Fluid Mechanics: Topic 10.1 - Lagrangian vs Eulerian descriptions of flow

Summary

This lesson distinguishes between Lagrangian and Eulerian approaches to fluid flow analysis. Lagrangian follows individual fluid elements over time, using dynamics' F=ma, but it's complex for large systems due to fluid element interactions. Eulerian observes fluid properties at fixed points, simplifying experimental observation. It employs scalar and vector fields, with weather forecasts as a scalar example. Both viewpoints have their applications, with Eulerian being more practical for stationary observations.

Takeaways

• 🔍 The lesson differentiates between Lagrangian and Eulerian descriptions of fluid flow, which are two distinct approaches to analyzing fluid dynamics.
• 🎯 In the Eulerian description, which is used for analyzing device performance with control volumes, only the flow details along the control volume surface are necessary, not the internal details.
• 🔬 Situations requiring knowledge of the entire flow at every point include drag on a racecar, mixing air and fuel in an engine cylinder, and ensuring medication delivery in the lungs.
• 🌊 A fluid element is a small, yet significant, portion of a fluid that contains many molecules but is invisible to the naked eye.
• 🕒 The Lagrangian viewpoint tracks individual fluid elements over time, using the fundamental equation of motion F = ma.
• 📍 The Eulerian viewpoint observes fluid elements at fixed points in space, which is more practical for experiments and aligns with how weather forecasts are made.
• 🌐 Eulerian description uses scalar and vector fields to describe fluid properties, such as temperature (a scalar field) and wind velocity (a vector field).
• 📊 Weather forecasts, including temperature and wind, are reported using the Eulerian viewpoint, showing how properties vary across space at a given time.
• 🔄 Fluid elements can deform, mix, and split, making the Lagrangian approach complex for tracking all elements over a large area like California.
• ⏱️ The Eulerian viewpoint treats space coordinates (x, y, z) and time (t) as independent, contrasting with the Lagrangian viewpoint where an element's position is a function of time.

Q & A

What is the main difference between the Lagrangian and Eulerian descriptions of fluid flow?

The Lagrangian description tracks individual fluid elements over time, while the Eulerian description observes fluid properties at fixed points in space.

When is it necessary to know the flow details at every point in a region?

It is necessary in situations like calculating drag on a racecar, enhancing air and fuel mixing inside an engine cylinder, or ensuring medication reaches deep into the lungs.

What is a fluid element, and how small is it typically considered?

A fluid element is a very small parcel of fluid, smaller than what can be seen with the naked eye but still made up of many molecules. There's no official size for a fluid element.

Why is it challenging to track all fluid elements using the Lagrangian viewpoint?

Tracking all fluid elements is challenging because fluid elements deform, mix, and split as they move through space, making it complex to follow every element.

What is one of the main benefits of using the Eulerian viewpoint in fluid flow analysis?

The Eulerian viewpoint allows for easier experimentation by observing specific points in space, rather than tracking numerous particles moving through space.

How does the Eulerian description use scalar and vector fields to describe fluid properties?

The Eulerian description uses scalar fields, like temperature, and vector fields, like wind velocity, to describe how fluid properties vary with position and time.

What is an example of a scalar field in the Eulerian viewpoint?

Temperature is an example of a scalar field, where it varies across different locations but is not tied to individual fluid particles.

What is an example of a vector field in the Eulerian viewpoint?

Wind velocity is an example of a vector field, with components u, v, and w, varying based on position and time.

How are coordinates described in the Lagrangian and Eulerian viewpoints?

In the Lagrangian viewpoint, coordinates (xA, yA, zA) are functions of time as they follow the fluid element. In the Eulerian viewpoint, coordinates (x, y, z) are independent of time, describing fixed points in space.

How does the velocity of a fluid element differ between the Lagrangian and Eulerian viewpoints?

In the Lagrangian viewpoint, the velocity is a function of time for a fluid element. In the Eulerian viewpoint, velocity is a function of both position and time at a fixed point in space.

