Development of the discontinuous Galerkin method for high-resolution, large scale CFD and acoustics in industrial geometries

First Edition

The main objective of this work is the practical development of the discontinuous Galerkin method, arguably the most mature high-order discretisation, for the scale resolving simulations of turbomachinery... Read More

New unstructured high order methods hold the promise of transforming industrial?CFD practices. Their combination of high precision, high geometrical flexibility?and parallel efficiency, make them much better suited for unsteady scale resolving?simulations (DNS, LES, …) of turbulent industrial flows than current industrial codes.?As industrial practices for this type of simulations have not yet been established,?there is an opportunity to increase simulation reliability and resolution capacity by?introducing these novel high-potential methods.?The main objective of this work is the practical development of the discontinuous?Galerkin method, arguably the most mature high-order discretisation, for the scaleresolving?simulations of turbomachinery flows.?First optimal interior penalty parameters, adapted to hybrid and high aspect ratio?meshes, typical for CFD, have been developed. This includes the development of a?sharp trace inverse inequality for all types of elements, as well as the generalisation?of the coercivity analysis.?Based on algebraic primitives performance, a flexible yet extremely efficient blockedmatrix?structure is defined, as well as reorganised assembly routines that compensate?the increase of computational complexity of the high-order quadrature. Different?iterative strategies have implemented and compared, and a generic framework is?presented that provides high-quality transfer operators for h- and p-multigrid.?Some extensions of the DGM towards low Reynolds incompressible flows and?frequential acoustics are furthermore developed and analysed.?The capacities of the developed platform are demonstrated by the direct numerical?simulation of the transitional flow on a low-pressure turbine.

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Presses universitaires de Louvain
Title Part
Numéro 422
Koen Hillewaert,
Thèses de l'École polytechnique de Louvain
BISAC Subject Heading
Onix Audience Codes
06 Professional and scholarly
CLIL (Version 2013-2019)
Title First Published
10 February 2013


Publication Date
10 February 2013
Main content page count : 172
Legal Copyright Date
D/2013/9964/6 Louvain-la-Neuve, Belgium
16 x 24 x 1 cm
291 grams
List Price
30.00 €
Version 2.1, Version 3

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1 Introduction 1
1.1 Motivation . 1
1.2 Overview 8
2 The discontinuous Galerkin method 11
2.1 Model equations11
2.2 Elements and functional spaces . 12
2.3 Variational formulation 14
2.4 Shape functions23
3 Extending the DG variational formulation 27
3.1 Stability of the interior penalty method on hybrid meshes27
3.2 Non-conformal formulation . 35
3.3 Frequential formulation of the homentropic LEE . 44
4 Iterative methods 57
4.1 Newton methods . 59
4.2 Multigrid methods 63
4.3 Concluding remarks . 71
5 Efficient data structures 75
5.1 Algebraic primitives on the computer . 76
5.2 Data Structures84
5.3 Efficient assembly . 89
5.4 Conclusions 102
6 noFUDGe: a first industrial application 105
6.1 Description of the flow 106
6.2 Computational setup . 106
6.3 Comparison of computed flow fields108
6.4 Validation . 114
6.5 Comparison of computational cost. 116
6.6 Scaling tests 116
6.7 Conclusions 118
7 Current status and prospects 119
7.1 Conclusions 119
7.2 Current status of the Argo group 120
7.3 Prospects 121
A Elements of functional analysis A.3
A.1 Hilbert spaces. A.3
A.2 Solvability of variational problems. A.4
A.3 The Lax-Milgram theoremA.5
A.4 The most simple exampleA.5
B Function spaces, reference elements and quadrature A.7
B.1 Construction of Lagrange interpolants . A.7
B.2 Interpolation on the boundary A.8
B.3 Specific elements . A.9
B.4 Quadrature rules . A.12
C Sharp values for the trace inverse inequality A.15
C.1 Simplices A.16
C.2 Outline ofWarburton's method . A.16
C.3 Tensor product elements. A.17
C.4 Wedges . A.19
C.5 Lagrange interpolation on pyramidsA.21
C.6 The Pascal space on the pyramid A.25
D Nonlinear instability of quadrature-free methods A.27
D.1 Original formulation . A.27
D.2 Extension to non-linear equations of state . A.30
D.3 Spurious modesA.30
Bibliography i