We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number (
$Ra\in [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies
$f_{N}\gg f_{c}$, where
$f_{c}$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like
$k_{\bot }^{4}\,f^{-13/2}$ for weakly damped waves (with
$k_{\bot }$ and
$f$ the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with
$z$, the distance from the convective region, like
$z^{-13/8}$.