The first main focus in the present project was to analyse the boundary treatment of the linearised Korteweg-de Vries equation. The second main focus was to derive a stable numerical solution using a high-order finite difference method. Since the model involved a third derivative in space, the numerical treatment of the boundaries was highly nontrivial. To aid the boundary treatment high-order accurate first and third derivative finite difference operators were employed. The boundaries are based on the summation-by-parts (SBP) framework, thereby guaranteeing linear stability. The boundary conditions were imposed using a penalty technique. A convergence study was performed where the derived numerical solution was compared with an analytical one. Fourth order accurate Runge-Kutta was used to time-integrate the numerical approximation. Measuring the rate of convergence, q, yielded q = 4 for 4th order accurate SBP-operators and q = 5.5 for 6th order accurate SBP-operators. Thus the convergence study proved the accuracy and stability of the numerical solution derived with the SBP-methodology.

The successful application of Computational Aeroacoustics (CAA) requires high accuracy numerical schemes with good dissipation and dispersion characteristics. Unstructured meshes have a greater geometrical flexibility than existing high order structured mesh methods. This work investigates the suitability of unstructured mesh techniques by computing a two-dimensionallinearised Euler problem with various discretisation schemes and different mesh types. The goal of the present work is the development of an unstructured numerical method with the high accuracy, low dissipation and low dispersion required to be an effective tool in the study of aeroacoustics. The suitability of the unstructured method is investigated using aeroacoustic test cases taken from CAA Benchmark Workshop proceedings. Comparisons are made with exact solutions and a high order structured method. The higher order structured method was based upon a standard central differencing spatial discretisation. For the unstructured method a vertex-based data structure is employed. A median-dual control volume is used for the finite volume approximation with the option of using a Green-Gauss gradient approximation technique or a Least Squares approximation. The temporal discretisation used for both the structured and unstructured numerical methods is an explicit Runge-Kutta method with local timestepping. For the unstructured method, the gradient approximation technique is used to compute gradients at each vertex, these are then used to reconstruct the fluxes at the control volume faces. The unstructured mesh types used to evaluate the numerical method include semi-structured and purely unstructured triangular meshes. The semi-structured meshes were created directly from the associated structured mesh. The purely unstructured meshes were created using a commercial paving algorithm. The Least Squares method has the potential to allow high order reconstruction. Results show that a Weighted Least gradient approximation gives better solutions than unweighted and Green-Gauss gradient computation. The solutions are of acceptable accuracy on these problems with the absolute error of the unstructured method approaching that of a high order structured solution on an equivalent mesh for specific aeroacoustic scenarios.

Les écoulements transsoniques rencontrés dans le cadre de la propulsion aéronautique et spatiale sont associés à l'apparition d'ondes de choc. En impactant la couche limite se développant sur une paroi, un gradient de pression adverse est généré qui conduit à l'épaississement ou au décollement de la couche limite. Lors de la vibration de la structure, l'onde de choc oscille et interagit avec la couche limite, générant une fluctuation de la pression statique à la paroi. Il s'ensuit alors un échange d'énergie entre le fluide et la structure qui peut être stabilisant ou au contraire conduire à une instabilité aéroélastique (flottement). La modélisation de la réponse instationnaire de l'interaction onde de choc / couche limite pour l'étude des interactions fluide-structure est l'objet de ce travail de recherche. Il s'appuie sur la résolution des équations de Navier-Stokes moyennées (RANS) et la modélisation de la turbulence. Les méthodes et modèles utilisés ont été validés à partir de résultats expérimentaux issus d'une tuyère transsonique dédiée à l'étude des interactions fluide-structure. Ces travaux sont ensuite appliqués à l'amélioration de la prédiction du flottement en turbomachine. Une méthode linéarisée en temps permettant la résolution des équations RANS dans le domaine fréquentiel est utilisée. Nous confirmons l'importance de la dérivation du modèle de turbulence lors de la prédiction d'une interaction forte entre une onde de choc et une couche limite décollée. Une méthode de régularisation est présentée puis appliquée aux opérateurs non dérivables du modèle de turbulence k-! de Wilcox (2006). La prédiction de la réponse instationnaire de l'interaction onde de choc / couche limite dans une tuyère est évaluée à partir de simulations bidimensionnelles et présente un bon accord avec les données expérimentales. En évaluant l'influence de la fréquence