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November 29, 2013, 23:48 
Minimum Time Step with Navier Stokes

#1 
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Hello,
I am a bit of a newbie at CFD. I am trying to determine what the minimum time step would be when solving the NS equations for a liddriven cavity flow (2D) in order to achieve stability. The flow is incompressible, viscous, and the results I am interested in are at steady state. I am using a finite difference method with a colocated grid. I'm really not sure how to even approach this problem and any advice on what I need to do to find the time step would be greatly appreciated. Thank you very much for any and all replies. 

November 30, 2013, 04:27 

#2  
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Filippo Maria Denaro
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I don't see your type of discretization, Re number and number of cells. I can suggest a rude approximation for evaluating a practical dt in case of Re sufficiently high: dt_p*(umax/dx+vmax/dy)<1 > dt = alpha*dt_p (alpha = 0.1  0.5) 

November 30, 2013, 10:18 

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I'm not sure exactly what you mean by dt_p? And the value of alpha is dependent on Re? 

November 30, 2013, 10:26 

#4 
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Filippo Maria Denaro
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what about the timeintegration scheme?


November 30, 2013, 10:40 

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I am using an Explicit Time Advance scheme.
I calculate the combination of the advective and viscous terms and its divergence from the initial velocity field. I solve Poisson's equation for pressure and then compute the velocity field at the next time step. I hope that answers your question. 

November 30, 2013, 11:42 

#6  
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Filippo Maria Denaro
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dt_d*(1/dx^2 + 1/dy^2)/Re <1/2. As your sistem is non linear such value should be furhter reduced 

November 30, 2013, 11:52 

#7 
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Well perhaps I have other errors in my code but this produces a far larger time step than I have been using and it is immediately unstable. From where did you derive this equation for stability?
If I have a square that I have divided into a grid of 100 points with delta x = delta y = 0.1 that gives me a time step of 1 second per your equation? 

November 30, 2013, 12:23 

#8  
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Filippo Maria Denaro
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The constraint comes from the linear stabilty analysis of the parabolic equation dphi/dt=Gamma* Lap Phi, discretized with the FTCS scheme Assuming a grid formed by 10x10 cells (dx=dy=0.1) you simply have dt_d < 0.25. Therefore dt =0.1 should work. If you see instability after few timesteps then there is an error in the code 

November 30, 2013, 12:27 

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Tags 
finite difference method, navierstokes, stability 
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