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# lecture2 - 2.20 - Marine Hydrodynamics, Spring 2005 Lecture...

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Lecture 2 Marine Hydrodynamics Lecture 2 Chapter 1 - Basic Equations 1.1 Description of a Flow To deﬁne a ﬂow we use either the ‘Lagrangian’ description or the ‘Eulerian’ description. Lagrangian description: Picture a ﬂuid ﬂow where each ﬂuid particle caries its own properties such as density, momentum, etc. As the particle advances its properties may change in time. The procedure of describing the entire ﬂow by recording the detailed histories of each ﬂuid particle is the Lagrangian description. A neutrally buoyant probe is an example of a Lagrangian measuring device. The particle properties density, velocity, pressure, . . . can be mathematically repre- sented as follows: ρ p ( t ) ,±v p ( t ) ,p p ( t ) ,... The Lagrangian description is simple to understand: conservation of mass and New- ton’s laws apply directly to each ﬂuid particle . However, it is computationally expensive to keep track of the trajectories of all the ﬂuid particles in a ﬂow and therefore the Lagrangian description is used only in some numerical simulations. ) ( t p υ r p Lagrangian description; snapshot 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

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Eulerian description: Rather than following each ﬂuid particle we can record the evolution of the ﬂow properties at every point in space as time varies. This is the Eulerian description. It is a ﬁeld description. A probe ﬁxed in space is an example of an Eulerian measuring device. This means that the ﬂow properties at a speciﬁed location depend on the location and on time. For example, the density, velocity, pressure, . . . can be mathematically represented as follows: ±v ( ±x, t ) ,p ( ) ( ) ,... The aforementioned locations are described in coordinate systems. In 13.021 we use the cartesian, cylindrical and spherical coordinate systems. The Eulerian description is harder to understand: how do we apply the conservation laws? However, it turns out that it is mathematically simpler to apply. this reason, in Fluid Mechanics we use mainly the Eulerian description. ) , ( t x r y x ) , ( t x r r υ Eulerian description; Cartesian grid 2
1.2 Flow visualization - Flow lines Streamline: A line everywhere tangent to the ﬂuid velocity ±v at a given instant (ﬂow snapshot). It is a strictly Eulerian concept. Streakline: Instantaneous locus of all ﬂuid particles that have passed a given point (snapshot of certain ﬂuid particles). Pathline: The trajectory of a given particle P in time. The photograph analogy would be a long time exposure of a marked particle. It is a strictly Lagrangian concept.

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## This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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lecture2 - 2.20 - Marine Hydrodynamics, Spring 2005 Lecture...

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