"Verification and validation of the DLR TAU-Code for spoiler aerodynamics at subsonic and transonic conditions"

(1)

1

List of figures

Fig. 2.1 Airfoil section……….12

Fig. 2.2 BACT Wing………....13

Fig. 2.3 Location of the pressure transducers………....13

Fig. 2.4 Installation of the BACT Wing……….14

Fig 2.5 Clean wing………..15

Fig. 2.6 NACA 0012 airfoil with spoiler……….15

Fig. 2.7 Wing tip and deflected spoiler………...16

Fig. 2.8 Parameter section of the Catia construction tree……….17

Fig. 2.9 Parameters………..17

Fig. 2.10 Wing with a 2° spoiler deflection……….18

Fig. 2.11 Wing with a 25° spoiler deflection………...18

Fig. 2.12 Large deflection of a spoiler……….20

Fig. 2.13 Small deflection of a spoiler……….20

Fig. 3.1 Example of a 3D structured mesh……….22

Fig. 3.2 Example of a 3D unstructured mesh……….23

Fig. 3.3 Example of a hybrid mesh……….24

Fig. 3.4 Location of the geometric sources……….29

Fig. 3.5 Spheres and frustrums on the spoiler……….29

Fig. 3.6 Surface mesh on the wing tip with a 20° spoiler deflection…………..30

Fig. 3.7 Surface mesh (20° of spoiler deflection)………30

Fig. 3.8 Mesh on the corner of the spoiler (20° deflection)……….31

Fig. 3.9 Tetrahedral baseline mesh (20° spoiler deflection)………...33

Fig. 3.10 Volume baseline mesh (cut on the spoiler, 20° deflection) including spoiler wake refinement………...33

(4)

4

Fig. 3.11 Wing refinement for the baseline mesh (25° deflection)………38

Fig. 3.12 Wing refinement for the First Family (25° deflection)………38

Fig. 3.13 Wing refinement for the Second Family (25° deflection)………38

Fig. 4.1 Representation of a generic finite control volume……….41

Fig. 4.2 Graphical representation of normal and shear stresses on a volume element………..45

Fig. 4.3 Example of a convergence plot for the case 8ESU01 (M = 0.770, α = 0°, δ = 0°, baseline)………53

Fig. 4.4 Surface pressure coefficient for the case 8ESU17 (M = 0.772 α = 0°, δ = 10°, baseline)………54

Fig. 4.5 Surface y+ distribution for the case 8ESU17 (M = 0.770, α = 0°, δ = 10°, baseline)………54

Fig. 4.6 Mach in the flow field for the case 8ESU26 (M = 0.820, α = 0°, δ = 10°, first family)………...54

Fig. 5.1 Convergence 8ESU01 (M = 0.648, α = 0°, δ = 0°, baseline)…………..68

Fig. 5.2 Convergence 8ESU01 (M = 0.648, α = 0°, δ = 0°, first family)……….68

Fig. 5.3 Convergence 8ESU01 (M = 0.648, α = 0°, δ = 0°, second family)……68

Figs. 5.4 and 5.5 Comparison 8ESU01, baseline grid……….70

Figs. 5.6 and 5.7 Comparison 8ESU01, first family………70

Figs. 5.8 and 5.9 Comparison 8ESU01, second family………..71

Figs. 5.10 and 5.11 Comparison 8ESU12, baseline grid………..71

Figs. 5.14 and 5.15 Comparison 8ESU12, second family……….72

Figs. 5.16 and 5.17 Comparison 8ESU22, baseline grid………..73

Fig. 5.22 Skin friction on the wing surface………..75

(5)

5

Fig. 5.24 x_velocity on the spoiler (plotted on the Mach contour)………76

Figs. 5.25 and 5.26 Comparison 8ESU24, baseline grid………77

Fig. 5.27 Convergence 8ESU32 (M = 0.897, α = 0°, δ = 3°, first family)……..79

Fig. 5.28 Convergence TEST1 (M = 0.897, α = 0°, δ = 2°, baseline grid)……...80

Fig. 5.29 Mach contour in the flow field for TEST1 ( M = 0.897, α = 0°, δ = 2°, baseline grid)………80

Fig. 5.30 Convergence TEST2 (M = 0.819, α = 0°, δ = 3°, baseline grid)……...81

Fig. 5.31 Mach contour in the flow field for TEST2………..82

Figs. 5.34 and 5.35 Comparison 8ESU03, first family………..86

Figs. 5.40 and 5.41 Comparison 8ESU17, first family………...88

Fig. 5.50 Convergence 8ESU20 (M = 0.770, α = 0°, δ = 25°, baseline grid)…..91

Fig. 5.51 Mach contour 8ESU20……….92

Fig. 5.52 Streamlines colored with the Mach number………92

Figs 5.53 and 5.54 Comparison 8ESU20, baseline grid……….94

(6)

6 Fig. 5.64 CL for the grid sensitivity analysis……….99 Fig. 5.65 Zoomed view on the convergence of 8ESU26_k-ω test case………102 Fig. 5.66 Correspondence 8ESU26 (M = 0.820, α = 0°, δ = 10°, first family)..103 Fig. 5.67 Correspondence 8ESU26 (M = 0.820, α = 0°, δ = 10°, first family)..103 Fig. 5.68 Convergence 8ESU32 (M = 0.897, α = 0°, δ = 3°, first family, k-ω-LEA model)………104 Figs. 5.69 and 5.70 Comparison 8ESU32, first family, k-ω-LEA model……106

(7)

7

List of tables

Tab. 2.1 Overview on the chosen experimental cases………19

Tab. 3.1 Overview on groups subdivision………....27

Tab. 5.1 IMS for the 10° spoiler………...61

Tab. 5.2 Flow computations overview……….62

Tab. 5.3 Variation in the coefficients for the clean wing test cases…………...66

Tab. 5.4 Variation in the coefficients for the 2° of spoiler deflection configuration………75

Tab. 5.5 Variation in the coefficients for the 10° of spoiler deflection configurations………...84

Tab. 5.6 Variation in the coefficients for the 25° of spoiler deflection configuration……….91

Tab. 5.7 Variation in the coefficients for the 20° of spoiler deflection configuration……….95

Tab. 5.8 Values of the coefficients for the grid sensitivity analysis………99

Tab. 5.9 Computational time for the different turbulence models………107

(8)

8

Abstract

Computational Fluid Dynamics can be considered as one of the most important means to investigate the flow over a body in order to obtain qualitative and sometimes even quantitative prediction of the main flow features.

