Large-Eddy Simulation of a Turbulent SWBLI over a

Technische Universität München

Term thesis Large-Eddy Simulation of a Turbulent Shock-Wave/Boundary-Layer Interaction over a Flexible Panel

Author: Matriculation Number: Field of Studies: Advisor:

Markus Zauner 03651638 Aerospace Engineering Vito Pasquariello† Georg Hammerl‡

Supervisor: Handindate:

Dr.-Ing. Stefan Hickel† 10.04.2015

† Institute of Aerodynamics and Fluid Mechanics Prof. Dr.-Ing. N. A. Adams

‡ Institute for Computational Mechanics Prof. Dr.-Ing. W. A. Wall

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel


Eidesstattliche Erkl¨ arung Ich versichere hiermit, dass ich meine Semesterarbeit mit dem Thema Large-Eddy Simulation of a Turbulent Shock-Wave/Boundary-Layer Interaction over a Flexible Panel selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe. Die Arbeit wurde bisher keiner anderen Pr¨ ufungsbehörde vorgelegt und auch nicht veröffentlicht.

M¨ unchen, den 10.04.2015

Markus Zauner


Abbreviations Re Pr Ma Kn Sr Cf δ0 xref d L h b λp pc t T∞ p∞ ρ∞ U∞ v γ µ F σij S bˆ0 ψ I E ρsolid ν β ϑ f Lsep LES P SD DF SW BLI F SI MR RR LF P

Descriptions Reynolds Number Prandtl number Mach number Knudsen number Strouhal number Skin-friction coefficient Boundary-layer thickness Position of reference point Displacement Panel length Panel thickness Panel width Dynamic pressure Cavity pressure Time Temperature Free-stream pressure Free-stream density Free-stream velocity component in wall direction Velocity component in wall-normal direction Heat capacity ration Dynamic viscosity Deformation gradient tensor Stress tensor Piola-Kirchhoff stress tensor Body force vector field Energy strain function Second-order identity tensor Young modulus of structure Density of structure Poisson number Shock angle Wedge angle Frequency Separation length Large-eddy simulation Power spectral density Digital filter Shock-wave/boundary-layer interaction Fluid structure interaction Mach reflection Regular reflection Low-frequency peak

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Figures

List of Figures 1.1. Restricted flow separation in a plane nozzle with transparent walls. Figure from [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Different coodinate systems used. . . . . . . . . . . . . . . . . . . . . .

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2.1. Sketch of shock generation. . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2. Basic shock-boundary layer interactions in supersonic flows. Figure from [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3. Schematic illustration of the strong turbulent shock-boundary layer interaction. Figure from [4] . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4. Schematic illustration of the regular reflection. Figure from [4] . . . . . 26 2.5. Schematic illustration of intersection of two shock waves with the occurrence of a Mach stem. Figure from [4] . . . . . . . . . . . . . . . . . 27 xref . . . . . . 29 2.6. Reτ over x. Reference point is marked at −100.4602 δ0,inf low xref 2.7. Cf over x. Reference Point is marked at −100.4602 δ0,inf low . . . . . . . 29 2.8. Distribution of the incompressible skin friction coefficient (solid). Reference data: Blasius Theory (dash-dotted), Cf,inc = 0.026 · Reδ−0.25 ; 2 ,w 1 Karman-Schoenherr Theory (medium-dashed), Cf,inc = 17.08 ·(log10 (Reδ2 ,w )2 + 25.11 · log10 (Reδ2 ,w )2 ) + 6.012)−1 ; Smith Theory (short-dashed), Cf,inc = 0.024 · Re−0.25 δ2 ,w ; Fernholz and Finley 1977 (); Guarini et al. 2000 (+); Komminaho and Skote 2002 ( ); Maeder et al. 2001 (×); Pirozzoli and Bernardini 2011 ( ); Pirozzoli, Grasso, et al. 2004 (D); Schlatter and ¨ u 2010 (4); Simens et al. 2009 (5). . . . . . . . . . . . . . . . . . . 30 Orl¨ 2.9. Van Driest transformed velocity profile compared with logarithmic wall ¨ u 2010. . . . . . . . . . . . . . . 31 law and DNS data of Schlatter and Orl¨ 2.10. Reynolds stresses in Morkovin scaling compared with DNS Data of ¨ u 2010. . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Schlatter and Orl¨ 2.11. Triangulation of an eight node hexahedral element face (gray) contributing to the fluid-structure interface Γ . Figure from [6] . . . . . . . . . . 32 2.12. Two-dimensional sketch of a cut-cell (i,j,k). Figure from [8] . . . . . . . 33

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Figures 2.13. Schematic of the staggered time integration of the coupled system. Figure from [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1. History of panel deflection in the 43 -chord point. U1 represents the freestream velocity, h the panel thickness and w the deflection of the 34 -chord point. Figure from [12] and modified . . . . . . . . . . . . . . . . . . . 38 3.2. Domain for simulation: Upper picture shows the start configuration and the lower one the end configuration after wedge rotation. Red are highlighted the shocks and the initial characteristic of the centered expansion. Transparent blue is marked the considered fluid domain. In dark, the flexible panel is shown between the two parts of the grey rigid base. 39 3.3. Measurements of set-up domain. . . . . . . . . . . . . . . . . . . . . . . 40 3.4. Oscillation frequency of panel center over the panel thickness. × remark simulation results for steel and + represent results for titanium. . . . . 42 3.5. Static displacement of panel center (time averaged displacement for the steady state after shock impingement δstatic ) over the ratio of panel thickness h and panel length L. × remark simulation results for steel and + represent results for titanium. . . . . . . . . . . . . . . . . . . . 42 3.6. Ratio of the oscillation amplitude around the static displacement (δamplitude ) and the static displacement (time averaged displacement for the steady state after shock impingement δstatic ) over the ratio of panel thickness h and panel length L, considering the panel center. x remark simulation results for steel and + represent results for titanium. . . . . . . . . . . 43 3.7. Oscillation amplitude around the static displacement (δamplitude ) over the ratio of panel thickness h and panel length L, considering the panel center. x remark simulation results for steel and + represent results for titanium. The vertical line indicates the final chosen set-up. . . . . . . . 43 3.8. Displacement (δy ) of the panel center over time (t). The vertical solid lines indicate rotation start of the wedge at τstart = 0.4 ms and rotation end at τend = 5.4 ms. The vertical dotted lines mark the considered sequence of 5 periods for comparison with PSD results of the final LES (starting point: 5.9 ms & ending point: 10.2 ms). The horizontal line show the time averaged deflection of the panel center during the oscillation, resulting in ystatic = −1.63 mm. . . . . . . . . . . . . . . . . . . 45 3.9. Normalized PSD using MATLAB’s pwelch function with 50% overlap. Vertical lines mark resonance frequencies at 1132 Hz, 2604 Hz, 4586 Hz and 6964 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Figures 3.10. Contour plot of first preliminary simulation of end-configuration. Instantaneous temperature distribution evaluated at the mid-plane is shown. 3.11. Comparison of temperature fields of LES with rotating wedge (top) and with 24.5 fixed wedge (bottom). Black iso line represents zero streamwise velocity. Temperature fields are averaged in time and span wise direction after reaching a final and stable end configuration with a wedge angle of 24.5◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12. Comparison of skin-friction coefficient Cf of LES with rotating wedge (red) and with 24.5◦ fixed wedge (blue). The green line additionally represents the physically wrong curve of the preliminary underresolved LES. Flow fields are averaged in time and span-wise direction after reaching a final and stable end configuration with an wedge angle of 24.5. The begin and end of the separation bubble is marked by the intersection of the curvatures with the y axis at x0red1 = −0.0433m and x0red2 = 0.0808m for the red curve and x0blue1 = −0.0414m and x0blue2 = 0.081m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13. Entropy difference between MR and RR (∆S = SM R −SRR ). Considered J are instantaneous mid-planes (z = 0). In the 2D-plot, the S = 5·10−8 kg iso-curves highlight the areas of maximum entropy in both simulations. In the 3D plot, the z-axis shows magnitude of ∆S for a better impression of the distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14. Time averaged pressure iso-lines for MR (red) and RR (blue). Streamlines at the border of the recirculation bubble for MR (pink) and RR (light blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15. Above: Contour plot showing distribution of turbulent kinetic energy. Below: Cross-sections at x0 = −0.02m (lhs) and x0 = 0.005m (rhs) showing the magnitude of velocity gradient |∇~v | which is calculated according to equation 3.8. Solid lines mark the sonic line where M a = 1 and dash-dotted lines mark the spots where the horizontal component of velocity is zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16. Streamlines for RR together with pressure iso-lines based on time averaged data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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50

