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June 8, 2007, 01:10 
Stability of finite diffs on a staggered grid

#1 
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Hey all,
This should be easy, but I just need a nudge in the right direction. I have an implementation of a simple 2D eulerian NS solver that uses a uniform staggered grid. It just computes a velocity field (and consequently a pressure field as well), but not temperature or anything else like that. I want to implement adaptive timestepping on this creature. So, the CLF conditions are easy enough. However, according to "Numerical Simulation in Fluid Dynamics: a practical introduction" by Griebel, Dornseifer, and Tilman Neunhoeffer, there is another condition that takes viscosity into account. Their condition looks like this: dt < ( Re / 2 ) ( 1/dx^2 + 1/dy^2 )^1 However, it seems intuitively wrong to me (and my experiments with the code seem to validate my skepticism). If that condition is right, then higher Reynolds numbers are more stable. So, what should this condition really be? Thanks. 

June 8, 2007, 08:03 
Re: Stability of finite diffs on a staggered grid

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This is a property of explicit time integration schemes when applied to the diffusion operator. Basically the domain of influence of a point at t = t_n is the whole domain at t=t_n + dt. However in an explicit method this domain of influence is only to the nearest neighbours and so the timestep dt needs to be sufficiently small that the influence this error in the domain of influence is "small". Increasing the viscosity (thus reducing Re) increases the effect of diffusive spreading and hence requires a reduction in timestep. Implicit methods don't have this problem.


June 8, 2007, 10:29 
Re: Stability of finite diffs on a staggered grid

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Hmm, so why is it that people talk about how hard it is to simulate large Reynolds numbers? According to that stability property, it should be easier than simulating lower ones.


June 8, 2007, 11:32 
Re: Stability of finite diffs on a staggered grid

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The classic condition is nu*(delta t)/(delta x)^2 < 1/2.
Combine this with the Courant condition, U*(delta t)/(delta x) < 1. What does the combination tell you about a Reynolds number limit? 

June 8, 2007, 11:35 
Re: Stability of finite diffs on a staggered grid

#5 
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Well, your condition says that higher Reynolds number is harder to simulate, which was what I had understood was true. The condition that I gave says the opposite.
Is your condition for a firstorder explicit method? Better yet, can you cite where you got it? Thanks. 

June 8, 2007, 11:41 
Re: Stability of finite diffs on a staggered grid

#6 
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Just because the explicit scheme is "more stable" doesn't mean that it's giving better answers!
(1) explicit methods don't do a good job of the boundary conidtions (basically they allow the fluid to slip along the noslip boundary) (2) at hight Reynolds number a thin boundary layer is formed near the surface (where noslip is satisfied) and the vertical grid spacing, h, needs to be small in this region; typically a lot less than (1/Re)**(1/2) for the boundary layer to be resolved. So, if h= a.(1/Re)**(1/2) with a<<1, then your stability condition becomes dt < a/2. If you want to resolve the instability of this layer (for transition) your horizontal length scales and time scales also need to be able to resolve fluctuations with wavenumbers and frequencies of O(Re^(1/2)). This requires a very fine grid and short timestep! 

June 8, 2007, 11:50 
Re: Stability of finite diffs on a staggered grid

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Just out of curiosity, how does an explicit method allow the flow to slip along a nonslip boundary? This is the first I'd heard of that.


June 8, 2007, 12:54 
Re: Stability of finite diffs on a staggered grid

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It's a classic:
C. W. Hirt, "Heuristic Stability Theory for FiniteDifference Equations," Journal of Computational Physics, Vol. 2, No. 4, June, 1968, pp. 339355. This was reprinted in Computer Fluid Dynamics  Recent Advances, edited by Francis H. Harlow, published by the AIAA, February, 1973. The examples (from the 60's remember) are on constant spatial increments and fully explicit, cartesian coordinate systems. However, the technique is robust and is still used for variable mesh implicit techniques. I don't know if the ideas are workable on distorted (skewed) meshes or not. 

June 8, 2007, 13:04 
Re: Stability of finite diffs on a staggered grid

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Because explicit methods do not rigourously enforce the boundary conditions at the new time level. This is eqivalent to letting it slip by a small amount proportional to the timestep.


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