Let $R=Z_4+uZ_4$, $R_n={R[x]}/{(x^n-(2u-1))}$, where $u^2=0$, $n=2^e$. By studying the structures of $(2u-1)$-constacyclic codes over $R$ of length $n$, we obtain the generators for these codes and classify all $(2u-1)$-constacyclic codes of length $n$ over $R$. Further, we study the Hamming distance distributions of $(2u-1)$-constacyclic codes of length $n$ over $R$. Finally, we give the structures of the duals of $(2u-1)$-constacyclic codes over $R$ of length $n$ and self-orthogonal and self-dual $(2u-1)$-constacyclic codes of length $n$ over $R$.