View source for Lagrangian and Eulerian specification of the flow field

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{{Short description|computational fluid dynamics tools}}
{{dablink| This article is about the continuum mechanics.  For the use of generalized coordinates in classical mechanics, see [[generalized coordinates]], [[Lagrangian mechanics]] and [[Hamiltonian mechanics]]}}
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In [[classical field theory|classical field theories]], the '''Lagrangian specification of the field''' is a way of looking at fluid motion where the observer follows an individual [[fluid parcel]] as it moves through space and time.<ref name=Batchelor>Batchelor (1973) pp. 71–73.</ref><ref name=Lamb>Lamb (1994) §3–§7 and §13–§16.</ref> Plotting the position of an individual parcel through time gives the [[Streamlines, streaklines, and pathlines|pathline]] 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.<ref name=Batchelor/><ref name=Lamb/> 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]], where "Eulerian" simulations employ a fixed [[Types of mesh|mesh]] while "Lagrangian" ones (such as [[Meshfree methods |meshfree simulations]]) feature simulation nodes that may move following the [[velocity field]].



==Description==

In the ''Eulerian specification'' of a [[Field (physics)|field]], it is represented as a function of position '''x''' and time ''t''. For example, the [[flow velocity]] is represented by a function 

:<math>\mathbf{u}\left(\mathbf{x}, t\right).</math>

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 '''x<sub>0</sub>'''. (Often, '''x<sub>0</sub>''' is chosen to be the position of the center of mass of the parcels at some initial time ''t<sub>0</sub>''. 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.)<ref name=Batchelor/> In the Lagrangian description, the flow is described by a function

:<math>\mathbf{X}\left(\mathbf{x}_0,t\right),</math>

giving the position of the particle labeled '''x<sub>0</sub>''' at time ''t''.

The two specifications are related as follows:<ref name=Lamb/>

:<math>\mathbf{u}\left(\mathbf{X}(\mathbf{x}_0,t),t \right) = \frac{\partial \mathbf{X}}{\partial t}\left(\mathbf{x}_0,t \right),</math>

because both sides describe the velocity of the particle labeled '''x<sub>0</sub>''' at time ''t''.

Within a chosen coordinate system, '''x<sub>0</sub>''' and '''x''' are referred to as the '''Lagrangian coordinates''' and '''Eulerian coordinates''' of the flow.

==Material derivative==
{{Main|Material derivative}}
The Lagrangian and Eulerian specifications of the [[kinematics]] and [[dynamics (physics)|dynamics]] of the flow field are related by the [[material derivative]] (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative).<ref name=Batchelor/>

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

:<math>\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t} = \frac{\partial \mathbf{F}}{\partial t} + \left(\mathbf{u}\cdot \nabla  \right) \mathbf{F},</math>

where &nabla; denotes the [[gradient]] with respect to '''x''', and the operator '''u'''&sdot;&nabla; 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]] since we are differentiating the function '''F'''('''X'''('''x<sub>0</sub>''',''t''),''t'') with respect to ''t''.

[[Conservation law]]s 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.<ref>{{harvtxt|Falkovich|2011}}</ref>

==See also==

* [[Conservation form]]
* [[Contour advection]]
* [[Equivalent latitude]]
* [[Generalized Lagrangian mean]]
* [[Lagrangian particle tracking]]
* [[Semi-Lagrangian scheme]]
* [[Streamlines, streaklines, and pathlines]]
* [[Trajectory (fluid mechanics)]]
* [[Stochastic Eulerian Lagrangian method]]

==Notes==
{{reflist}}

==References==
* {{cite book| ISBN=978-3-319-59694-5 |last1=Badin|first1=G.|last2=Crisciani|first2=F.| title=Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws - | publisher=Springer| year=2018 | pages=218 | doi= 10.1007/978-3-319-59695-2}}
*{{citation
 | first=G.K. | last=Batchelor | authorlink=George Batchelor
 | title=An introduction to fluid dynamics
 | publisher=Cambridge University Press
 | year=1973
 | isbn=978-0-521-09817-5
}}
*{{citation
 | first=Lev | last=Landau| authorlink=Lev Landau
 | first2=E.M.| last2=Lifshitz| author2link=Evgeny Lifshitz
 | title=Fluid Mechanics, 2nd Edition (Course of Theoretical Physics, Volume 6)
 | publisher= Butterworth-Heinemann
 | year=1987
 | isbn=978-0750627672
}}
*{{citation
 | first=H. | last=Lamb | authorlink=Horace Lamb
 | title=Hydrodynamics
 | edition=6th
 | publisher=Cambridge University Press
 | year=1994
 | origyear=1932
 | isbn=978-0-521-45868-9
}}
*{{citation 
 | last=Falkovich | first=Gregory 
 | year=2011 
 | title=Fluid Mechanics (A short course for physicists)
 | publisher=Cambridge University Press 
 | isbn=978-1-107-00575-4 
}}

[[Category:Fluid dynamics]]
[[Category:Aerodynamics]]
[[Category:Computational fluid dynamics]]