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Figures 4.1. Panel deflection over simulation time of panel center. Exact measurement position is on the panel surface at x0 = 1.3975 · 10−4 m and z = 9.109 · 10−5 m. The dash-dotted vertical lines mark rotation start at 0.4 ms and rotation stop at 5.4 ms. The solid vertical line indicates the start time for evaluating the displacement for frequency analysis of the final LES. The dotted vertical line represents the evaluation start time of the inviscid simulation. The dashed vertical line marks the time step, by which the behavior of the recirculation bubble is considered as 00 steady state00 at 6.7 ms. The horizontal lines show the dimensionless mean deflection of −1.45 for the final LES (solid) and −1.42 for the inviscid simulation (dotted). . . . . . . . . . . . . . . . . . . . . . . . . 4.2. For statistical evaluations considered oscillation around the mean deflection over simulation time of panel center (solid line), 1/3-chord point (dashed line) and 1/4-chord point (dotted line). Exact measurement position of all considered points is on the panel surface at z = 9.109·10−5 m and at x0 = −0.024867 m (25.2%), x0 = −0.01675 m (33.333%) and x0 = −1.3975 · 10−4 m (50.003%). . . . . . . . . . . . . . . . . . . . . . 4.3. Evolution of the amplitude of the panel center. Vertical line marks 6.7 ms of simulation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. PSD of the panel oscillation at the panel center (solid line), 1/3-chord point (dashed line) and 1/4-chord point (dash-dotted line) for the final LES. The dotted curve represents the PSD of the center point gained by the inviscid simulation. The solid vertical lines highlight the peaks for the results of the final LES, whereas the dotted lines indicate the peaks for the panel center of the inviscid simulation considering a longer oscillation period. Exact measurement position of all considered points is on the panel surface at z = 9.109 · 10−5 m and at x0 = −0.024867 m (25.2%), x0 = −0.01675 m (33.333%) and x0 = −1.3975·10−4 m (50.003%). 4.5. Temperature field of coupled simulation for various time steps, considering the section −0.05 < x0 < 0.037. The cell values are averaged in spanwise direction for each time step. Dashed line marks the undeformed structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dy ) · 360 at 6.7 ms 4.6. Local panel deflection angle according to αdef = atan( dx 2·π after rotation start. Vertical dashed lines mark points with zero-deflection at x0 = −0.0499635 m, x0 = 0.0043571 m and x0 = 0.128884 m . . . . .

56

59 60

60

63

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Figures 4.7. Temperature field of both, coupled and uncoupled simulations for different states during the rotation. The angle β represents the angle of the incident shock. The cell values are averaged in spanwise direction for each time step. The coupled plot is restricted to positive values in wall-normal direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Pressure evolution along wall for coupled (blue) and uncoupled (red) simulation, using spanwise averaged flow fields at 6.7 ms simulation time. Dashed vertical lines indicate the leading and trailing edge of the panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. Iso-lines for both, coupled (blue) and uncoupled (red) simulation, with zero streamwise velocity. Considered is a spanwise averaged flow field at 6.7 ms simulation time. Dashed vertical line indicates the leading edge of the panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10. Pressure elevation along the wall for coupled (blue) and uncoupled (red) simulation. Dashed vertical line indicates the leading edge of the panel. The parameter s represents the run length. . . . . . . . . . . . . . . . . 4.11. Iso-Mach lines for M a = 1, M a = 1.2 and M a = 2.4 for coupled (blue) and uncoupled (red) simulation at β = 43.4454◦ right after rotation stop (equals 5 ms after rotation start). . . . . . . . . . . . . . . . . . . 4.12. Iso-mach lines for M a = 1, M a = 1.2 and M a = 2.4 for coupled (blue) and uncoupled (red) simulation at 6.7ms simulation time. . . . . . . . . 4.14. PSD of a pressure probes near separation point of the uncoupled LES (red) at x0 = −0.0417 m and coupled LES (blue) at x0 = −0.0482 m. The vertical lines indicate the resonance frequencies of the panel at 1132 Hz, 2601 Hz and 4586 Hz. . . . . . . . . . . . . . . . . . . . . . 4.13. Local deflection angles of streamlines according to atan( uv ). Please mind that the color map for the angles is not equally spaced. . . . . . . . . . 4.15. Pressure signal of a pressure probe of the coupled LES (blue) at x0 = −0.0482 m and uncoupled LES (red) at x0 = −0.0417 m near separation point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16. Plot showing time-averaged Reynolds stresses for coupled and uncoupled LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Tables

List of Tables 2.1. Summary of incoming turbulent boundary-layer data. . . . . . . . . . .

22

3.1. Overview of smallest cells of preliminary simulations. . . . . . . . . . . 3.2. Summary of set-up parameters. . . . . . . . . . . . . . . . . . . . . . .

47 55

4.1. Summary of structure results. . . . . . . . . . . . . . . . . . . . . . . .

58

5.1. Summary of simulations. . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel List of Tables

Abstract The occurrence of a fluid-structure interaction in a Mach 3.0 flow over a flexible panel is investigated by performing a large-eddy simulation (LES) coupled with a simulation of the structure. The turbulent shock-wave/boundary-layer interaction, exciting the panel, is caused by an incident oblique shock, generated by a 24.5◦ wedge and leads to a recirculation area near the theoretical impingement point. The Reynolds number at the inflow based on the incoming boundary-layer thickness of 0.0005 m is Reδ0 = 22800. As a weak partitioned coupling is chosen, the fluid and structure domain can be calculated separately. The Adaptive-Local-Deconvolution-Method (ALDM) is used to solve the compressible Navier-Stokes-Equations for the discretization of the convective fluxes, which in turn explicitly establishes a undefined framework between subgridscale model and truncation error. For solving the balance of linear momentum in the structure domain, a hyperelastic Saint Venant-Kirchhoff material model is applied. The interface motion within the Eulerian flow solver is accounted for by means of a conservative cut-cell Immersed Boundary method. To obtain physically correct results of the oscillation characteristics, the inflow boundary conditions and the development of the turbulent boundary-layer are treated carefully. Beside several preliminary simulations, the final LES is performed, using 3264 cores of the SuperMUC Petascale System in Munich. For investigating 10 ms of the turbulent shock-wave/boundary-layer interaction over the flexible panel, 3370674 iteration steps are necessary with an average loop-time of 1.29 s, resulting in a total CPU time of about 3.9 · 106 hours. Preliminary tests with uncoupled LES, for getting an impression of the behaviour of the recirculation bubble and coupled simulations considering an inviscid fluid, to determine resonance frequencies of the deformed panel are performed. The idea is to design the domain of fluid and structure in a way, to increase the probability for obtaining resonance, induced by the oscillating recirculation bubble, which would result in an excited panel oscillation. With the chosen set-up it is indeed possible to observe an increase of the oscillation amplitude of the panel with half of the oscillation frequency of the recirculation bubble. The influence of a flexible panel can be clearly seen in the change of the separation length, which seems to strongly depend on the size of the panel. Further panel deflection has also a big impact on the oscillation behaviour of the recirculation bubble.

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel Contents

Contents List of Figures

5

List of Tables

10

1. Introduction 14 1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2. Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3. Conventions in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. Numerical Approach 2.1. Structure . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Physical equations . . . . . . . . . . . . . . . . 2.1.2. Solver . . . . . . . . . . . . . . . . . . . . . . . 2.2. Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Governing physical equations . . . . . . . . . . 2.2.2. Solver . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Boundary Layer . . . . . . . . . . . . . . . . . . 2.2.3.1. Shock-wave/boundary-layer interaction 2.2.3.2. Profile . . . . . . . . . . . . . . . . . . 2.3. FSI coupling and single field solver . . . . . . . . . . . 3. Approach for Set-up 3.1. Requirements . . . . . . . . . . . . . . . . . . 3.2. Basic domain architecture . . . . . . . . . . . 3.2.1. Coupled simulation with inviscid fluid . 3.2.2. Preliminary uncoupled LES . . . . . . 3.2.2.1. End configuration . . . . . . 3.2.2.2. Rotating wedge . . . . . . . . 3.3. Final LES Set-up . . . . . . . . . . . . . . . . 4. Interpretation of the LES results

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel Contents 4.1. Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 63

5. Conclusion 73 5.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 A. Domain Optimizer (Matlab)

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B. Structure-displacement analysis

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C. Probes Reader

86

D. Modification of fourier transform tool (1D)

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E. Modification of fourier transform tool (2D)

