This is the second part of a two-part special issue of the Philosophical Transactions of the Royal Society A, which recognizes, and hopefully encourages, the growing convergence of interests amongst mathematicians and physicists to scale the turbulence edifice. This convergence is explained in more detail in the editorial which accompanies the first part (Bec et al . 2022 Phil. Trans. R. Soc. A 380 , 20210101. ( doi:10.1098/rsta.2021.0101 )) and includes a tribute to our friend, collaborator and mentor Uriel Frisch, to whom these special issues are dedicated. Uriel, the principal architect of the Nice School of Turbulence, remains the finest example of this synthesis of mathematics and physics in tackling the outstanding problem of turbulence. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.