Session E23: Flow Instability: Global Modes

Global stability analysis of flow in eccentric annular channels

A temporal linear stability analysis of laminar flow in eccentric annular channels has been performed. Spectrally-accurate algorithms using Fourier and Chebyshev series have been applied to solve the Navier-Stokes equations for fully developed laminar flow in annular channels with various eccentricities. Time-dependent, three-dimensional (radial, axial and azimuthal) perturbations were superimposed to the basic flow and the resulting linearised perturbation equations were also solved using spectrally-accurate techniques. For concentric annuli, the most unstable modes of instability were found to be axisymmetric travelling waves, when the diameter ratio was larger than 0.771 or smaller than 0.115, and spiral travelling waves otherwise; the critical Reynolds number was in all cases larger than 11544, in agreement with the literature. Unlike previous work, the present eccentric annulus analysis took into account the effect of azimuthal variation of the base flow. The critical Reynolds number for flow in eccentric annuli was much smaller than the concentric value and its value depended on the diameter ratio and the eccentricity; the streamwise wavenumber of the most unstable disturbances was much smaller than values in concentric annuli.   

Receptivity of a precessing vortex core to open-loop forcing in a swirling jet and its predictability by linear stability adjoint theory

The precessing vortex core (PVC) is a dominant coherent structure which occurs in swirling jets such as in swirl-stabilised gas turbine combustors. It stems from a global hydrodynamic instability caused by an internal feedback mechanism within the jet core. In this work, open-loop forcing is applied to a generic non-reacting swirling jet to investigate its receptivity to external actuation regarding lock-in behaviour of the PVC for different streamwise positions and Reynolds numbers. The forcing is periodically exerted by zero net mass flux synthetic jets which are introduced radially through slits inside the duct walls upstream of the swirling jet's exit plane. Time-resolved pressure measurements are conducted to identify the PVC frequency and stereo PIV combined with proper orthogonal decomposition in the duct and free field is used to extract the mean flow and the PVC mode. The data is used in a global linear stability framework to gain the adjoint of the PVC which reveals the regions of highest receptivity to periodic forcing based on mean flow input only. This theoretical receptivity model is compared with the experimentally obtained receptivity results and the validity and applicability of the adjoint model for the prediction of optimal forcing positions is discussed.   

Emergence of three-dimensional flow structures in shock boundary layer interactions

Experiments and computations point to the emergence of three-dimensional (3D) flow structures in laminar shock boundary layer interactions in various configurations. We examine a Mach $5$ flow over a double compression ramp and reveal the presence of a bifurcation from a steady 2D to a steady 3D flow state. This is done by varying the relative angle of the two ramps which increases the interaction strength. We employ global linear stability analysis and direct numerical simulation to characterize this bifurcation and demonstrate that global instability induces 3D flow structures. We use the direct and adjoint linear equations to further investigate the origin of this instability and examine the influence of uncertainty (including the effect of geometric irregularities in the ramp and free-stream disturbances in wind tunnel) on this bifurcation.   

Resolvent analysis of shear flows using One-Way Navier-Stokes equations

For three-dimensional flows, questions of stability, receptivity, secondary flows, and coherent structures require the solution of large partial-derivative eigenvalue problems. Reduced-order approximations are thus required for engineering prediction since these problems are often computationally intractable or prohibitively expensive. For spatially slowly evolving flows, such as jets and boundary layers, the One-Way Navier-Stokes (OWNS) equations permit a fast spatial marching procedure that results in a huge reduction in computational cost. Here, an adjoint-based optimization framework is proposed and demonstrated for calculating optimal boundary conditions and optimal volumetric forcing. The corresponding optimal response modes are validated against modes obtained in terms of global resolvent analysis. For laminar base flows, the optimal modes reveal modal and non-modal transition mechanisms. For turbulent base flows, they predict the evolution of coherent structures in a statistical sense. Results from the application of the method to three-dimensional laminar wall-bounded flows and turbulent jets will be presented.   

Low-rank behavior of turbulent jets: spectral analysis and resolvent model

We show that unforced turbulent jets exhibit a low-rank behavior that is predicted by a global resolvent analysis of the mean flow. The low-rank character of the flow is unveiled in the energy spectra of three large-eddy simulations of turbulent jets representing the incompressible, transsonic and supersonic regime. Spectral proper orthogonal decomposition (SPOD) is used to extract coherent parts of the flow in the form of orthogonal modes ranked by their energy. The low-rank behavior manifests itself in large separation between the modal energies of the leading structure from the rest. In other words, a single dominant coherent structure makes up a considerable portion of the total flow energy. This behavior is most pronounced at low azimuthal wavenumbers. Despite its limitation to optimal linear perturbations, a resolvent model accurately predicts the observed trends, and allows for the physical interpretation of the empirical results. The low-rank behavior is the result of one dominant mechanism: the Kelvin-Helmholtz (K-H) type instability of the near-nozzle shear-layer. The sub-dominant modes correspond to a second, distinct, mechanism that is active downstream of the potential core.   

Role of symmetry in inertial wave and mode excitations

The flow in a rapidly rotating cylinder split in half, with the rotation in the two halves modulated harmonically with a small amplitude is studied numerically. We consider modulation frequencies ranging from zero to twice the background rotation frequency, so that the system supports inertial waves. The split in the cylinder at mid-height provides a localized perturbation from which inertial wave beams emanate, but so too do the corners where the thin modulated endwall and sidewall boundary layers meet providing localized perturbations to the rapid background rotation due to the mismatch in their fluxes. Inertial wave beams from the corners are more intense than those from the split at the cylinder mid-height. Due to finite viscosity and nonlinear flow conditions, the wave beams produce intricate patterns formed by constructive and destructive interference as they self intersect and reflect off cylinder boundaries and the axis. A phase difference between the modulations of the two cylinder halves is also imposed. The phase difference impacts the symmetries of the system and its response to the modulations. In particular, some low-order Kelvin modes are driven resonantly, and their selection depends not only on the frequency but also on the phase of the differential modulation.