The main objective of this work, as the title suggests, was to verify and validate the DLR TAU Code for spoiler aerodynamics at subsonic and transonic flow conditions.

The dissertation starts with an Introduction (Chapter 1) which clarifies the main objectives and how the work has been carried out.

In Chapter 2 the parametric design of the wing geometry is briefly explained, since a good geometry is the basic step to proceed with the mesh generation. The geometry reproduces as accurately as possible the real wing geometry, described in the experimental setup. The choice of the test cases to investigate is also made. The selection of the most proper kind of grid together with the mesh generation process and the subsequent scaling of the baseline mesh for some selected configurations are accurately described in Chapter 3. The generated grids have been used to conduct a sensitivity analysis to the resolution of the mesh.

An overview on the mathematical models behind CFD, on the flow equations and on how CFD resolves the flow equations in order to model the flow is given in

Chapter 4, where a description of the three turbulence models used in the

performed flow computation is also present.

Chapter 5 presents all the conclusions and the findings about the grid resolution

sensitivity analysis, about the turbulence model sensitivity analysis together with the findings which were considered most interesting. Since CFD cannot completely prescind from the wind tunnel tests, the numerical results are compared to the experimental data. An investigation on the effect on the accuracy of the solution of the different flow parameters is also presented, revealing how the increase of the Mach number can be a crucial point in the quality of the solution.

An overlook on the most important conclusions and about how this work should proceed in the future is given in Chapter 6.

(9)

9

Chapter 1 Introduction

A spoiler can be defined as a device which “spoils” the flow around an aerodynamic body. Usually it is mounted on a wing and used when an increase in drag is required with the result of a flow separation, a loss in lift and a change in pitching moment, with the sign of the pitching moment depending on the chordwise location of the spoiler [24].

Spoilers can be considered as effective control devices because they can be used to obtain a broad range of effects on the aerodynamics of an aircraft. If they are deployed independently on either aircraft wing, they provide lateral control during the flight and they can be used to assist or replace ailerons, with the further advantage of not presenting the significant pitching moment values which are typical of the ailerons mounted on swept wings. If deployed symmetrically on either wing, they provide the spoiling action and the sequential lift dumping that can be useful during the landing phase.

The main effects related to the spoiler deployment are the following [2]:  spanwise reduction of lift and drag increase;

 load alleviation;  direct force control;

 reduction of the effects of turbulence and ability to produce rapid changes in lift.

Of course, turbulence is something always present in the atmosphere. A spoiler may intervene modifying the flow and subsequently producing a rapid variation of the aerodynamic loads, thus reducing the effects of turbulence on the aircraft. Unfortunately, spoilers deployment presents many issues. The rapid deflection causes a strong vortex downstream of the spoiler and this vortex induces local suction that initially produces an increase in lift, even though the desired effect is opposite. This phenomenon is known as adverse lift generation. Then, the vortex is convected downstream, shed and replaced by the separation bubble and the lift starts to adjust approaching the final steady value corresponding to the new static angle of deflection [2].

(10)

10 All these features make the implementation of spoilers on wings and the analysis of the overall flow very challenging. In this context of constant research, one of the reference experiments is the BACT (Benchmark Active Control Technology) Experiment, that was conducted in the NASA Langley Transonic Dynamics Tunnel (TDT).

First objective of this thesis is to analyze with CFD methods some test cases selected from the database of the BACT Experiment. The BACT database includes multiple data about static and dynamic pressures (minimum, maximum and mean values and standard deviation) and aerodynamic loads for various configurations of deflection of spoilers, ailerons and flaps.

The main purpose is the study of the phenomena related to the three-dimensional aerodynamics of spoilers and of the effects connected to its static deflection, thus performing steady computations. A geometry exactly reproducing the experimental wing but having the spoiler as only implemented control surface was used as a reference for all the flow computations discussed in this report. The work consisted in various sequential phases: first of all, a 3D reference geometry responding to some important requirements directly derived from the experimental setup was realized in a CAD environment. The geometry has later been exported with different angles of deflection of the spoiler and each one of the exported geometries was used for mesh generation.

Selected cases were computed with three different levels of refinement of the mesh, in order to verify the sensitivity of the solution to the quality of the grid and the resulting variation of the convergence behavior. The grid families were obtained by refining a first “coarser” mesh which was, however, considered enough accurate for the correct analysis of the problem.

From the broad available experimental database, eleven test cases were selected, using the simple principle of being in the condition to be able to analyze a broad range of angles of deflection (from 0° up to 25°) and of Mach numbers (from subsonic to strongly transonic flow).

A sensitivity analysis of the solution to the turbulence model was also conducted. Starting from the study of the results of the first turn of flow computations (led with Spalart-Allmaras turbulence model), the configurations which showed a problematic convergence behavior were computed with three different turbulence models: two two-equations model (the k-ω model in its default version and the LEA model) and the RSM model.

(11)

11 Finally, the BACT Experiment database was used to validate CFD results. The comparison between experimental data and numerical results was the real fulcrum of this thesis project. It has led to some interesting results and has underlined the differentiated potentials of the application of the CFD analysis to the study of the spoilers aerodynamics.

(12)

12

Chapter 2 Description of experiment and geometries

2.1 Experimental setup and simplifications

The BACT model was used as a reference for all the flow computations presented in this work and consists of a rectangular wing with a NACA 0012 airfoil section.

Fig. 2.1: Airfoil section

According to the BACT documentation [1], the wing has the following geometric characteristics:

 Planform: rectangular

 Aspect Ratio: 2 for the panel (neglecting the tip of rotation)  Leading Edge Sweep: unswept

 Trailing Edge Sweep: unswept  Taper Ratio: 1

 Twist: none

 Wing centerline chord: 406.4 mm  Semi-span of the model: 812.8 mm

 Area of the planform: 0.3303 m2

 Definition of profiles: NACA 0012 airfoil throughout, except for flat spoiler surface

 Form of the wing tip: tip of rotation  Upper surface spoiler details:

o Span: 243.84 mm

o Hinged at: 243.84 mm (60% chord)  Material: aluminum, very smooth.

(13)

13 Fig. 2.2: BACT Wing [1]

Two rows of pressure transducers were located on the upper and lower surface of the wing, one at 40% span and one at 60% span.

(14)

14 These specifications were used as general guidelines in drawing the geometry and also to realize two cuts in the flow surface solution (corresponding to the location of the pressure transducers) in order to compare the numerical results with experimental data.

