The primary challenge of sensitivity analysis in LES is its chaotic dynamics. Sensitivity information, most obviously the gradient of the quantity-of-interest with respect to all design variables, is essential to optimization algorithms and analyses of how robust a design or prediction is. The finite difference method to obtain sensitivities is popular and easy to implement, but is surprisingly problematic in LES. This method approximates derivatives by selecting a pair of adjacent design points, then evaluating the quantity-of-interest at these points, and dividing their difference by the distance between the design points. When both objective functions are polluted by sampling errors due to finite time-averaging, the resulting derivative is polluted by a large error, specifically twice the sampling error divided by the distance between the design points. The sampling error decreases slowly as the averaging-time, and thus reducing it sufficiently for
accurate sensitivity information could require impractically long simulations.
In addition to the sampling error, the more standard problem with finite difference sensitivities is that they require N+1 simulations for N design variables. To overcome this limitation, Jameson introduced the adjoint method into CFD. By simultaneously computing the derivative to many design variables, the adjoint method has become widely used in aerodynamic design optimization. In LES and DES, however, the adjoint method encounters a fundamental challenge: the “butterfly effect” of chaotic dynamics.
By resolving the large scales of turbulence, LES inherits the chaotic dynamics characteristic of turbulence. A small perturbation, perhaps a slightly different mesh, can utterly change the time history of a simulation. This sensitivity to small perturbations (the “butterfly effect”) causes the adjoint method to diverge when integrated over a long time period T. The divergent adjoint solutions, observed in a variety of flow solvers.
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