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In
classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual
fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constan ...
as it moves through space and time.
Plotting the position of an individual parcel through time gives the
pathline
Streamlines, streaklines and pathlines are field lines in a fluid flow.
They differ only when the flow changes with time, that is, when the flow is not steady.
Considering a velocity vector field in three-dimensional space in the framework of ...
of the parcel. This can be visualized as sitting in a boat and drifting down a river.
The Eulerian specification of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes.
[ This can be visualized by sitting on the bank of a river and watching the water pass the fixed location.
The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the Lagrangian and Eulerian frame of reference. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's frame of reference, and in any coordinate system used within the chosen frame of reference.
These specifications are reflected in ]computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
, where "Eulerian" simulations employ a fixed mesh
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, exp ...
while "Lagrangian" ones (such as meshfree simulations) feature simulation nodes that may move following the velocity field
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
.
Description
In the ''Eulerian specification'' of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, the field is represented as a function of position x and time ''t''. For example, the flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
is represented by a function
On the other hand, in the ''Lagrangian specification'', individual fluid parcels are followed through time. The fluid parcels are labelled by some (time-independent) vector field x0. (Often, x0 is chosen to be the position of the center of mass of the parcels at some initial time ''t''0. It is chosen in this particular manner to account for the possible changes of the shape over time. Therefore the center of mass is a good parameterization of the flow velocity u of the parcel.)[ In the Lagrangian description, the flow is described by a function
giving the position of the particle labeled x0 at time ''t''.
The two specifications are related as follows:][
because both sides describe the velocity of the particle labeled x0 at time ''t''.
Within a chosen coordinate system, x0 and x are referred to as the Lagrangian coordinates and Eulerian coordinates of the flow respectively.
]
Material derivative
The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
(also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative).[
Suppose we have a flow field u, and we are also given a generic field with Eulerian specification F(x, ''t''). Now one might ask about the total rate of change of F experienced by a specific flow parcel. This can be computed as
where ∇ denotes the ]nabla Nabla may refer to any of the following:
* the nabla symbol ∇
** the vector differential operator, also called del, denoted by the nabla
* Nabla, tradename of a type of rail fastening system (of roughly triangular shape)
* ''Nabla'' (moth), a ge ...
operator with respect to x, and the operator u⋅∇ is to be applied to each component of F. This tells us that the total rate of change of the function F as the fluid parcels moves through a flow field described by its Eulerian specification u is equal to the sum of the local rate of change and the convective rate of change of F. This is a consequence of the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
since we are differentiating the function F(X(x0, ''t''), ''t'') with respect to ''t''.
Conservation laws for a unit mass have a Lagrangian form, which together with mass conservation produce Eulerian conservation; on the contrary, when fluid particles can exchange a quantity (like energy or momentum), only Eulerian conservation laws exist.
See also
* Brewer-Dobson Circulation
* Conservation form Conservation form or ''Eulerian form'' refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term i ...
* Contour advection
Contour advection is a Lagrangian method
of simulating the evolution of one or more contours or isolines of
a tracer as it is stirred by a moving fluid.
Consider a blob of dye injected into a river or stream: to first order it could be modelled ...
* Equivalent latitude
In differential geometry, the equivalent latitude is a Lagrangian coordinate .
It is often used in atmospheric science,
particularly in the study of stratospheric dynamics.
Each isoline in a map of equivalent latitude follows the flow velocity ...
* Generalized Lagrangian mean
In continuum mechanics, the generalized Lagrangian mean (GLM) is a formalism – developed by – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the flo ...
* Lagrangian particle tracking
* Semi-Lagrangian scheme The Semi-Lagrangian scheme (SLS) is a numerical method that is widely used in numerical weather prediction models for the integration of the equations governing atmospheric motion. A Lagrangian description of a system (such as the atmosphere) foc ...
* Streamlines, streaklines, and pathlines
Streamlines, streaklines and pathlines are field lines in a fluid flow.
They differ only when the flow changes with time, that is, when the flow is not steady.
Considering a velocity vector field in three-dimensional space in the framework of ...
* Trajectory (fluid mechanics) In fluid mechanics, meteorology and oceanography, a trajectory traces the motion of a single point, often called a parcel, in the flow.
Trajectories are useful for tracking atmospheric contaminants, such as smoke plumes, and as constituents to Lag ...
* Stochastic Eulerian Lagrangian method In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM)
is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while introducing approximations which facilitate analysi ...
* Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectorie ...
Notes
References
*
* {{cite book
, first1 = Lev , last1 = Landau , author-link = Lev Landau
, first2 = E.M. , last2 = Lifshitz , author2-link = Evgeny Lifshitz
, title = Fluid Mechanics
, series = Course of Theoretical Physics, Volume 6
, edition = 2nd
, publisher = Butterworth-Heinemann
, year = 1987
, isbn = 978-0750627672
External links
Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity.
Fluid dynamics
Aerodynamics
Computational fluid dynamics