The designed geometry tries to reproduce closely the BACT model, but some simplifications were necessary in order to adapt the wing to the main purposes of our work.

The first simplification regarded the control devices. The experimental wing presented also a flap and a lower surface spoiler but, since the study focuses on the aerodynamics of the upper spoiler, the choice of not modeling the other two control surfaces was made. No simplification was made about the spoiler geometry: as in the original experimental wing, the spoiler has flat external surfaces, so it was not built to conform to the airfoil shape.

Secondly, during the experiment, the model was mounted on a large splitter plate set out approximately 1.02 m from the tunnel sidewall. The model had an end plate fixed to its root that was used to move the model within a recessed or undercut section of the splitter plate. A large fairing behind the splitter plate insolated the equipment between the splitter plate and the tunnel sidewall from the free air stream. As can be read on the reference [1], some recent tests performed just on the splitter plate equipment without the wing have shown that there are some non-uniformities in the flow field around M = 0.8 and that the data can be affected by them, but the amount of this influence cannot be easily quantified.

The splitter plate and the wind tunnel walls have not been represented in the considered geometry.

(15)

15

2.2 Design

The geometry design started from a “clean” NACA 0012 2D airfoil without a spoiler. CATIA V5 in its R20 version was the main design instrument used for this work.

Firstly, the basic geometry (clean wing) was designed. This was a very important phase because the clean wing has then been used as a simple reference configuration and as the examined layout to study the effects of zero or very small (<2°) angles of deflection of the spoiler.

Fig. 2.5: Clean wing

In order to build the wing with the implemented spoiler, a NACA 0012 airfoil with a 2D spoiler was extruded.

(16)

16 As no information about the tip of rotation was available, the wing tip was obtained as a revolution surface simply making the upper shell of the airfoil rotate around its midchord.

Fig. 2.7: Wing tip and deflected spoiler

Once the preliminary design was completed, the whole wing was subdivided in a part including the spoiler and a part without the spoiler and this was really crucial to generate meshes locally more refined.

Every modification or adjustment of the geometry was made in CAD environment, as a way to prevent the user to modify the geometry using other software (for example, the mesh generator) and to ensure that the grid generation could be as robust as possible.

The Parametric Generative Shape Design technique was used as CATIA workbench. In general, parametric design is based on some user defined parameters that can be considered as extra data added to the CATIA document. They can take any value or can be constrained by formulas, which can be used to define equalities between sets of parameters. Formulas can be easily written using an editor and their effect is then propagated to all the terms belonging to the equivalence [23].

Using parameters has many advantages:

 parameters can pilot the geometry and change it;  they allow to edit formulas;

 they allow to centralize useful information so that a new user on the model can deal with them immediately, thus increasing the designer productivity.

(17)

17 In this specific case, parametric modeling consists in setting some important geometric variables as parameters that can be modified by simply clicking on the corresponding voice on the CATIA construction tree.

The parameters are:  chord length  airfoil thickness

 relative spoiler position  relative spoiler chord length  spoiler deployment angle

 relative position of blunt trailing edge

Fig. 2.8: Parameters section of the CATIA construction tree

Fig. 2.9: Main parameters

The sizes of whole wing and the spoiler angle of deflection are thus changed modifying the value of the corresponding parameter and the geometry just adapts to the typed information.

(18)

18 Precise information about the thickness of the trailing edges was missing. In order to obtain a reliable model, blunt trailing edges for both the wing and the spoiler were designed. The reasons for this choice are two. First of all, because of manufacturing limits, it is impossible to practically create perfectly sharp edges. Secondly, a study [11] proofs that sharp edges are detrimental to the convergence of some turbulence models of the TAU code (DLR flow solver, see next chapters). For these reasons, the sharp edges are always cut off.

Examples of resulting wings (with a spoiler deflection of 2 and 25° degrees) can be seen in the images below.

Fig. 2.10: Wing with a 2° spoiler deflection

(19)

19

2.3 Test cases

The BACT Experiment database includes a large amount of experimental data regarding static and dynamic deflection of the spoiler and of the trailing edge control device.

In order to proceed with the steady analysis, eleven test cases were selected from the BACT database regarding the steady spoiler deflection:

Test Case No.

M Re α (deg) δ (deg) Grids

8ESU01 0.648 4480000 0 0 3 8ESU03 0.649 4480000 0 10 3 8ESU04 0.648 4480000 0 20 1 8ESU09 0.649 4490000 4 20 1 8ESU12 0.775 3900000 0 0 3 8ESU17 0.772 3910000 0 10 3 8ESU20 0.770 3900000 0 25 3 8ESU22 0.817 3910000 0 0 3 8ESU24 0.819 4170000 0 2 1 8ESU26 0.820 4170000 0 10 3 8ESU32 0.897 4170000 0 3 3

Tab. 2.1: Overview on the chosen experimental cases

where α is the angle of incidence of the flow and δ is the angle of deflection of the spoiler, M is the Mach number and Re is the Reynolds number based on the chord length.

The clean configuration was considered to analyze the flow over the BACT wing without a spoiler. This was a good way to obtain a reference to whom compare the configurations with the deflected spoiler, for every kind of flow. The clean wing and the wings with 10° and 25° of deflection of the spoiler were studied for subsonic, transonic and strongly transonic flow and on three levels of refinement of the mesh, in order to discover the influence of the variation of the Mach number and of the number of grid points on the accuracy of the solution and to explore the flow field past the spoiler .

The minimum and maximum angle of deflection investigated in the present work were respectively 2° and 25°. These cases, together with the case 8ESU32 (3° deflection of the spoiler and highest Mach number in the entire experiment), were considered very challenging and they were analyzed to study the impact of the small and big deflection of the spoiler on the resulting flow. More precisely, in the case of a big deflection of the spoiler, a fully separated flow is expected. If the deflection of the spoiler is small, the resulting geometric discontinuity on the

(20)

20 surface of the wing, can produce a separation followed by a re-attachment of the flow past the spoiler.

Fig. 2.12: Large deflection of a spoiler

(21)

21 The configuration with a 20° spoiler deflection was studied in order to investigate the impact on the flow solution of very small variations of the number of Mach and to analyze the flow when the angle of incidence is not zero.

Besides these basic steady computations, other computations with different turbulence models (k-ω model and k-ω LEA model) and computations with the only aim of verification were run. Further information about the computational context will be found in the following chapters.

