The Froude number for solitary waves is calculated by dividing the velocity of the fluid (V) by the wave speed (c). This ratio helps predict how the wave will behave in relation to the flow of the fluid. If the wave speed is greater than the fluid speed (c > V), the waves can travel upstream, indicating a subcritical flow. Conversely, if the fluid speed exceeds the wave speed (V > c), the waves cannot travel upstream, indicating a supercritical flow.
Imagine a boat on a river. If the current of the river (V) is slower than the speed of the boat (c), the boat can sail upstream against the current. However, if the river current is faster than the boat, it’s swept downstream regardless of the boat’s ability. Similarly, the Froude number tells us about the relative strengths of waves and the fluid currents.
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If c is greater than V, so what is going to be the Froude number, less than 1, so that means subcritical flow, very simple to see. If c is less than V, so that means V is greater than c waves do not travel upstream and because of this definition V / c, because V is greater than c Froude number is greater than one and this falls into regime of supercritical flow.
In fluid dynamics, flow can be classified into subcritical and supercritical based on the Froude number. A Froude number less than 1 indicates subcritical flow, where the wave speed (c) is greater than the fluid speed (V), allowing waves to propagate upstream. When the Froude number is greater than 1, it indicates supercritical flow, where the fluid speed exceeds the wave speed, and waves cannot travel upstream. This classification is crucial for understanding wave behavior in channels.
Consider a water slide at a park. When the slide is steep and fast (supercritical flow), if you try to roll back up by swimming, you won’t succeed because the slope is too steep (fluid speed greater than wave speed). But on a gentle slope (subcritical flow), you have a chance to paddle back upstream since the slide isn't pushing you down as fast.
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But if the waves are of finite size that means, they are no longer small amplitude, no longer small amplitude, then the wave speed actually is given by this, you can try to remember this formula, but, the derivation is not in the scope of this course. So, c is given by under the root of g y multiplied by one + delta y / y to the power half. And this is equation number 5.
For small amplitude waves, the speed is independent of amplitude. However, when the waves become of finite size, the speed depends on their amplitude. The new formula indicates that as the amplitude increases, the wave speed also increases. This adjustment factor (delta y / y) accounts for the effect of wave amplitude on speed, showing distinctive behavior as the wave scales change.
Think of a child on a swing. When the swing is pushed gently (small amplitude), it swings at a certain speed. But if the swing is pushed with more energy (larger amplitude), it swings faster. Similarly, for waves, smaller amplitudes result in consistent speed, but larger amplitudes lead to faster wave propagation.
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
Froude Number: A dimensionless parameter used to assess flow behavior.
Subcritical Flow: Flow condition characterized by wave speed greater than fluid velocity.
Supercritical Flow: Flow condition characterized by fluid velocity exceeding wave speed.
Wave Speed (c): Speed of solitary waves determined by water depth.
Amplitude Effect: Larger wave amplitudes result in increased wave speeds.
See how the concepts apply in real-world scenarios to understand their practical implications.
In a channel flowing with a speed of 3 m/s and a depth of 2 m, the wave speed calculated using c = √(gy) yields approximately 4.43 m/s, illustrating a subcritical condition as Froude < 1.
If the fluid speed is 5 m/s and wave speed is 4 m/s, the flow is supercritical (Fr > 1), indicating the waves cannot propagate upstream.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
Froude's flow can show you right, how waves move in day and night.
Imagine a river where waves try to dance upstream, but if the water flows faster than the waves, they just cannot keep their dream.
Use 'Subcritical' for 'Slower Underwater and Better Waves' to recall that they can travel upstream.
Review key concepts with flashcards.
Review the Definitions for terms.
Term: Froude Number
Definition:
A dimensionless number defining the ratio of inertial forces to gravitational forces in fluid flow.
Term: Subcritical Flow
Definition:
A flow condition where the wave speed exceeds the fluid velocity (Fr < 1).
Term: Supercritical Flow
Definition:
A flow condition where the fluid velocity exceeds the wave speed (Fr > 1).
Term: Solitary Wave
Definition:
A wave packet that retains its shape while traveling at a constant speed.
Term: Small Amplitude Wave
Definition:
Waves whose amplitudes are small enough to maintain linear behavior.
Term: Finite Amplitude Wave
Definition:
Waves where the amplitude is significant enough to deviate from linear behavior.