  In this paper we study the long time existence of the Ricci-harmonic flow in
terms of scalar curvature and Weyl tensor which extends Cao's result
\cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral
bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a
recently result of Simon \cite{Simon2015}.