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Bibliography

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 1. Introduction

1. Introduction 1.1. Motivation Nowadays, there are many reasons to launch rockets up to space: Beside political aspects, satellites are playing more and more an important role in everyday life. Scientists of various fields see huge potential in zero-gravity experiments and space exploration is still advancing progressively. As space flight is demanding high reliability, but is facing on the other hand economic interests, every kilogram of weight counts. This is just one reason, why big efforts are taken, to reduce weight of the rocket-structure. For doing this without effecting the high standard of safety and reliability, it is important to understand critical physical effects that could lead to loss of mission, spacecraft or even life. Effects like shock-wave/boundary-layer interaction (SWBLI) and fluidstructure interaction (FSI), which could lead to severe structural damage of the rocket nozzle extension. FSI occurs if an oscillating disturbance in the flow excites a surrounding flexible structure. This can happen for example in an overexpanded nozzle-extension of a spacecraft engine, if the fluid expands below surrounding pressure. At some point, the fluid in the boundary-layer has not enough energy to overcome the pressure gradient and cannot follow the divergent nozzle contour anymore. The separation generates a shock impinging on the opposite wall. Nowadays it is well known that SWBLI are unsteady, which reveals oneself by a breathing motion of the separation bubble. In Figure 1.1, the shock interacts with the boundary-layer and causes a detachment of the flow and thus an oscillating recirculation area. As the pressure level in the recirculation bubble is much higher than in an attached boundary-layer, high side-loads on the nozzle structure can occur due to the unsteady, nonuniform circumferential pressure distribution on the inner surface of the nozzle extension, while on the outer surface the atmospheric pressure can have much lower constant values. This can lead to deformations and damage of the structure.

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 1. Introduction

Figure 1.1.: Restricted flow separation in a plane nozzle with transparent walls. Figure from [9]

1.2. Objective In this thesis, a Mach 3.0 flow over a flexible panel will be analyzed by performing a large-eddy simulation. The shock-wave/boundary-layer interaction without a flexible panel will be compared to similar precursor simulations with different shock strengths. Finally the effect of the FSI on the SWBLI and structure will be analyzed and compared to other simulations and theoretical considerations. These simulations should contribute to better understanding of correlations between shock strength, oscillations of the turbulent boundary-layer and material parameters of the excited panel. By gaining more experience and better comprehension of the influence of incident shocks, it will be possible to benefit from these induced mechanisms as tools for passive or active flow control.[12] The structure of this thesis is as follows: The next chapter will provide an overview of the numerical approach, followed by a chapter that gives an impression of the procedure and considerations that lead to the final set-up, after which results will be presented. In the end, there is a short summary and outlook.

1.3. Conventions in this thesis Coordinate systems: There are two different kinds of coordinate systems used in this thesis. For describing the domain set-up, basically the right coordinate system of Figure 1.2 is used. For discussing the results, a coordinate system, which has its origin in the theoretical impingement point of the shock is used (see left coordinate system in Figure 1.2). The boundary-layer thickness at a reference point is often used to obtain dimensionless

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 1. Introduction 0

measurement ( δ0y = δ0y ). Both systems have its origin in the symmetry plane of the ref ref domain and the positive x-direction points downstream and is measured in meters. y' y x

x'

8m

0.0

z' z

Figure 1.2.: Different coodinate systems used. Reference values: As already mentioned, the boundary-layer thickness at the reference point x = 0.0297699 m after the inlet is chosen as a reference value for describing the flow field. For describing flow quantities, the free-stream parameters are chosen as reference values. Non-dimensional quantities are denoted as ’*’ or ’+’.

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Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach

2. Numerical Approach 2.1. Structure 2.1.1. Physical equations The structure has to fulfill the local form of the balance of linear momentum, according to equation 2.1. ρsolid · d¨ = 50 · (F · S) + bˆ0

in Ωsolid ,

(2.1)

where Ωsolid is the undeformed structural domain and ρsolid the density of the structure. The displacement is denoted as d, d˙ the velocity and d¨ the acceleration, which are actually unknown. The vector field bˆ0 stands for the body force. The tensor S denotes second Piola-Kirchhoff stress tensor, representing the stresses in the structure and F is the deformation gradient tensor. To obtain the stresses, a hyperelastic Saint Venant-Kirchhoff material model is defined for the strain energy function Ψ per unit reference volume (see equation 2.3). For initialization the panel is undeformed. 1 · λsolid · (E : I)2 2 1 ∂Ψ with E = · (F T · F − I) and S = 2 ∂E P =F ·S Ψ(E) = µsolid · (E : E) +

(2.2)

(2.3)

In this relation, µsolid and λsolid are Lamè constants and I is the second-order identity tensor. The first Piola-Kirchhoff stress tensor is defined by P .

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2.1.2. Solver As the strong form of nonlinear solid mechanics problems presented above can only be solved analytically for simple cases, a numerical method is needed. The discretization in space is done by a Finite Element Method (FEM) and the temporal discretization using a Finite Difference Method (FDM). For doing so, the weak form of nonlinear solid mechanics problems has to be introduced, which fulfills the equations only in an integral sense by applying the principle of virtual work. The principle of FEM is to split up the spatial continuum of the domain into geometrical finite elements (sub domains), defined by so-called nodes. The time still stays continuous. For the time integration, the One-step-theta (generalized trapezoidal) scheme is applied. The solution in time is obtained by a linear combination of a forward and backward Euler time integration scheme. The blending factor θ is set to 0.5, obtaining the Crank-Nicolson scheme. For more detailed discussion of the numerical approach for the structural solver, please see the reference literature [6] and [10].

2.2. Fluid 2.2.1. Governing physical equations Kn =

λ , L

U∞ Ma = √ , γRT

Reδ =

U∞ δ0 , ν

Pr =

ηcp λc

(2.4)

Postulating continuum hypothesis, justified by a sufficiently small Knudsen number (first definition in equation 2.4) Kn 1, the fluid can be described by the unsteady, compressible Navier-Stokes equations (NSE): ∂ρ ∂(ρuj ) + = 0, ∂t ∂xj ∂(ρui ) ∂(ρui uj ) ∂p ∂σij + + − = 0, ∂t ∂xj ∂xi ∂xj ∂(ρE) ∂(ρEuj + puj ) ∂(σij ui − qi ) + − =0 ∂t ∂xj ∂xj

(2.5)

The index i shows the components of the three-dimensional fluid domain. For general considerations, it is often useful to transform the NSE to dimensionless equations with following relations using adequate reference values:

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u∗i =

ui , U∞

ρ T p T∗ = , p∗ = 2 , ρ∞ T∞ U∞ ρ∞ E U∞ xi E∗ = , t∗ = t · x∗i = 2 ρ∞ U∞ δ99 δ99 ρ∗ =

(2.6)

= 0.99. By transforming Here, δ99 is defined as the boundary-layer thickness, where u(y) U∞ the NSE using the relations of equation 2.6, the Mach number (Ma), Reynolds number (Reδ ) and Prandtl number (Pr) represent the governing flow parameters (equation 2.4). For Newtonian fluids, implying proportional behavior between stresses and velocity gradients, the stress tensor σij is defined as 2 σij = µ 2 Sij − δij Skk 3

(2.7)

where Sij represents the strain rate tensor 1 Sij = 2

∂ui ∂uj + ∂xj ∂xi

.

(2.8)

The dynamic viscosity is depending on the temperature and can be modeled through the Sutherland law: TS + S µ = µS T +S with TS = 273.15 K,

S = 122 K,

T TS

32 (2.9)

µS = 17.10 · 10−6 P a · s

For completion, it is mentioned that it is often useful to combine inner energy E, pressure and density by introducing the enthalpy H. The heat flux through conduction qi in the fluid follows from the Fourier law (equation 2.11). H=

p ρ

qi = −λc ·

19

(2.10)

∂T ∂xi

(2.11)

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach The conductivity coefficient λc follows from the definition of a constant Prandtl number. The shock is treated analytically according to the oblique shock theory, using the Hugoniot equation [2] for a calorically perfect gas: 1 1 p1 + p2 · ( − ), 2 ρ1 ρ2 p with E = cv T, =R·T ρ

E2 − E1 =

(2.12)

where the index 1 represents the gas condition before the shock and 2 after the shock. For the application on oblique shocks, only the component of the upstream Mach number normal to the shock wave is considered, assuming the tangential component of the velocity vector constant over the shock. For detailed derivation, please see page 133 ff. of [2].

