Weakly nonlinear modelling of a forced turbulent axisymmetric wake

A theory is presented where the weakly nonlinear analysis of laminar globally unstable flows in the presence of external forcing is extended to the turbulent regime. The analysis is demonstrated and validated using experimental results of an axisymmetric bluff-body wake at high Reynolds numbers, Re_D ∼1.88×10^5, where forcing is applied using a zero-net-mass-flux actuator located at the base of the blunt body. In this study we focus on the response of antisymmetric coherent structures with azimuthal wavenumbers m = ±1at a frequency St_D = 0.2 S, responsible for global vortex shedding. We found experimentally that axisymmetric forcing (m = 0) couples nonlinearly with the global shedding mode when the flow is forced at twice the shedding frequency, resulting in parametric subharmonic resonance through a triadic interaction between forcing and shedding. We derive simple weakly nonlinear models from the phase-averaged Navier–Stokes equations and show that they capture accurately the observed behaviour for this type of forcing. The unknown model coefficients are obtained experimentally by producing harmonic transients. This approach should be applicable in a variety of turbulent flows to describe the response of global modes to forcing

MODEL BASED CONTROL OF NONLINEAR COMBUS- TION INSTABILITIES

Lean premixed combustion chambers are susceptible to combustion instabilities arising from the coupling between the heat release rate perturbations and the acoustic disturbances. These instabilities are generally harmful. Active feedback control can be used to interrupt the coupling between the acoustic waves and unsteady heat release and prevent or suppress instability. The design of most types of controller requires prior knowledge of how the combustor responds to actuation -the "open loop transfer function" (OLTF). This includes the flame response to oncoming flow disturbances, which becomes non-linear at larger amplitudes. Saturation of the heat release rate amplitude or a phase lag change relative to the acoustic pressure as the modulation level increases has been confirmed to occur experimentally in many cases. This may cause the dominant unstable frequency to change, which in turn alters the OLTF and makes the design of controller more complicated. A flame describing function model is proposed in this paper to account for the non-linearities of the heat release rate amplitude. This flame model is applied to a model combustor. The ν-gap metric is used to quantify the deviation of the set of OLTFs for different disturbance levels from the selected transfer function for the controller design. This provides us a bound on the minimum required "robustness stability margin" for an H ∞ loop shaping controller. Such a controller is synthesized, and it is confirmed that as long as the ν-gap is smaller than the stability margin, the robust controller is guaranteed to stabilize the combustor for all possible flame responses

Investigation of nonlinear flame response to dual-frequency disturbances

The two-way interaction between the unsteady flame heat release rate and acoustic waves can lead to combustion instability within combustors. To understand and quantify the flame response to oncoming acoustic waves, previous studies have typically considered the flame dynamic response to pure tone forcing and assumed a dynamically linear or weakly nonlinear response. In this study, the introduction of excitation with two distinct frequencies denoted $St_1$ and $St_2$ is considered, including the effect of excitation amplitude in order to gain more insight into the nature of flame nonlinearities and these associated with combustion instabilities. Corresponding results are obtained by combining a low-order asymptotic analysis (up to third order in normalised excitation amplitude) with numerical methods based on the model framework of the $G$-equation. The influence paths of the disturbance at $St_2$ on the flame dynamic response at $St_1$ are studied in detail. Due to the flame propagating forward normally to itself (named flame kinematic restoration), the perturbation at $St_2$ acts together with that at $St_1$ to induce a third-order nonlinear interaction in the flame kinematics, impressively suppressing the spatial wrinkling of the flame at $St_1$. Additionally, introducing the perturbation at $St_2$ alters the effective flame displacement speed, which is responsible for the calculation of the flame heat release rate and further affects the global response at $St_1$. Taking into account the above two factors, the nonlinear response of the flame at $St_1$ is completely quantified and the corresponding characteristics are clearly interpreted

