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Old   February 25, 2012, 19:59
Default nonlinear/linear stability
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Shenren Xu
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Hello all,

I have a quite basic CFD question to ask: How does a steady flow solver manage to get a steady flow solution when the flow is intrinsically unsteady? For example, when I run a simple case of Re=50 laminar flow around a naca airfoil with a blunt trailing edge. If the mesh is coarse enough, my NS solver can converge to machine precision easily; but if the mesh is refined enough, the solver goes to a LCO, although the convergence level is also relatively low. Could anyone give me some hint on how to look at this phenomenon?

I find it hard to ask the "right" question here, but my first question would be: how does the coarse mesh suppress the unsteadiness and help with the convergence to "steady state"? Then, how is linear/nonlinear stability play a role in this?

Thanks in advance.

Shenren
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Old   February 26, 2012, 04:21
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one way to look at this is the following: a coarse discretization (in your case a coarse mesh) will bring with a large numerical error, often called numerical dissipation. In case you are using a FV scheme, your Riemann solver will see high jumps, and thus introduce higher dissipation to stabilize the scheme. For other schemes, slightly different mechanism exist.
Underresolution of a problem leads to a loss of physics plus a need for numerical stabilization via dissipation. In short, a coarse grid just damps away all the unsteady stuff and makes an essentially unsteady flow steady through numerical damping.
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Old   February 26, 2012, 08:18
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Shenren Xu
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Quote:
Originally Posted by cfdnewbie View Post
one way to look at this is the following: a coarse discretization (in your case a coarse mesh) will bring with a large numerical error, often called numerical dissipation. In case you are using a FV scheme, your Riemann solver will see high jumps, and thus introduce higher dissipation to stabilize the scheme. For other schemes, slightly different mechanism exist.
Underresolution of a problem leads to a loss of physics plus a need for numerical stabilization via dissipation. In short, a coarse grid just damps away all the unsteady stuff and makes an essentially unsteady flow steady through numerical damping.

Thanks, cdfnewbie. This makes sense and is indeed consistent with my experience in checking mesh convergence. I guess a similar but in principle same phenomenon is that for an internal flow case, unless I use fine enough mesh, some secondary flow features wouldn't appear in the solution field.

I have a further question: in the case of low Re # flow around a 2D cylinder, suppose the mesh is fine enough for a certain Re # and a steady flow solver normally will converge to a low residual, but falls into a limit cycle oscillation, instead of converging to machine precision. even though the physical time accuracy is destroyed due to preconditioning (local time stepping, etc), the LCO manifests some unsteadiness in the physical flow. I heard people say in this case the solution has "linear instability" but "nonlinear stability", could anyone please elaborate a little bit on these terms "linear/nonlinear stability/instability"?

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