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4. Lagrangian and Eulerian Reference Frames — MUDE
Lagrangian and Eulerian
Reference Frames
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Contents
4.1. Material Derivative
In physics-based modelling, a reference frame describes a coordinate system that is defined
relative to specific points of interest. Lagrangian and Eulerian reference frames are two
approaches used in fluid mechanics that are useful for describing the motion and behavior of
a fluid over time. Although they are completely different perspectives on the same
phenomenon, both are necessary to fully model the movement of particles or elements
within the fluid mathematically.
Some concepts and notation on this page are explained further on the notation page.
 MUDE Exam Information
This page is to provide background information to support the flow-focused
conservation equations that are used in FVM. You are not expected to reproduce
this for the exam, but reading through it will give you a better understanding for
why and how the equations are derived, as well as different modelling perspectives.
Lagrangian Frame
In a Lagrangian frame, one follows the particles as they move through space; Fig. 4.1
illustrates how a particle is described in Cartesian coordinates for the Lagrangian frame using
position
and velocity
. The reference frame is illustrated by the small Cartesian
coordinate system attached to the particle. Because the frame is moving with the particle it
will not witness any relative motion. Therefore, position and velocity are functions of time
only, not space.
A Lagrangian frame is helpful for understanding the history and trajectory of individual
particles. In fluid mechanics it is suitable for studying issues like particle dispersion, tracer
movement or tracking the paths of specific objects in a flow. For solid mechanics it is useful
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for describing deformation.
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4. Lagrangian and Eulerian Reference Frames — MUDE
Fig. 4.1 Lagrangian (left) and Eulerian (right) reference frames.
Eulerian Frame
Instead of following individual particles, the Eulerian frame observes what is happening at
fixed points in space; Fig. 4.1 illustrates how particle velocity is described in Cartesian
coordinates for the Eulerian frame using
. The reference frame is illustrated by the
small unit volume surrounding the particle. Because the frame stays fixed, the Eulerian frame
is often described as “watching the particles move through the box.”
A Eulerian frame is helpful suitable for studying the overall properties of a fluid, for example:
flow patterns, pressure distribution or heat transfer. It provides a global view of what is
happening in the volume at various locations.
In summary, the Lagrangian frame follows individual elements as they move through space,
providing a particle-focused view, whereas the Eulerian frame makes observations from fixed
points in space, offering a location-based perspective. The choice between these frames
depends on the specific problem one is trying to solve and in many cases, a combination of
both frames may be used to gain a complete understanding of a fluid.
4.1. Material Derivative
As the reference frames are can be related by a mapping (one-to-one transformation)
between two Cartesian coordinate systems,
the same for a small change in time,
, the velocity and location vectors are
. This is called Lagrangian-Eulerian equivalence, and is
illustrated in Fig. 4.2.
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4. Lagrangian and Eulerian Reference Frames — MUDE
Fig. 4.2 Displaced particle in Lagrangian (left) and Eulerian (right) reference frames.
Consider a system of particles in motion using the reference frames in Fig. 4.2 from time to
. In the Lagrangian frame a new velocity is observed for the particle of interest at time
(a function of time only). The reference frame has moved with the particle (black
circle), and a new particle is now in the location of the frame at time (white circle). On the
other hand, in the Eulerian frame the original particle (black circle) has simply left the cube
after time
has passed and a new particle (white circle) is observed with a different velocity
(a function of both time and space). In this case, the reference frame does not move.
For small
and if the spatial scale at which the particles are being observed is much larger
than the scale of the particles themselves (the molecular scale), the particle system can be
viewed as a continuous mass (i.e., a fluid or deformable solid). Therefore, the change in
velocity in the Lagrangian frame is the same as the change observed in the Eulerian frame. In
other words, the two descriptions of motion of the black and white circles are equivalent.
Mathematically:
The right-hand side is a function of 4 variables; application of the chain rule of differentiation
leads to:
Dividing by
while recalling that
, which is the -component of velocity, one gets:
This is the mathematical representation of Lagrangian-Eulerian equivalence, and implies that
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the quantities on both sides are the
same
in thecontent
Cartesian frame. Rearranging and combining
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4. Lagrangian and Eulerian Reference Frames — MUDE
terms results in:
where an upright capital symbol
is used to represent the total derivative in the Lagrangian
reference frame, and the entire equation above is called the material derivative or Lagrangian
derivative.
While we will not use the material derivative often in this textbook, it is important for work in
continuum mechanics. When conservation laws are used to model real-world phenomenon,
especially flow applications, measurements are being performed at specific locations defined
in a Lagrangian frame, for example, probes (points) or electromagnetic or sound waves (lines),
but the phenomenon is being measures in an Eulerian frame. In other words, observing
changes in a volume (or multiple volumes) rather than the movement of a particle in the fluid
of interest. The Lagrangian derivative is the only way to convert the Eulerian spatial
experimental data into a (Lagrangian) description of the way that the fluid moves through
space and time.
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