Session L25: Flow Instability: Global Modes

Weakly nonlinear behavior of transonic buffet on airfoils

The occurrence of large scale buffeting flow can result in a reduced operating envelope for an aircraft.  In practice, the buffet-onset boundary is defined in terms of finite-amplitude lift fluctuations exceeding a threshold value.  For transonic flow conditions, the buffeting flow is associated with oscillations in the shock position that result in large-amplitude lift fluctuations.  For both airfoils and wings, these oscillations have been linked to global flow instabilities that arise from a Hopf bifurcation.  We employ a combination of numerical simulations and global stability analysis to investigate the near-critical behavior of the oscillatory buffet-onset instability on airfoils.  The flow is governed by the unsteady RANS equations, with a basic state provided by a steady RANS solution. In the weakly nonlinear formulation, the disturbance amplitude is described by the Landau equation.  The linear growth rate can be determined from either the simulations or the stability analysis, and the Landau constant is derived from simulations resulting in finite-amplitude equilibrium states.  The results show that the Landau constant is nearly independent of Mach number and angle of attack for a given airfoil.  Using the Landau constant derived from a small number of simulations, the stability analysis can be employed to efficiently capture the essential finite-amplitude behavior needed to estimate the buffet-onset boundary.   

On the Stability of a Separation Bubble in a Supersonic Compression Ramp Flow

Linear stability of supersonic flows over short compression corners has been investigated using direct simulation Monte Carlo (DSMC) and global linear modal stability theory. Supersonic free stream at M=3, Re=1.1E4, and large ramp angles between 30 and 42° have been considered. Two-dimensional steady laminar base flows were generated with DSMC and exhibited large separation bubbles extending to the plate leading edge at all ramp angles. The recirculation strength is found to be higher than 10% for all the cases, and scaled angles calculated using triple deck theory (Egorov et al. DOI:10.2514/6.2011-730) are higher than 6.0. Both indicators could suggest self-excitation of steady three-dimensional global instabilities (Theofilis et al. DOI: 10.1098/rsta.2000.0706). However, solution of the BiGlobal eigenvalue problem that the known stationary three-dimensional global mode of separation is stable, but a previously unknown leading edge (LE) mode is unstable over a range of spanwise wavenumbers at the highest ramp angle. Close to the leading edge, the amplitude function of the LE mode peaks along the leading edge shock and the dividing streamline of laminar separation, while closer to the compression corner the two branches merge in a single periodic structure. The spatial structure and amplification rate of the LE mode are confirmed independently by three-dimensional (spanwise periodic) DSMC simulations. To the best of the authors' knowledge, the leading edge global mode of compression corners is identified for the first time in the present work.   

Transient growth in accelerating and decelerating laminar planar flows

Although the stability of planar flows with constant pressure gradient or wall motion is well understood, little focus has been put towards understanding the stability properties of unsteady baseflows. Two major challenges when investigating these flows include: 1) determining the laminar flow about which to analyze stability and 2) determining how to incorporate the time-varying linear operator. Here we overcome these challenges by first deriving an analytical solution for laminar profiles of planar flows with arbitrary wall motion and pressure gradient. Then, we study the stability of specific flows by investigating the nonnormal growth of perturbations through the time-varying linearized equations of motion. In particular, we investigate exponentially decaying acceleration and deceleration of wall motion and flow rate. The accelerating cases exhibit growth comparable to stationary flows, while the decelerating cases exhibit massive nonnormal growth -- at a Reynolds number of 800 growth of the decelerating flow is O(10⁵) times larger than growth of the stationary flow. As the rate of deceleration and Reynolds number increase the perturbations become further amplified, and the largest growth rate moves from a spanwise disturbance to a streamwise disturbance upon increasing these values.   

The influence of the secondary mean on linear analyses of turbulent square duct flow

Turbulent flow through square and rectangular ducts features Prandtl's secondary flow of the second kind, where the mean flow presents streamwise vortex pairs located near each corner. While the magnitude of these secondary flows are small (often a few percent of the streamwise component), here we show that they can have a substantial effect on stability and resolvent analyses of the mean-linearized system. In particular, the inclusion of such secondary mean components leads to the emergence of leading modes that are not otherwise present, particularly for streamwise-elongated structures. We show why such modes emerge mathematically, but through a comparison with direct numerical simulation data, argue that they are not necessarily physically relevant. When performing linear analyses with such secondary mean components removed, we instead can obtain leading (almost stationary) modes that resemble the secondary flow itself, potentially providing an alternative lens through which to study the emergence of such secondary flows. We lastly discuss the broader implications of these findings for other flows that feature secondary mean components.   

