the reason w->w0 gets in here and not w->0 is because I was getting errors due to a divide by zero error

It's too bad you haven't told us what answer you expect to get, because then I would know whether this is right:

Limit[-((CD U)/Sqrt[((U^2 + w0^2)/U^2)]) - 
  CL\[Alpha] U^2 (1/Sqrt[w0^2] -
  w0/(U^2 Sqrt[(U^2 + w0^2)/U^2])) ArcSin[Sqrt[w0^2]/U],
 w0 -> 0]

(*  -((CD + CL\[Alpha]) U)  *)

Mathematica doesn't automatically eliminate Abs[w0]/Sqrt[w0^2] from an expression

The elimination can be done if w0 is a real number, but not if it is complex. Tell MMA that w0 is real and it will do the elimination.

Simplify[Abs[w0]/Sqrt[w0^2], Assumptions -> Element[w0, Reals]]
(* 1 *)

Wild guess: Are you after this?

Normal[Series[{
   {-((U^2 w0 (CD w0 + CL\[Alpha] U ArcSin[Abs[w0]/U]))/(U^2 + w0^2)^(3/2))},
   {0},
   {(U^2 w0 (CD U - CL\[Alpha] w0 ArcSin[Abs[w0]/U]))/(U^2 + w0^2)^(3/2)}},
  w0 -> 0, Assumptions -> w0 > 0 && U > 0]]

(*  {{-(((CD + CL\[Alpha]) w0^2)/U)}, {0}, {CD w0}}  *)

It's slightly different than the result your substitutions imply, so my guess may be wrong. However, asymptotic expansions are common operations, and you seem to be treating w0 as if it were small and approaching zero. I thought it worth suggesting.

In my experience, algebraic manipulation via substitution rules is tricky and bug-prone. Rules of the form variable -> expr that replace all instances of the variable by the same expr are generally safe. But I find it's more often better to ask Mathematica to calculate the mathematical operation I want. There are exceptions, though.

First off, thank you for the suggests and help. The following is information in case somebody has an idea. I think I need to do a little work to isolate the issue, and that may resolve it.

In response to Gianluca's suggestion to move the Simplify before the ReplaceAll's, I found that all of the substitutions fail entirely now. In other words, // Simplify /. u -> U /. v -> 0 /. w -> w0 yields exactly the same output as // Simplify. In response to Eric's suggestion to post my code, I agree that's a good idea, but rather than dump the whole thing, I feel I have an obligation to try and find the simplest case that recreates the issue. The first iteration of that was to just see if Mathematica failed to be able to do the substitution into ArcSin[Abs[w0]/U], and it does not fail. Given what I said in the opening two sentences, I think I have an idea of where to focus. If I can't figure it out after I find time to spend another hour or two on it, I'll just upload the whole notebook. Both U>0, u, v, w, and w0 are declared to be Reals in an $Assumption statement.

This is something of an aside, but the reason w->w0 gets in here and not w->0 is because I was getting errors due to a divide by zero error when I tried to evaluate even though things cancel. This could be related to the above. w!=0 is physically reasonable, so might as well include it.

Thanks for the suggestions. I thought attaching a snippet from a notebook was the best way to show what was going on. When I cut-and-paste, the result seems to be not intelligible. I tried the suggestion of moving the simplify, but the substitution still doesn't work. One interesting thing that I noticed is elsewhere in the full notebook Mathematica doesn't automatically eliminate Abs[w0]/Sqrt[w0^2] from an expression, but it does use the substitution in what is attached to this post.

What I am trying to do is an order-of-magnitude reduction on the full expression (there are more). Based on limited prior experience, I did not expect Mathematica to do all of the work for me, and figured manual substitutions would be necessary. I'm a little surprised that this one doesn't seem to work.

After trying a few other things, I think it looks like the Abs[] function is maybe a bit of a blind spot for Mathematica. It does recognize and substitute for Sqrt[w0^2] but not Abs[w0].

I've added an update to the notebook snippet.

Attachments:

simp.nb

You need to provide the expression you're applying ReplaceAll to.