## Section: New Results

### Numerical schemes and algorithms for fluid mechanics.

Participants : Rémi Abgrall [Corresponding member] , Guilaume Baurin, Pietro-Marco Congedo, Cécile Dobrzynski, Marc Duruflé, Dante De Santis, Algyane Froehly, Gianluca Geraci, Robin Huart, Arnaud Krust, Pascal Jacq, Cédric Lachat, Mario Ricchiuto, Christelle Wervaecke.

#### Residual distribution schemes

This year, many developments have been conducted and implemented in
the `RealfluiDS ` software after [36] which has
opened up many doors.

A three dimensional and parallel version of the second order RD scheme on hybrid meshes (tetrahedral-hexahedral) for the steady Euler equation is now working. It has been
validated on several M6 wing meshes given by ONERA. The largest mesh has
5.5 10^{6} vertices and the simulation has been done on 256 processors. Meanwhile
the third order version of the same scheme (for tetrahedrons) was run succesfully on a busines jet configuration in supersonic conditions. The mesh was given by GAMMA3.
This last version is also able to run the Navier Stokes equation, but the approximation is not fully satisfactory.

For this reason, we have put a lot of effort in understanding the correct way of
approximation advection-diffusion like problems that degenerate to standard RD scheme when the viscous term disapear. The main difficulty was to have
a scheme that work well in the largest possible range of Peclet numbers. This goal has been achieved, ane we have two versions of
the method (one using a gradient reconstruction and one using a relaxation interpretation of the steady advection diffusion.) These methods will be implemented in `RealfluiDS ` in the coming months for IDIHOM.

Mario Ricchiuto has conducted work on more efficient RD discretizations for time dependent problems. This has led to two formulations. The first is a genuinely explicit variant of the method based on high order mass lumping and on a temporally shifted stabilization (upwinding) operator [7] , [22] . This formulation is well suited for problems where the time scales of interest are small. The higher order variant of the methodology, currently limited to second order, is under development. In parallel, in collaboration with M. Hubbard of the university of Leeds, an unconditionally stable space-time formulation has been proposed [18] , [32] . This variant allows arbitrarily large time steps to be taken while preserving the accuracy and monotonicity of the results. Further work is under way to extend the accuracy to more than second order. The PhD of Guillaume Baurin has started to implement the third order version of our methods in a real industrial platform (SAFRAN). He has started the implementation of the RD scheme for the Navier Stokes equation in that platform.

Results on curved meshes and non-Lagrangian elements (Bézier and NURBS) have been obtained by Algiane Froehly. The method is now third and fourth order accurate in 2D. Cécile Dobrzynski has worked on the construction of “high order” meshes using Bézier and Nurbs elements. Numerical results using these meshes and A. Froehly's development has been obtained in 2D for subsonic, transonic and hypersonic problems. Extension in 3D is underway, one of the main difficulty is to generate meshes.

#### A Stabilized Finite Element Method for Compressible Turbulent Flows.

C. Waervaecke's PhD thesis has been a collaborative work between BACCHUS, MC2 (Héloise Beaugendre) and PUMAS (Boniface Nkonga). The main weakness of the classical finite element method (Galerkin) is its lack of stability for advection dominated flows. We consider in this work a compressible Navier-Stokes equations combined with the one equation Spalart-Allmaras turbulence model. These equations are solved in a coupled way. The numerical stability is achieved thanks to the Streamline Upwind Petrov-Galerkin (SUPG) formulation. Within the framework of SUPG method, artificial viscosity is anisotropic and the principal component is aligned with streamlines. The aim is to put sufficient viscosity to get rid of instability and unphysical oscillations without damaging the accuracy of the method. The amount of artificial viscosity is controlled by a stabilization tensor. Since an optimal way to choose this tensor is still unknown, several ways have been investigated. Beside SUPG method is also used in combination with a shock-parameter term which supplied additional stability near shock fronts. Numerical results show that the method is able to reproduce good turbulent profiles with less numerical diffusion than a finite volume method. Even in the case of almost incompressible flow, the numerical strategy is robust and gives good results.

#### Uncertainty quantification

S. Galéra, P. Congedo and R. Abgrall have made a detailed comparison between the semi-intrusive method developed last year with more classical non intrusive polynomial chaos methods, and Monte Carlo results. These results have been presented in part during the ECCOMAS CFD conference in June 2010. We have also adapted the SUPG method for turbulent flows to this method, so that we are able to produce turbulent simulations including one and two uncertainties (here on the inflow mach number and the velocity angle).

During E. Mbinki's internship, we have tried to understand the algorithms behind the Smoliak algorithm and how they can be adapted to the semi intrusive method using local conditional expectancy.

G. Geraci has started his PhD and one of the goals is to handle as many as possible uncertainties for fluid problems.

#### Discontinuous Galerkin schemes, New elements in DG schemes

Explicit schemes may become very expensive because of a restrictive stability condition (small CFL), especially when the computational mesh comprises some very small elements. A solution, known as local time-stepping, consists of considering different time steps for each element of the mesh. These kinds of solutions can be applied to continuous finite element but are more natural when applied to Discontinuous Galerkin methods. Marc Duruflé with S. Imperiale (PhD at project POEMS) have proposed a new local time-stepping strategy and validated the approach for wave problems.

Rémi Abgrall and Pierre-Henri Maire, with François Vilar (PhD at CELIA funded by a CEA grant started in October 2009), have started to work on Lagrangian schemes within the Discontinuous Galerkin schemes. The idea is to start from the formulation of the Euler equation in full Lagrange coordinates: the spatial derivative are written in Lagrangian coordinates. This has led to a publication in Computers and Fluids where our results in 1D are described. Currently, we are developing the method in 2 dimensions. The main difficulty is to understand the role and the structure of the Geometric Conservation law.

#### Mesh adaptation

C. Dobrzynski has developed an efficient tool for handling moving
2D and 3D meshes. Here, contrarily to most ALE methods, the
connectivity of the mesh is changing in time as the objects within the
computational domain are moving. The objective is to guaranty a high
quality mesh in terms of minimum angle for example. Other criteria,
which depend on the physical problem under consideration, can also
been handled. Currently this meshing tool is being coupled with
`RealfluiDS ` in order to produce CFD applications. One target example is
the simulation of the 3D flow over helicopter blades.

A work on high order mesh generation has begun. We are
modifying the classic mesh operators to take into account the curve
edges. Beginning with a derefined valid curve mesh, we would to be
able to generate an uniform refined curve mesh and also to adapt the
mesh density in certain region (boundary layer).
Moreover, starting with a P^{1} (triangle) mesh and some information
on the boundary, we are able to generate a valid three order curved mesh.
The algorithm is based on edge swaps and is
similar to a boundary enforcement procedures.