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Table of contents
Nomenclature
General Introduction
Presentation of the Main Existing Methods of Numerical Drag Prediction
1 Historical presentation of the main thermodynamic methods
1.1 Steady methods
1.1.1 Betz
1.1.2 Jones
1.1.3 Oswatitsch
1.1.4 Maskell
1.1.5 Van der Vooren and Destarac
1.2 First unsteady generalization of Van der Vooren’s formulation
1.2.1 Theoretical developments
1.2.2 Results on a pitching case
2 Presentation of the formulations based on the velocity vector
2.1 Formulation for steady incompressible flows
2.2 Breakdown into induced and profile components for steady incompressible cases .
2.3 Extension to steady compressible flows
2.4 Breakdown in the steady compressible case
2.5 Generalization to unsteady flows
2.5.1 Noca
2.5.2 Wu
2.5.3 Marongiu
2.5.4 Xu
2.5.5 Other contributions to the unsteady generalization
I Development of an Unsteady Formulation starting from Van der Vooren’s Formulation
1 What is difficult about a generalization to unsteady flows?
1.1 To account for all the additional terms due to unsteadiness and relate them to phenomenological components
1.2 To avoid applying the steady theory as such
1.3 To take the propagation delays into account
2 Derivation of a directly generalizable proof of Van der Vooren’s steady formulation
2.1 Derivation of the far-field equation in the steady case
2.2 Thermodynamic breakdown
2.2.1 Breakdown of vector f
2.2.2 Derivation of the irreversible axial velocity
2.3 Volume splitting using streamtubes
2.3.1 Wave drag
2.3.2 Viscous drag
2.3.3 Another justification of the use of streamtubes
2.4 Derivation of the final steady formulation
2.4.1 A first « raw » formulation
2.4.2 Numerical deviations from the theory
2.4.3 Practical refinements of the theoretical formulation
3 Generalization of Van der Vooren’s formulation to unsteady flows
3.1 Implementation of the additional unsteady terms in the far-field equation
3.2 Derivation of the four components unsteady formulation
3.2.1 Unsteady wave drag expression
3.2.2 Unsteady viscous drag expression
3.2.3 Unsteady induced drag expression
3.2.4 Final decomposition
3.3 Criteria used in practice for the integration volumes definition
4 Discussion
4.1 Robustness of the formulation
4.1.1 Domain of definition of the irreversible axial velocity
4.1.2 Physical criteria used for the definition of the integration volumes
4.2 Physical background for the definition of the unsteady induced drag
4.3 Comparison with Gariépy’s formulation
II Study of Improvement Axes for the Robustness and the Physical Background
1 Study of an alternative expression for the irreversible axial velocity to improve the robustness
1.1 Derivation of the expression developed by Méheut
1.2 Domain of definition of the reversible axial velocity
1.3 Study of its theoretical validity
1.4 Analysis of the variant suggested by Gariépy
1.5 Comparison of the three expressions on several steady test cases
1.5.1 Airfoil in a transonic inviscid flow: assessment of CDw
1.5.2 Airfoil in a subsonic viscous flow: assessment of CDv
1.5.3 Wing in a subsonic inviscid flow: assessment of CDi
1.5.4 Wing in a transonic viscous flow: assessment of all three drag components
2 Study of new criteria for the robustness of the volume definitions
2.1 Expression of the unsteady criterion
2.2 Evaluation on an unsteady subsonic test case
2.3 Evaluation on an unsteady transonic test case
2.4 Filtering
2.5 Conclusions on the validity of the unsteady wave criterion
3 Study of the physical interpretation of the volume term in the unsteady induced drag component
3.1 Link between surface and volume terms
3.2 Acoustic effects
3.3 Breakdown of the unsteady induced drag component
4 Description of the final method used for the unsteady applications
4.1 Final formulation with five components
4.2 Good practice recommendations
III Assessment of the Wave, Viscous, and Acoustic Drag Components on Naturally Unsteady Cases
1 Application to a vortex shedding case
1.1 Quick literature review
1.2 Description of the test case
1.3 Convergence study
1.4 Analysis of the flow field resulting from the simulation
1.5 Application of the drag extraction method
1.6 Analysis of the drag breakdown results
1.7 Comparison with Gariépy’s formulation
1.8 Comparison between steady and time-averaged unsteady results
2 Application to a buffet case simulated by a URANS method
2.1 Quick literature review
2.2 Description of the test case
2.3 Convergence study
2.4 Analysis of the flow field resulting from the simulation
2.5 Application of the drag extraction method
2.6 Analysis of the drag breakdown results
2.7 Comparison with Gariépy’s formulation
2.8 Comparison between steady and time-averaged unsteady results
3 Conclusions regarding the validity of the method
IV Assessment of the Motion, Induced, and Propagation Drag Components on Mobile Cases
1 Application to a pitching airfoil in an inviscid flow
1.1 Quick literature review
1.2 Description of the test case
1.3 Convergence study
1.4 Analysis of the flow field resulting from the simulation
1.5 Application of the drag extraction method
1.6 Analysis of the drag breakdown results
1.7 Comparison with Gariépy’s formulation
1.8 Influence of the reduced frequency
1.9 Comparison between steady and time-averaged unsteady results
2 Application to a pitching airfoil in a viscous flow
2.1 Quick literature review
2.2 Description of the test case
2.3 Convergence study
2.4 Analysis of the flow field resulting from the simulation
2.5 Application of the drag extraction method
2.6 Analysis of the drag breakdown results
2.7 Comparison with Gariépy’s formulation
2.8 Influence of the reduced frequency
2.9 Comparison between steady and time-averaged unsteady results
3 Conclusions regarding the validity of the method
V Application of the Unsteady Formulation to Complex Cases
1 Application to a pitching wing in an inviscid flow
1.1 Quick literature review
1.2 Description of the test case
1.3 Convergence study
1.4 Analysis of the flow field resulting from the simulation
1.5 Application of the drag extraction method
1.6 Analysis of the drag breakdown results
2 Application to a buffet case simulated by the ZDES method
2.1 Quick literature review
2.2 Description of the test case
2.3 Convergence study
2.4 Analysis of the flow field resulting from the simulation
2.5 Application of the drag extraction method
2.6 Analysis of the drag breakdown results
2.7 Spectral analysis
2.8 Comparison with URANS results
General Discussion
Conclusion and Perspectives
Appendices
A Numerical tools used for the applications
A.1 Modeling of aerodynamics
A.1.1 RANS approach
A.1.2 LES Approach
A.1.3 Hybrid RANS/LES approaches
A.2 Codes used
B Grid studies
B.1 Airfoil in a steady transonic inviscid flow
B.2 Airfoil in a steady subsonic viscous flow
B.3 Wing in a steady subsonic inviscid flow
B.4 Pitching airfoil in a viscous flow
B.5 Pitching wing in an inviscid flow
C Time evolution figures
Bibliography
