Session Q23: Control and Nonlinear Dynamics

The effects of tonal and broadband acoustic excitation on the transition process within a laminar separation bubble

The effects of controlled acoustic excitation on the transition process in a laminar separation bubble formed on the suction side of a NACA 0018 airfoil at a chord Reynolds number of 125,000 and an angle of attack of 4 degrees are studied experimentally. The investigation is carried out using time-resolved, planar, two-component Particle Image Velocimetry. Two types of excitation are considered: (i) tonal excitation at the frequency of the most unstable disturbances in the natural flow, and (ii) broadband excitation consisting bandpass filtered to the natural unstable frequency range, modelling two common types of airfoil self-noise production. For equal energy input levels, the results show that tonal and broadband types of excitation have equivalent effects on the mean flow field. Specifically, both cause the streamwise extent and height of the bubble to decrease. However, further analysis reveals notable differences in the underlying physics. For the tonal case, the transition process is dominated by the growth of disturbances at the excitation frequency that damps the growth of all other disturbances, leading to the formation of strongly coherent vortices in the aft portion of the separation bubble. On the other hand, broadband excitation promotes more moderate growth of all disturbances within the unstable frequency band, producing less coherent shear layer structures that experience earlier breakdown. Thus, the frequency content of acoustic excitation has a strong influence on the transition process in laminar separation bubbles.   

Control of three-dimensional waves on thin liquid films. I - Optimal control and transverse mode effects

We consider a weakly nonlinear model for interfacial waves on three-dimensional thin films on inclined flat planes -- the Kuramoto---Sivashinsky equation. The flow is driven by gravity, and is allowed to be overlying or hanging on the flat substrate. Blowing and suction controls are applied at the substrate surface. In this talk we explore the instability of the transverse modes for hanging arrangements, which are unbounded and grow exponentially. The structure of the equations allows us to construct optimal transverse controls analytically to prevent this transverse growth. In this case and the case of an overlying film, we additionally study the influence of controlling to non-trivial transverse states on the streamwise and mixed mode dynamics. Finally, we solve the full optimal control problem by deriving the first order necessary conditions for existence of an optimal control, and solving these numerically using the forward---backward sweep method.   

Control of three-dimensional waves on thin liquid films. II -- Point actuated control

We consider the application of point actuated blowing and suction controls in the two-dimensional Kuramoto---Sivashinsky equation; this is a weakly nonlinear model for interfacial waves of three-dimensional thin films on inclined flat surfaces. The flow is driven by gravity, and is allowed to be overlying or hanging the flat substrate. In this talk, the controls are modelled using Dirac delta functions. We first study the case of proportional control, where the actuation at a point depends on the local interface height alone. Here, we study the influence of control strength and number/location of actuators on the possible stabilization of the zero solution. For hanging films, we observe that there are critical parameters above which we obtain bounded solutions, and in general there is another set of critical parameters above which the zero solution is stabilized exponentially. We also consider the full feedback problem, which assumes that we can observe the full interface and allow communication between actuators. Using these controls we can obtain exponential stability where proportional controls fail, and stabilize non-trivial solutions.   

Airfoil stall interpreted through linear stability analysis

Although airfoil stall has been widely investigated, the origin of this phenomenon, which manifests as a sudden drop of lift, is still not clearly understood. In the specific case of static stall, multiple steady solutions have been identified experimentally and numerically around the stall angle. We are interested here in investigating the stability of these steady solutions so as to first model and then control the dynamics. The study is performed on a 2D helicopter blade airfoil OA209 at low Mach number, $M\sim0.2$ and high Reynolds number, $Re\sim1.8 \times 10^6$. Steady RANS computation using a Spalart-Allmaras model is coupled with continuation methods (pseudo-arclength and Newton's method) to obtain steady states for several angles of incidence. The results show one upper branch (high lift), one lower branch (low lift) connected by a middle branch, characterizing an hysteresis phenomenon. A linear stability analysis performed around these equilibrium states highlights a mode responsible for stall, which starts with a low frequency oscillation. A bifurcation scenario is deduced from the behaviour of this mode. To shed light on the nonlinear behavior, a low order nonlinear model is created with the same linear stability behavior as that observed for that airfoil.   

Ground effects on the stability of separated flow around an airfoil at low Reynolds numbers

We perform a BiGlobal stability analysis on the separated flow around a NACA 4415 airfoil at low Reynolds numbers ($Re=300-1000$) and a high angle of attack $\alpha=20^\circ$ with a focus on the effect of the airfoil's proximity to a moving ground. The results show that the most dominant perturbation is the Kelvin--Helmholtz mode and that this traveling mode becomes less unstable as the airfoil approaches the ground, although this stabilizing effect diminishes with increasing Reynolds number. By performing a Floquet analysis, we find that this ground effect can also stabilize secondary instabilities. This numerical--theoretical study shows that the ground can have a significant influence on the stability of separated flow around an airfoil at low Reynolds numbers, which could have implications for the design of micro aerial vehicles and for the understanding of natural flyers such as insects and birds.   

Shape optimisation for linear stability with a RANS base-flow

A linear stability analysis of a steady flow yields a series of mode shapes and their corresponding growth rates and frequencies. The presence of modes with positive growth rates indicates that the flow will transition to another steady state or develop unsteady behaviour. Targeting the growth rate, we demonstrate shape optimisation for RANS flows. We examine the flow over a cylinder at a Reynolds number of 1000 with the Spalart-Allmaras turbulence model. A linear stability analysis yields a mode shape analogous to the low Reynolds number vortex shedding mode. Through the introduction of an adjoint global mode and an adjoint base-flow we derive the Hadamard form, a surface integral representation of the shape gradient. This gradient information is then used to modify the shape and reduce the growth rate of the mode. The cost of this approach is independent of the number of parameters and equivalent to an additional eigenvalue problem together with a linear flow calculation. In addition to the model problem of flow over a cylinder an industrial application is also presented.   

