A space-time adaptive algorithm is presented to solve the incompressible Navier-Stokes equations. Time discretization is performed with the BDF2 method while continuous, piecewise linear anisotropic finite elements are used for the space discretization. The error indicator in time is justified through an posteriori upper bound of the error due to time discretization. The error indicator in space is taken from [1]. Numerical experiments confirm the sharpness of the error indicator and the efficiency of the adaptive algorithm. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).