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Table of contents
Introduction
I RANS solver implementation for turbulent applications
Introduction
1 Main solver description
1.1 Main flow solver description
1.1.1 Modeling equations
1.1.2 Spatial discretization
1.1.3 Boundary Conditions
1.1.4 Rotating Flow
1.1.5 Time integration
1.2 Linear System Resolution
1.2.1 Standard linear solvers
1.2.2 SGS – Line solver
1.2.3 Multigrid solver
1.2.4 Multigrid in Wolf
1.2.5 Residual computation and limiter freezing
1.3 Adjoint
1.3.1 Functional sensitivity to the solution
1.3.2 Jacobian computation and system resolution
1.4 Verification and validation
1.4.1 Verification
1.4.2 Validation
2 RANS solver description
2.1 Turbulence modeling
2.1.1 Spalart-Allmaras Negative Model
2.1.2 Spalart-Allmaras Quadratic Constitutive Relation (SA-QCR)
2.2 Spatial discretization
2.2.1 Di↵erentiation
2.3 Wall Functions
2.3.1 RANS solution initialization
2.3.2 RANS adjoint
3 Periodic computations for turbomachinery applications
3.1 Implementation
3.2 Periodic Adjoint
3.3 Applications
3.3.1 LS89 case
3.3.2 RO37 case
4 Solution correction for error estimation
4.1 Non-linear Corrector computation
4.1.1 1D RANS Example
4.1.2 Formal discretization
4.1.3 Finite Volume Corrector
4.1.4 Finite Element Corrector
4.2 Numerical results: manufactured solutions
4.2.1 Finite elements 1D nonlinear advection di↵usion problem
4.2.2 Finite volume 2D Navier-Stokes equations
4.3 Numerical results: applied cases
4.3.1 2D Laminar NACA0012
4.3.2 2D Turbulent NACA0012
4.3.3 3D ONERA M6 wing turbulent Case
4.3.4 3D inviscid supersonic case: Sonic Boom Prediction Workshop
4.4 Conclusion and perspectives
Conclusion
II Anisotropic mesh adaptation for RANS applications
Introduction
5 Continous mesh framework for mesh adaptation
5.1 Continous mesh
5.1.1 A simple 1D example
5.2 Extension to higher dimensions
5.2.1 Euclidian and Riemannian metric spaces
5.2.2 Unit mesh
5.2.3 Operations on metrics
5.3 The continuous mesh framework
5.3.1 Duality discrete-continous: a new formalism
5.3.2 Continuous linear interpolation
5.3.3 Summary
5.4 Multiscale mesh adaptation
5.4.1 The non-linear adaptation loop
5.4.2 Optimal control of the interpolation error and optimal meshes
5.4.3 General error control
5.4.4 Error equidistribution
5.4.5 Application to numerical solutions
5.4.6 Control of the error in Lp norm
5.4.7 Conclusion
6 RANS mesh adaptation
6.1 RANS goal-oriented mesh adaptation
6.1.1 General non-linear error estimation
6.1.2 RANS goal-oriented error estimation: Linearization
6.1.3 RANS goal-oriented error estimation: Case of Spalart-Allmaras equations
6.1.4 Second derivatives error estimation
6.1.5 Hessian computation
6.2 Mesh adaptation cycle for applied cases
6.3 Applied cases: NASA Rotor
6.3.1 Geometry description
6.3.2 Mesh comparison
6.4 Applied cases: M6 wing
6.4.1 Case description
6.4.2 Feature-based mesh adaptation
6.4.3 Comparison of hessian reconstructions
6.4.4 Goal-oriented mesh adaptation
6.4.5 Comparison with other codes
6.5 Applied cases: Trap wing
6.5.1 Case description
6.5.2 Results
6.6 Applied cases: High-Lift Common Research Model
6.7 Improvement of RANS mesh adaptation
6.7.1 Adaptation with Wall-Laws: Subsonic NACA0012 validation
6.7.2 Hessian boundary correction: impact on a 2D High-Lift configuration
6.7.3 Conclusion
6.8 Conclusion
7 Periodic mesh adaptation
7.1 Periodic mesh adaptation
7.1.1 Handling the metric
7.1.2 Periodicity recovery
7.2 Application to turbomachinery
7.2.1 LS89 case
7.2.2 RO37 case
Conclusion
Conclusion and perspectives
Acknowledgments
Bibliography
