  In this paper, we introduce two discrete curvature flows, which are called
$\alpha$-flows on two and three dimensional triangulated manifolds. For
triangulated surface $M$, we introduce a new normalization of combinatorial
Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming
at evolving $\alpha$ order discrete Gauss curvature to a constant. When
$\alpha\chi(M)\leq0$, we prove that the convergence of the flow is equivalent
to the existence of constant $\alpha$-curvature metric. We further get a
necessary and sufficient combinatorial-topological-metric condition, which is a
generalization of Thurston's combinatorial-topological condition, for the
existence of constant $\alpha$-curvature metric. For triangulated 3-manifolds,
we generalize the combinatorial Yamabe functional and combinatorial Yamabe
problem introduced by the authors in \cite{GX2,GX4} to $\alpha$-order. We also
study the $\alpha$-order flow carefully, aiming at evolving $\alpha$ order
combinatorial scalar curvature, which is a generalization of Cooper and Rivin's
combinatorial scalar curvature, to a constant.
