A flow meter well tool includes a volumetric well flow meter to be positioned in a wellbore. The volumetric well flow meter includes a hollow cylindrical tubing to be positioned in the wellbore. The tubing splits well fluid through the wellbore into an internal fluid through the volumetric well flow meter and an external fluid through an annulus between the wellbore and the volumetric well flow meter. The hollow cylindrical tubing includes an internal portion, where an internal cross-sectional area of the internal portion is less than an internal cross-sectional area of the hollow cylindrical tubing.