Least Squares Shadowing for Sensitivity Analysis of Turbulent Fluid Flows

Abstract

Computational methods for sensitivity analysis are invaluable tools for aerodynamics research and engineering design. However, traditional sensitivity analysis methods break down when applied to long-time averaged quantities in turbulent fluid flow fields, specifically those obtained using high-fidelity turbulence simulations. This is because of a number of dynamical properties of turbulent and chaotic fluid flows, most importantly high sensitivity of the initial value problem, popularly known as the "butterfly effect".The recently developed least squares shadowing (LSS) method avoids the issues encountered by traditional sensitivity analysis methods by approximating the "shadow trajectory" in phase space, avoiding the high sensitivity of the initial value problem. The following paper discusses how the least squares problem associated with LSS is solved. Two methods are presented and are demonstrated on a simulation of homogeneous isotropic turbulence and the Kuramoto-Sivashinsky (KS) equation, a 4th order chaotic partial differential equation. We find that while LSS computes fairly accurate gradients, faster, more efficient linear solvers are needed to apply both LSS methods presented in this paper to larger simulations.

Multiple Shooting Shadowing for Sensitivity Analysis of Chaotic Systems and Turbulent fluid flows

“…This adjoint multiple shooting algorithm is derived in great detail in the appendices of Blonigan et al 16 The 4th step of the algorithm calls for an update of the initial and terminal conditions v i and w i . Current implementations use MINRES, a Krylov subspace method for solving symmetric, sparse matrices.…”

“…15,20 The second approach is multiple shooting LSS, or LSS type III. 16 This approach uses a multiple shooting method 21 to solve equations (5), (6), and (7), with α = 0. In multiple shooting, the simulation time span T 0 < t < T 1 is divided into K segments, as shown in figure 1.…”

“…The first approach is called transcription LSS or LSS type II. 16 Transcription LSS is when equations (5), (6), and (7) are discretized with a forward euler scheme and solved simultaneously for all time T 0 < t < T 1 . See Wang et al 13 for detailed derivations of the system of equations, referred to as the "KKT system".…”

“…The following paper discusses the ergodicity breaking error in LSS and proposes a potential solution using a hybrid of the multiple shooting implementation of LSS (referred to as LSS type III in previous literature 16 ) and Lea and Eyink's ensemble adjoint method. 6,8 This hybrid method, Multiple Shooting Shadowing (MSS), can reduce or eliminate the ergodicity breaking error in LSS.…”

Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NI-LSS)

“…24 LSS avoids many of the shortcomings of previous methods and has been successfully applied to some dynamical systems such as the Lorenz 63 system and a modified Kuramoto-Sivashinsky equation. 14,25 LSS has also been applied by Blonigan et al to engineering applications such as sensitivity analysis for airfoils. 26 The tangent version of the LSS theorem was first proven for discrete maps by Wang.…”