Measurements of exoplanetary masses and radii have revealed a population of massive super-Earths --- planets sufficiently large that, according to one dimensional models, they should have turned into gas giants. To better understand the origin of these objects, we carry out hydrodynamical simulations of planetary cores embedded in a nascent protoplanetary disk. In this first paper of a series, to gain intuition as well as to develop useful diagnostics, we focus on two-dimensional simulations of the flow around protoplanetary cores. We use the pluto code to study isothermal and adiabatic envelopes around cores of sub- to super-thermal masses, fully resolving the envelope properties down to the core surface. Owing to the conservation of vortensity, envelopes acquire a substantial degree of rotational support when the core mass increases beyond the thermal mass, suggesting a limited applicability of one-dimensional models for describing the envelope structure. The finite size of the core (relatively large for super-Earths) also controls the amount of rotational support in the entire envelope. Steady non-axisymmetric shocks develop in the supersonic envelopes of high-mass cores, triggering mass accretion and turbulent mixing in their interiors. We also examine the influence of the gas self-gravity on the envelope structure. Although it only weakly alters the properties of the envelopes, the gas gravity has significant effect on the properties of the density waves triggered by the core in the protoplanetary disk.