We study the evolution of strong Kahler with torsion (SKT) structures via the pluriclosed flow on complex nilmanifolds, i.e., on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces, we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow, and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving symplectic forms taming a complex structure, by considering the evolution of symplectic forms via the flow induced by the Bismut Ricci form.