A new method to seamlessly simulate the continuous growth of droplets advected by turbulent flow inside a cumulus cloud was developed from first principle. A cubic box ascending with a mean updraft inside a cumulus cloud was introduced and the updraft velocity was self-consistently determined in such a way that the mean turbulent velocity within the box vanished. All the degrees of freedom of the cloud droplets and turbulence fields were numerically integrated. The box ascended quickly inside the cumulus cloud due to the updraft and the mean radius of the droplets grew from 10 to 24 μm for about 10 min. The turbulent flow tended to slow down the time evolutions of the updraft velocity, the box altitude and the mean cloud droplet radius. The size distribution of the cloud droplets in the updraft case was narrower than in the absence of the updraft. It was also found that the wavenumeber spectra of the variances of the temperature and water vapor mixing ratio were nearly constant in the low wavenumber range. The future development of the new method was argued.