2.2.2. Solver As it hasn’t even been proven yet that the NSE can be solved analytically, we rely on numeric solutions of this system of nonlinear partial differential equations. The direct numeric solution (DNS) requires a very fine grid and leads to expensive simulations. Another approach would be averaging the NSE, causing unclosed terms, which have to be closed by turbulence models. For this method, it is also impossible to resolve turbulent structures. The approach for explicit large-eddy simulations is to make a trade-off by dividing the scales of turbulent structures in the flow according to the energycascade in non-universal grid scales (GS) and universal sub-grid scales (SGS), which will be modeled. For the simulations performed in this thesis, an implicit LES (ILES) method, more precisely the adaptive local deconvolution method (ALDM)[3], is used. Here, the SGS tensor is modeled implicitly using an adequate discretization scheme for the convective fluxes. The numerical truncation error of the spatial discretization scheme is used as a physically motivated model for the SGS, whereas classic explicit SGS models assume, that the truncation error has no impact on the SGS physics. By doing so, no additional model has to be considered [15]. For initializing a fully developed turbulent boundary-layer (see next chapter), a socalled Digital Filter (DF) procedure is used [16]. A random dataset gets filtered for

20

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach obtaining required statistics and autocorrelation functions. These are based on integral length and time scales and the mean values and Reynold stresses of all three components of the velocity vector at the inlet. It is also possible to define zones of the boundary-layer with different length or time scales, for gaining more realistic conditions. The thermodynamic fluctuations are gained by using the Strong-ReynoldsAnalogy (SRA) for linking the already obtained velocity fluctuations with the corresponding temperature fluctuations. As the pressure is assumed constant in wall-normal direction, the density is obtained by the ideal gas relation. [13]

2.2.3. Boundary Layer In this chapter, the generated boundary-layer profile is discussed and the physical correctness of the velocity profile is investigated. In advance, table 2.1 gives an overview of parameters, which are relevant for defining and evaluating the velocity profile of the turbulent boundary layer for the simulation. For detailed information, please see the following discussion and reference literature. For the LES, an adiabatic wall is considered. The adiabatic wall temperature Twall is calculated by equation 2.13.

Twall = T∞ · (1 + r ·

1 γ−1 2 · M∞ ), assuming a recovery-factor r = P r 3 2

(2.13)

In the evaluation the relations 2.14 are used for obtaining non-dimensional values, allowing general comparison with other results. ∂hui hτwall i = µwall · |wall ∂y s µwall · ∂u | µwall ∂y wall uτ = ( ) l+ = ρwall uτ · ρwall s y ρwall y+ = + = y · l µwall · ∂u | ∂y wall hui+ =

21

hui uτ

(2.14)


Parameter

Dimensions

U∞ T∞ ρ∞ p∞

[ ms ] [K] kg [m 3] [P a]

δ0,inf low

[m]

0.0005

Ma Pr Reδ0,inf low

[-] [-] [-]

3 0.72 22800

δ0inf low xref

[m] [m]

0.0005 0.0297699

[-]

-100.4602

[-] [-] [m] [m] [m] [-] [-] [-] [-] [-] [-] [ mN2 ] [ ms ]

450 52400 0.00115 0.000348 0.0000705 4.93 14349.5 2908.8 2516.4 510.1 0.00222 223.9 32.2

x0ref δ0inf low

Reτ Reδ0,ref δ0ref δ1ref δ2ref H12,ref Reδ1,ref Reδ2,ref Reδ1,w,ref Reδ2,w,ref Cfref τwallref uτ,ref

Values 601.0 100 0.558138 16021.7967

for obtaining the incompressible numbers with index w, a multiplication by µρww has to be performed. Table 2.1.: Summary of incoming turbulent boundary-layer data.

2.2.3.1. Shock-wave/boundary-layer interaction (SWBLI) In this thesis, the induced shock is strong and leads to an abrupt increase of the pressure. This has a crucial impact on the turbulent boundary-layer, where velocity and kinetic energy decrease significantly near the wall according to the no-slip condition (2.15) and therefore is sensitive to an adverse pressure gradient. At the area close to

22

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach the impingement point, there will occur an interaction between the shock-wave and the turbulent boundary layer. In this chapter, the different types will be outlined. The here presented background is based on [1], [2] and [4].

(2)

β12

θ 12 (1)

Ut1

Un1

U2

U∞

Ut2

Ut1 = Ut2

Un2

Figure 2.1.: Sketch of shock generation.

uy=0 = 0 vy=0 = 0 wy=0 = 0

sin(βmach ) =

1 M a(T, γ, R, u)

(2.15)

(2.16)

A shock occurs due to a disturbance in a supersonic flow. In this case, a wedge is used to generate a shock. As the air particles can not penetrate the wedge, they are forced to change the direction without transmitting information upstream, as information can just spread according to the Mach-Cone downstream, described by the half-cone angle βmach (equation 2.16). For fulfilling still the conservation laws of mass, impulse and energy, this change of the velocity vector parallel to the contour of the wedge (Figure 2.1) leads to a compression of the fluid. On the other hand, a divergent contour would force the fluid to expand by means of expansion waves (Figure 2.2,d). Near the impingement point of the shock, a combination of basic mechanisms and reflections occurs, which is sketched in Figure 2.3, where the pressure disturbance propagates upstream through the subsonic layer of the boundary-layer. As the impinging shock has to adapt to decreasing upstream Mach number in the boundary layer, it becomes very weak and vanishes, reaching the sonic line of the boundary-layer. The thickness of this subsonic layer can be described by the shape factor H12 , as a small H12 represents a fuller velocity profile and therefore a thin subsonic layer, which also results in a shorter upstream influence length l0 [1].

23


C1

ϑ

C1

ϑ

ϑ M∞

M∞ δ0

C2

ϑ

δ0

ϑ (a) Ramp fl ow

(b) Shock refl ection

C1

M∞ δ0

M∞ δ0 (c) Forw ard step

(d) Backw ard step

Figure 2.2.: Basic shock-boundary layer interactions in supersonic flows. Figure from [4] As Figure 2.3 shows, strong shocks lead to significant detachment of the boundary layer. The theoretical inviscid impingement point ximp is our reference position of the coordinate system and l0 represents the distance, a disturbance propagates upstream influencing the boundary-layer thickness. This disturbance can be clearly identified in the pressure distribution (in the middle of Figure 2.3). The pressure gradient leads to an increasing boundary-layer thickness and generates compression waves, which coalesce and form the separation shock C2 . The deflection angle ϑ120 is generally not equal to the deflection angle ϑ12 of C1 , but if ϑ12 = ϑ120 , the whole pattern would be symmetric. The intersection of C1 and C2 generates additional shocks. In the case of Figure 2.3, two shocks, C3 and C4 , occur. Other shock configurations due to this intersection will be discussed later. As shown for an inviscid model in the bottom subplot of Figure 2.3 (bottom subplot), the velocities in section (3) and (30 ) are different and divided by a slip-line, along which the pressure stays constant. The shear layer is in viscid flows basically responsible for energy transfer between high speed flow and separation bubble. As already mentioned before, there can occur different shock configurations depending on deflection angles, free-stream parameters and gas properties. The type, presented in Figure 2.3 is called Regular Reflection (RR). Another form is the so called Mach Reflection (MR). As shown in chapter 3.2.1 the relations between shock and deflection

24


Figure 2.3.: Schematic illustration of the strong turbulent shock-boundary layer interaction. Figure from [4]

25

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach angle are described by non-linear equations. Therefore, as sketched in Figure 2.4 for an inviscid shock reflection, the deflection angles are equal (ϑ12 = ϑ23 ). If the deflection angle of C1 is equal to that one of C2 the reflection point R in Figure 2.4 for an inviscid flow is comparable to the intersection point of C1 and C2 in a SWBLI. Moreover, it can be shown that a ϑmax exists, limiting the possibility of a RR. For a perfect gas with γ = 1.4 the max. deflection equals ϑmax = 45.5◦ for M a → ∞. If the shock reaches a level where the RR cannot be achieved anymore (called detachment criterion), a MR occurs.

Figure 2.4.: Schematic illustration of the regular reflection. Figure from [4] The MR can be described by the analytical model of Neumann, called three-shock theory. Configurations exceeding the detachment criterion are characterized by the almost vertical Mach-Stem (m) which must be actually curved, as the velocity vectors in each triple point have different orientations (see Figure 2.5). These two triple points (tp1 and tp2 ) occurring make a prediction of the shock architecture highly complex an thus making it very interesting to investigate numerically[4]. For the case treated in this thesis, a configuration is chosen, for which generally a RR as well as a MR can occur in a supersonic channel flow. A change of the shock architecture would be a good indicator for highlighting the effects of a FSI, as due to the panel deflection a change of the separation shock generated by the recirculation bubble is expected. It is also very important to mention that the SWBLI in viscous fluids is highly complex and instationary. The periodic movement of the separation bubble (namely lowfrequency phenomena) is mentioned in many numerical and experimental investigations on SWBLI [1]. The frequency, in which the separation bubble is ’breathing’, is

26


Figure 2.5.: Schematic illustration of intersection of two shock waves with the occurrence of a Mach stem. Figure from [4] a factor of 103 smaller than the frequency of pressure fluctuations in the arriving turbulent boundary-layer [1]. Hypothesis for describing these phenomena are still under research. These periodic oscillations are supposed to be the driving mechanism of the excited panel in the simulation treated in this thesis.

