Jean-Christophe Robinet

Spatially convective global modes in a boundary layer

The linear stability of a weakly nonparallel flow, the case of a flat plate boundary layer, is revisited by a linear global stability approach where the two spatial directions are taken as inhomogeneous, leading to a fully nonparallel stability method. The resulting discrete eigenvalues obtained by the fully nonparallel approach seem to be in agreement with classical Tollmien–Schlichting waves. Then the different modes are compared with classical linear stability approach and weakly nonparallel method based on linear parabolized stability equations (PSEs). It is illustrated that the nonparallel correction provided by the linear global stability approach is well matched by linear PSE. Furthermore, physical interpretation of these spatio-temporal global modes is given where a real pulsation, which has more physical interest, is considered. In particular the use of a Gaster transformation and the pseudospectrum illustrate the local and global properties of these Tollmien–Schlichting modes. Finally, the contr...

Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow

Subcritical transition in plane Poiseuille flow is investigated by means of a Lagrange-multiplier direct-adjoint optimization procedure with the aim of finding localized three-dimensional perturbations optimally growing in a given time interval (target time). Space localization of these optimal perturbations (OPs) is achieved by choosing as objective function either a p-norm (with ) of the perturbation energy density in a linear framework; or the classical (1-norm) perturbation energy, including nonlinear effects. This work aims at analyzing the structure of linear and nonlinear localized OPs for Poiseuille flow, and comparing their transition thresholds and scenarios. The nonlinear optimization approach provides three types of solutions: a weakly nonlinear, a hairpin-like and a highly nonlinear optimal perturbation, depending on the value of the initial energy and the target time. The former shows localization only in the wall-normal direction, whereas the latter appears much more localized and breaks the spanwise symmetry found at lower target times. Both solutions show spanwise inclined vortices and large values of the streamwise component of velocity already at the initial time. On the other hand, p-norm optimal perturbations, although being strongly localized in space, keep a shape similar to linear 1-norm optimal perturbations, showing streamwise-aligned vortices characterized by low values of the streamwise velocity component. When used for initializing direct numerical simulations, in most of the cases nonlinear OPs provide the most efficient route to transition in terms of time to transition and initial energy, even when they are less localized in space than the p-norm OP. The p-norm OP follows a transition path similar to the oblique transition scenario, with slightly oscillating streaks which saturate and eventually experience secondary instability. On the other hand, the nonlinear OP rapidly forms large-amplitude bent streaks and skips the phases of streak saturation, providing a contemporary growth of all of the velocity components due to strong nonlinear coupling.

Development of Numerical Schemes for Hybrid Turbulence Modelling in an Industrial CFD Solver

Many turbulent flows are dominated by large turbulent structures caused by massive boundary layer separation, i.e : bluff body flows past launchers and missiles, cars, turbomachines. These flows represent a challenge for numerical simulation methods since models based on Reynolds averaged equations (URANS) are not able to give an accurate representation of unsteady turbulent structures. On the other hand, Large Eddy Simulation (LES) is generally too expensive for an industrial use. In the attempt to find a compromise between computation cost and accuracy, it becomes useful to consider hybrid approaches between RANS and LES modelling. The aim of such simulations is to resolve only that portion of the turbulent spectrum responsible for the aerodynamics forces acting on an immersed body, and to model the rest. A lot of simulation strategies hybrid exist (for example SAS, PANS, PITM, ZDES..) but exploring them is beyond the scope of this paper. Here we focuse instead on the impact of the numerical properties of the resolution scheme on hybrid RANS/LES simulations and investigate a simple but effective strategy to improve numerical accuracy of the solver for a given mesh resolution.

