Instead of the pressure-correction scheme used in the FEniCS tutorials 07 & 08, what about one of the operator-splitting schemes described by Elman, Silvester, & Wathen (2014, §10.2.2), e.g. that of Smith & Silvester (1997).
Elman, H. C., Silvester, D. J.,, Wathen, A. J. (2014). Finite Elements and Fast Iterative Solvers, Oxford University Press
Smith, A. & Silvester, D. (1997). Implicit algorithms and their linearization for the transient incompressible Navier–Stokes equations. IMA Journal of Numerical Analysis, 17, 527-545. doi: 10.1093/imanum/17.4.527
Instead of the pressure-correction scheme used in the FEniCS tutorials 07 & 08, what about one of the operator-splitting schemes described by Elman, Silvester, & Wathen (2014, §10.2.2), e.g. that of Smith & Silvester (1997).