Numerical hydrodynamic simulations of three-dimensional, turbulent, fully compressible thermal convection have been performed for an internally heated layer with varying degrees of density and temperature stratification as a preliminary study of realistic convection in the primordial solar nebula and other protostellar disks. Gravity is made to vary linearly as distance from the midplane, approximating the conditions in thin, gaseous protostellar disks. These simulations have been performed in a periodic channel at unrealistically low Reynolds numbers in order to resolve all relevant scales of turbulent motion; and differential rotation, another important feature of protostellar disks, has not been included. In this paper we describe the numerical techniques used to perform the hydrodynamic simulations; we examine the convective now structure and turbulence statistics; and we discuss the effects of compressibility and stratification on the turbulent convection. The turbulence (rms) Mach numbers M(t) in the central convective regions are found to be less than or similar to 0.25, and acoustic terms, scaling as M(t)(2), are found to be negligible except near (unrealistic) solid boundaries. Near these walls, there is enhanced compression and rapid tangential flow, which sometimes becomes supersonic and exhibits weak shocks. Temperature deviations of less than 25% from the mean in horizontal planes are typically observed. Density stratification is shown to have significant effects on the convection, primarily due to increased thermal diffusivity in the outer, more rarefied regions, which (1) reduces the efficiency of convective heat transport, and (2) can stabilize the gas against convection in the outermost regions. Some simulation results are presented in which the convection is realistically bounded by convectively stable regions that effectively buffer the interior convective flow from the physically unrealistic boundaries. Significant overshooting of convective motions into the stable region is observed, but, because of the inefficient nature of this convection, little penetration of convective layer into the stable layer is observed. The stable layer acts as a ''soft wall'' with less enhancement of compression and horizontal velocities, making supersonic flow and shocks unlikely in turbulent now driven solely by convection in protostellar disks.