Intermittency and transition to chaos in the cubical lid-driven cavity flow

Transition from steady state to intermittent chaos in the cubical lid-driven flow is investigated numerically. Fully three-dimensional stability analyses have revealed that the flow experiences an Andronov-Poincare-Hopf bifurcation at a critical Reynolds number

Automatic Hybrid RANS/LES Strategy for Industrial CFD

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Multiple-correction hybrid k-exact schemes for high-order compressible RANS-LES simulations on fully unstructured grids

= 1914. As for the 2D-periodic lid-driven cavity flows, the unstable mode originates from a centrifugal instability of the primary vortex core. A Reynolds-Orr analysis reveals that the unstable perturbation relies on a combination of the lift-up and anti lift-up mechanisms to extract its energy from the base flow. Once linearly unstable, direct numerical simulations show that the flow is driven toward a primary limit cycle before eventually exhibiting intermittent chaotic dynamics. Though only one eigenpair of the linearized Navier-Stokes operator is unstable, the dynamics during the intermittencies are surprisingly well characterized by one of the stable eigenpairs.

Assessment of automatic Hybrid RANS/LES Models for Industrial CFD

An automatic HRL (Hybrid RANS/LES) strategy is investigated in FLUSEPA, a finite-volume solver developed by Airbus Defense and Space. A HRL turbulence model is coupled to a high-order hybrid numerical approximation method. Concerning the turbulence model, the well-known \(k-\varepsilon \) two equations RANS turbulence model is sensitized to the grid as suggested by Perot and Gadebusch (Phy Fluids 19:1–11, 2007). Concerning the numerical strategy, a third-order accurate upwind approximation method is locally re-centered in vortex dominated regions to achieve non-dissipative fourth-order accuracy. Results are presented for a 2D backward facing step and an an axisymmetry backward facing step, which represent good prototypes of after body flows.

Aerodynamic rotor blade optimization at Eurocopter - a new way of industrial rotor blade design

Abstract A Godunovs type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.

Characterisation of Synthetic Turbulence Methods for Large-Eddy Simulation of Supersonic Boundary Layers

The objective of this paper is to assess and compare several self-adapting hybrid RANS/LES models for a backward facing step flow, in conjunction with a high-order finite volume numerical scheme well suited for scale-resolving simulations. The most promising model is then applied to an axisymmetric backward facing step, representative of the base flow behind a launcher. Present computations provide a satisfactory representation of the characteristic flow frequencies. Finally, the proposed methodology is applied to the computation of the flow behind the Ariane 5 space launcher to demonstrate its applicability to complex flow configurations of industrial interest.

INVESTIGATION OF TWO MECHANISMS GOVERNING CLOUD CAVITATION SHEDDING: EXPERIMENTAL STUDY AND NUMERICAL HIGHLIGHT

Industrial aerodynamic design optimization of helicopter rotor blades requires employment of multi-objective optimization methods to account for the two distinct objectives in hover and forward flight. Genetic algorithms are preferred for finding the Pareto Optimal Front, as they allow the engineer to select out of optimal designs. An optimization loop is created, coupling the Dakota optimization library and two simulation methods for objective function evaluation: comprehensive rotor code HOST and CFD solver elsA. For low-cost rotor simulations by HOST, a genetic algorithm is employed to maximize hover and forward flight rotor performance in single- and two-point optimizations. Twist and chord laws of the 7A blade are optimized separately and simultaneously. As genetic algorithms require too many cost function evaluations for CFD-based optimizations, Surrogate Based Optimization (SBO) is employed. SBO is initialized by a preliminary Design of Experiment (DoE). The results are used to generate a metamodel for estimation of cost function evaluations in the optimization algorithm. The metamodel is updated using information from subsequent simulations. Validation of HOST-based SBO against full genetic algorithm optimizations shows that the Pareto Optimal Front is correctly represented by SBO, while requiring 88% less cost function simulations. Several sizes of initial DoE, number of update cycles and number of simulations added per cycle are tested. Then, a similar SBO optimization is carried out by replacing the HOST code by CFD for hover performance simulation. The results demonstrate the ability of both solvers and both optimization techniques to perform aerodynamic design optimization of helicopter rotor blades.