  In this paper, we concern a generalized fully nonlinear curvature flow
involving $k$-th elementary symmetric function for principal curvature radii in
Eulidean space $\rnnn$, $k$ is an integer and $1\leq k\leq n-1$. For $1\leq k<
n-1$, based on some initial data and constrains on smooth positive function
defined on the unit sphere $\sn$, we obtain the long time existence and
convergence of the flow. Especially, the same result shall be derived for
$k=n-1$ without any constraint on the smooth positive function.