Early detection of global instability via recurrence plots and neural networks

We present a data-driven approach for the early detection of global instability in an axisymmetric low-density jet, a prototypical open shear flow. Under certain conditions, such jets are known to undergo a Hopf bifurcation to global instability in the form of self-excited limit-cycle flow oscillations. Such oscillations are undesirable in many situations, especially when they couple with structural or acoustic modes. Our solution combines the topological visualization capabilities of recurrence plots (RPs) with the classification capabilities of neural networks to create a hybrid framework for early detection of global instability. Specifically, we construct two-dimensional unbinarized RPs from time traces of the local jet velocity measured experimentally in the unconditionally stable fixed-point regime. Using these RPs, we train a residual neural network (ResNet), a deep learning model that uses residual connections to overcome the vanishing gradient problem. Our results indicate that this hybrid framework can generate early warning indicators of global instability using only data collected before the bifurcation point, providing a useful tool for avoiding limit-cycle oscillations in open shear flows.   

Bi-global stability and forced response analysis of reacting, swirling jets.

Swirling jets are canonical flow fields used to stabilize flames in combustion systems. In this study, a bi-global hydrodynamic stability analysis is used to model the unsteady structures and vortical hydrodynamic modes of a harmonically forced, reacting swirling flow. The base state of this study is an axisymmetric, swirling and reacting mean flow computed with LES, based upon a commercial nozzle. First, the unforced, natural stability eigenmodes are analyzed for the linearized Navier Stokes equations around the base state using finite elements in COMSOL. Then, an empirical velocity transfer function is calculated by determining the flow response to a harmonically varying 150-1500 Hz velocity disturbance imposed normal to the inflow boundary. A key result of this study shows that 1050 Hz inlet-forced disturbance is dominant in amplification of axial velocity disturbances. Excitations around this St = 0.375 mode lead to axial velocity disturbances with amplification factors varying between 20 and 50 times at axial locations 0.24 < z/d_Sw < 1.5  and over 60 times at downstream axial locations  z/d_Sw > 1.5. Lastly, this study also considers comparison of stability solutions in a fully or tri-global framework for a 3-D, cartesian base flow as a future direction.   

Effect of downstream wall on linear global instability of hypersonic flow over axi-symmetric open cavities

This study follows upon the experimental findings of (Das & Cohen - DOI: 10.2514/1.J055895), which demonstrated the attenuation of unsteadiness by modifying the downstream wall of an open cavity at transonic conditions. Qualitatively the effect was attributed to the disruption of the Rossiter resonance mechanism. Global stability analysis associated Rossiter modes with global modes of the steady subsonic (Bres & Colonius - DOI: 10.1017/s0022112007009925) and supersonic (Sun et al. - DOI: 10.1017/jfm.2017.416) 2D laminar baseflow. In this work, we extend the analysis to the hypersonic regime, employing a BiGlobal linear stability analysis of compressible open cavity flows. The impact of varying downstream wall geometry on the breakdown of self-sustaining oscillation mechanism is explored. A multi-domain matrix forming method, utilizing high-order accurate finite difference and spectral discretization in generalized coordinates, is used for LNSE-based stability analysis. A direct sparse parallel solver is employed, and the potential for matrix-free iterative schemes for eigendecomposition is examined. Future research will incorporate non-modal stability analysis and novel WENO schemes, further expanding the scope of this study.   

Spatial-temporal harmonic resolvent analysis of asymmetric bluff-body wakes

We extend the use of resolvent analysis for base flows that exhibit both temporal and spatial periodicity. This allows us to investigate frequency crosstalk and energy transfer across different spatial and temporal scales. We demonstrate this analysis on bluff-body wakes with the presence of periodic vortex shedding. Here, a streamwise slice of the wake downstream of the bluff body is considered as the base flow while assuming spatial periodicity in the streamwise direction, conceptually similar to the traditional parallel flow assumption. First, a transverse slice of a two-dimensional wake downstream a circular cylinder is considered, where the interaction between the fundamental frequency and its harmonics are identified. The analysis is also applied to a spanwise asymmetric wake of a finite-span wing at 10$^circ$ angle of attack, where only the frequency-wavenumber combinations that result in the same phase velocity for the perturbations are considered. This identifies the interactions between tip vortex and the wake at different time and space scales.   

The influence of viscosity stratification and counter-flow on instability in free shear layers

In this study, we undertake a comprehensive examination of the instability exhibited by free shear layers (specifically, mixing layers ) subject to viscosity stratification and counter-flow. We construct a base state where the two fluids share an identical density, employing boundary layer assumptions to model both the high and low-speed streams that form a mixing layer. Using self-similar base state profiles that satisfy the boundary layer equations, we perform a local linear stability analysis of the perturbed planar Navier-Stokes equations. While temporal stability of such flows has been examined previously, we demonstrate that for sufficiently high viscosity contrast between the high-speed and low-speed streams, absolutely unstable profiles can result even in the absence of counterflow, which is a new result. In addition, we have thoroughly examined absolute/convective instability transition boundary as a function of Reynolds number, viscosity ratio, and velocity ratio, which determines the nature of the mixing layer (co-flow and counter-flow). Our investigation delineates the intricate dynamics of free shear layers under viscosity stratification, and it is hoped that these numerical results stimulate an experimental realization that investigates instabilities in variable viscosity-free shear layers.