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This book offers detailed insights into new methods for high-fidelity CFD, and their industrially relevant applications in aeronautics. It reports on the H2020 TILDA project, funded by the European Union in 2015-2018. The respective chapters demonstrate the potential of high-order methods for enabling more accurate predictions of non-linear, unsteady flows, ensuring enhanced reliability in CFD predictions.

The book highlights industrially relevant findings and representative test cases on the development of high-order methods for unsteady turbulence simulations on unstructured grids; on the development of the LES/DNS methodology by means of multilevel, adaptive, fractal and similar approaches for applications on unstructured grids; and on leveraging existent large-scale HPC networks to facilitate the industrial applications of LES/DNS in daily practice. Furthermore, the book discusses multidisciplinary applications of high-order methods in the area of aero-acoustics. All in all, it offers timely insights into the application and performance of high-order methods for CFD, and an extensive reference guide for researchers, graduate students, and industrial engineers whose work involves CFD and turbulence modeling.

### The TILDA Project—Objectives, Project and Activities

Abstract
This chapter offers a short introduction to the background motivations and organization of the TILDA project.
C. Hirsch

### Implicit Methods

Abstract
This chapter describes implicit time integration methods developed by four TILDA partners for the application in scale-resolving simulations. While explicit time integration methods like explicit Runge-Kutta methods are traditionally used in scale-resolving simulations they are slowed down significantly for high Reynolds number flows due to their restrictive CFL conditions. Recent developments in implicit solver techniques increased the efficiency of implicit time integration schemes making them competitive to explicit schemes. In this chapter the effectiveness is investigated of implicit time integration methods in the framework of high-order Discontinuous Galerkin methods.
Ralf Hartmann, Francesco Bassi, Igor Bosnyakov, Lorenzo Botti, Alessandro Colombo, Andrea Crivellini, Matteo Franciolini, Tobias Leicht, Emeric Martin, Francescocarlo Massa, Florent Renac, Alexey Troshin, Vladimir Vlasenko, Marcel Wallraff, Andrey Wolkov

### Multi-level Approach

Abstract
In the framework of the TILDA project, Dassault Aviation and ONERA have worked on the development of variational multiscale (VMS) approaches to LES in the context of high-order, respectively continuous and discontinuous Galerkin, Finite-Element methods. Dassault Aviation presents the implementation of VMS in its higher-order industrial stabilized continuous finite-element code AeTher, whereas ONERA investigates the performance of the VMS approach based on a high-order DG method. Both authors assess the numerical developments with the Taylor-Green vortex test case at different Reynolds numbers ranging from 1500 to 20,000. VMS results are compared with reference data provided by direct numerical simulation and with simulations obtained using various subgrid scale models and numerical fluxes. LES simulations of the sub-critical flow past a circular cylinder at Reynolds 3900 and 20,000 have also been considered by ONERA.
M. de la Llave Plata, F. Chalot, E. Lamballais, F. Naddei, P. Yser

Abstract
This chapter describes space adaptive approaches developed by six TILDA partners for the application in scale-resolving simulations. They are designed to provide sufficient spatial resolution in regions where required and to allow a lower resolution elsewhere for efficiency reasons. Adaptation techniques considered include mesh (h-refinement), order refinement of the spatial discretization (p-refinement) or a combination of both (hp-refinement). Furthermore, near-wall local mesh refinement, refinement using feature-based indicators and indicators obtained from the Variational Multiscale Method are considered.
R. Hartmann, A. Balan, F. Bassi, J.-F. Boussuge, A. de Brauer, J.-S. Cagnone, A. Colombo, V. Couaillier, O. Coulaud, A. Crivellini, M. Franciolini, A. Ghidoni, K. Hillewaert, M. de la Llave Plata, G. Manzinali, F. Naddei, G. Noventa, G. Puigt, B. C. Vermeire, P. E. Vincent

### Wall-Modeled LES

Abstract
This chapter describes the approaches to wall-modeled LES proposed by three TILDA Partners. DLR adopts a classic treatment of the near wall region with wall functions. University of Bergamo and Dassault Aviation rely on hybridization of LES with a RANS model near the wall. University of Bergamo relies on the switch of X-LES and Dassault Aviation adopts a DDES methodology coupled with their VMS.
F. Chalot, P. Yser, R. Hartmann, F. Bassi, A. Colombo, A. Ghidoni, G. Noventa

### Quality Measures for Curvilinear Finite Elements

Abstract
We present a method for computing robust shape quality measures defined for finite elements of any order and any type, including curved pyramids. The measures are heuristically defined as the minimum of the pointwise quality of curved elements. Three pointwise qualities are considered: the ICN that is related to the conditioning of the stiffness matrix for straight-sided simplicial elements, the scaled Jacobian that is defined for quadrangles and hexahedra, and a new shape quality that is defined for triangles and tetrahedra. Based on previous work presented by Johnen et al. (Journal of Computational Physics 233:359–372, 2013, [1]); Johnen and Geuzaine (Journal of Computational Physics 299:124–129, 2015, [2]), the computation of the minimum of the pointwise qualities is efficient. The key feature is to expand polynomial quantities into Bézier bases which allows to compute sharp bounds on the minimum of the pointwise quality measures.
A. Johnen, C. Geuzaine, T. Toulorge, J.-F. Remacle