2.2.3.2. Profile To avoid affecting the periodic low-frequent oscillation of the reflected shock and the separation area, the inflow boundary conditions are defined carefully. For initiating directly a fully developed turbulent boundary-layer without transition, a DF generates at the inlet fluctuations of the flow parameters near the wall according to a predefined turbulent boundary-layer velocity profile. This profile is the result of a temporary calculation, where a LES is performed with periodic boundary conditions in flow direction and an adiabatic wall condition, started with a laminar boundary-layer profile superimposed by white noise fluctuations to initiate laminar-turbulent transition. At

27

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach the inflow, a minor disturbance is generated by the initialization of the turbulent boundary-layer, but the effect is negligible. ρ ∞ u ∞ δ0 µ∞

(2.17)

ρw uτ δ δ = + µw l

(2.18)

Reδ0 =

Reτ =

The boundary-layer thickness of the applied profile is δ0,inf low = 0.0005 m at the inflow with a referring Reynolds number based on the boundary-layer thickness at the inflow of Reδ0,inf low = 22800. For validating the velocity profile and other characteristics of a physical correct turbulent boundary-layer, a reference position is defined upstream of the SWBLI within the undisturbed turbulent boundary-layer. As there is data of a direct numerical simulation (DNS) of a turbulent boundary-layer with a friction Reynolds number of Reτ = 450 (equation 2.18), the reference position is defined at xref = −100.4602, if the theoretical impingement point of the incident shock would δ0,inf low ximp be at δ0,inf low = 0 (see Figure 2.6 ). Figure 2.7 shows, that the reference position is still in the linear section of the friction coefficient (Cf ) curve, which is an indicator to justify general comparison with theoretical considerations and test results of boundary-layer profiles. This Figure also shows transient behavior of the turbulent boundary-layer at the inflow, caused by the DF [13]. As long as the profile downstream develops to a physically correct profile before reaching the investigated section of the domain, this transient zone is not problematic. The sudden collapse of the Cf in the end of the graph is caused by the separation of the boundary-layer due to the shock impingement. The data of the simulation discussed in this chapter is gained by averaging the flow in time and in spanwise direction. To be also able to compare the friction coefficient with test results within the incompressible regime, a van Driest II transformation according to equations 2.19 is done [1]. It can be seen that the evaluated friction coefficient correlates to the results of Guarini et al. (2000) and the theory of Blasius in Figure 2.8.

28


500

Reτ

400 300 200 −160

−140

−120

−100

x δ0

Figure 2.6.: Reτ over x. Reference point is marked at −100.4602

xref δ0,inf low

hCf i [·10−3 ]

2.5

2

1.5 −160

−140

−120

−100

−80

x δ0

Figure 2.7.: Cf over x. Reference Point is marked at −100.4602

Reδ2 ,w = Reδ2

xref δ0,inf low

µ∞ ρ∞ u∞ δ2 = µw µw

Cf i = Fc · Cf Taw − 1 Fc = (arcsin(A) + arcsin(B))2 Taw + Tw − 2 Taw − Tw A= p , B=p 2 (Taw − Tw ) − aTw (Taw − Tw )2 − aTw

(2.19)

For comparison with the logarithmic wall law (equation 2.20), a Van Driest transformation is performed according to equation 2.21. In 2.9 the profile is also compared

29


hCf, inc i [·10−3 ]

5

4

3 1000

2000

3000

4000

5000

Reδ2,w Figure 2.8.: Distribution of the incompressible skin friction coefficient (solid). Reference data: Blasius Theory (dash-dotted), Cf,inc = 0.026 · Reδ−0.25 ; Karman2 ,w 1 Schoenherr Theory (medium-dashed), Cf,inc = 17.08 · (log10 (Reδ2 ,w )2 + 25.11 · log10 (Reδ2 ,w )2 ) + 6.012)−1 ; Smith Theory (short-dashed), Cf,inc = 0.024 · Re−0.25 δ2 ,w ; Fernholz and Finley 1977 (); Guarini et al. 2000 (+); Komminaho and Skote 2002 ( ); Maeder et al. 2001 (×); Pirozzoli and Bernardini 2011 ( ); Pirozzoli, Grasso, et al. 2004 (D); Schlatter and ¨ u 2010 (4); Simens et al. 2009 (5). Orl¨ ¨ u 2010 at a Reτ = 450. As the deviation is to DNS test data of Schlatter and Orl¨ minimal, it is also indicating the physically correctness of the profile. 1 log y + + C + κ (typically κ = 0.41 and C + = 5.2 for boundary-layers on flat plates) hui+ =

huVD i+ =

Zu+s

hρi dhui+ ρw

(2.20)

(2.21)

0

As a last investigation, the Reynolds stresses are compared with with the same data ¨ u 2010 at Reτ = 450, using the similar Morkovin scaling (see of Schlatter and Orl¨ equation 2.22). Figure 2.10 shows deviations, but according to Hickel and Adams [5] these deviations are typical and acceptable for comparing results of LES and DNS data. s hρi ξ= (2.22) ρw

30

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach 30

huVD i+

20

10

0 100

101

102

103

y+ Figure 2.9.: Van Driest transformed velocity profile compared with logarithmic wall ¨ u 2010. law and DNS data of Schlatter and Orl¨

2 1

ξ

q

|hu0i u0j i+ |

3

0 −1 100

101

102

103

y+ Figure 2.10.: Reynolds stresses in Morkovin scaling compared with DNS Data of ¨ u 2010. Schlatter and Orl¨

2.3. FSI coupling and single field solver This chapter provides an overview for the numerical approach for coupling the fluidand structure-solver. The content is based on the publication [6].

31

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach For describing the interface between the FEM solver of the structure and the FVM Solver of the fluid, a conservative cut-cell Immersed Boundary Method is used. The big advantage of this approach is, that the Cartesian mesh of the fluid domain does not have to adapt to the deformed structure and the problem can be treated with Eulerian approach without mesh movement steps. The cut-cells deliver the position of the structure for the fluid and the load, like pressure forces and heat flux for solid. As the mesh near the wall is very fine in an LES, the position of the plate can be determined and resolved accurately enough for the fluid domain. The interface conditions are described by equations 2.23. u~f = u~s Tf = Ts σf · n~f = σs · n~s ⇔ τ n~f + p n~f = σ s n~s

(2.23)

The index f represents fluid- and s structure-side. The variable ~u denotes the velocities, T the temperatures, ~n the normal-vectors, τ the shear stress tensor and σ the total stress tensor. Γ stands for the fluid-structure interface. [6] For enhancing the accuracy

Figure 2.11.: Triangulation of an eight node hexahedral element face (gray) contributing to the fluid-structure interface Γ . Figure from [6] of the representation of the time varying solid interface, a subdivision of these cells is done. The cut plane is divided into four triangles by introducing an additional node (see Figure 2.11). A linear approximation of the possibly non-linear structure interface is now possible. For modelling the interaction of fluid and structure, an interface exchange term is introduced. It accounts for the pressure and pressure work of each triangle, obtained by the procedure described above. The pressure on the elements is obtained by solving an one-sided face-normal Riemann problem with an appropriate solver. The non-cut cells, adjacent to the interface, which would actually be in the structure domain, remain as ghost cells. These cells are used for enabling the finite volume scheme also for fluid cells near the interface. They contain ghost fluid states which

32

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach allow to set the required boundary condition along the interface without changing the interpolation scheme. For doing so, following steps are done:

1. Ghost cells contributing to the interpolation stencil are identified. 2. For each cut-cell (in Figure 2.12 exemplary xGP ), an averaged normal vector (nravg ) is calculated, which is an average of all contained triangles, weighted by their area. 3. An auxiliary point (called boundary intercept point xBI ) on the interface surface is defined, where the extended averaged normal vector of the cut-cell meets the cell center of the ghost cell. 4. The tangential plane in this auxiliary point serves now quasi as a mirror plane. An image point (xIP ) in the fluid region is obtained, having the same minimum distance to the auxiliary point as the cell center of the ghost cell, which means that xGP xBI = xBI xIP . 5. A trilinear interpolation between the adjacent cell centers is done to obtain the values of quantity for the image point. 6. Based on the obtained values, the ghost cell values (here for example ϕ) are set according to equation 2.24, so that the auxiliary point would quasi fulfill wall boundary conditions. xIP

ΩF

Γ

2∆ l xBI

nΓavg

xGP ΩS

Figure 2.12.: Two-dimensional sketch of a cut-cell (i,j,k). Figure from [8]

33

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 2. Numerical Approach ϕGP = 2 · ϕBI − ϕIP + O(∆l2 ) f or Dirichlet boundary conditions ϕGP = ϕIP − 2 · ∆l(5ϕ · ~nΓavg )|xBI

f or N eumann boundary conditions

(2.24)

For more detailed information about this procedure, please see [6].

