## Final Report Summary - ALORS (Advanced Lagrangian Optimization, Receptivity and Sensitivity analysis applied to industrial situations)

In engineering design, numerical simulations are often used on a trial-and-error basis, combined with engineering judgement by the user. This neglects a remarkable fact: with a single extra simulation (the adjoint calculation), it is possible to calculate how the performance of a design is affected by any small change to any of the design parameters. This is called receptivity and sensitivity analysis, and it arises from the application of constrained optimization using Lagrangian techniques, hence the title of this project.

In this project, receptivity and sensitivity analysis has been extended to turbulent flows, flows with complex geometry, and reacting flows. The applications considered were gas turbine fuel injectors, microfluidic mixers, stenotic arteries, and oscillations in rockets and gas turbines. One of these is described below.

Rocket engines and gas turbines are designed to operate in steady flow, in the sense that the time average flow is the same as the ensemble average flow. Sometimes, however, heat release oscillations lock into acoustic oscillations and amplify into a violent oscillation that can cause excessive heat transfer, structural vibration, and even catastrophic failure.This is known as a thermoacoustic oscillation. The mechanism is similar to that of a car piston engine but with an acoustic wave taking the place of the piston: when more heat release than average occurs during moments of high pressure, and less heat release than average occurs during moments of low pressure, then more work is done during the expansion phase than is absorbed during the compression phase of a cycle, causing oscillations to grow. A key component in this mechanism is the flame's response to acoustic perturbations. As the ratio of air to fuel in gas turbines is increased, in order to reduce nitrous oxide emissions, flames tend to become more responsive to acoustics. As a consequence, thermoacoustic oscillations are increasingly problematic for gas turbines and a systematic procedure to eliminate them early in the design process is required. The techniques developed in this project will provide such a mechanism.

The goal of rocket and gas turbine manufacturers is to design an engine that is linearly stable to thermoacoustic oscillations over the entire operating regime. Currently, this is achieved through (i) extensive testing, which is expensive, (ii) the avoidance of certain engine operating regimes, which reduces flexibility, and (iii) the retro-fitting of passive dampers such as Helmholtz resonators, which add weight. A systematic method to identify and then stabilize thermoacoustic oscillations would speed up testing, eliminate dangerous operating points, and either determine the optimal placement of passive dampers or remove the need for them entirely. Such a method can exploit three convenient facts: (i) the requirement is for linear (rather than nonlinear) stability, meaning that the tools of linear analysis can be used; (ii) usually only a handful of thermo-acoustic modes are unstable; (iii) usually many parameters of the system can be altered. The challenge is therefore to identify the most influential parameters for each mode. An ideal solution is to combine adjoint-based receptivity and sensitivity with a conventional thermoacoustic stability analysis. The stability analysis identifies the handful of unstable eigenvalues. The adjoint methods then show, in a single calculation for each eigenvalue, how each eigenvalue is affected by every parameter in the system. While this can be achieved with finite difference calculations, this would require a single calculation for each parameter, which is too expensive. This gradient information enables efficient optimization. e.g. change of boundary conditions, change of flame shape, change of geometry, optimal addition of feedback devices such as helmholtz resonators. This technique will provide manufacturers with an automated method to design a linearly stable engine.

In this project, receptivity and sensitivity analysis has been extended to turbulent flows, flows with complex geometry, and reacting flows. The applications considered were gas turbine fuel injectors, microfluidic mixers, stenotic arteries, and oscillations in rockets and gas turbines. One of these is described below.

Rocket engines and gas turbines are designed to operate in steady flow, in the sense that the time average flow is the same as the ensemble average flow. Sometimes, however, heat release oscillations lock into acoustic oscillations and amplify into a violent oscillation that can cause excessive heat transfer, structural vibration, and even catastrophic failure.This is known as a thermoacoustic oscillation. The mechanism is similar to that of a car piston engine but with an acoustic wave taking the place of the piston: when more heat release than average occurs during moments of high pressure, and less heat release than average occurs during moments of low pressure, then more work is done during the expansion phase than is absorbed during the compression phase of a cycle, causing oscillations to grow. A key component in this mechanism is the flame's response to acoustic perturbations. As the ratio of air to fuel in gas turbines is increased, in order to reduce nitrous oxide emissions, flames tend to become more responsive to acoustics. As a consequence, thermoacoustic oscillations are increasingly problematic for gas turbines and a systematic procedure to eliminate them early in the design process is required. The techniques developed in this project will provide such a mechanism.

The goal of rocket and gas turbine manufacturers is to design an engine that is linearly stable to thermoacoustic oscillations over the entire operating regime. Currently, this is achieved through (i) extensive testing, which is expensive, (ii) the avoidance of certain engine operating regimes, which reduces flexibility, and (iii) the retro-fitting of passive dampers such as Helmholtz resonators, which add weight. A systematic method to identify and then stabilize thermoacoustic oscillations would speed up testing, eliminate dangerous operating points, and either determine the optimal placement of passive dampers or remove the need for them entirely. Such a method can exploit three convenient facts: (i) the requirement is for linear (rather than nonlinear) stability, meaning that the tools of linear analysis can be used; (ii) usually only a handful of thermo-acoustic modes are unstable; (iii) usually many parameters of the system can be altered. The challenge is therefore to identify the most influential parameters for each mode. An ideal solution is to combine adjoint-based receptivity and sensitivity with a conventional thermoacoustic stability analysis. The stability analysis identifies the handful of unstable eigenvalues. The adjoint methods then show, in a single calculation for each eigenvalue, how each eigenvalue is affected by every parameter in the system. While this can be achieved with finite difference calculations, this would require a single calculation for each parameter, which is too expensive. This gradient information enables efficient optimization. e.g. change of boundary conditions, change of flame shape, change of geometry, optimal addition of feedback devices such as helmholtz resonators. This technique will provide manufacturers with an automated method to design a linearly stable engine.