(22)

22

Chapter 3 Mesh generation

3.1 Elements of mesh generation

The following crucial phase of the work was the mesh generation process, which is a very important step towards the CFD solution. To frame the problem, a general overview about this process can be derived from the reference [3] and it is here presented.

A mesh can be defined as a mean to discretize the three-dimensional flow into small geometric shapes and can be structured or unstructured.

Structured meshes consist of hexahedra. They have a rather simple inherent

structure consisting of regular repeating elements that does not require a lot of memory storage since every single cell of the grid is directly addressed by the index (i, j, k). Furthermore, they require relatively simple computational resources and visualization tools. In spite of these advantages, structured meshes also present many issues. The quality of the resulting grids can be high but they can be not suitable to describe 3D geometry with high complexity.

Fig. 3.1: Example of a 3D structured mesh. [4]

Unstructured meshes consist of tetrahedra. The elements of this kind of meshes

(23)

23 and stored in some extra memory, implying more computational efforts. They can be divided into two main categories:

 tetrahedral grids  Cartesian grids.

Tetrahedral grids provide a great flexibility and automation in 3D grid

generation and refinement/coarsening with an approach that is suitable for inviscid flows. The geometrical complexity of the cells composing the unstructured grids make them not suitable for Navier-Stokes computations, because the construction of numerical algorithms for solving the flow equations on this kind of elements is not immediate. Above all, unstructured tetrahedral

grids are known to be not well suited for resolving turbulent flow computations,

whose accuracy require the capturing of the fine flow features. Computations on grids with tetrahedral elements show that, due to numerical damping, more computational elements are required compared to computations on structured grids to reach the same level of accuracy.

Cartesian grids are aligned with Cartesian axes. A master hexahedron

surrounding the body is built and then subdivided to create other elements that become progressively smaller as the body surface is approached. Even in this case, generation is simple and automatic.

The use of Cartesian grids presents multiple problems: the nodes forming at the interface between the small hexahedra require for a special numerical treatment and the quality of the mesh close to the surface is very poor. Furthermore, since in Cartesian grids the edges of the cells are oriented in parallel to the Cartesian coordinates, an accurate numerical treatment of the boundaries which are either curved or not oriented as the Cartesian axis, can be really hard to achieve [4]. This is the reason why Cartesian grids are good to simulate potential and inviscid flows but not the boundary layer.

(24)

24 Thus, it appears clear that the employment of a single mesh type is not sufficient in order to properly resolve three-dimensional flows around very complex geometries. This leads to the use of hybrid meshes that provide considerable flexibility, especially in modeling boundary layers, shock waves, wakes and vortices.

A hybrid mesh can be defined as a grid consisting of prisms and tetrahedra that combines advantages of both structured and unstructured grids. In specific, anisotropic triangular faces capture the anisotropy of the flow on the surface of the geometry especially in those regions, like leading and trailing edges, which have strong directionality. Prisms capture the features in normal direction to the body surface and the wake surface. Tetrahedra just fill the remaining domains, capturing those features that are away, as shock waves and vortices.

Fig. 3.3: Example of a hybrid mesh [25]

When involved in a mesh generation process, there are some general elements that must be taken into account.

Firstly, the spatial scales involved in the grid generation vary of many orders of magnitude. The flow scales are of course imposed by the main flow features (such as boundary layers, wakes, shocks and vortices) and by the geometry and they can also have a prominent directionality. For example, the small scale that characterizes the boundary layer is really pronounced only in the direction normal to the surface. This feature must be taken into account when the sizes of the elements for the discretization are chosen.

Secondly, Navier-Stokes solvers place strict requirements on the mesh in term of uniformity, elements shape and resolution, so the marching step sizes and the stretching factors must be constrained. Furthermore, as the cited paper [3] states, the smooth transition from prisms to tetrahedra is very important for the accuracy and the robustness of the numerical method.

(25)

25 The quality of the mesh is thus very important to gain a good CFD solution and it depends on various factors, such as the number of hexahedral elements (which give more accurate solutions compared to tetrahedra), the mesh density that has to be sufficient in order to capture all the relevant flow features and has to be even more accurate for the part of the mesh adjacent to the wall to resolve the boundary layer flow.

3.2 Mesh generator

Centaur software was used as a mesh generator. Centaur was developed by

CentaurSoft. It is widely applied to perform computational flow simulations in many fields such aerospace, turbomachinery, automotive, biomedical and topography [25].

Centaur is a hybrid grid generator that creates either hexahedral or prismatic

grids. In this work a prismatic mesh was created, in order to capture in a good way what happens in the boundary layer and an outer tetrahedral mesh to resolve the flow in the volume at a certain distance from the surface of the body and to see, for instance, what happens in the wake. The unstructured tetrahedral meshes, generated with the Marching Method, are then used for the solution of the flow equations.

The Marching Method (also known as Advancing Front Method), formulated and described by Kallinderis in the reference [5], is based on the idea that the mesh is created step by step and, in each step, new elements are added to the existing ones, sweeping out a front across the entire. So, basing on some settings that are specific for each case, Centaur creates an unstructured triangular surface grid, which is used as the starting surface to generate the prismatic mesh that completely surrounds the geometry. The grid covering the surface is marched away from the body surface in distinct steps resulting in the generation of prismatic layers in the marching direction. Finally, tetrahedra fill the remaining volume [19].

The single elements must be intended as single control volumes and on each one of them, the flow solver will resolve the flow equations. The goal of this generation mechanism is to reduce the curvature of every single marching surface at each step while ensuring smooth grid spacing to avoid the surface overlap.

(26)

26 1. determination of the marching vectors that identify the directions along

which the nodes will march;

2. determination of the distance by which the nodes will march along the marching vectors;

3. smoothing operations on positioning the nodes on the new layer.

More information about the single phases of the Marching Method can be found in the reference [5].

To make mesh generation reliable, the most important thing to do is to make the prismatic layer thickness sufficiently high. In fact, the accuracy of the prisms in resolving the flow is much higher than that of tetrahedra, thus it is a good choice to extend the prismatic layer so that the boundary layer completely resides in it. According to the analyzed case, the generation of prisms can also be much more flexible and faster compared to tetrahedra.

The vortices, besides, tend to quickly dissipate in the tetrahedral region so it is of primary importance to resolve them in the prismatic domain, associating the maximized prismatic layer to a quite refined mesh. With regard to this, the study [6] demonstrates that a thick prismatic layer and a fine grid are of capital importance for the correct prediction of the vortices strength and position.