In this thesis, a loosely coupled algorithm is used, which does not satisfy the coupling conditions at each time step. A drawback of this method is numeric instability, if the fluid volume fraction of cut-cells gets too small. Therefore a so-called mixing procedure (described in [7] and [8]) is used. For the fluid side, the equation 2.25 has to be solved, where K(ω) represents the flux-tensor in dependency of the state-vector ω containing the conserved variables density, momentum and total energy. ∂ω + 5 · K(ω) = 0 ∂t

(2.25)

By integrating K(ω) for the whole cell and a division by the fluid volume fraction of the cut-cell, a new vector w is formed. Now the face-averaged numerical fluxes over the surfaces connected to other fluid cells and finally also the flux across the structure interface are added to obtain the w for the next time step.[6] For spatial discretization, an adaptive central upwind scheme of 5th order is used. The integration in time is done according to a strong stable Runge-Kutta scheme of 3rd order. Balance of linear momentum discretized in space with FEM leading to formula 2.26 M d¨ + fS;int (d) − fS;ext (d) − fSΓ = 0

(2.26)

Here M represents the mass matrix, d the displacement and the derivatives of d, the velocity and acceleration vectors. The internal and external forces are considered by introducing fs;int and fs;ext . The term representing the interface is denoted by fSΓ . For avoiding shear locking phenomena, the method of enhanced assumed strains is used. As already mentioned, the time integration of this equation is done by the one-steptheta scheme, which is a combination of forward and backward Euler time integration. The FEM is used for spatial discretization of the structure domain. Figure 2.13 shows the schematic sequence of coupling steps, as described in the fol-

34


Figure 2.13.: Schematic of the staggered time integration of the coupled system. Figure from [6] lowing lines:

1. The cut-cells are defined for time step tn by the known displacement of the structure. 2. For the fluid, the next time step is calculated. An interpolation is done to transfer solid velocities to fluid velocities for the no-slip condition at the wall. 3. The load for the structure is defined by the fluid pressure. Again an interpolation is needed to transfer fluid forces to structure forces. 4. The next time step for the structural domain is calculated. 5. Ready for next loop. As usually the time steps are very small for LES due to the small cells and the CFL condition, the fluid-structure coupling is not done for every time step (called subcycling), as the relevant time scales of the fluid are much smaller than the relevant time scales of the structure.

35

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up

3. Approach for Set-up For optimizing the fluid domain, which has a significant impact on the computationaltime, the dimensions of the flexible panel play an important role. On the one hand, the panel should be as short as possible, on the other hand, it must be big enough to fully contain the recirculation area and should be able to reach amplitudes in the order of the boundary-layer thickness to enhance the FSI. To be able to compare this simulation with results of experimental tests, it was decided to chose the material properties corresponding to physical materials. Regarding the analytical solution for a both-ends-clamped panel (see below), the ratio of thickness and length ( Lh ) is used to characterize the geometry of our panel. Other important factors are material properties of the panel, like the modulus of elasticity (Young’s Modulus) and the density. In the simulation of the structure, also the Poissonratio will be considered. But as the ratio of thickness/length is small and longitudinal stresses are in this case not high enough, the Poisson-effect will be minor.

3.1. Requirements The main idea is to set up a simulation with physically realistic parameters and a configuration, which enables also experimental investigations on the scenario presented here. Therefore it is payed attention to publications of already performed experiments. The fact of the big numerical efforts associated with coupled LES is especially for the domain size a limiting factor and leads to trade-offs. In order to enable MR as well as RR, the wedge angle in the end-configuration is set to 24.5◦ . The area where both kinds of reflections are possible is called dual-solutiondomain. For defining the flow-parameters, it must be ensured that modern wind tunnels can provide the postulated flow and the corresponding values of pressure, temperature, density and velocity. For orientation, Willems’ experiments at the Trisonic Test Section (TMK) in Cologne are chosen [11]. One experiment was performed at Mach 3.0,

36

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up 102.1 K and 608 ms . Finally the free-stream parameters are chosen accordingly a set-up kg m proposed by the DLR, to T∞ = 100 K, ρ∞ = 0.558138 m 3 and U∞ = 601 s . In Figure 3.1a, which is based on results of Visbal’s simulations of FSI considering a laminar boundary layer ([12]), the history of the panel deflection in the 34 -chord point is shown. Visbal considers a Mach 2.0 flow generating a shock with a constant pressure ratio of 1.8 impinging at the panel center. It clearly points out the transient until the final state is reached at about t·UL∞ = 600. For a flow with 600 ms and a panel of 10 cm length, an extrapolation of the transient duration would result in 10 seconds, which is not acceptable for a LES with explicit time integration regarding computation power and prices per CPU-hour yet. Of course, this approximation is very rough and is not proven to be valid for this case, but anyhow, to avoid long transient behaviour of the panel oszillations for the LES, the approach of a rotating wedge resulting in a transient FSI is chosen. Additionally it is also easier to set up a test-rig of this configuration being able to control the shock parameters properly. As already mentioned, it is important that the turbulent boundary layer, reaching the panel, has a physically correct velocity profile and corresponding Reynold stresses, which is already discussed in the last chapter. To enhance the probability of gaining a proper FSI, a deflection of about the order of the arriving boundary-layer thickness is targeted.

37


Figure 3.1.: History of panel deflection in the 34 -chord point. U1 represents the freestream velocity, h the panel thickness and w the deflection of the 43 -chord point. Figure from [12] and modified

38


3.2. Basic domain architecture

Figure 3.2.: Domain for simulation: Upper picture shows the start configuration and the lower one the end configuration after wedge rotation. Red are highlighted the shocks and the initial characteristic of the centered expansion. Transparent blue is marked the considered fluid domain. In dark, the flexible panel is shown between the two parts of the grey rigid base.

Considering the, so far mentioned, requirements and boundary conditions, a Matlab script (appendix A) is set up to find the optimum domain size for a basic set-up. To avoid a simulation of the rotating wedge, in order to save computation time, the fluid domain ends at the trailing edge of the wedge, which is set as a fixed rotation

39

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up axis. The shock is initiated for the whole domain, while prescribing the position of the mentioned trailing edge, the wedge length and the wedge angle. For initializing the fluid domain properly, rotation angle is set to 0◦ and the disturbances caused by the wedge are supposed to impinge behind the panel (see upper graphic in Figure 3.2). In the end-configuration both, the shock-wave generated by the leading edge and the PrandtlMeyer expansion fan are meant to impinge on the middle of the panel (see lower graphic in Figure 3.2). To guarantee a fully developed and physically correct boundary-layer profile, the domain starts 30 mm before and ends 30 mm after the panel, to be also able to see the wake after the shock and avoid reflections of the shock at the outlet. For the panel, a length of 100 mm is chosen, referring to preliminary coupled inviscid simulations (please see chapter 3.2.1). Finally the Matlab code returns an optimized domain with an accuracy of 0.01mm for the domain dimensions corresponding to the postulated optimization criteria which is shown in Figure 3.3.

Figure 3.3.: Measurements of set-up domain.

40


3.2.1. Coupled simulation with inviscid fluid As a first step, the basic input file is validated by conducting a set of simulations for a both-ends-clamped panel for material parameters of steel with a panel thickness of 0.5 mm. The length is varied from 3 cm to 10 cm, resulting in a variation of the first mode oscillation frequency (f1 ). The obtained values are compared to the analytical solution of the Euler-Bernoulli-Equation (equation 3.1) for a thin panel. Here L is the length of the panel, E the Young’s-Modulus, ρsolid the material density, A the profile area and Izz the second moment of area relative to the z-axis. As the simulations deliver convenient results, the analytical solution is shown being acceptable for further considerations of resonance frequencies of the undeflected panel. Consequently, coupled simulations considering inviscid fluid are set up for investigating now the behaviour of deflected panels. The fluid grid consists of 9900 cells. The smallest cells adjacent to the panel surface are cubic and measure 0.5 mm. The panel grid consists of 50 elements in flow direction, 2 elements in wall normal direction and one element in span wise direction. s ω1 = κ21 ·

EIzz ρsolid A

with κ1 · L = 4.73

ω1 2π b · h2 Izz = 12 δy = δstatic + δamplitude f1 =

(3.1) (3.2) (3.3) (3.4)

The fluid domain parameters are set accordingly the result returned by the Matlabtool for a panel with 10 cm (3.3). The shock is generated by a rotating wedge and propagates within 5 ms from the trailing edge of the panel to the panel center. A variation of several parameters for the flexible panel is done. The most relevant results are obtained for steel and titanium. The effect of the panel thickness on the panel oscillation is shown in Figure 3.4, where the solid line represents a titanium panel and the dashed line a steel panel. Figure 3.5 shows the static deflection of the panel center (δstatic ) over the panel thickness. Again, dashed lines for steel and solid lines for titanium. The non-linear behavior of the panel for variations of the panel thickness can be also seen in the oscillation amplitude of the panel center (δamplitude ) around the static deflection due to the increase of the pressure load induced by the impinging shock (Figure 3.6).

