Augmented resolution of linear hyperbolic systems under nonconservative form

Hyperbolic systems under nonconservative form arise in numerous applications modeling physical processes, for example from the relaxation of more general equations (e.g. with dissipative terms). This paper reviews an existing class of augmented Roe schemes and discusses their application to linear nonconservative hyperbolic systems with source terms. We extend existing augmented methods by redefining them within a common framework which uses a geometric reinterpretation of source terms. This results in intrinsically well-balanced numerical discretizations. We discuss two equivalent formulations: (1) a nonconservative approach and (2) a conservative reformulation of the problem. The equilibrium properties of the schemes are examined and the conditions for the preservation of the well-balanced property are provided. Transient and steady state test cases for linear acoustics and hyperbolic heat equations are presented. A complete set of benchmark problems with analytical solution, including transient and steady situations with discontinuities in the medium properties, are presented and used to assess the equilibrium properties of the schemes. It is shown that the proposed schemes satisfy the expected equilibrium and convergence properties

Alternative Modal Basis Selection Procedures for Nonlinear Random Response Simulation

Three procedures to guide selection of an efficient modal basis in a nonlinear random response analysis are examined. One method is based only on proper orthogonal decomposition, while the other two additionally involve smooth orthogonal decomposition. Acoustic random response problems are employed to assess the performance of the three modal basis selection approaches. A thermally post-buckled beam exhibiting snap-through behavior, a shallowly curved arch in the auto-parametric response regime and a plate structure are used as numerical test articles. The results of the three reduced-order analyses are compared with the results of the computationally taxing simulation in the physical degrees of freedom. For the cases considered, all three methods are shown to produce modal bases resulting in accurate and computationally efficient reduced-order nonlinear simulations

Optimal control theory : a method for the design of wind instruments

It has been asserted previously by the author that optimal control theory can be a valuable framework for theoretical studies about the shape that a wind instrument should have in order to satisfy some optimization criterion, inside a fairly general class. The purpose of the present work is to develop this new approach with a look at a specific criterion to be optimized. In this setting, the Webster horn equation is regarded as a controlled dynamical equation in the space variable. Pressure is the state, the control being made of two parts: one variable part, the inside diameter of the duct and one constant part, the weights of the elementary time-harmonic components of the velocity potential. Then one looks for a control that optimizes a criterion related to the definition of an {oscillation regime} as the cooperation of several natural modes of vibration with the excitation, the {playing frequency} being the one that maximizes the total generation of energy, as exposed by A.H. Benade, following H. Bouasse. At the same time the relevance of this criterion is questioned with the simulation results.Comment: To appear in Acta Acustica united with Acustica, 201

2.5D singular boundary method for exterior acoustic radiation and scattering problems

In this paper, a numerical methodology based on a two-and-a-half-dimensional (2.5D) singular boundary method (SBM) to deal with acoustic radiation and scattering problems in the context of longitudinally invariant structures is proposed and studied. In the proposed 2.5D SBM, the desingularisation provided by the subtracting and adding-back technique is used to determine the origin intensity factors (OIFs). These OIFs are derived by means of the OIFs of the Laplace equation. The feasibility, validity and accuracy of the proposed method are demonstrated for three acoustic benchmark problems, in which detailed comparisons with analytical solutions, the 2.5D boundary element method (BEM) and the 2.5D method of fundamental solutions (MFS) are performed. As a novelty of the present study, it is found that the 2.5D SBM provides a higher numerical accuracy than the 2.5D linear-element BEM and lower than the 2.5D quadratic-element BEM. Although the results obtained depict that a nodal approximation of the boundary geometry leads to a significant reduction in the accuracy of the 2.5D SBM, the delivered errors are still acceptable. For complex geometries, the 2.5D SBM is found to be simpler and more robust than the 2.5D MFS, since no optimization procedure is required.Peer ReviewedPostprint (published version

Signal Analysis Algorithms for Optimized Fitting of Nonresonant Laser Induced Thermal Acoustics Damped Sinusoids