The Centaur software is accurate, highly automatic and stable but, from time to time, simulations may also crash. Furthermore, the computational time can be very long depending on the available computational resources, on the CAD tolerance and on the mesh parameters. Furthermore, in Centaur as in every mesh generator, there is the necessity of CAD and geometric sources to locally refine the mesh.

3.3 Preparatory steps and generation of the baseline

mesh

Before proceeding with the mesh generation, the geometry was imported into

Centaur from the CAD environment into an .IGS format.

The geometry was then analyzed using the Centaur CAD Diagnostics tool, that highlights potential CAD problems, and cleaned up. This operation gave as a result a clean geometry ready and optimized for mesh generation.

The operation of importing the geometry was repeated for every chosen angle of deflection of the spoiler, so the generation process was carried out independently for each analyzed configuration.

(27)

27 After this first step, the wing was encased into a bounding box having a size of 75 times the chord length and having a panel as a symmetry plane for the wing. The imported geometry was then subdivided into two zones: a “spoiler zone” and a “main zone” that included everything else but the spoiler.

When the geometry was loaded into Centaur, two groups were automatically created: the visible panels group and the inactive panels group. These groups are related to the CAD work but they can be modified or new groups can be created using the group manager tool. As long as new groups are created, the panels are shifted from the old groups to the new ones and it is possible to assign to every group its precise boundary condition for the mesh generation and the following flow computation. The assignment of the panels to groups is also very useful to be able to localize the grid refinement.

A full overview about the groups (for a 20° deflection of the spoiler, but this scheme is general) and their boundary conditions can be found in the following table:

Group name

Zone 1 Type Zone 2 Type Boundary

condition

Panels Wing out Main Prisms <None> <None> Viscous

wall

21 Wing in Spoiler

20 deg

Prisms <None> <None> Viscous wall

8 Farfield

boundary

Main Tetrahedra <None> <None> Farfield 5 Symmetry Main Hybrid <None> <None> Viscous

wall

1 Spoiler Spoiler

20 deg

Prisms <None> <None> Viscous wall

5 Inactive

panels

Main Prisms <None> <None> Viscous wall

0 Visible

panels

Main Prisms <None> <None> Viscous wall

0 Tab. 3.1: Overview on groups subdivision.

Eleven test cases, six different angles of deflection (including the clean wing configuration) of the spoiler for a total amount of 33 flow simulations were computed in the present work.

Every computation was carried out first of all using the baseline mesh. The

(28)

28 The makegrid tool was used to produce a first surface mesh made of triangles. Of course, the so obtained mesh was too coarse, so a refinement of the most critical parts of the geometry using CAD and geometric sources was carried out.

In literature many guidelines about mesh generation are specified, but there are no general rules to choose the correct size of elements composing the sources. Most of the times the size is just chosen by experience and the user has to perform some attempts until a visually satisfactory mesh is obtained. As can be read in [3], we can base on the length scales which have proved to be most successful:

 for those regions with surfaces in close proximity, the distances between surfaces or the local surface curvature are used;

 for the tetrahedral portions, the local thickness of the last prismatic layer is used and this ensures that the size of the tetrahedra in the direction normal to the other prismatic surfaces is the same as the height of neighboring prisms.

CAD sources were chosen in order to achieve the basic discretization of the wing. The chosen size of the baseline elements is 4mm, about 1/100 of the chord length. CAD sources were also used for the basic discretization of the spoiler and of the small panel at the end of the wing tip of rotation.

To capture all the flow features, a more specific refinement was necessary thus some geometric sources (with fixed or variable sizes of elements) were added:

 a cylinder for the leading edge;

 one frustrum in order to capture the wing tip vortex;

 two frustrums for the side panels of the spoiler, because of their small size;

 two spheres, one for each point in which the spoiler intersects the wing surface;

 one hexahedral box to capture the vortices at the spoiler’s trailing edge. The geometric sources can be visualized in the following images.

(29)

29 Fig. 3.4: Location of the geometric sources

Fig. 3.5: Spheres and frustrums on the spoiler (geometry drawn as a wireframe model). A complete overview on the location of the sources and on the sizes of their constitutive elements is given in the end of this chapter.

It is important to guarantee a smooth transition between the size of the triangles located at the interface between zones with different discretization to avoid the failure of the grid generation and to obtain a good quality grid able to resolve the flow.

The same sources were used for every configuration with the deflected spoiler. The size of the source elements remained the same, the only thing that was adjusted for every single configuration was the location in space of some of the

(30)

30 geometrical sources, to ensure the vortices detection for each examined angle of attack of the flow investing the wing.

After the refined procedure, the makegrid tool was used again until a satisfactory surface mesh was produced.

Fig. 3.6: Surface mesh on the wing tip with a 20° spoiler deflection

(31)

31 Fig. 3.8: Mesh on the corner of the spoiler (20° deflection).

In order to construct the prismatic mesh starting from the surface one, some more input parameters were necessary and they were obtained using the Viscalc software, an internal tool of the department AS-TFZ of the DLR of Braunschweig developed by Stefan Melber.

To gain useful information using Viscalc, some settings such as the Reynolds number, the Reynolds reference length and the dimensionless wall-distance y+ must be imposed by the user .

y+ is an essential parameter connected to the problem of computing the turbulent flows near a wall. The presence of the wall produces a boundary layer. Because of the no-slip condition at the wall, in the boundary layer the flow velocity changes from zero to its free-stream value. The gradients are stronger in this region, thus it is really important to capture the near-wall variation. The standard procedure is to deeply refine the mesh close to the wall in order to resolve the flow in that region. This is the reason why the dimensionless wall-distance is introduced. This function is defined as:

where is the friction velocity at the nearest wall, y is the distance to the nearest wall and is the local kinematic viscosity of the fluid.

(32)

32 However, the modeling of this critical region can be very difficult and instead of modeling the flow behavior, the common practice is to set y+ equal to 1 according to the flat plate theory and Viscalc gives back the geometric characteristics of the prism-layer with respect of a turbulent boundary layer of a flat plate for high Reynolds numbers.

y+ determines the Initial Marching Step of the first prismatic layer (which reaches the height of the flat plate boundary layer). The value of the Initial Marching

Step given by Viscalc was inlaid in the Initial Layer Thickness tab in the Centaur

dialogue window that allows to configure input parameters. The other superimposed settings were:

 Depth factor: 1  Thickness Prism: 2  Stretching Factor: 1.28

Viscalc gave back the following information:

 number of layers;  height of the last layer;  thickness of the prism-layer;  thickness of the boundary-layer;

 number of points in the boundary layer.

