Online Gradient-Flow Optimization Over the Statistical Steady-State of Unsteady Turbulent Flows

Developing practical techniques for optimizing over the steady-state statistics of unsteady turbulent flow simulations is an important but challenging problem. The chaoticity of turbulence causes adjoint gradient calculations (the usual tool of choice for optimization) to diverge exponentially. Although some methods have been proposed to stabilize this, they are generally not scalable to representative grid sizes.

We present a new online gradient-flow (OGF) method that forward-propagates an unbiased finite-difference estimator for the gradient of the objective function while simultaneously optimizing the parameters. Unlike in adjoint methods, the gradient estimate does not explode. The online nature of OGF enables faster and more robust optimization compared to iterative finite-difference gradient descent.

The method is applied to DNS of compressible forced homogeneous isotropic turbulence with up to 17M grid points. OGF reduces the objective function by several orders of magnitude and successfully recover the optimal stochastic forcing amplitude for a target mean-squared velocity fluctuation to within 1%. Given that the usual quantities of interest from unsteady turbulent flow simulations are steady-state statistics, OGF has the potential to enable engineers and scientists to develop more accurate turbulence models, more efficient flow controllers, and better design geometries at a significantly reduced cost compared to existing methods.