Since $\operatorname{cf}(\aleph_\omega)=\omega$, $\aleph_\omega<\aleph_\omega^\omega$. However, can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$? I am especially interested in models for which $2^{\aleph_n}<\aleph_\omega$ for all $n<\omega$ in addition to our other condition.

Traditional ways of adding subsets of $\aleph_\omega$ seem unreliable, as they appear to collapse $\aleph_\omega$ to $\aleph_0$ or raise $2^{\aleph_n}$ at the same time.