The information about the number of layers was then inserted in Centaur and the

makegrid tool was used to produce the final prismatic and tetrahedral meshes.

This operation was repeated for every configuration and each value of the Reynolds number.

Centaur includes some tools to automatically improve the mesh quality and

smoothing. The quality of this work prismatic meshes was object of various attempts of improvement. Unfortunately, because of the complexity of the part of the geometry in which the spoiler intersects the wing surface, two holes in correspondence of the spoiler lateral edges are inevitable because of the chopping of the prismatic layers. Chopping can occur in correspondence of critical regions of a geometry (i.e. small cavities) and it can be defined as the generation of a different number of prismatic layers compared to other regions. In small cavities such as the one beneath the spoiler, growing a large number of prisms results in very fine cells which substantially increase the computational time. Furthermore, as a result of the small final layer thickness, the tetrahedra are forced to match these scales producing huge meshes even when it is not required. Thus, in the

(33)

33 mesh generation process, a minimum allowable prism layer thickness is specified. If the cavity requires any layer thickness smaller than this value, instead of reducing the layer thickness further, the number of layers is reduced in order to decrease the overall prismatic layer thickness [25]. This transition between different values of the layer thickness is responsible for the above mentioned holes, but many efforts have been made in order to make their surface as smooth as possible and to obtain a mesh that was suitable for the flow computations.

Fig. 3.9: Tetrahedral baseline mesh (20° spoiler deflection).

Fig. 3.10: Volume baseline mesh (cut on the spoiler, 20° deflection) including spoiler wake refinement.

(34)

34 Some clarifications are necessary.

Not all the Centaur parameters were modified, just the ones indicated in the following overview. Some of these settings were obtained with Viscalc, others were calculated with formulas or recursive methods, others were inserted arbitrarily based on best practice, guided by the appearance of the mesh and the required accuracy.

Some parameters (such as the Stretching factor) are slightly different from one mesh to the other. These parameters can have a severe impact on the solution, but previous experiences have shown that changing the second digit only does not significantly affect the final result, so the meshing process is still coherent. They were just adjusted in order to face some meshing difficulties and obtain higher quality meshes.

The main modified parameters are subsequently listed.

 Stretching ratio (surface): controls the rate of change of elements size from one element to its neighbors. This ratio is based on the average edge lengths of the cells.

 Number of prism/hex layers (prism/hex): number of

prismatic/hexahedral layers to be generated in the grid.

 Initial layer thickness (prism/hex): normal layer thickness for the prismatic grid generation. It can be modified basing on the desired level of accuracy.

 Stretching factor (prism/hex): normal stretching to be used in successive layers of prisms.

 Minimum layer thickness (prism/hex): minimum allowable initial layer thickness for the geometry. If the cavity or source algorithm requires any layer thicknesses smaller than this value, instead of reducing the layer thickness further, the number of layers is reduced in order to reduce the overall layer thickness.

 Stretching ratio (tetrahedra): controls the rate of change in size from one element to its neighbors.

3.4 Grid scaling and generation of families of meshes

One of the objectives of this work was to carry out a study aiming to determine the sensitivity of the flow solution to the grid refinement, to see if the baseline grid resolutions was sufficient to properly describe the flow or a finer grid was necessary to capture all the significant flow features.

(35)

35 In order to do so, two new families of grids were created and studied on the cases:

 8ESU01 (M = 0.648, Re = 4480000, α = 0°, δ = 0°)  8ESU20 (M = 0.770, Re = 3900000, α = 0°, δ = 25°)  8ESU26 (M = 0.820, Re = 4170000, α = 0°, δ = 10°)  8ESU32 (M = 0.897, Re = 4170000, α = 0°, δ = 3°)

The test cases above were selected because analyzing the clean wing, one of the smallest angles of deflection at a maximum value of the Mach number, the maximum angle of deflection and an intermediate angle of deflection for transonic flow was considered of primary interest.

The First Family and the Second Family were generated as a global refinement of the baseline mesh. The three families of grids have consistently refined spacing. The special location of the sources or their zone of influence were not changed at all, only the sizes of the elements in CAD and geometric sources were reduced following a particular procedure that is described below.

To scale the baseline mesh a factor of 2 was chosen:  Baseline mesh: n points

 First Family: 2n points  Second Family: 4n points

The scaling procedure was the one suggested by Crippa, see [7].

When it comes to grid generation, it is important to be aware of the fact that in

Centaur the number of layers is chosen so that the cell aspect ratio in the last

layer is 1. Thus, refining the surface grid reduces the total height of the prismatic layer. In order to obtain an accurate solution and to be sure that the whole boundary layer flow is resolved with the same type of elements (i.e. prisms), the total height of the prismatic layer has to remain constant between the different grid levels.

Thus, the sizes of the sources elements, the Initial Marching Step and the

Stretching factor were scaled in such a way to achieve the constant height of the

prism layer using the procedure described below.

The refinement procedure consisted of the following steps. 1. The baseline mesh was chosen as the starting mesh.

(36)

36 2. The elements composing the CAD and geometric sources were scaled of a

factor of √ .

3. The prismatic layer height of the prismatic mesh of the baseline configuration was then computed using the formula:

where: IMS is the Initial Marching Step, q is the Stretching Ratio and N is the number of prismatic layers.

The objective was trying to keep nearly constant for all the other meshes.

4. The Initial Marching Step for the two finer meshes was derived: ( ) ( ) ⁄

( ) ( ) ⁄

where f is the scaling factor and its value is 2.

5. Then, the number of prism layers for the two finer meshes was calculated: ( ) ( )

( ) ( )

6. q was then modified for (fine1) and (fine2) in such a way that the equation

1 still yields , using the recursive formulas:

( ) ( ) ( ) ( )

In the Appendix A all the relevant settings regarding the two finer families of meshes are shown.

(37)

37 Roughly, the baseline mesh has 7,000,000 nodes, the First Family grids have around 14,000,000 nodes and the Second Family meshes have around 28,000,000 nodes.

(38)

38 Visual comparison of the three levels of refinement.

Fig. 3.11: Wing refinement for the baseline mesh (25° deflection).

Fig. 3.12: Wing refinement for the First Family (25° deflection).

