A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification

Sensitivity analysis is a key element in a design optimization procedure. Although the related theory and numerical implementation are well known for steady problems, the application to unsteady partial differential equations, in particular in fluid mechanics, is still an active research area.In this report, a sensitivity equation method is described, in the context of compressible Navier-Stokes equations, and an efficient numerical implementation is proposed. The resulting approach is verified for two- and three-dimensional problems of increasing complexity

On the use of second-order derivatives and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics

International audienceThis article adresses the delicate issue of estimating physical uncertainties in aerodynamics. Usually, flow simulations are performed in a fully deterministic approach, although in real life operational uncertainty arises due to unpredictable factors that alter the flow conditions. In this article, we present and compare two methods to account for uncertainty in aerodynamic simulation. Firstly, automatic differentiation tools are used to estimate first- and second-order derivatives of aerodynamic coefficients with respect to uncertain variables, yielding an estimate of expectation and variance values (Method of Moments). Secondly, metamodelling techniques (radial basis functions, kriging) are employed in conjunction with Monte-Carlo simulations to derive statistical information. These methods are demonstrated for 3D Eulerian flows around the wing of a business aircraft at different regimes subject to uncertain Mach number and angle of attack

Comparison and Assessment of some Synthetic Jet Models

A synthetic jet is an oscillatory jet, with zero time-averaged mass-flux, used to manipulate boundary layer characteristics for flow control applications such as drag reduction, detachment delay, etc. The objective of this work is the comparison and assessment of some numerical models of synthetic jets, in the framework of compressible flows governed by Reynolds- averaged Navier-Stokes (RANS) equations. More specifically, we consider three geometrical models, ranging from a simple boundary condition, to the account of the jet slot and the computation of the flow in the underlying cavity. From numerical point of view, weak and strong oscillatory boundary conditions are tested. Moreover, a systematic grid and time-step refinement study is carried out. Finally, a comparison of the flows predicted with two turbulence closures (Spalart-Allmaras and Menter SST k − ω models) is achieved

Adaptive Refinement for Compressible Flow Analysis using an Isogeometric Discontinuous Galerkin Method

International audienceThis work aims at developping an efficient isogeometric approach to simulate compressible flows including shocks. The use of the classical isogeometric analysis method based on a finite-element formulation [1] is tedious in this context, due to the necessity to introduce suitable terms for stabilization and shock capturing. Therefore, an alternate formulation is explored, based on a Discontinuous Galerkin formulation , better suited to hyperbolic conservation laws [2]. This approach relies on the Bézier extraction technique to transform NURBS patches into a set of discontinous rational elements without altering the geometry. It has been shown [3] that the resulting method exhibits optimal convergence rates for regular solutions, allows a sharp capture of discontinuities and preserves CAD-based geometries. On this basis, the present work focuses on the use of adaptive refinement techniques to improve the computational efficiency. An error indicator based on the measure of the solution jump at element interfaces is introduced, in conjunction with a dynamic refinement and coarsening technique for rational Bézier elements based on multiple knot insertion. We show that this approach allows a significant reduction of the computational time and preserves the convergence rates. In particular, a local refinement is achieved in regions where rapid variations of the solution are observed, while a coarse discretization is maintained in other regions without altering the geometry. Illustrations are provided for unsteady Euler equations, with and without shocks, and compressible Navier-Stokes equations

Coupling Local and Global Shape Optimization in Aerodynamic Design

Wing design in aerodynamics requires the definition of global geometrical characteristics, such as span, root/tip length ratio, angle of attack, twist angle, sweep angle, etc, as well as local geometrical features that determine the wing section. The objective of this study is to propose an efficient algorithm to achieve the optimization of both global and local shape parameters. We consider as testcase the drag minimization, under lift constraint, of the wing shape of a business aircraft in transonic regime, the flow being modeled by the compressible Euler equations. We show that a straightforward optimization of all parameters fails, due to multimodality of the optimization problem. Then, some alternative strategies are proposed. Among them, the use of virtual Nash games yields the best results, in terms of cost function value obtained as well as computational efficiency

Aerodynamic Shape Optimization with Uncertain Operating Conditions using Metamodels

In this paper, we address aerodynamic shape optimization problems including uncertain operating conditions. After a review of the possible approaches to take into account uncertainty, we propose to use meta-modeling techniques in order to develop a two-level modeling procedure for statistics estimation. Radial basis functions are employed to approximate the aerodynamic coefficients as operating conditions vary. Then, a Monte-Carlo method is employed to estimate statistics using the approximate model. The proposed approach is applied to the robust optimization of the wing shape of a business aircraft, by minimizing the mean and the variance of the drag coefficient with uncertain free-stream Mach number

An Introduction to Isogeometric Analysis with Application to Thermal Conduction

In this report, we present a new method, namely isogeometric analysis, proposed by T. Hughes and his collaborators, whose objective is to combine Computer Aided Design (CAD) tools and Finite-Element (FE) solvers into a single software entity. This consists in replacing grids by parametric surfaces and volumes to define the integration elements used in a variational formulation. This approach allows to define a computational domain that matches exactly the geometry of the problem, whatever the number of degrees of freedom. High-order, h- and p-adaptive schemes, as well as hierarchical solving strategies can be constructed. After presenting the method and comparing with classical finite-element approach, some numerical experiments are carried out using a thermal conduction test-case

Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization

Parameterization techniques commonly used in aerodynamic shape optimization are essentially general and multi-purpose approaches. As a consequence, they cannot be well suited to a particular shape optimization problem. The present study proposes a new method that adapts an initial and perhaps naïve parameterization of an aerodynamic shape by the Free-Form Deformation (FFD) technique, to the particular optimization problem to solve, according to a first approximation of the solution. This parameterization adaption method is included in a three-dimensional aerodynamic shape optimization procedure for Eulerian flows and is demonstrated for the design optimization of the wing of a business aircraft

Isogeometric analysis for compressible flows using a Discontinuous Galerkin method

International audienceThe objective of this work is to investigate a Discontinuous Galerkin (DG) method for compressible Euler equations, based on an isogeometric formulation: the partial differential equations governing the flow are solved on rational parametric elements, that preserve exactly the geometry of boundaries defined by Non-Uniform Rational B-Splines (NURBS), while the same rational approximation space is adopted for the solution. We propose a new approach to construct a DG-compliant computational domain based on NURBS boundaries and examine the resulting modifications that occur in the DG method. Some two-dimensional test- cases with analytical solutions are considered to assess the accuracy and illustrate the capabilities of the proposed approach. The critical role of boundary curvature is especially investigated. Finally, a shock capturing strategy based on artificial viscosity and local refinement is adapted to this isogeometric context and is demonstrated for a transonic flow

Uncertainty quantification in a macroscopic traffic flow model calibrated on GPS data

International audienceThe objective of this paper is to analyze the inclusion of one or more random parameters into the deterministic Lighthill-Whitham-Richards traffic flow model and use a semi-intrusive approach to quantify uncertainty propagation. To verify the validity of the method, we test it against real data coming from vehicle embedded GPS systems, provided by Autoroutes Trafic