Role of anisotropy and inhomogeneity on the instability due to viscosity stratification of Poiseuille flow in a porous channel

Understanding of stability characteristics of two-fluids system in confined complex geometries is crucial in industrial applications and natural phenomena. This study is motivated by the necessity to understand possible drag reduction using superhydrophobic surfaces or liquid-infused surfaces or surfaces with complex features which can be modeled as porous substrates with appropriate properties. A linear stability analysis of Poiseuille flow of viscosity-stratified two-layer immiscible fluids system in a porous channel with anisotropic and inhomogeneous permeability is analyzed. The flow in the porous medium is governed by the generalized Darcy model with Beavers-Joseph condition at the interface of the liquid-porous layers. The resulting generalized eigenvalue problem is solved numerically and the temporal linear stability analysis shows the existence of three distinct modes of instability; a porous mode, an interface mode and a shear mode. The influence of the variations in anisotropic and inhomogeneous properties of the porous medium on the interface and shear mode instabilities is assessed.The study reveals a possibility of controlling instabilities in two-layer flows in a rigid channel by designing a wall of the channel as a porous surface with appropriate properties.   

A weakly nonlinear approach to predict the dynamics of precession of vortex core in a strongly swirling flow

Swirling flows at large swirl numbers, ie the ratio of the axial flux of tangential momentum to the axial flux of axial momentum, the central recirculation zone (CRZ) precesses about the streamwise normal axis causing precession of the vortex core (PVC). In the present study swirling flow field at various inflow swirl numbers are obtained from a variable swirl experimental facility. At large inflow swirl numbers, a PVC was observed from the experiment. We perform a weakly non-parallel linear stability analysis on the time averaged flow field at the different swirl numbers to identify all the unstable global modes. The linear stability analysis predicts that the flow becomes globally unstable through m=1 helical instability and eventually with the increase in swirl m=2 double helical mode also becomes unstable. A Fast Fourier Transform of the velocity time series data where PVC is present shows a dominant m=1 oscillation at 1060Hz. But the linear stability analysis predicts a globally unstable m=1 mode at the first subharmonic of the PVC frequency and globally unstable m=2 mode at the PVC frequency. Hence, in the present study, a weakly nonlinear model is developed to predict the PVC frequency and the amplitude variation due to nonlinear interaction of m=1 and m=2 modes.   

Transverse acoustic forcing of a round hydrodynamically self-excited jet

Hydrodynamically self-excited jets can readily synchronize with longitudinal acoustic forcing, but their response to transverse acoustic forcing is less clear. In this experimental study, we apply transverse acoustic forcing to an axisymmetric low-density jet at frequencies around its natural global frequency. We place the jet in a rectangular box containing two loudspeakers, one at each end, producing nominally one-dimensional standing pressure waves. By traversing the jet across this box, we subject it to a range of acoustic modes, from purely longitudinal (streamwise) modes at the pressure anti-node to purely transverse (cross-stream) modes at the pressure node. Using time-resolved Background-Oriented Schlieren (BOS) imaging and hot-wire anemometry, we characterize the jet response for different forcing frequencies, amplitudes and mode shapes, providing new insight into the way transverse acoustic oscillations interact with axisymmetric hydrodynamic oscillations.   

An intermittency route to global instability in low-density jets

Above a critical Reynolds number ($Re$), a low-density jet can become globally unstable, transitioning from a steady state (i.e. a fixed point) to a self-excited oscillatory state (i.e. a limit cycle) via a Hopf bifurcation. In this experimental study, we show that this transition can sometimes involve intermittency. When $Re$ is just slightly above the critical point, intermittent bursts of high-amplitude periodic oscillations emerge amidst a background of low-amplitude aperiodic fluctuations. As $Re$ increases further, these intermittent bursts persist longer in time until they dominate the overall dynamics, causing the jet to transition fully to a periodic limit cycle. We identify this as Type-II Pomeau-Manneville intermittency by quantifying the statistical distribution of the duration of the aperiodic fluctuations at the onset of intermittency. This study shows that the transition to global instability in low-density jets is not always abrupt but can involve an intermediate state with characteristics of both the initial fixed point and the final limit cycle.   

Coherence resonance in low-density jets

Coherence resonance is a phenomenon in which the response of a stable nonlinear system to noise exhibits a peak in coherence at an intermediate noise amplitude. We report the first experimental evidence of coherence resonance in a purely hydrodynamic system, a low-density jet whose variants can be found in many natural and engineering systems. This evidence comprises four parts: (i) the jet’s response amplitude increases as the Reynolds number approaches the instability boundary under a constant noise amplitude; (ii) as the noise amplitude increases, the amplitude distribution of the jet response first becomes unimodal, then bimodal, and finally unimodal again; (iii) a distinct peak emerges in the coherence factor at an intermediate noise amplitude; and (iv) for a subcritical Hopf bifurcation, the decay rate of the autocorrelation function exhibits a maximum at an intermediate noise amplitude, but for a supercritical Hopf bifurcation, the decay rate decreases monotonically with increasing noise amplitude. It is clear that coherence resonance can provide valuable information about a system’s nonlinearity even in the unconditionally stable regime, opening up new possibilities for its use in system identification and flow control.