(39)

39

Chapter 4 Flow computation

4.1 CFD: purposes and basic equations

Computational methods have revolutionized the aircraft design process. Prior to the CFD exploit, aircraft were designed and built combining the analytic theory and a large amount of wind tunnel experiments. With the increase of the complexity of the considered geometries and problems, there was a need for more accurate and faster designing methods and the rapid development of numerical analysis and of powerful computational resources were the starting point for the computational fluid dynamics.

The most important advantage of the computational methods is that they provide a fair amount of information that can be very helpful in the aircraft design process with a cost much lower than the cost of wind tunnel tests.

CFD provides a qualitative and, increasingly, even quantitative prediction of fluid flows [8] by means of:

 mathematical modeling using partial differential equations;

 numerical methods, which discretize these equations and give some solution techniques;

 software tools, such as flow solvers and visualization programs. Computational methods ought to:

 allow the simulation of the behavior of complex systems, especially in the presence of nonlinearities (such as transonic flow, shock waves, large structural deformations);

 predict the behavior of the engineering systems;  provide the tools for multidisciplinary design;

 give the possibility to study some flow conditions which are very difficult or impossible to investigate with experimental techniques.

There are many issues that can derive from the misuse of the computational methods:

 a solution can be considered good only if the right model is chosen (for example, a problem with highly non-linear content solved using a

(40)

40 computational method designed for linear problems will give a wrong result);

 the accuracy of the numerical solution depends strictly on the discretization procedure and on the mesh resolution;

 cumulative effects of various kinds of errors, like discretization errors, implementation errors, physical errors due to the fact that sometimes a large amount of simplifications is made and iterative convergence errors, which depend on the stopping criteria [8];

 difficulty in analyzing and understanding the large quantity of results;  high time to result, if the geometry slightly changes.

A complete CFD analysis consists of three main phases: 1. Preparatory steps

 Problem statement: identification of the problem, of the physical phenomena present into the flow, of the geometry and of the type of analysis that must be performed.

 Mathematical model: definition of the computational domain, formulation of the governing equations and of their simplifications, specification of initial and boundary conditions.

 Mesh generation: discretization of the computational domain into cells.

2. Solving

3. Post-processing  Data analysis

 Visualization of the results

 Validation: comparison between CFD results and experimental data.

 Verification: comparison between the numerical results of different CFD simulations.

Governing equations.

Fulcrum of the CFD analysis are the equations that allow to model the flow behavior, by describing it in terms of the main fluid properties: density, viscosity, pressure, velocity and temperature.

The derivation of the main equations of fluid dynamics is based on the fact that the dynamical behavior of a fluid fulfills three conservation laws:

(41)

41 2. the conservation of momentum;

3. the conservation of energy.

The conservation of a certain flow quantity means that its total variation inside an arbitrary volume can be expressed as the net effect of the amount of the quantity being transported across the boundary, any internal forces and sources, and external forces acting on the volume [4]. The amount of quantity crossing the boundary is called flux. Thus, the flux entering the control volume must be equal to the flux leaving the control volume. The flux can be generally decomposed into two different contributions: one due to the convective transport and one due to the molecular motions present in the fluid at rest, which is of diffusive nature and this means that it is proportional to the gradient of the quantity considered. The basic idea is to divide the flow field into a number of finite volumes and to concentrate on the modeling of the behavior of the fluid in one of these finite regions.

A generic flow field can be represented by the streamlines in the following figure and we can consider a boundary that defines a closed control volume Ω and indicate the volume element as Ω and the surface element as dS and its associated outward pointing unit normal vector ⃗ [4]:

Fig. 4.1 Representation of a generic finite control volume [4].

The conservation law for a general scalar quantity U says that its variation in time within the control volume is equal to the sum of the contributions due to the

convective flux (that is to say, the amount of the quantity U entering the control

volume through the boundary with velocity ), to the diffusive flux and due to the volume and the surface sources. These quantities are listed in the following equation:

(42)

42 where k is the thermal diffusivity coefficient and QV and ⃗⃗⃗⃗ are the volume and surface sources, - ∮ ( ⃗ ) is the convective flux and ∮ ( ⃗ ) is the diffusive flux.

If the quantity U is a vector, the above equation would still be valid but the convective and diffusive fluxes would become tensors: ̿̿̿ would be the

convective flux tensor and ̿̿̿ the diffusive flux tensor. The conservation law in a

vector form would be:

∫ ⃗⃗ ∮ [( ̿̿̿ ̿̿̿) ⃗ ] ∫ ⃗⃗⃗⃗⃗ ∮ ( ̿̿̿ ⃗ )

These two equations are known as the integral formulation of the conservation law. This form has two important properties [4]:



if there are no volume sources, the variation of U depends only on the flux across the boundary ;



this formula remains valid in presence of discontinuities in the flow field like shocks waves.

These characteristics are the reason why the majority of CFD codes is based on the integral form of the governing equations. In the following pages, a brief overview on the equations will be given. The details on the mathematical background can be found in the reference [4].

1. Conservation of mass.

This equation expresses the fact that mass cannot be created or destroyed in a fluid system.

Considering the same control volume, if there are no volume or surface sources present, the continuity equation can be expressed in the following form [4]:

∫ ∮ ( ⃗ )

The conserved quantity is of course the density ρ, the first term of the equation is the time rate of change of the total mass inside the control volume and the second term is the contribution from the convective flux across each surface element dS (its sign depends on if the mass flow is leaving the control volume or entering it).

(43)

43 2. Conservation of momentum.

The momentum equation can be derived manipulating the Newton’s second law which states that the variation of momentum is caused by the net force acting on a mass element. Thus, the variation in time of momentum within the control volume is equal to the sum of the contribution of the convective flux tensor (the diffusive term is zero since the diffusion of momentum for a fluid at rest is not possible), of the contribution of the external forces acting on the volume and of the surface forces. The surface forces consist of two parts: an isotropic pressure component (due to the pressure distribution imposed by the outside fluid surrounding the volume) and a viscous stress tensor ̿ (that takes into account the shear and normal stresses resulting from the friction between the fluid and the surface of the examined control volume) [4]. Into detail, the expression of the surface sources is:

̿̿̿ ̿ ̿

Summing the above listed contributions, the general expression of the momentum conservation law for a control volume fixed in space is the following:

∫ ∮ ( ⃗ ) ∫ ∮ ⃗ ∮ ( ̿ ⃗ )

3. Conservation of energy.

The energy equation derives from the first law of thermodynamics, that states that any change in time of the total energy inside the volume is caused by the rate of work of forces acting on the volume and by the net heat flux into it [4].

