A Chern-Calabi flow on Hermitian manifolds

We study an analogue of the Calabi flow in the non-K\"ahler setting for compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern-Ricci-flat metric in the $\partial\bar{\partial}$-class of the initial metric.