Analysis of data on the relation between eddies and streaky structures in turbulent flows using the placebo method

An artificially synthesized velocity field with known properties is used as a test data set in analyzing and interpreting the turbulent flow velocity fields. The objective nature of this approach is utilized for studying the relation between streaky and eddy structures. An analysis shows that this relation may be less significant than is customarily supposed

Optimal Inverse Method for Turbomachinery Design

An adjoint optimization method based on the solution of an inverse problem is proposed. In this formulation, the distributed control is a flow variable on the domain boundary, for example pressure. The adjoint formulation delivers the functional gradient with respect to such flow variable distribution, and a descent method can be used for optimization. The flow constraints are easily imposed in the parametrization of the controls, thus those problems with many strict constraints on the flow solution can be solved very efficiently. Conversely, the geometric constraints are imposed either by additional partial differential equations, or by penalization. Constraining the geometric solution, the classical limitations of the inverse problem design are overcome. Two examples pertaining to internal flows are give

RANS closure approximation by artificial neural networks

Turbulence modelling remains a challenge for the simulation of turbomachinery flows. Reynolds Averaged Navier-Stokes (RANS) equations will still be used for high-Reynolds number flows for several years and so there is interest in improving their prediction capability. Machine learning techniques offer several strategies which could be exploited for this purpose. In this work, an approach to improve the Spalart-Allmaras model is investigated. In particular, the model is used to predict the flow around the T106c low pressure gas turbine cascade. As a first step, an Artificial Neural Network (ANN) is trained on the data generated by the original model. Then, an optimisation procedure is applied in order to find the weights of the network which minimise the error between the predicted results and the available experimental data. The new model is tested at different Reynolds numbers on the T106c cascade and on a wind turbine airfoil in post-stall conditions. Significant improvements are observed in the condition chosen for the optimisation. Future work will be devoted to the generalisation of the approach by including multiple working conditions optimisations and adding new physical variables as inputs of the ANN

Efficient asymptotic preserving schemes for BGK and ES-BGK models on cartesian grids

This work is devoted to the study of complex flows where hydrodynamic and rarefied regimes coexist. This kind of flows are found in vacuum pumps or hypersonic re-entries of space vehicles where the distance between gas molecules is so large that their microscopic behaviour differ from the average behaviour of the flow and has be taken into account. We then consider two models of the Boltzmann equation viable for such flows: the BGK model dans the ES-BGK model. We first devise a new wall boundary condition ensuring a smooth transition of the solution from the rarefied regime to the hydrodynamic regime. We then describe how this boundary condition (and boundary conditions in general) can be enforced with second order accuracy on an immersed body on Cartesian grids preserving the asymptotic limit towards compressible Euler equations. We exploit the ability of Cartesian grids to massive parallel computations (HPC) to drastically reduce the computational time which is an issue for kinetic models. A new approach considering local velocity grids is then presented showing important gain on the computational time (up to 80$\%$). 3D simulations are also presented showing the efficiency of the methods. Finally, solid particle transport in a rarefied flow is studied. The kinetic model is coupled with a Vlasov-type equation modeling the passive particle transport solved with a method based on remeshing processes. As application, we investigate the realistic test case of the pollution of optical devices carried by satellites due to incompletely burned particles coming from the altitude control thrusters

Iterative Methods for Model Reduction by Domain Decomposition

We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including the Laplace equation with non-linear boundary conditions and the compressible Euler equations

Shape optimization governed by the Euler equations using an adjoint method

A numerical approach for the treatment of optimal shape problems governed by the Euler equations is discussed. Focus is on flows with embedded shocks. A very simple problem is considered: the design of a quasi-one-dimensional Laval nozzle. A cost function and a set of Lagrange multipliers are introduced to achieve the minimum. The nature of the resulting costate equations is discussed. A theoretical difficulty that arises for cases with embedded shocks is pointed out and solved. Finally, some results are given to illustrate the effectiveness of the method

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components' interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component -- associated with the solution on interior boundaries -- and a bubble component that vanishes at ports: this decomposition allows the static condensation of the bubble degrees of freedom and ultimately allows to recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-square problem and can be solved using the Gauss-Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our proposal; we further discuss the well-posedness of the mathematical formulation and the \emph{a priori} error analysis for linear coercive problems

Remarks on the Approximation of the Euler Equations by a Low Order Model

Fluid flows are very often governed by the dynamics of a small number of coherent structures, i.e., fluid features which keep their individuality during the evolution of the flow. The purpose of this paper is to present a way to simulate the Euler equations on the basis of the evolution of such coherent structures. One way to extract from flow simulations some basis functions which can be interpreted as coherent structures is by Proper Orthogonal Decomposition (POD). Then, by means of a Galerkin projection, it is possible to find the system of ODEs which approximates the problem in the finite dimensional space spanned by the basis functions found. Issues concerning the stability and the accuracy of such an approximation are discussed. It is found that a straight-forward Galerkin method is unstable. Some features of discontinuous Galerkin methods are therefore incorporated to achieve stability, which is proved for a linear scalar case. In addition, we propose a way to reduce the cost of the computation and to increase accuracy at the same time. Some one-dimensional computational experiments are presented, including shock tube simulations and rarefaction fans