Link between reluctance and inductance

[rs20]

I'll start off with my question:
-- Is my understanding of inductors wrong, or
-- Is my $20 eBay component identifier failing me as an L meter?

Here's what I did: I wrapped a (presumably steel) nail with Kapton tape, and then wrapped a whole bunch (est. 250) of windings of enamelled copper wire around it. De-enammeled the ends with a flame, plugged it into my cheap meter, and it reads 0.7 ohms + 0.09mH. So, then I add a steel screwdriver touching both ends of the nail (yes, the nail was bent to enable this). I was expecting the screwdriver to "complete the magnetic circuit", increasing the permeance* and thus increasing the inductance. Yet, I take another reading with the meter, and it reads 0.7 ohms + 0.09mH... exactly the same as before. Am I down in the noise floor of the component identifier, or am I wrong to expect a change in inductance? I feel like gapped transformers have a lower mutual inductance because the gap sorta ""breaks"" the magnetic circuit, and was expecting a similar effect here.

* (I am in no way qualified to use this term with such confidence)

[johansen]

no, you're down below the noise floor of eddy current and hysteresis eating up the energy.

try the same experiment again using a ferrite core and you'll find yourself back in reasonable territory.

[Circlotron]

The circumference of the nail is probably acting as a shorted turn secondary winding. That's why transformers have strips of laminations with an insulating coating on each.