Eigenmode Analysis of Boundary Conditions for the One-Dimensional Preconditioned Euler Equations

The effect of local preconditioning on boundary conditions is analyzed for the subsonic, one-dimensional Euler equations. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions and different preconditioners whose intent is to accelerate low Mach number computations. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective when used with preconditioning, and asymptotically, at low Mach numbers, initial disturbances do not decay. Other boundary conditions are shown to be perfectly non-reflective in conjunction with preconditioning. Two-dimensional numerical results confirm the trends predicted by the one-dimensional analysis

Eigenmode analysis for turbomachinery applications

This paper discusses the numerical computation of unsteady eigenmodes superimposed upon an annular mean flow which is uniform axially and circumferentially, but non-uniform in the radial direction. Both inviscid and viscous flows are considered, and attention is paid to the separation of the eigenmodes into acoustic, entropy and vorticity modes. The numerical computations are validated by comparison to analytic test cases, and results are presented for more realistic engineering applications, showing the utility of the approach for post-processing and for the construction of non-reflecting boundary conditions.\ud \ud This work was supported by Rolls-Royce and DTI funding

Multigrid Renormalization

We combine the multigrid (MG) method with state-of-the-art concepts from the variational formulation of the numerical renormalization group. The resulting MG renormalization (MGR) method is a natural generalization of the MG method for solving partial differential equations. When the solution on a grid of $N$ points is sought, our MGR method has a computational cost scaling as $\mathcal{O}(\log(N))$, as opposed to $\mathcal{O}(N)$ for the best standard MG method. Therefore MGR can exponentially speed up standard MG computations. To illustrate our method, we develop a novel algorithm for the ground state computation of the nonlinear Schr\"{o}dinger equation. Our algorithm acts variationally on tensor products and updates the tensors one after another by solving a local nonlinear optimization problem. We compare several different methods for the nonlinear tensor update and find that the Newton method is the most efficient as well as precise. The combination of MGR with our nonlinear ground state algorithm produces accurate results for the nonlinear Schr\"{o}dinger equation on $N = 10^{18}$ grid points in three spatial dimensions.Comment: 18 pages, 17 figures, accepted versio

The Influence of Unstructured Mesh Type on the Prediction of Convoluted Shear Layers

Lobed mixers are used in gas turbine engines to enhance mixing between hot and cold streams and to reduce noise. Computational modelling of such systems has previously been carried out on structured meshes, although mesh generation difficulties have encouraged the use of unstructured tetrahedral meshes. However, the ability of numerical schemes to predict the mixing behaviour correctly on tetrahedral meshes has not been studied and is the subject of this work. Three different mesh types for the mixing region resolution have been studied: purely hexahedral, purely tetrahedral, and a mixed mesh combining hexahedra, tetrahedra and pyramids. Results are presented for the evolution of both a planar and a convoluted turbulent shear layer. In regions of high shear, misalignment of control volume faces has a major influence on spurious numerical spreading of the shear layer. For the tetrahedral mesh, there is an initial rapid mixing, followed by a reduction in mixing rate. The smoothing terms present are triggered by the combination of a high gradient across a control volume face and a velocity normal to that face; this occurs on the diagonal edges of tetrahedral meshes. The magnitude of the spurious smoothing is diminished by increasing the aspect ratio of the cells. For lobed mixer predictions, a mixed mesh with aligned high aspect ratio hexahedral elements in the shear layer region and pyramids and tetrahedra linking to the outer domain provides a good compromise between ease of mesh generation and quality of solution.\ud \ud The authors would like to acknowledge funding from the Engineering and Physical Sciences Research Council (UK), Grant No. GR/L17863. Financial assistance and technical review monitoring was also provided by Rolls-Royce and DERA; the authors would like to thank in particular Dr Leigh Lapworth (RR), Dr Jens-Dominik Müller (Oxford University) and Prof Mike Giles (Oxford University)

Variational quantum algorithms for nonlinear problems

We show that nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing. We achieve this by utilizing multiple copies of variational quantum states to treat nonlinearities efficiently and by introducing tensor networks as a programming paradigm. The key concepts of the algorithm are demonstrated for the nonlinear Schr\"{o}dinger equation as a canonical example. We numerically show that the variational quantum ansatz can be exponentially more efficient than matrix product states and present experimental proof-of-principle results obtained on an IBM Q device.Comment: 8 pages, 3 figures + Supplemental Material (7 pages, 7 figures), accepted versio

Over-Tip Choking and Its Implications on Turbine Blade Tip Aerodynamic Performance

At engine representative flow conditions a significant portion of flow over a high pressure turbine blade tip is transonic. In the present work, the choking flow behavior and its implications on over-tip leakage flow loss generation are computationally analyzed. An extensively developed RANS code (HYDRA) is adopted. Firstly a high speed linear cascade validation case is introduced, and the computations are compared with the experimental data to identify and establish the capability of the code in predicting the aerodynamics losses for a transonic turbine blade tip. The computational studies are then carried out for the blading configuration at different flow conditions ranging from a nearly incompressible to a nominal transonic one, enabling to establish a qualitatively consistent trend of the tip leakage losses in relation to the exit Mach number conditions. The results clearly show that the local choking sets a limiter for the over tip leakage mass flow, leading to a different leakage flow structure compared to that in a low speed and/or unchoked condition. The existence of tip choking effectively blocks the influence of the suction surface side on the over-tip flow, and hence leads to a breakdown of the pressure-driven mechanism, conventionally used in tip treatment and designs. The decoupling between blade loading and over tip leakage mass flow is clearly identified and highlighted. Furthermore, the realization of the loading-leakage flow decoupling indicates a possibility of a high-load blading design with a relatively low tip leakage loss. A high load blading is generated and analyzed to demonstrate the feasibility of such designs with a reduced tip leakage loss

Madame de Lafayette, Oeuvres complètes, édition établie, présentée et annotée par Camille Esmein-Sarrazin, Gallimard "Bibliothèque de la Pléiade", 2014

Moinier Bernard. Madame de Lafayette, Oeuvres complètes, édition établie, présentée et annotée par Camille Esmein-Sarrazin, Gallimard "Bibliothèque de la Pléiade", 2014. In: Cahiers Saint Simon, n°44, 2016. Saint-Simon et Proust. Journée d'études du samedi 12 mars 2016. pp. 148-151

À La rencontre de Saint-Simon

Moinier Bernard. À La rencontre de Saint-Simon. In: Cahiers Saint Simon, n°47, 2019. Saint-Simon et les égarements du Langage. Journée d’études du samedi 16 mars 2019. pp. 145-152