  In the dynamical systems approach to describing turbulent or otherwise
chaotic flows, an important quantity is the Lyapunov exponents and vectors that
characterize the strange attractor of the flow. In particular, knowledge of the
Lyapunov exponents and vectors will help identify perturbations that the system
is most sensitive to, and quantify the dimension of the attractor. However,
reliably computing these Lyapunov quantities requires robust nu- merical
algorithms. While several perturbation-based techniques are available in
literature, their application to commonly used turbulent flow solvers as well
the numerical convergence properties have not been studied in detail. The goal
of this work is two-fold: a) develop a robust algorithm for obtaining Lyapunov
exponents and vectors for low-Mach based sim- ulation of turbulent flows, b)
quantify the spatial and temporal convergence properties of this algorithm
using a series of progressively complex flow problems. In particular, a manu-
factured solutions approach is devised using the Orr-Sommerfeld (OS)
perturbation theory to extract Lyapunov exponents and vectors from OS eigen
solutions. While individual test cases show interesting results, overall the
spatial convergence rates of Lyapunov exponents follow the truncation order for
the discretization scheme. However, temporal convergence rates were found to be
only a weak function of time step used. Additionally, for some config-
urations, the convergence properties are found to depend on the Lyapunov
exponent itself. These results indicate that convergence properties do not
follow universal rates, and require careful analysis for specific
configurations considered.
