Hydrodynamic Stability
Linear tendencies of flowfields, including their unforced and forced responses, can provide us valuable information on large-scale dynamics, inherent instability mechanisms, controllability, and effective actuator-placement strategies. Our emphasis is on generalizing the fundamental constructs of stability theory utilizing matrix-free approaches, to make it applicable to a wide spectrum of flows encompassing compressibility and three-dimensional inhomogeneity.
When the base flow is 3D and time-invariant
In compressible 3D flows, when the basic state is a laminar or a time-averaged solution of the Navier-Stokes equations, we employ the Mean Flow Perturbation (MFP), in conjunction with Krylov subspace iterations. MFP implicitly linearizes the governing equations using a constant body-force constraint
When the base flow is 3D and time-periodic
When the basic state is time-periodic, we utilize a periodic body-force constraint to implement a Floquet-type analysis. This also accounts for the mean flow distortion induced due to the nonlinear saturation of the primary instability. This approach is named as Unsteady Flow Perturbation (UFP).
The image to the right shows the utility of UFP in efficiently predicting the spectral and spatial signatures of transition in a hypersonic boundary layer. A direct numerical simulation is used as the truth model here. In comparison, the linear response of the laminar basic state does not identify transition accurately.
When the base flow is unsteady and stochastic
This is a general scenario, where simplifications are not straightforward. We utilize a linear perturbation tracking technique termed Synchronized Large-Eddy Simulations (SLES) to follow the linear evolution of small-perturbations in turbulent flows. This provides us time-accurate linear tendencies of the system. Shown below is the implementation of SLES, and its application to identify scale-specific intermittency and directivity of linear perturbations in a turbulent jet.
