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We perform numerical simulations of gravitational waves (GWs) induced by hydrodynamic and hydromagnetic turbulent sources that might have been present during the cosmological QCD phase transition, with typical turbulent scales as large as a sizable fraction of the Hubble scale at that time. The resulting GWs are found to have an energy fraction of about $10^{-9}$ of the critical energy density in the nHz range today, which means they might have been detected by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration already. In summary, we find the following: 1) At low frequencies, nonhelical hydromagnetic turbulence generates GWs with a spectrum shallower than expected, providing a better chance of detection; 2) At higher frequencies, the spectral GW energy exhibits a sharp drop by up to five orders of magnitude for vortical turbulence, but decays with a steep power law for acoustic turbulence; 3) GW energy spectrum becomes steeper at low frequencies with nonzero graviton mass, providing a potential constraint on massive gravity theories.