What does the Navier-Stokes equation look like after time discretization?

  • #1
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Hi,

I know the general form of the Navier Stokes Equation as follows.
upload_2016-12-2_12-30-26.png


I am following a software paper of "Gerris flow solver written by Prof. S.Popinet"
[Link:http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.374.5979&rep=rep1&type=pdf]
and he mentions after time discretization he ends with the following equation:
upload_2016-12-2_12-31-35.png

where n-1 is the previous time step, n+1 is the next time step and n+0.5 is mid time for the present time step.

Solving equation implicitly/ explicitly in time means solving for next time data however in the equation there are rather two unknowns un+0.5 and
un+1.

Not sure why he uses different terms at different time intervals. Density at n+0.5, velocity at n, n-1, n+0.5 etc..

Can anyone point me or explain me how he arrives at this specific sort of discretized equation.
 
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Answers and Replies

  • #2
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The link to the paper doesn't work.
 
  • #3
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I'm not familiar with this particular finite difference scheme, but presumably un+0.5 is already know when you are calculating un+1
 
  • #4
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Sorry for the link.
http://www.sciencedirect.com/science/article/pii/S002199910900240X

@Chestermiller guess thats true. It is being solved by the time step projection method which means an intermediate velocity is computed and later updated to a divergence free velocity by solving the laplace of pressure term as in the mentioned paper.

The notation is a bit strange for me as he uses density at time n+0.5 without solving any advection equation. From what I see density terms at n and n+0.5 should be the same.
 

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