Hydrodynamical instabilities are likely the main source of turbulence in weakly ionized regions of protoplanetary disks. Among these, the vertical shear instability (VSI) stands out as a rather robust mechanism due to its few requirements to operate, namely a baroclinic stratification, which is enforced by the balance of stellar heating and radiative cooling, and short thermal relaxation timescales. Our goal is to characterize the transport of angular momentum and the turbulent heating produced by the nonlinear evolution of the VSI in axisymmetric models of disks around T Tauri stars, considering varying degrees of depletion of small dust grains resulting from dust coagulation. We also explore the local applicability of both local and global VSI-stability criteria. We modeled the gas-dust mixture in our disk models by means of high-resolution axisymmetric radiation-hydrodynamical simulations including stellar irradiation with frequency-dependent opacities. This is the first study of this instability to rely on two-moment radiative transfer methods. Not only are these able to handle transport in both the optically thin and thick limits, but also they can be integrated via implicit-explicit methods, thus avoiding the employment of expensive global matrix solvers. This is done at the cost of artificially reducing the speed of light, which, as we verified for this work, does not introduce unphysical phenomena. Given sufficient depletion of small grains (with a dust-to-gas mass ratio of $10<!PCT!>$ of our nominal value of $10^ $ for $<0.25$ mu m grains), the VSI can operate in surface disk layers while being inactive close to the midplane, resulting in a suppression of the VSI body modes. The VSI reduces the initial vertical shear in bands of approximately uniform specific angular momentum, whose formation is likely favored by the enforced axisymmetry. Similarities with Reynolds stresses and angular momentum distributions in 3D simulations suggest that the VSI-induced angular momentum mixing in the radial direction may be predominantly axisymmetric. The stability regions in our models are well explained by local stability criteria, while the employment of global criteria is still justifiable up to a few scale heights above the midplane, at least as long as VSI modes are radially optically thin. Turbulent heating produces only marginal temperature increases of at most $0.1<!PCT!>$ and $0.01<!PCT!>$ in the nominal and dust-depleted models, respectively, peaking at a few (approximately three) scale heights above the midplane. We conclude that it is unlikely that the VSI can, in general, lead to any significant temperature increase since that would either require it to efficiently operate in largely optically thick disk regions or to produce larger levels of turbulence than predicted by models of passive irradiated disks.
