In 2011 J.Streets and G.Tian introduced a family of metric flows over a complex Hermitian manifold. We consider one particular member of this family and prove that if the initial metric has Griffiths positive Chern curvature, then this property is preserved along the flow.  On a manifold with Griffiths non-negative Chern curvature this flow has nice regularization properties, in particular, for any t>0 the zero set of Chern curvature becomes invariant under certain torsion-twisted parallel transport. If time permits, we discuss applications of the results to some uniformization problems.