Low order Carleman simulation of steady state flows at mild Reynolds numbers

Abstract

It is shown that the second-order Carleman truncation (C2) captures the steady state of the decaying forced logistic with second-order accuracy in the forcing parameter, provided such forcing remains below a critical value of 1/2. We then investigate the extension of these results to spatially extended systems by means of Carleman Lattice Boltzmann simulations of a two-dimensional periodically forced flow. It is found that C2 correctly reproduces the low-Reynolds-number (between 1 and 10) quasi-linear dynamics, well above the critical value of the forced logistic equation. However, C2 fails to capture more complex steady states characterised by persistent vortical structures at Reynolds numbers around 20. Interestingly, the divergence observed in the latter case shows qualitative similarities with the runaway behaviour of the C2 linearization of the growing logistics, suggesting that suitable closures and/or change of variables may improve the convergence of the C2 procedure.