réduite, une instabilité aéroélastique de type flottement transsonique est identifiée. Un dispositif de contrôle, reposant sur la génération d'ondes de pression rétrogrades à l'aval de la tuyère, est proposé puis validé numériquement. Enfin, une méthodologie est proposée pour comprendre les mécanismes aérodynamiques conduisant au flottement. Pour cela, il a été réalisé un dessin provisoire d'une soufflante transsonique à fort taux de dilution. Cette soufflante, l'ECL5, est destinée à l'étude expérimentale des instabilités aérodynamiques et aéroélastiques. La méthodologie proposée repose sur la simulation 2D d'une coupe de tête et met à profit la linéarisation pour analyser la contribution de sources locales en fonction de la fréquence réduite, du diamètre nodal et de la déformée modale / Transonic flows, which are common in aeronautical and spatial propulsion systems, produce shock-waves over solid boundaries. When a shock-wave impacts the boundary layer, an adverse pressure gradient is generated and a thickening or even a separation of the boundary layer is induced. If the solid boundary vibrates, the shock-wave oscillates, interacts with the boundary layer and produce a fluctuation of the static pressure at the wall. This induces an exchange of energy between the fluid and the structure which can be stabilising or lead to an aeroelastic instability (flutter).The main objective of this PhD thesis is the modelling of the unsteady behaviour the simulation of the shock-wave/boundary layer interaction for fluid-structure interaction studies. To this end, simulations have been carried out to solve Reynolds-Averaged Navier-Stokes equations using two equations turbulence model. The method is validated thanks to experimental data obtained on a transonic nozzle dedicated to aeroelastic studies. This method is then use to increase the predictability of flutter events in turbomachinery.A time linearised frequency-domain method is applied to RANS equations. It is shown that the unsteady behaviour of the turbulent boundary-layer contributes to the fluctuating static pressure when the shock-wave boundary layer interaction is strong. Hence, the frozen turbulence assumption is not valid and the turbulence model must be derivated. Thus, the regularisation of the non derivable operators is proposed and applied on k-? Wilcox (2006) turbulence model.The unsteady behaviour of the shock-wave/boundary-layer interaction in a transonic nozzle is evaluated thanks to 2D numerical simulations and shows good agreement with experimental data. When varying the reduced frequency an aeroelastic instability is found, known as transonic flutter. An active control device generating backward travelling pressure waves is then designed and numerically validated.Finally, a methodology is proposed to understand the aerodynamic onsets of transonic flutter. To this end, a preliminary design of a high bypass ratio transonic fan has been carried out. This fan, named ECL5, is dedicated to experimental aerodynamic and aeroelastic studies. The methodology relies on 2D simulations of a tip blade passage and uses linearisation to analyse the contribution of local sources as a function of reduced frequency, nodal diameter and mode shape

Since the inverse problem in Diffusive Optical Tomography (DOT) is nonlinear and severely ill-posed, only low resolution reconstructions are feasible when noise is added to the data nowadays. The purpose of this thesis is to improve image reconstruction in DOT of the main optical properties of tissues with some novel mathematical methods. We have used the Landweber (L) method, the Landweber-Kaczmarz (LK) method and its improved Loping-Landweber-Kaczmarz (L-LK) method combined with sparsity or with total variation regularizations for single and simultaneous image reconstructions of the absorption and scattering coefficients. The sparsity method assumes the existence of a sparse solution which has a simple description and is superposed onto a known background. The sparsity method is solved using a smooth gradient and a soft thresholding operator. Moreover, we have proposed an improved sparsity method. For the total variation reconstruction imaging, we have used the split Bregman method and the lagged diffusivity method. For the total variation method, we also have implemented a memory-efficient method to minimise the storage of large Hessian matrices. In addition, an individual and simultaneous contrast value reconstructions are presented using the level set (LS) method. Besides, the shape derivative of DOT based on the RTE is derived using shape sensitivity analysis, and some reconstructions for the absorption coefficient are presented using this shape derivative via the LS method.\\Whereas most of the approaches for solving the nonlinear problem of DOT make use of the diffusion approximation (DA) to the radiative transfer equation (RTE) to model the propagation of the light in tissue, the accuracy of the DA is not satisfactory in situations where the medium is not scattering dominant, in particular close to the light sources and to the boundary, as well as inside low-scattering or non-scattering regions. Therefore, we have solved the inverse problem in DOT by the more accurate time-dependant RTE in two dimensions.