This study seeks a numerical algorithm which optimizes frequency precision for the damped sinusoids generated by the nonresonant LITA technique. It compares computed frequencies, frequency errors, and fit errors obtained using five primary signal analysis methods. Using variations on different algorithms within each primary method, results from 73 fits are presented. Best results are obtained using an AutoRegressive method. Compared to previous results using Prony s method, single shot waveform frequencies are reduced approx.0.4% and frequency errors are reduced by a factor of approx.20 at 303K to approx. 0.1%. We explore the advantages of high waveform sample rates and potential for measurements in low density gases

Actuator selection and placement for linear feedback control of compressible flows

Actuator and sensor placement for active control of high-Reynolds number flows is largely based on experience and trial-and-error because of the system’s large dimensionality and complexity. A novel strategy for estimating how to select and place a linear feedback control system using co-located actuator(s)/sensor(s) suitable for affecting the dynamics of compressible, viscous flows is developed. The methodology uses the flow’s gain and receptivity information from the forward and adjoint global modes of the baseflow obtained from direct/large eddy simulations. The baseflow can be an equilibrium (steady-state) or a time-averaged solution of the compressible Navier Stokes equations. The method uses structural sensitivity arguments to determine regions of the flow-field with high dynamical sensitivity, and a search procedure determines effective actuator/sensor locations. The control algorithm is flexible, and different types of control and feedback can be considered. The efficacy of the method is demonstrated with three different flow control problems: flow stabilization in a Mach 0.65 diffuser, noise reduction of an axisymmetric Mach 1.5 jet, and noise reduction of a turbulent Mach 0.9 jet. For the diffuser, global stabilization is achieved for low Reynolds numbers resulting in complete suppression of vortex shedding. For longer domains and higher Reynolds number flows in the diffuser, although significant reduction in growth rates of the unstable modes was achieved, complete stabilization could not be attained. For the axisymmetric Mach 1.5 jet, equilibrium and time-averaged configurations are compared to examine the differences in global stability. The jet’s optimal transient response that leads to the largest pressure fluctuations away from the jet is used to relate the global modes needed for the control methodology to the radiated sound. The spectrum also contains modes that are hydrodynamically bound to the jet, without significant sound field contributions. Direct numerical simulations using the control show significant noise reduction, with additional reduction with increase in control gain. Eigenanalysis of the controlled mean flows reveal fundamental changes in the spectrum at frequencies lower than that used by the control, with the quieter flows having unstable eigenvalues that correspond to eigenfunctions without significant support in the acoustic field. Analysis of the mean flow quantities shows that the control induced mean flow changes only become obvious beyond 15 radii from the nozzle. Reduced order analysis using Proper Orthogonal Decomposition (POD) shows flow regularization in the quieter flows. The active control strategy is then applied to a Mach 0.9 turbulent jet. The global analysis of the time-and-azimuthal averaged baseline flow showed that the flow supports acoustically efficient super-directive and multi-directive global modes. Significant noise reduction was obtained and, similar to the axisymmetric case, the global analysis of the time-and-azimuthal averaged flow for the quiet jet show the existence of an unstable mode at a low Strouhal number, that lacks any significant sound-field support. The variation of mean quantities at the centerline and the lipline for the loud and quiet jets also showed trends similar to the axisymmetric case

On non-normality and classification of amplification mechanisms in stability and resolvent analysis

We seek to quantify non-normality of the most amplified resolvent modes and predict their features based on the characteristics of the base or mean velocity profile. A 2-by-2 model linear Navier-Stokes (LNS) operator illustrates how non-normality from mean shear distributes perturbation energy in different velocity components of the forcing and response modes. The inverse of their inner product, which is unity for a purely normal mechanism, is proposed as a measure to quantify non-normality. In flows where there is downstream spatial dependence of the base/mean, mean flow advection separates the spatial support of forcing and response modes which impacts the inner product. Success of mean stability analysis depends on the normality of amplification. If the amplification is normal, the resolvent operator written in its dyadic representation reveals that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes. If the amplification is non-normal, then resolvent analysis is required to understand the origin of observed flow structures. Eigenspectra and pseudospectra are used to characterize these phenomena. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with normal mechanisms and quantification of non-normality using the inverse inner product of the leading forcing and response modes agrees well with the product of the resolvent norm and distance between the imaginary axis and least stable eigenvalue. In turbulent channel flow, structures result from both normal and non-normal mechanisms. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how non-normality is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures