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Current theories on planetary formation suggest that the formation of giant planets is related to their quick migration towards the central star as a result of protoplanet–disc interactions on a time‐scale of the order of 10 5  yr, for objects of nearly 10 terrestrial masses. Such a time‐scale should be smaller by an order of magnitude than that of gas accretion on to the protoplanet during the hierarchical growth of protoplanets from collisions with other minor objects. These arguments have recently been analysed using  N ‐body and/or fluid‐dynamics codes or a mixture of the two. In this work, we have performed inviscid two‐dimensional simulations, using the smoothed particle hydrodynamics (SPH) method, to study the migration of one protoplanet, in order to evaluate the effectiveness of the accretion disc in the protoplanet dragging towards the central star, as a function of the mass of the planet itself and of the disc tangential kinematics. For this purpose, the SPH scheme is considered suitable to study the roles of turbulence, kinematic and boundary conditions, because of its intrinsic advective turbulence, especially in two‐ and three‐dimensional codes. Simulations are performed in both sub‐Keplerian and Keplerian disc kinematic conditions as a parameter study of protoplanetary migration if moderate and consistent deviations from Keplerian kinematics occur. Our results show the migration times of a few orbital periods for Earth‐like planets in sub‐Keplerian conditions. For Jupiter‐like planets, estimates show that about 10 4  orbital periods are needed to half the orbital size. Time‐scales of planet migration are strongly dependent on the relative position of the planet with respect to the shock region near the centrifugal barrier of the disc flow.

How initial and boundary conditions affect protoplanetary migration in a turbulent sub‐Keplerian accretion disc: 2D non‐viscous SPH simulations