this might seem, out of the way for you, i am troubled by the terms lagrangian coordinate system, and eulerian coordinate system. i have no problems with lagrangian or eulerian representation. can someone throw light on it, please explain fully. thanx

In a Lagrangian coordinate system you move along with a particule about which the initial position is known.

In an Eulerian coordinate system you stand still at certain position and look at the particule when it reaches you.

I have never particularly liked this terminology. A Lagrangian system is of course one where the description of the motion of the particles is in terms of a fixed reference configuration of fluid particles, whereas in an Eulerian system no such configuration is required. Strictly speaking this description is independent of the coordinate system which could be either stationary or moving, and in particular it could be moving with the flow (though one might ask why on earth one would do that!)

CFD however is not classical continuum mechanics, and the terms Lagrangian and Eulerian are used rather loosely to indicate whether the mesh points are part of the structure of the fluid or not: In an Eulerian grid, the mesh points are essentially a passive structure or framework through which fluid flows, i.e. they have no kinetic (mechanical) properties. In a Lagrangian coordinate system, the mesh points are actually part of the fluid, with intrinsic properties, temperature, velocity etc. In so-called arbitrary Lagrangian codes, at one stage of the algorithm, the mesh is treated as being passive, but at another stage is re-zoned based on computed nodal values of velocity.

thanks, i came across a treatment in which, the author talks of, lagrangian and eulerian coordinate systems, but i think, there is only one coordinate system, and eulerian is just a way of looking at the system, and if we were to consider the fluid, for purposes like, finding velocity, what we actually use are the same coordinates. if u get a feel for what i am saying give ur openion.

"but i think, there is only one coordinate system,". Actually, there is no "one" coordinate system. (x,y,z) or (r,theta,phi) or any other coordinate system is as arbitrary as any other. Usually the physical laws are stated in a coordinate system. However, the form of the law is invariant to the coordinate system. In fact, this is a necessary condition for any law to be a valid physical law. Many laws can be stated in a form which is independent of the coordinate system. For example, the equation

div grad phi = 0 is independent of a coordinate system. The div and grad operators can be given a general definition which does not involve any coordinates.

When velocity is involved, we use the principle of galilean invariance, which states that Newton's laws are invariant under a Galilean transformation. So

F = dp/dt transforms to

F' = dp'/dt (Of course, t is invariant under Newtonian Mechanics) The velocity in two coordinate systems may be different, the force may be different, but as long as they are inertial coordinate frames, the functional form of the laws remains same. I do not know how relevant all this is to your question. Please ignore if I am digressing from your question.

thanks for ur time, u addressed a question i didn't mean to ask, what i meant is this, when we take one co-ordinate system and speak of eulerian and lagrangian approaches, we are reffering to the same co-ordinate system in two different ways. assigning different meaning to the co-ordiantes.

In a Lagrangian system the motion of every fluid particle is described in terms of a fixed reference configuration of fluid particles. This could (but need not be) the position in space occupied by each fluid particle at t=0. In an Eulerian description no such reference configuration is required and the motion of the fluid is described in terms of the current configuration (at time t) of fluid particles, hence the convection terms in the material derivative.

Considering (1) the coordinate system (2) the "control volume" and (3) you, the motion of the fluid is definitely independent of all of the above, each of which could also be moving. In particular it is the description of the motion of the fluid which is Lagrangian/Eulerian, not the coordinate system used to obtain the solution.

here i see, u r addressing my question, thanks i am taking a graduate course for which, Dr.Warsi's book on Fluid Dynamics, is the text book, in which i came across, the usage, lagrangian co-ordiantes, and eulerian coordiantes, i am not trying to get him by words, but, i had some unresolved things, so, i posted this note, to seek openion, i did work on problems involving transformation from one approach to another. math is not the problem for me. if u have the book, can u read the first 6 or 8 pages and see what i am seeing.