Outlines

00:00

🌊 Introduction to Lagrangian and Eulerian Descriptions of Fluid Flow

This paragraph introduces the concepts of Lagrangian and Eulerian descriptions in fluid dynamics. The Lagrangian description involves tracking individual fluid elements over time, using the fundamental equation of motion F = ma. However, it's challenging to apply this method to complex fluid systems due to the potential for fluid elements to deform, mix, and split. The Eulerian description, on the other hand, observes fluid properties at fixed points in space, which is more practical for conducting experiments and is used in weather forecasting. This viewpoint employs scalar and vector fields to describe fluid properties, such as temperature and wind velocity, which are reported for specific locations at a given time. The paragraph also explains the difference in how coordinates are treated in both viewpoints, with the Lagrangian description focusing on the position of a fluid element as a function of time, while the Eulerian description treats space and time coordinates as independent.

Keywords

💡Lagrangian description

The Lagrangian description is a method of analyzing fluid flow by tracking individual fluid elements over time. It is named after the mathematician Joseph-Louis Lagrange. In the context of the video, this approach is used to understand how each small parcel of fluid moves and changes. The script mentions that in the Lagrangian viewpoint, the equations of motion for fluid elements are governed by the fundamental principle of dynamics, F = ma, which simplifies the analysis to the motion of these individual elements.

💡Eulerian description

The Eulerian description, named after the mathematician Leonhard Euler, is an alternative approach to studying fluid flow where the focus is on observing the fluid properties at fixed points in space rather than following individual fluid elements. The video script explains that this viewpoint is often more convenient for experiments and is used in weather forecasting, where temperature and wind velocity are reported at various locations at specific times.

💡Control volume

A control volume is a region in space through which fluid flows and is used in the analysis of fluid dynamics problems. The script mentions that when analyzing the performance of devices using control volumes, only the flow details along the surface of the control volume are necessary, not the details inside it. This concept is crucial for understanding how fluid dynamics can be applied to real-world engineering problems.

💡Fluid element

A fluid element refers to a very small parcel of fluid that is still made up of many molecules. The script describes it as a chunk of fluid that is smaller than what the eye can see. This concept is essential in both Lagrangian and Eulerian descriptions, as it helps to conceptualize the fluid's behavior at a microscopic level within the larger context of fluid flow.

💡Scalar field

A scalar field is a type of field in physics where each point in space is associated with a single value, such as temperature or pressure. The video script uses the example of temperature distribution across California to illustrate how scalar fields can be used in the Eulerian description of fluid flow, where temperature is reported for many locations at a specific time.

💡Vector field

A vector field is a field in physics where each point in space is associated with a vector quantity, which has both magnitude and direction. The script mentions wind velocities across California as an example of a vector field, where the wind's components can vary with position and time, and are reported for various locations.

💡Deformation

Deformation in fluid dynamics refers to the change in shape or volume of a fluid element as it moves. The script notes that fluid elements can deform, mix with other fluid elements, and split into new fluid elements as they move through space, which complicates the analysis when using the Lagrangian description.

💡Mixing

Mixing in the context of fluid dynamics is the process by which different fluid elements combine and distribute their properties throughout a volume. The script gives an example of enhancing the mixing of air and fuel inside an engine cylinder prior to combustion, which is crucial for efficient combustion and is better analyzed using the Eulerian description.

💡Inhaler medication

The script uses the example of inhaler medication to illustrate the importance of understanding fluid flow in ensuring that medication penetrates deeply into the lungs. This application highlights the practical relevance of fluid dynamics in the medical field and the need for detailed knowledge of fluid behavior at every point in the region of interest.

💡Observation stations

Observation stations are fixed points in space where measurements are taken to analyze fluid properties. The script suggests building multiple observation stations in California to record the properties of fluid elements as they travel past, which is a practical application of the Eulerian viewpoint in meteorology and environmental science.

Highlights

Discussing the difference between Lagrangian and Eulerian descriptions of fluid flow.

In control volume analysis, only flow details along the surface are needed, not inside the volume.

Situations requiring knowledge of the entire flow at every point include drag on a racecar and mixing air and fuel in an engine cylinder.

A fluid element is a small, invisible chunk of fluid made up of many molecules.

The Lagrangian viewpoint tracks individual fluid elements over time using F = ma.

The Eulerian viewpoint observes fluid elements at fixed points in space, making experiments more convenient.

Eulerian description uses scalar and vector fields to describe fluid properties, unlike Lagrangian's focus on individual elements.

Weather forecasts use the Eulerian viewpoint, reporting temperature and wind as scalar and vector fields respectively.

In the Eulerian viewpoint, coordinates x, y, z, and t are independent, contrasting with Lagrangian's time-dependent position function.

Velocity of a fluid element (VA) from the Lagrangian viewpoint is the same as the velocity (V) seen by an Eulerian observer at a specific moment.

