Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow

Abstract

This paper uses compressible flow simulation to analyze the hyperbolicity, shadowing directions, and sensitivities of a weakly turbulent three dimensional cylinder flow at Reynolds number 525 and Mach number 0.1.By computing the first 40 Covariant Lyapunov Vectors (CLVs), we find that unstable CLVs are active in the near-wake region, whereas stable CLVs are active in the far-wake region. This phenomenon is related to hyperbolicity since it shows that CLVs point to different directions; it also suggests that for open flows there is a large fraction of CLVs that are stable. However, due to the extra neutral CLV and the occasional tangencies between CLVs, our system is not uniform hyperbolic.By the Non-intrusive least-squares shadowing (NILSS) algorithm, we compute shadowing directions and sensitivities of long-time-averaged objectives. Our results suggest that shadowing methods may be valid for general chaotic fluid problems.

Adjoint sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)

“…For example, the 3D cylinder flow we investigate later in this paper has at least two neutral CLVs, corresponding to translations in time and in the span-wise directions, due to the periodic boundary condition. In fact, in [27] we also showed that the smallest angle between tangent CLVs depends on meshes and may fall below a threshold value: this further violates our assumptions. However, we did found the trend that angles between tangent CLVs gets larger when their indices are further apart: this property is related to hyperbolicity, but has not been well investigated yet.…”

“…First, our system has at least two neutral CLVs: the first one corresponds to the common time translation of continuous dynamical systems, and the second corresponds to span-wise translations due to the periodic boundary conditions. Second, due to the similarity of this fluid problem with the one investigated in [27], whose tangent CLVs appear to have occasional tangencies, it is reasonable to assume that adjoint CLVs in our current system may also have occasional tangencies. Still, like NILSS did in [27], NILSAS computes a correct sensitivity: this en- courages us to test NILSS and NILSAS on more general chaotic systems.…”

“…A major variant of NILSS is the Finite Difference NILSS (FD-NILSS) algorithm [26], whose implementation requires only primal solvers, but not tangent solvers. FD-NILSS has been applied to several complicated flow problems [27,26] which were too expensive for previous sensitivity analysis methods.…”

Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)

“…The LEs, CLVs and shadowing directions of the same physical problem, on both the current mesh and a finer mesh with twice as many cells, are studied with more details in [40], which shows that for both meshes, (1) there are only a few unstable CLVs, (2) CLVs are active at different area in the flow field, indicating angles are large between CLVs whose indices are far-apart, and (3) shadowing directions exists and can give accurate sensitivities. Moreover, [40] also plots snapshots of CLVs and shadowing directions. Using above settings, the cost of FD-NILSS is from integrating the primal solution over 400 × 200 × 42 = 3.36 × 10 6 time steps.…”

“…For very chaotic systems, if we still have m us m, then NILSS and NILSAS may still be competitive in computational efficiency; at least, the idea of the 'non-intrusive' formulation, that is, reducing the computation to unstable subspace, will still be important. Current investigation on some computersimulated fluid systems all have m us ≤ 0.1%m [32,36,39,40], but we do not yet have a good estimation for very chaotic systems. On the other hand, there are systems with m us ≈ m, such as Hamiltonian systems, which has equally many stable and unstable CLVs; for these systems, NILSS or NILSAS may not be faster than other methods.…”