41


f requency [Hz]

1800 1600 1400 1200 1000 800 0

0.001

0.002 h L

0.003 [−]

0.004

0.005

Figure 3.4.: Oscillation frequency of panel center over the panel thickness. × remark simulation results for steel and + represent results for titanium.

|δstatic | [m · 10−3 ]

2.5 2 1.5 1 0.5 0 0

0.001

0.002 h L

0.003 [−]

0.004

0.005

Figure 3.5.: Static displacement of panel center (time averaged displacement for the steady state after shock impingement δstatic ) over the ratio of panel thickness h and panel length L. × remark simulation results for steel and + represent results for titanium. Regarding these results, a titanium panel with a thickness of 0.3 mm is chosen, as the inviscid simulations show a local maximum of the amplitudes in this area (see Figure 3.7). The reason for this peak can be the fact, that a thinner panel means also less flexural stiffness, while it will also cause a higher static deflection, which leads to internal stresses, that would increase the stiffness. That means for negligible static deflections, the amplitude will increase for thinner panels, but at a certain point, the deflection is getting stronger and the latter mentioned effect dominates and causes again a decrease of the amplitude.

42


δamplitude δstatic

[−]

0.15 0.1 0.05 0 0

0.001

0.002 h L

0.003 [−]

0.004

0.005

Figure 3.6.: Ratio of the oscillation amplitude around the static displacement (δamplitude ) and the static displacement (time averaged displacement for the steady state after shock impingement δstatic ) over the ratio of panel thickness h and panel length L, considering the panel center. x remark simulation results for steel and + represent results for titanium.

δamplitude [m · 10−4 ]

2.5 2 1.5 1 0.5 0 0

0.001

0.002 h L

0.003 [−]

0.004

0.005

Figure 3.7.: Oscillation amplitude around the static displacement (δamplitude ) over the ratio of panel thickness h and panel length L, considering the panel center. x remark simulation results for steel and + represent results for titanium. The vertical line indicates the final chosen set-up. In Figure 3.8 the history of the deflection of the penal center (δy ), obtained by an inviscid coupled simulation over a titanium panel, is shown. Here, the shock propagates from the trailing edge to the panel center as it will later also do in the final LES. At 5.44818 ms the panel center reaches its maximum deflection of δy = −1.851946 mm. An analysis of the steady oscillation after the maximal deflection returns a mean deflection over time of δstatic = 1.637061 mm. It is clearly shown that the panel is not excited and the panel center performs a damped oscillation. At this point it should

43

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up be mentioned that according to Dolling [17], also in inviscid supersonic flows, panel flutter can occur as the panel deflection can cause periodically shocks and expansion waves which can lead to resonance. For investigations on the oscillation behaviour of the panel, a PSD is performed by a Matlab-tool (B) using the implemented pwelch function with an overlap of 50% using Hanning windows. Tests, by varying the considered simulation time using pwelch schemes with and without overlap showed, that a 50% overlap delivers results of the same precision with a smaller considered time sequence than the method without overlap. Therefore, for further PSD analysis, the pwelch method with 50% overlap is chosen. In this chapter, the presented frequency analysis is done by using the complete data after 5.8 ms simulation time of the inviscid simulation, when a free, harmonic oscillation of the panel center around a time averaged static deflection is expected. The fourth peak of the oscillation is chosen as the starting point for the PSD analysis of the inviscid simulation as well as the viscous coupled simulation. This point is marked for the inviscid simulation in Figure 3.8 by the first vertical dotted line at 5.8 ms, only 0.4 ms after the end of the wedge rotation. Later, only 5 oscillation periods will be considered for the analysis, to be comparable with the final LES, where the simulation time is much shorter. As indicated by the vertical lines in Figure 3.9, the oscillation frequency of the panel center with the highest amplitude (mode 1) is 1132 Hz, while further frequencies with much lower amplitudes (higher modes) would be at 2604 Hz and 4643 Hz. Considering another point on the panel close to the 14 chord line shows additional modes only with frequencies even higher than the already mentioned ones. As these oscillations contain much less energy, the focus is on the first two mentioned oscillation frequencies. A more detailed discussion on the obtained results of the inviscid simulation will follow in the chapter 4.

44


δy [m · 10−3 ]

0 −0.4 −0.8 −1.2 −1.6 −2 0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 t [s · 10−3 ]

1e − 07 1e − 08 1e − 09 1e − 10 1e − 11 1e − 12 1e − 13 1e − 14

R f ·P SD (f ·P SD)df

[−]

Figure 3.8.: Displacement (δy ) of the panel center over time (t). The vertical solid lines indicate rotation start of the wedge at τstart = 0.4 ms and rotation end at τend = 5.4 ms. The vertical dotted lines mark the considered sequence of 5 periods for comparison with PSD results of the final LES (starting point: 5.9 ms & ending point: 10.2 ms). The horizontal line show the time averaged deflection of the panel center during the oscillation, resulting in ystatic = −1.63 mm.

1000

10000 f requency [Hz]

Figure 3.9.: Normalized PSD using MATLAB’s pwelch function with 50% overlap. Vertical lines mark resonance frequencies at 1132 Hz, 2604 Hz, 4586 Hz and 6964 Hz.

3.2.2. Preliminary uncoupled LES For testing the chosen domain parameters for the treated case, two simulations are set up. Firstly, a simulation with a fixed 24.5◦ wedge, representing the end configuration, is performed, to make sure that the domain is capable of containing all relevant phenomena of the SWBLI. Furthermore it is important that the boundary-conditions are not causing shock reflections, effecting the SWBLI in an unphysical way. It is also

45

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up a test for the solver, as the strong shocks and turbulent flow are very challenging for a stable simulation. For a first draft, a mesh for the fluid domain consisting of 107 to 108 cells with smallest cells of x+ ∼ 30, y + < 1, z + ∼ 15 is targeted. Secondly, the domain will be adapted and refined accordingly for performing a last test with a rotating wedge. This simulation has the same grid as the final coupled LES in the flow domain and also the settings for the shock generator are the same to check the feasibility of the wedge rotation and the resulting shock propagation. The result of the final LES will be also compared to this simulation for pointing out the effects and influence of a flexible panel.

3.2.2.1. End configuration The grid consists of 5120000 cells divided to 320 blocks. The smallest cell measures 4 · 10−4 m in x-, 2.5984 · 10−6 m in y- and 1.5625 · 10−4 m in z-direction and for non-dimensional scale: x+ = 127.734, y + = 0.829 and z + = 49.8963. In wall-normal direction, a hyperbolic grid-stretching is performed according to equation 3.5 where j denotes the specified point in the grid, Ly the domain height and Ny represents the number of cells in y-direction. For the stretching factor b, a value of 3.5 is used. y(j) = Ly ·

) sinh( b(j−1) Ny −1

(3.5)

sinh(b)

As it is shown in Figure 3.10, a Mach-stem occurs. Unfortunately, problems with 34.8

y δ0ref

17.4

0 −0.08

−0.06

T emperature[K] :

−0.04 120

−0.02 160

0 x0 [m]

0.02 200

0.04 240

0.06 280

Figure 3.10.: Contour plot of first preliminary simulation of end-configuration. Instantaneous temperature distribution evaluated at the mid-plane is shown. the numeric stability occurred. In the region of the Mach-stem, the grid is too coarse.