Abstract
This paper aims at addressing the following issue. Assume a unit square: $$\varOmega = \{(x^1,x^2) \in [0,1] \times [0,1]\}$$ and a Riemannian metric $$g_{ij}(x^1,x^2)$$ defined on U. Assume a mesh $$\mathscr {T}$$ of U that consist in non overlapping valid quadratic triangles that are potentially curved. Is it possible to build a unit quadratic mesh of U i.e. a mesh that has quasi-unit curvilinear edges and quasi-unit curvilinear triangles? This paper aims at providing an embryo of solution to the problem of curvilinear mesh adaptation. The method that is proposed is based on standard differential geometry concepts. At first, the concept of geodesics in Riemannian spaces is quickly presented: the geodesic between two points as well as the unit geodesic starting at a given point with a given direction are the two main tools that allow us to address our issue. Our mesh generation procedure is done in two steps. At first, points are distributed in the unit square U in a frontal fashion, ensuring that two points are never too close to each other in the geodesic sense. Then, a simple isotropic Delaunay triangulation of those points is created. Curvilinear edge swaps as then performed in order to build the unit mesh. Notions of curvilinear mesh quality is defined as well that allow to drive the edge swapping procedure. Examples of curvilinear unit meshes are finally presented.
Ruili Zhang, Amaury Johnen, Jean-François Remacle

### Parallelisation to Several Tens-of-Thousands of Cores

Abstract
In this Section a detailed and quantitative understanding is provided of how algorithms should be designed and implemented to effectively target a range of existing and emerging ‘massively parallel’ hardware platforms. The goal set up in the TILDA project was to demonstrate the capability and efficiency of the high-order methods developed by the partners on up to 50,000 cores.
F. Bassi, L. Botti, L. Verzeroli, R. Hartmann, J. Jägersküpper, E. Martin, M. Lorteau, P. E. Vincent, F. D. Witherden, B. C. Vermeire, J. S. Park, A. Iyer, K. Puri, D. Gutzwiller, C. Hirsch, F. Chalot

### I/O Post- and Co-Processing for High-Order Methods

Abstract
While the exascale computing era is approaching, the growing gap between computing resources and IO bandwidth for massively parallel simulations has already become a major bottleneck for the scientific discovery process. In this context, various strategies to enable and accelerate the analysis of data produced by massively parallel high-order methods are presented, with an emphasis on in-situ visualization and co-processing techniques. First, a library of parallel procedures is presented for an efficient collection of turbulence statistics within the framework of a modal discontinuous Galerkin method. Afterwards, an acoustic co-processing strategy is presented whereby sound radiation calculations are performed concurrently with CFD calculations in order to avoid the need to store prohibitive amount of data. Finally, an open-source and scalable post-hoc visualization and processing tool dedicated to the analysis of large data sets produced by high order methods is first presented. This post-hoc processing tool has then been extended to a co-processing interface which enables live in-situ visualization and analysis of high-order solutions produced by massively parallel simulations.
M. Rasquin, K. Hillewaert, F. Bassi, A. Colombo, F. Massa, G. Rahier, E. Martin, F. Renac

### Periodic Hill

Abstract
In this chapter, we focus on the use of high-order discontinuous Galerkin (DG) methods to perform simulations of the turbulent flow over periodically arranged hills, which corresponds to configuration TC-F1 in the TILDA project (Periodic Hill).
M. de la Llave Plata, F. Bassi, R. Hartmann, F. Massa

### Taylor-Green Vortex

Abstract
This chapter collects and synthesizes the computational results obtained by seven TILDA partners for the Taylor-Green Vortex, a standard test case for testing the accuracy and the performance of high-order methods on direct numerical simulations. It includes details of the numerical setup chosen by each of the partners, the computational cost as well as a comparison of the numerical results with respect to accuracy. An additional investigation analyzes the dependency of the accuracy of the results on the numerical flux and the polynomial function space employed.
R. Hartmann

### Implementation of High Order Discontinuous Galerkin Method and Its Verification Using Taylor-Green Vortex and Periodic Hills Test Cases

Abstract
An implementation of Discontinuous Galerkin method designed for unsteady computations is presented. Explicit 4-th order 5-stage strong stability-preserving Runge-Kutta scheme is used together with global time stepping technique. Shape functions are taken to be orthonormal polynomials in physical space. Polynomial orders K of up to 5 are used, formally giving a K + 1 accuracy order. Bassi and Rebay 2 approximation of viscous fluxes is adopted. Test cases presented are DNS of Taylor-Green Vortex at Re = 1600 and ILES/DDES computations of ERCOFTAC Periodic Hills test case at Re = 10 595. The results include accuracy versus computational cost comparisons.
Igor Bosnyakov, Sergey Mikhaylov, Vladimir Podaruev, Alexey Troshin, Andrey Wolkov