Evaluating aerodynamic noise from aircraft engines is a design stage process, so that it conform to regulations at airports. Aerodynamic noise is also a principal source of structural vibration and internal noise in short/vertical take off and landing and rocket launches. Acoustic loads may be critical for the proper functioning of electronic and mechanical components. It is imperative to have tools with capability to predict noise generation from turbulent flows. Understanding the mechanism of noise generation is essential in identifying methods for noise reduction. Lighthill (1952) and Lighthill (1954) provided the first explanation for the mechanism of aerodynamic noise generation and a procedure to estimate the radiated sound field. Many such procedures, known as acoustic analogies are used for estimating the radiated sound field in terms of the turbulent fluid flow properties. In these methods, the governing equations of the fluid flow are rearranged into two parts, the acoustic sources and the propagation terms. The noise source terms and propagation terms are different in different approaches. A good description of the turbulent flow field and the noise sources is required to understand the mechanism of noise generation. Computational aeroacoustics (CAA) tools are used to calculate the radiated far field noise. The inputs to the CAA tools are results from CFD simulations which provide details of the turbulent flow field and noise sources. Reynolds-Averaged Navier Stokes (RANS) solutions can be used as inputs to CAA tools which require only time-averaged mean quantities. The output of such tools will also be mean quantities. While complete unsteady turbulent flow details can be obtained from Direct Numerical Simulation (DNS), the computation is limited to low or moderate Reynolds number flows. Large eddy simulations (LES) provide accurate description for the dynamics of a range of large scales. Most of the kinetic energy in a turbulent flow is accounted by the large-scale structures. It is also the large-scale structures which accounts for the maximum contribution towards the radiated sound field. The results from LES can be used as an input to a suitable CAA tool to calculate the sound field. Numerical prediction of turbulent flow field, the acoustic sources and the radiated sound field is at the focus of this study. LES based on explicit filtering method is used for the simulations. The method uses a low-pass compact filter to account for the sub-grid scale effects. A one-parameter fourth-order compact filter scheme from Lele (1992) is used for this purpose. LES has been carried out for four different flow situations: (i) round jet (ii) plane jet (iii) impinging round jet and (iv) impinging plane jet. LES has been used to calculate the unsteady flow evolution of these cases and the Lighthill’s acoustic sources. A compact difference scheme proposed by Hixon & Turkel (1998) which involves only bi-diagonal matrices are used for evaluating spatial derivatives. The scheme provides similar spectral resolution as standard tridiagonal compact schemes for the first spatial derivatives. The scheme is computationally less intensive as it involves only bi-diagonal matrices. Also, the scheme employs only a two-point stencil. To calculate the radiated sound field, the Helmholtz equation is solved using the Green’s function approach, in the form of the Kirchhoff-Helmholtz integral. The integral is performed over a surface which is present entirely in the linear region and covers the volume where acoustic sources are present. The time series data of pressure and the normal component of the pressure gradient on the surface are obtained from the CFD results. The Fourier transforms of the time series of pressure and pressure gradient are then calculated and are used as input for the Kirchhoff-Helmholtz integral. The flow evolution for free jets is characterised by the growth of the instability waves in the shear layer which then rolls up into large vortices. These large vortical structures then break down into smaller ones in a cascade which are convected downstream with the flow. The rms values of the Lighthill’s acoustic sources showed that the sources are located mainly at regions immediately downstream of jet break down. This corresponds to the large scale structures at break down. The radiated sound field from free jets contains two components of noise from the large scales and from the small scales. The large structures are the dominant source for the radiated sound field. The contribution from the large structures is directional, mainly at small angles to the downstream direction. To account for the difference in jet core length, the far field SPL are calculated at points suitably shifted based on the jet core length. The peak value for the radiated sound field occurs between 30°and 35°as reported in literature. Convection of acoustic sources causes the radiated sound field to be altered due to Doppler effect. Lighthills sources along the shear layer were examined in the form of (x, t) plots and phase velocity pattern in (ω, k) plots to analyse for their convective speeds. These revealed that there is no unique convective speeds for the acoustic sources. The median convective velocity Uc of the acoustic sources in the shear layer is proportional to the jet velocity Uj at the center of the nozzle as Uc ≈ 