This ref. may get you close, or at least give you some more references to look at...
Driver, D.M. and Seegmiller, H.L., "Features of a Reattaching Turbulent Shear Layer in Divergent Channel Flow," AIAA Journal, Vol. 23, No. 2, Feb. 1985, pp. 163-171.
The paper is referenced by some numerical studies of the problem you are looking at (the numerical studies are for air flow).
"and without the use of numerical modeling? "
The numerical models I've seen (NASA/AF NPARC and WIND CFD codes) do a fairly poor job of predicting reattachment distance. You (will) do yourself credit to find as much published experimental data as possible, and hope to correlate to your specific case.