### Large Eddy Simulation of Single Stream Jet Using High-Order Methods

Abstract
This chapter deals with the prediction of turbulent jet simulations, which are seen to be efficient flow cases to investigate further compressible LES solvers regarding their ability to simulate jet noise.
J. F. Boussuge, R. Biolchini, M. Lorteau, M. de la Llave Plata, N. Lupoglazoff, F. Vuillot, V. Couaillier

### Boeing Rudimentary Landing Gear Configuration

Abstract
This chapter includes computational details and results obtained in the EU-project TILDA on the Boeing Rudimentary Landing Gear.
R. Hartmann, F. Bassi, A. Colombo, A. Crivellini, M. Franciolini, F. Massa

### LAGOON Landing Gear

Abstract
This chapter describes the results obtained during the TILDA Project on the additional LAGOON landing gear test case.
F. Chalot, P.-E. Normand, P. Yser, S. Barré, J.-M. Hasholder

### Generic Falcon Business Jet in Landing Configuration

Abstract
This chapter describes the results obtained during the TILDA Project on the Generic Falcon in Landing Configuration test case.
F. Chalot, P.-E. Normand, S. Barré, J.-M. Hasholder, N. Réau

### High-Lift Low Pressure Turbines T106-A and T106-C

Abstract
This chapter presents the high order simulations of the T106 performed by SAFRAN (T106-A) and NUMECA (T106-C). The T106 cascade is a low-pressure turbine cascade. The suction side is subject to separation-induced transition in the aft-part of the blade, with sensitivity to the passing wakes and pressure gradient. The configuration was experimentally characterized by [15, 16] and has become a widely studied numerical configuration. The configuration is thus part of the High Order Workshop validation cases [1]. Simulations of both partners are performed with clean inlet conditions. Results are assessed in terms of pressure coefficient and loss coeffecient. Comparisons with numerical results from literature and experiments are discussed.
S. Mouriaux, K. Puri

### Computational Campaign on the MTU T161 Cascade

Abstract
Preliminary high-fidelity simulations of the MTU T161 low pressure turbine cascade with diverging end walls have been performed on massively parallel computational resources with four different high-order methods at outlet isentropic Mach number $$M_{2s}=0.601$$ and two outlet isentropic Reynolds numbers, namely $$Re_{2s}=90\,\text {K}$$ and $$Re_{2s}=200\,\text {K}$$. First the flow regime and the boundary conditions are thoroughly described. The implementation of each method is then briefly introduced before the main results are presented. The main flow features of this test case have been qualitatively highlighted by these simulations. However, discrepancies have been observed quantitatively in terms of separation point on the suction side of the blade, especially at the lowest Reynolds. These simulations relied mainly on a laminar boundary layer at the inlet of the domain, which is likely the root cause of the observed discrepancies. Additional simulations with turbulent boundary layer imposed at the inlet are required to characterize the flow separation based on the turbulence intensity at the inlet.
M. Rasquin, K. Hillewaert, A. Colombo, F. Bassi, F. Massa, K. Puri, A. S. Iyer, Y. Abe, F. D. Witherden, B. C. Vermeire, P. E. Vincent

### Detached-Eddy Simulation of Dual Stream Nozzle Jet Using High Order Discontinuous Galerkin Method

Abstract
Dual stream nozzle jet computations are presented in the chapter. The test case was prepared by TsAGI and ITAM SB RAS for TILDA project. The jet issued from a coaxial convergent nozzle, the outer flow being supersonic and the inner one being subsonic. The computation was performed using Discontinuous Galerkin method implemented in computer code Zoom (TsAGI). Piecewise-cubic representation of solution was adopted in the method. Spalart–Allmaras DDES approach was used for this scale-resolving simulation. The details of numerical techniques are discussed. Computational results including Pitot pressure profiles and mass flow rate spectra at several points in the jet are presented and compared with the experimental data.
Sergey Bosnyakov, Sergey Mikhaylov, Vladimir Podaruev, Alexey Troshin, Andrey Wolkov

### NASA Rotor 37

Abstract
The NASA Rotor 37 is an isolated transonic axial compressor rotor. This case was initially included in a wider research program to cover a range of design parameters typical of high pressure compressor inlet stage of aircraft engines. Most numerical studies fail at predicting with accuracy the overall performance, e.g., the adiabatic efficiency and the losses distribution downstream of the blade. This case presents indeed several phenomena which are challenging to capture: laminar-to-turbulent transition on the blade, interaction of the boundary layer with the shock, secondary and tip-leakage flows. If LES appears a more adequate tool than RANS to predict such inherently unsteady phenomena, it remains delicate, especially because wall modeling is required. This section presents results obtained by Safran and UniBG of WMLES using the Discontinuous Galerkin approach.
S. Mouriaux, F. Bassi, A. Colombo, A. Ghidoni

### Conclusion and Prospects

Abstract
A main outcome of the TILDA project is the demonstration of a reliable road towards high-fidelity simulations of scale resolving turbulence, based on High-Order methods at the LES or DNS levels, depending on Reynolds number and available HPC capacity.
C. Hirsch