(1). I thought I had seen the book sometimes ago. (2). I wasn't interested in the book, because he was from the Math Department. (I always had this feeling. But since I have not done any Internet search on him, I could be wrong.) (3). From my point of view, if there is a multi-level robot arm each connected to another, then there are two ways to specify the motion or location of each arm (or sub-arm). (4). The world coordinates system will be equivalent to the Eulerian coordiante system. In this system, the location and the motion of each sub arms are specified based on the fixed reference frame, that is, there is only one origin. (5). In this system, you can see the absolute location and motion of each arms, in time. (6). But, sometimes, you are interested in the motion of one particular arm relative to its parent arm. This will make the control and specification of the arm position much easier to do. The local coordinates system attached to that particular arm is equivalent to the Lagrangian coordinates system. We normally call this the local coordinates system. In this way, you can rotate the sub arm 10 degrees around the local x-axis using the local coordiantes system. The parent arm can be either stationary or moving. (7). If you replace the arm system by a system of fluid particles, then you have the similar situations, which can be described by either systems. (8). In fluid dynamics, the fixed coordiantes or the mesh is sometimes called Eulerian description or formulation. On the other hand, if you place a moving grid on top of the fluid in addition to the existing fixed reference frame, then it is usually called the Lagrangian formulation. Researchers at the Los Alamos National Lab were famous about the use of Lagrangian and Eulerian formulation. This requires the re-zoning of the mesh at the end of each calculation. (9). You can read their publications in 70's for the more precise definition of the terms. I think, it is important to know that, by using the moving grid, the formulation will generate extra terms related to the motion of the grid. In other words, it is no longer quasi-steady state. (10). So, if you use the robot arm case as an example, it should be much easier to understand the difference and the advantage of using the Lagrangian formulation. In short, the Lagrangian formulation must re-calculate and re-zone the mesh from time to time, while the Eulerian mesh is essentially fixed, and requires no internal mesh movement.

Herein lies my discomfort with using the terms Eulerian and Lagrangian in CFD (as opposed to continuum mechanics). Since it is the description of the flow, not the coordinate system which is Lagrangian/Eulerian, and as was rightly pointed out in this thread, both formulations must be coordinate independent in order to satisfy the requirements of superposed rigid body motion.

For any real problem though we do need a coordinate system, and in most CFD methods we also need a mesh. For structured BFC meshes it is tempting to associate the mesh with a coordinate system; although you can also treat the mesh as a series of finite 'control' volumes, of more use perhaps with unstructured or octree approaches. Does moving the mesh with the flow in a transient manner make it Lagrangian? I suppose it effectively does, if only due to common usage of the terminology. I debated this with some of the Los Alamos T3 group, and we talked about coining the term "Arbitrary Eulerian" methods. I am sure that will never catch on, and the terms Lagragnian and Eulerian are here to stay in CFD, even though it is somewhat imprecise. In practice nowadays, we refine meshes in all kinds of ways for all kinds of reasons. Thoughts, comments?...

(1). It is easier to use the author's name to identify the method used in the paper. (2). Stationary mesh, or moving mesh would be much better than Eulerian or Lagrangian formulation. (3). My suggestion is: stop using the names Eulerian, or Lagrangian, unless, it is defined clearly in each paper the exact meaning or processes. (4). Arbitrary Eulerian will give an ordinary person the impression of something related to random numbers. (5). Perhaps, it is time for the T3 group fellows to start using "stationary mesh" and "moving mesh", instead of "Eulerian" or "Lagrangian" . Unless they can dig out the original paper by these old masters, that they were indeed the persons who invented the formulation. (6). So, the case is not closed. There is still possibility to use the names. (At the same time using T3-Eulerain, T3-Lagrangian would be acceptable )

I like stationary and moving mesh. It's plain and to the point, but I doubt it will catch on. Incidentally, it was Lagrange who first solved for a 2-D stream function, generally considered the opposite of (Lagrangian) particle tracking.

So given the above, what terminology would one use for so-called "Eulerian-Eulerian" and "Eulerian-Lagrangian" methods used in two-phase flow (and in the latter case also for visualization of stream-lines)? Perhaps mesh-mesh and mesh-particle based methods? Obviously particle-in-cell is out...

when i posted this message, i was looking for some thing like this, I think that it is only the approach (as u say the description) of the flow, that is eulerian or lagrangian, not the co-ordinate system.

i couldn't get any of my class mates to discuss this, they quickly jump into a conclusion that i dont understand the two approaches.

in the process of following this discussion, i have come to know these terms have relevance in mesh generation too. and i see the problem much the same in this arena too.

thanks for all the postings cox

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[Marcus081]

Hi, I have been using the version of ansys 12.1, and the coordinate system appears with these component (v, w, x) , I have not found the form to change it a (x, y, z), since I can do it?