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

  1. Dec 2, 2016 #1
    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.
     
  2. jcsd
  3. Dec 2, 2016 #2
    The link to the paper doesn't work.
     
  4. Dec 2, 2016 #3
    I'm not familiar with this particular finite difference scheme, but presumably un+0.5 is already know when you are calculating un+1
     
  5. Dec 2, 2016 #4
    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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