I recently came across this reaction in my undergraduate labs. The experiment is taken from this article: J. Chem. Educ., 2008, 85, 413.
The condensation of 2-methoxynaphthalene with cinnamoyl chloride leads to the formation of 9-hydroxyphenalenone:
I think it's fairly clear that the first step is a Friedel-Crafts acylation. Subsequently to form the new C-C bond, there must be a Michael addition of some sort; however this would imply that the phenyl group leaves as "$\ce{Ph-}$". There are two things I'm not very sure about:
How does the phenyl group even leave?
How does the oxygen get demethylated? I am guessing that the mechanism should be similar to that with $\ce{BBr3}$, i.e. coordination of oxygen to the Lewis acid followed by nucleophilic attack of $\ce{Cl-}$ on the methyl carbon. Would that be right?
I looked up the original synthesis, which was reported quite a long time ago, in J. Org. Chem., 1941, 6, 558. The authors write that the reaction proceeds via the intermediacy of this compound:
which suggests that the initial nucleophile in the Michael addition is not the aromatic ring but rather the -OMe (or -OH) group. Unfortunately, that just makes things more confusing.