The general integral expression for the conservation of the energy is: ∫ ∮ ( ⃗ ) ∮ ( ⃗ ) ∫ ( ̇ ) ∮ ( ̿ ) ⃗ where: | ⃗ | = total energy = total enthalpy = body forces

= thermal conductivity coefficient

The terms in the equation are respectively: the variation in time of the total energy within the control volume a gathered term which includes

(44)

44 the convective contribution and the pressure contribution, the diffusive heat flux, a term which collects the rate of work done by the body forces and the volumetric heating ̇ due to the absorption or emission of radiations or to chemical reactions and one last term that is the time rate of work done by the pressure as well as the shear and the normal stresses on the fluid elements.

Viscous stresses.

The viscous stresses originate from the friction between the fluid and the surface of the control volume, they depend on the dynamical properties of the medium and they are described by the stress tensor ̿ which has the following form in Cartesian coordinates: ̿ [ ]

The notation means that that particular stress component affects a plane perpendicular to the i-axis in the direction of the j-axis. The components represent the normal stresses and the other components represent the shear stresses.

The expression of the components of the stress tensor can be simplified if a

Newtonian fluid is considered. In this kind of fluid, the shear stress is

proportional to the velocity gradient and the components of ̿ are:

( ) ( ) ( ) ( ) ( ) ( )

(45)

45 where: λ is the second viscosity coefficient and µ represents the dynamic

viscosity.

Fig. 4.2 Graphical representation of normal and shear stresses on a volume element. These equations were derived by Stokes in the beginning of the 19th century. The terms multiplied by µ in the previously listed equations represent the rate of

linear dilatation and thus a change in shape, while the terms multiplied by λ

represent the rate of change in volume and so, in density.

In order to eliminate λ and simplify the equations, Stokes introduced the bulk

viscosity equation:

Except for very high pressure and temperatures, this hypothesis is always valid and yields to a very important simplification of the normal stresses expressions, which become even easier for an incompressible fluid (because ( ) ) More information about the full derivation of the equations can be found in [4]. All these equations can be collected into one system of equations called the

Navier-Stokes system which is the base of the analysis of most of the fluid

dynamics problems. This system describes the flux of mass, momentum and energy through the boundary of a control volume Ω which is fixed in space. Applying the Gauss theorem, these equations can also be written in differential form.

In three dimensions, the Navier-Stokes system is composed of five equations with seven unknown field variables (ρ, u, v, w, E, p and T). Therefore, two more equations of thermodynamic nature between the state variables are necessary.

(46)

46 Plus, the values of µ and k must be given and they are a function of the state fluid as well.

Thus, the form of this equations changes according to the state and the kind of the considered gas. The full expression of the Navier-Stokes system for a perfect gas and a real gas can be also found in [4].

This system cannot be solved in a close form. In some special cases and for very simple geometries, such as incompressible flow, steady flow, inviscid flow and thin shear layer, they can be considerably eased and an analytical solution can be reached, at least for fully laminar flows.

In most of the cases and in the case of this work, the Navier-Stokes system cannot be solved analytically so the solution to the equations must be numerically approximated by using a discretization method to transform the differential equations into a system of algebraic equations which can be solved digitally. This is the goal of the CFD.

Discretization techniques.

The discretization is based on the method of lines [4]: a separate discretization in space and time which allows to use different levels of accuracy for the spatial and temporal derivatives, providing large flexibility compared to coupled space and time discretization techniques.

 Space discretization.

The chosen discretization method is the Finite Volume Method, that uses the integral formulation of the Navier-Stokes equations, which are discretized first by dividing the physical space into a number of polyhedral control volumes by a grid and then by applying the conservation principles over each small control volume. The volume integrals of the equations are converted to surface integrals by the Gauss’ theorem and they are evaluated as fluxes at the cell walls. The flux between all the neighboring cells must be calculated and the values of the variables at the faces of the control volumes are determined by interpolation in terms of nodal values using different methods.

The numerical dissipation is inevitable and it can affect the accuracy of the solution. Two different numerical schemes can be used to complete the spatial discretization: the central schemes and the upwind schemes.

The upwind schemes can be very suitable in order to describe the physics of certain kinds of flows (i.e. strongly transonic flows) because they are

(47)

47 able to predict more accurate shock waves positions and gradients. Unfortunately, they require a long computational time and they are not apt to describe subsonic flows, because they use flow information coming just from one direction.

In this work, a central scheme was used. The principle is to average the conservative variables in the Navier-Stokes equations in order to evaluate the flux at a side of a control volume. Thus, these methods use information coming from more directions and from the neighboring control volumes. This scheme cannot avoid the even-odd decoupling of the solution, so an artificial dissipation must be added in order to gain more stability and accuracy [4].

The central scheme can be modified to be more similar to the upwind

scheme way of resolving discontinuities by using the matrix dissipation scheme. The idea is to use a matrix instead of a scalar to scale the

dissipation terms, thus each equation is scaled with a different value. In this way, it is possible to gain better accuracy than the upwind scheme with just a slightly bigger computational effort compared to the scalar dissipation. Unfortunately, this dissipation scheme is less robust than the

upwind but the results obtained with it is the closest to what happens in

reality.

Once the discretization procedure is completed, the result is a discrete algebraic system per each control volume that can be solved digitally by different iterative methods.

The Finite Volume Method has many advantages that make it the most used discretization technique. Firstly, the resulting solution satisfies the conservation laws and this is valid independently on the number of cells. Secondly, the spatial discretization is carried out directly in physical space [4], thus there are no problems with any transformation between coordinate systems. Lastly, it is suitable for every kind of flow and for very complex geometries.

 Time discretization.

Time derivatives are approximated using finite difference discretization. Considering a short time interval ( _{), where} _{is the}

so-called time step, the time discretization methods can be divided into

explicit and implicit methods.

In the explicit methods the values at the time n+1 are computed from the values at time n. These methods are easy to implement and have a low

"Verification and validation of the DLR TAU-Code for spoiler aerodynamics at subsonic and transonic conditions"

Contents

List of figures

List of tables

Abstract

Chapter 1

Introduction

Chapter 2

Description of experiment and geometries

2.1 Experimental setup and simplifications

2.2 Design

2.3 Test cases

Chapter 3

Mesh generation