Digraph: Flow with length constraint

I'm considering the following problem: Given a directed graph $$G = (V,A)$$ with unit capacities, determine if there exists a flow from $$x$$ to $$y$$ ($$x = y$$ is allowed) with a specific length $$k$$, i.e., if $$f(a)$$ denotes the flow on arc $$a \in A$$, we want to find a flow from $$x$$ to $$y$$ such that $$\sum_{a \in A} f(a) = k$$.

Do you know if this problem is of interest in research and if yes, what it is called?

(Note that this problem is equivalent to finding a trail (i.e. a closed walk without repeating edges) from $$x$$ to $$y$$ of length $$k$$).