Finding equilibrium solutions of two-dimensional Kolmogorov turbulence via selective modification

We discover equilibrium solutions in two-dimensional Kolmogorov turbulent ﬂow by selectively altering the time rate of change of energy and dissipation rate of the ﬂow. The strategy automatically identiﬁes additional forcing terms added to the Navier–Stokes equation that direct the turbulent ﬂow to a steady state. Switching oﬀ the additional forcing terms from this state allows the Kolmogorov ﬂow to settle near the stable attractor of certain unstable equilibrium solutions. Selectively increasing the energy of the ﬂow and selectively decreasing the dissipation rate have similar eﬀects in terms of ﬂow modiﬁcation and obtained steady-state forcing ﬁelds. We recover some known equilibrium solutions from Chandler & Kerswell [8] and Farazmand [10] for 𝑅𝑒 = 40 and discover a new solution for 𝑅𝑒 = 60. The steady states of the modiﬁed governing equation accelerate the convergence of the Newton searches to ﬁnd equilibrium solutions of Kolmogorov ﬂow.