A low-storage adjoint lattice Boltzmann method for the control of incompressible flows

In this paper, we present a low-storage adjoint lattice Boltzmann method (LSALBM) for the control of unsteady incompressible flows. The core of the method is to adopt a velocity-independent approximate equilibrium for the adjoint variable. Thus, the space-time history of the flow field is not needed to be stored for the adjoint variable and the well-known drawback of the existing adjoint methods is overcome. The new method is validated through both initial-value and boundary control problems. Numerical examples demonstrate that the LSALBM agrees well with the standard adjoint lattice Boltzmann method when the terminal time of the problem is not too long. For long-time unsteady flow problems, the standard method loses its effectiveness, while the LSALBM still produces ideal results. Moreover, the efficiency of our method for steady-state problems is shown by simulating a lid-driven grooved cavity flow.