After time passes, the fluid element's Lagrangian coordinates no longer coincide with the Eulerian position, and velocities may change.

Fluid elements can deform, mix, and split, complicating the Lagrangian tracking of all elements over a region.

Observation stations in the Eulerian method record properties of fluid elements as they pass, unlike tracking individual particles in the Lagrangian method.

The Eulerian equations of motion are more complex than F = ma and will be developed in future videos.

Temperature as a scalar field and wind as a vector field are examples of how Eulerian viewpoint describes fluid properties.

The Eulerian viewpoint is beneficial for practical applications like weather forecasting, where tracking individual fluid particles is impractical.

Transcripts

In this lesson, we will discuss the difference between the Lagrangian and Eulerian descriptions

of fluid flow.

When analyzing the performance of devices using control volumes, we only need to know

the flow details along the control volume surface and are not required to know the details

of what happens inside the control volume.

However, there are many situations where knowing the details of the entire flow at every point

in the region of interest is necessary.

Some examples include, trying to drag on a racecar, trying to enhance the mixing of air

and fuel inside of an engine cylinder prior to combustion, and trying to ensure inhaler

medication penetrates deeply into the lungs.

A large body of fluid consists of many fluid elements.

A fluid element as a very small parcel of fluid.

Although there is no official size of a fluid element, think of it as a chunk of fluid that

is smaller than your eyes can see, but is still made up of many molecules.

There are two points of view from which we can examine the motion of fluid elements.

The first viewpoint is called Lagrangian, in which each individual fluid element is

tracked in time.

Here we have four fluid elements off the west coast of California.

In the Lagrangian viewpoint, the equations of motion of these four fluid elements are

well-known from dynamics, simply F = ma.

However, it would be too difficult to try to keep track of all the fluid elements that

would be present over California at a given time.

To make matters more complicated, the fluid elements can deform, mix with other fluid

elements, and split into new fluid elements as they move through space.

The second viewpoint is called Eulerian, in which fluid elements are observed at fixed

points in space.

For example, we could build multiple observation stations in California and record the properties

of the fluid elements as they travel past the stations.

At the moment, we do not know the equations of motion at every point in space from the

Eulerian point of view.

We will develop these equations in future videos, and they will turn out to be more

complicated than F = ma.

One of the main benefits of using the Eulerview viewpoint is that it is usually much more

convenient to conduct experiments by observing specific points of space compared to tracking

numerous particles as they flow through space, which is the Lagrangian viewpoint.

The Eulerian description of flow uses scalar fields and vector fields to describe fluid

properties.

Weather forecasts are reported using the Eulerian viewpoint.

For example, on the right is temperature distribution in Fahrenheit across California at a specific

time.

Temperature is a scalar quantity and it is reported for many locations, not for individual

fluid particles which move from location to location.

So temperature can be thought of as a scalar field that varies with both position and time.

On the left we see wind velocities across California at a specific time.

Wind is reported for various locations, not for various fluid particles, which move through

space.

Wind can be thought of as a vector field and its components, lowercase u, lowercase v,

and lowercase w, all can vary with position and time.

Notice that in the Eulerian point of view, the coordinates x, y, z, and t are all independent

of each other.

However, in the Lagrangian point of view, a fluid element’s position is described

as function of time.

Here we see a fluid element labeled A in a Cartesian coordinate system.

From the Lagrangian viewpoint, the fluid element’s current location is (xA, yA, zA).

Since the fluid element’s location can change in time, xA, yA, and zA are functions of time.

This means the velocity of the fluid element, VA, can be thought of as just a function of

time.

Now let’s imagine someone is taking the Eulerian point of view and observing a specific

location in space, which we label as (x,y,z).

At this moment in time both (x,y,z) and (xA,yA,zA) coincide with each other.

This means the velocity of fluid element A from the Lagrangian point of view, which we

call VA, is the same velocity seen by an observer from the Eulerian point of view, which we

call V.

As discussed earlier, in the Eulerian viewpoint velocity is a function of position and time.

After a small time dt passes, the fluid element has moved to a new location and has a new

velocity.

We still refer to the fluid element’s coordinates as (xA, yA, zA) in the Lagrangian description,

but these coordinates no longer coincide with the original location (x,y,z) in the Eulerian

description.

The velocity at (x,y,z) may change as well due to new fluid elements entering the region.