If \psi\colon M^n\to \mathbb{R}^{n+1} is a smooth immersed closed hypersurface, we consider the functional \mathcal{F}_m(\psi) = \int_M 1 + |\nabla^m \nu |^2 \, d\mu, where \nu is a local unit normal vector along \psi , \nabla is the Levi-Civita connection of the Riemannian manifold (M,g) , with g the pull-back metric induced by the immersion and \mu the associated volume measure. We prove that if m>\lfloor n/2 \rfloor then the unique globally defined smooth solution to the L^2 -gradient flow of \mathcal{F}_m , for every initial hypersurface, smoothly converges asymptotically to a critical point of \mathcal{F}_m , up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional \mathcal{F}_m .