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 quarkz November 17, 2010 09:40

Confusion about non-dimensionalization of NS eqns

Hi,

It seems that depending on how the NS equations are non-dimensionalized, I can end up with 2 different eqns.

(*) denotes non-dimension

If I use

1. u(*)=u/(u infinity)
2. x(*)=x/c, c=chord length
3. p(*)=p/(rho*(u infinity)^2)

and either t(*)=t*(u infinity)/c or t(*)=t*f where f=frequency

I get 2 different forms of the final eqns ending with either Re or St and Re.

where St=f*c/(u infinity) and Re=(u infinity)*c/v

So how will this 2 eqns influence my simulation results? If I use different St, it will not appear in the 1st eqn because there is no St in the equation.

Or it does not matter? The results will be different but as long as I'm comparing against the same non-dimensionalization, it is fine.

Thanks!

 sbaffini November 17, 2010 19:58

In non-dimensional analysis you express the dependence of your problem from a set of parameters and then try to make this parameters nondimensional by a proper choice of 3 (in dynamics) of them as a dimensional basis.

In your second case you are explicitly saying that there is some frequency f which influences your problem, hence it has to be nondimensionalized by the proper parameters from your basis (u_infinity and c).

However, if you use f as parameter for your dimensional basis you can't still use both u_infinity and c.

I'm more comfortable in thinking f as a parameter appearing in the B.C. of your problem (which still need to be nondimensionalized)...in that case the St is going to appear in your B.C. and the momentum eq.s assume their classical form.

In contrast, if there is no such a frequency affecting your problem, why would you introduce it?

Be aware of the fact that phenomena like the Von Karman Vortex Street do exhibit a characteristic frequency but this is not an independent parameter (hence useful to form a dimensional basis). Instead it strictly depends on the shape of the body, u_infinity, c and the fluid.

 quarkz November 18, 2010 09:21

Thanks sbaffini. I saw the NS eqns with St in a JCP paper which talks about simulating flapping wing in cartesian grid. In the paper, St= fL/U. f is an imposed freq, L and U are the length and velocity. So they are using U and L and f.

So in this case, Von Karman vortex street will be appear and hence St is included. But if it isn't, how will take influence the analysis? Will it be erroneous?

Moreover, the BC used in the paper is dp/dn = -St(Du/dt).n

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