0.6Uj. Simulations of the round jet at Mach number 0.9 were used for validating the LES approach. Five different cases of the round jet were used to understand the effect of Reynolds number and inflow perturbation on the flow, acoustic sources and the radiated sound field. Simulations were carried out for an Euler and LES at Reynolds number 3600 and 88000 at two different inflow perturbations. The LES results for the mean flow field, turbulence profiles and SPL directivity were compared with DNS of Freund (2001) and experimental data available in literature. The LES results showed that an increase in inflow forcing and higher Reynolds number caused the jet core length to reduce. The turbulent energy spectra showed that the energy content in smaller scale is higher for higher Reynolds number. LES of plane jets were carried out for two different cases, one with a co-flow and one without co-flow. LES of plane jets were carried out to understand the effect of co-flow on the sound field. The plane jets were of Mach number 0.5 and Reynolds number of 3000 based on center-line velocity excess at the nozzle. This is similar to the DNS by Stanley et al. (2002). It was identified that the co-flow leads to a reduction in turbulence levels. This was also corroborated by the turbulent energy spectrum plots. The far field radiation for the case without co-flow is higher over all angles. The contribution from the low frequencies is directional, mainly towards the downstream direction. The range of dominant convective velocities of the acoustic sources were different along shear layers and center-line. The plane jet results were also used to bring out a qualitative comparison of flow and the radiation characteristics with round jets. For the round jet, the center-line velocity decays linearly with the stream-wise distance. In the plane jet case, it is the square of the center-line velocity excess which decays linearly with the stream-wise distance. The turbulence levels at any section scales with the center-line stream-wise velocity. The decay of turbulence level is slower for the plane jet and hence the acoustic sources are present for longer distance along the downstream direction. Subsonic impinging jets are composed of four regions, the jet core, the fully developed jet, the impingement zone and the wall jet. The presence of the second region (fully developed free jet) depends on the distance of the wall from the nozzle and the length of the jet core. In impinging jets, reflection from the wall and the wall jet are additional sources of noise compared to the free jets. The results are analysed for the contribution of the different regions of the flow towards the radiated sound field. LES simulations of impinging round jets and impinging plane jet were carried out for this purpose. In addition, the results have been compared with equivalent free jets. The directivity plots showed that the SPL levels are significantly higher for the impinging jets at all angles. For free jets, a typical time scale for the acoustic sources is the ratio of the nozzle size to the jet velocity. This is ro/Uj for round jets and h/Uj for plane jets. For impinging jets, the non-dimensionlised rms of Lighthill’s source indicates that the time scale for acoustic sources is the ratio of the height of the nozzle from the wall to the jet velocity be L/Uj. LES of impinging round jets was carried out for two cases with different inflow perturbations. The jets were at Reynolds number of 88000 and Mach number of 0.9, same as the free jet cases. The impingement wall was at a distance L = 24ro from the nozzle exit. For impinging round jets, the SPL levels are found to be higher than the equivalent free jets. From the SPL levels and radiated noise spectra it was shown that the contribution from the large scale structures and its reflection from the wall is directional and at small angles to the wall normal. The difference in the range of angles where the radiation from the large scale structures were observed shows the significance of refraction of sound waves inside the flow. The rms values of the Lighthill’s sources indicate two dominant regions for the sources, just downstream of jet breakdown and in the impingement zone. The LES of impinging plane jet was done for a jet of Mach number 0.5 and Reynolds number of 6000. The impingement wall was at a distance L = 10h from the nozzle exit. The radiated sound field appears to emanate from this impingement zone. The directivity and the spectrum plots of the far field SPL indicate that there is no preferred direction of radiation from the impingement zone. The Lighthill’s sources are concentrated mainly in the impingement zone. The rms values of the sources indicate that the peak values occur in the impingement zone. The results from the different flow situations demonstrates the capability of LES with explicit filtering method in predicting the turbulent flow and radiated noise field. The method is robust and has been successfully used for moderate Reynolds number and an Euler simulation. An important feature is that LES can be used to identify acoustic sources and its convective speeds. It has been shown that the Lighthill source calculations, the calculated sound field and the observed radiation patterns agree well. An explanation for these based on the different turbulent flow structures has also been provided.