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August 15, 2023, 12:55 |
2D version of mass co-ordinate
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#1 |
Senior Member
Matthew
Join Date: Mar 2022
Location: United Kingdom
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It is possible to simplify the Euler equations by introducing a new set of Lagrangian co-ordinates.
, by t=t' and h is defined by: This method has the nice property of fixing the amount of mass in an interval(this can be demonstrated by integrating the definition of h, and using the Lagrangian conservation of mass. My question is this: Is there something similar for 2D? |
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August 15, 2023, 16:47 |
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#2 |
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Filippo Maria Denaro
Join Date: Jul 2010
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Quote:
http://depts.washington.edu/clawpack...system_2d.html |
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August 15, 2023, 20:06 |
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#3 |
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Lucky
Join Date: Apr 2011
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Recognize that x and h can be generalized to vectors and that d/dx is actually a divergence operator. In other words, it works in n-d! See Noether's second theorem.
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August 16, 2023, 05:15 |
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#4 |
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Matthew
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I thought that it would be generalised from the fact that mass is conserved for the cells, the integral of the density would give the amount of mass in a particular direction.
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August 16, 2023, 07:18 |
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#5 |
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Lucky
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You're literally assuming the premise when you say the question you are trying to ask is explained by itself
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August 17, 2023, 06:28 |
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#6 |
Senior Member
Matthew
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I think that there is an easy generalisation, you just have to use:
Then this gives the following linear equations for the conservation of mass and momentum: This transformation is good because it makes the convective terms disappear (like any lagrangian co-ordinate) but (hopefully) allows a fixed set of co-ordinates. like Eulerian co-ordinates. |
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