Feedback control of oscillations in combustion and cavity flows

This thesis considers the control of combustion oscillations, motivated by the susceptibility of lean premixed combustion to such oscillations, and the long and expensive development and commissioning times that this is giving rise to. The controller used is both closed-loop, employing an actuator to modify some system parameter in response to a measured signal, and adaptive, meaning that it is able to maintain control over a wide range of operating conditions. The controller is applied to combustion systems with annular geometries, where instabilities can occur both longitudinally and azimuthally, and which require multiple sensors and multiple actuators for control. One of the requirements of Lyapunov-based adaptive control which is particularly troublesome for combustion systems is then addressed: that the sign of the high-frequency gain of the open-loop system is known. We address it by using an adaptive controller which employs a Nussbaum gain, and successfully apply it experimentally to combustion oscillations in a Rijke tube. Another type of fluid-acoustic resonance is then considered: the compressible flow past a shallow cavity. We start by finding a linear model of the cavity flow's dynamics, or its 'transfer function', which we identify from direct numerical simulations. We compare this measured transfer function to that given by a conceptual model which is based on the Rossiter mechanism, and which models each component of the flow physics separately. We then look at using closed-loop control to eliminate these cavity oscillations. We start by designing a robust H₂ controller based on a balanced reduced order model of the system, the model being provided by the Eigensystem Realization Algorithm (ERA). The robust controller provides closed-loop stability over a much wider Mach number range than seen in previous studies. Finally, we look at the suitability of the adaptive controller, earlier developed for combustion oscillations, for the cavity. Based on some general properties of the cavity flow, and by using collocated control, the oscillations are eliminated at all Mach numbers tested in the range 0.4 ≤ M ≤ 0.8.EThOS - Electronic Theses Online ServiceEPSRCRolls-Royce plcGBUnited Kingdo

The effect of an axial mean temperature gradient on communication between one-dimensional acoustic and entropy waves

This work performs a theoretical and numerical analysis of the communication between one-dimensional acoustic and entropy waves in a duct with a mean temperature gradient. Such a situation is highly relevant to combustor flows where the mean temperature drops axially due to heat losses. A duct containing a compact heating element followed by an axial temperature gradient and choked end is considered. The proposed jump conditions linking acoustic and entropy waves on either side of the flame show that the generated entropy wave is generally proportional to the mean temperature ratio across the flame and the ratio ( F - 1 ) , where F is the flame transfer function. It is inversely proportional to the Mach number immediately downstream of the flame M 2 . The acoustic and entropy fields in the region of axial mean temperature gradient are calculated using four approaches: (1) using the full three linearised Euler equations as the reference; (2) using two linearised Euler equations in which the acoustic and entropy waves are assumed independent (thus allowing the extent of communication between the acoustic and entropy wave to be evaluated); (3) using a Helmholtz solver which neglects mean flow effects and (4) using a recently developed analytical solution. It is found that the communication between the acoustic and entropy waves is small at low Mach numbers; it rises with increasing Mach number and cannot be neglected when the mean Mach number downstream of the heating element exceeds 0.1. Predictions from the analytical method generally match those from the full three linearised Euler equations, and the Helmholtz solver accurately determines the acoustic field when M 2 ≤ 0 . 1

Variation of acoustic energy accross a sudden cross section area increase sustaining a subsonic mean flow

International audienceThe variation of acoustic energy across a given geometrycan be quantified using the acoustic absorption coefficient.A quasi-steady analytical model predicting the absorptioncoefficient across an axisymmetric sudden area expansionsustaining a subsonic mean flow is derived in this study.This dimensionless parameter is shown to depend on theinlet Mach numberMu, cross section area ratioθ, and up-stream acoustic reflection coefficientRu. Numerical sim-ulations are then used to validate the analytical model fora wide range of parameters. It is found that depending on(Mu,θ,Ru), the acoustic energy can be either completelydamped, partially damped, unaffected, or even amplifiedacross the sudden area expansion