46

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up As the simulation breaks even with lower Courant numbers (CFL number) regularly down, a grid refinement is intended. Referring to [1], the target for the refinement is to achieve non-dimensional measurements of the smallest cell (based on the flow parameters of the preliminary LES) of approximately x+ = 30, y + = 1 and z + = 15. The decision for a higher y + than before is made respectively to the unnecessarily high resolution of the boundary-layer of the preliminary LES to save cells but still not exceeding the y + = 1 limit. For the stretching in wall-normal direction, the same formula as in the preliminary uncoupled LES (formula 3.5) is used with a stretching factor of 3.39053 now. The refined grid contains now 2000 blocks and 114967600 cells in total (table 3.1. In order to obtain this number of blocks that can be perfectly distributed to 125 nodes (each node has 16 processors), the domain measurements are adapted: The domain length is now set to 0.15813961 m instead of 0.16 m and the flexible panel starts at 0.0300465259 m and ends at 0.1296744802 m. Now the panel has a length of 0.099627954 m instead of the 0.1 m. The smallest cells measure now 9.30233 · 10−5 m in x-, 3.13146 · 10−6 m in y- and 4.54545 · 10−5 m in z-direction. The simulation is restarted using the last result of the coarse simulation. To do so, an interpolation of the cell values of the coarse simulation is done to obtain values for initializing the additional cells of the refined grid. This procedure reduces the transient phase of the LES and that saves simulation time. Simulation x+

y+

z+

cells

first draft refined

0.8 50 1 15

5.1 · 106 1.1 · 108

130 30

Table 3.1.: Overview of smallest cells of preliminary simulations.

A frequency analysis of pressure probes along the wall (y = 0) in the x-y-plane at z = 0 shows that at first, the low-frequency peak is allocated at an oscillation frequency of the recirculation bubble of 788 Hz (with falling tendency as the simulation time until the occurrence of numeric instability is not long enough for providing a statistically significant result), while after the refinement this frequency raises by 68.4% to 1327 Hz. At this point, it has to be mentioned that due to the numeric instability of the coarse simulation, the frequency analysis considers a shorter time section than for the refined simulation. As seen later in Figure 3.12, there are indicators of an incorrect turbulent boundary-layer profile, caused by a too high x+ . For a simulation with a

47

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up similar set-up in [1], a low-frequency peak is found at a Strouhal Number based on the separation length (see equation 3.6, where f represents the oscillation frequency) of SrLsep = 0.013, while the separation length in that case is much shorter than in the simulations discussed here and therefore also has a different dynamic behaviour due to smaller moments of inertia. For the two simulations, mentioned before, the coarse simulation’s peak can be found at SrLsep = 0.0469 (with Lsep = 0.03577m beginning at x0 = −0.033) and the refined has its peak at SrLsep = 0.097 (with Lsep = 0.04412 m beginning at x0 = −0.0433 m). This significant deviation of the Strouhal numbers is a result of the coarse mesh of the preliminary simulation. More detailed investigation would be necessary to find out the exact mechanism causing a lower Cf value and a smaller recirculation bubble using a coarse mesh, which is not part of this thesis. But this clearly shows the sensibility of SWBLI to the setup and boundary and initial conditions.

SrLsep = f ·

Lsep U∞

(3.6)

3.2.2.2. Rotating wedge With the same settings of the refined simulation of the end configuration, a simulation considering a propagating shock wave is performed. The ramp-up time for the wedge rotation is set to 0.005 s, referring to experiments of Naidoo ([14]), who performed a rotation over 25◦ within 0.006 s. The Mach number ME is defined as the ratio of the average velocity of the leading edge and the local speed of sound. For our setting, this average value is 0.017 while Naidoo reaches an average of 0.01 and a maximum instantaneous value of about 0.033. The ramp-up is performed according to a time curve defined by the function 3.7, where t represents the time passed since rotation start, trot the ramp-up time, βend the shock angle of the final wedge position (in this case 43.4454◦ , corrsponding to the wedge angle of 24.5◦ ) and βdeg the Mach angle (for Mach 3.0 it is 19.47122◦ ). π·t · (0.5 · (βend − βdeg )) + βdeg f (t) = 1 − cos trot

(3.7)

Although the previous simulation revealed a Mach-stem, in the simulation with a propagating shock and therefore a continuously increasing pressure gradient, the RR remains stable. In Figure 3.11, a temperature plot is shown for comparing both flow

48


34.8

y δ0ref

0

34.8 −0.05

T emperature[K] : 120

−0 x0 [m] 160

200

0.05

240

Figure 3.11.: Comparison of temperature fields of LES with rotating wedge (top) and with 24.5 fixed wedge (bottom). Black iso line represents zero streamwise velocity. Temperature fields are averaged in time and span wise direction after reaching a final and stable end configuration with a wedge angle of 24.5◦ .

49

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up 35 30 25 20 Cf · 104 15 10 5 0 −5 −0.05

0 x [m]

0.05

0

Figure 3.12.: Comparison of skin-friction coefficient Cf of LES with rotating wedge (red) and with 24.5◦ fixed wedge (blue). The green line additionally represents the physically wrong curve of the preliminary underresolved LES. Flow fields are averaged in time and span-wise direction after reaching a final and stable end configuration with an wedge angle of 24.5. The begin and end of the separation bubble is marked by the intersection of the curvatures with the y axis at x0red1 = −0.0433m and x0red2 = 0.0808m for the red curve and x0blue1 = −0.0414m and x0blue2 = 0.081m. fields. It can be seen that basically the recirculation area looks similar, but the iso lines for u = 0 (which give an impression of the shape of the recirculation area) are slightly different. In Figure 3.12, the skin-friction coefficient is shown. The reattachment point for both results is quasi the same. Only the separation of the turbulent boundary layer takes place earlier for the LES considering a fixed wedge angle (blue line) so that the separation length increases from 0.0425 m to 0.0441 m. This is an increase of approximately 1.5 times the boundary-layer thickness at the reference position right in front of the leading edge of the panel. In Figure 3.12, the green line shows the physically wrong behavior of the skin-friction coefficient caused by the too coarse mesh. Without going further into detail, why the MR actually occurs in the coarse LES, it is interesting to note, that the MR remains stable after the refinement of the grid, even if the simulation with the rotating wedge can sustain the RR. As already shown for the chosen flow parameters, both RR and MR can exist. In Figure 3.13, the entropy of both simulations is compared. For doing so, the absolute value of the entropy is calculated for every cell. The plot actually shows the difference of entropy of each cell. Red colors imply at the corresponding

50

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up location higher entropy for the MR and blue colors higher entropy for the RR. In the 2D plot, areas of maximum entropy are spotted by a black iso-line. It can be already seen, that the plot is mostly red colored and if all values of the entropy difference are summed up, a total difference of entropy for the observed instantaneous planes of 0.0105 KJ is obtained. As also mentioned in [18], the MR ’produces’ more entropy than the RR and is therefore more stable once this configuration is reached. As a consequence, restarting the fine LES with the results of the coarser on sustains the MR.

|dS| y x

17.4

y δ0ref

17.4

0 −0.05

0 x0 [m]

∆S[ KJ ] : −2 · 10−8

0

0.05 2 · 108

Figure 3.13.: Entropy difference between MR and RR (∆S = SM R − SRR ). Considered J are instantaneous mid-planes (z = 0). In the 2D-plot, the S = 5 · 10−8 kg iso-curves highlight the areas of maximum entropy in both simulations. In the 3D plot, the z-axis shows magnitude of ∆S for a better impression of the distribution. As this case is some kind of reference solution for comparison with the coupled LES, following plots are for general discussion of occurring shock structures as actually RR

51

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up 34.8

y δ0ref

17.4

0 −0.04

−0.02

x0 [m]

0

Figure 3.14.: Time averaged pressure iso-lines for MR (red) and RR (blue). Streamlines at the border of the recirculation bubble for MR (pink) and RR (light blue). as well as MR could basically occur within the Dual-solution domain. Figure 3.14 shows in blue the shock system for RR and in red the development of a MR. It can be clearly seen that in case of a MR, the Mach-stem proceeds along the incident shock until a stable state is reached. The recirculation area grows and therefore the separation shock also propagates upstream. The reattachment point and the height of the recirculation bubble do not really change, but the rear end of the recirculation area becomes less steep, as indicated by the streamlines. In general, the areas with higher pressure increase for a MR. Figure 3.15 shows a contour plot of the turbulent kinetic energy (TKE) for a RR. The solid line marks the curve along M a = 1. It can be seen that along this curvature, the TKE is in general higher. The reason is that along this curvature, a shear layer exists, which is responsible for the exchange of energy between free-stream and recirculation bubble. As already mentioned, the incident shock will also dissipate at this line. The dashed line represents the curve where no stream-wise (horizontal) velocity component exists. This line can be seen as an indicator for the shape and size of the recirculation area (similar to the chamber line for airfoils). In the cross-sections it can be seen that in the area between these two lines, the shear stresses are increased. The maximum of the TKE can be found in the region after the reattachment of the recirculation bubble.

52


T KE[J] :

2000

20000

20000

13 y δ0ref

8.7 4.3 −0.04

−0.03

−0.02

−0.01 0.01 x0 [m]

x0 = −0.02m

0.02

0.03

x0 = −0.005m

0.04 |∇~v | 8 · 105

8.7

6 · 105

y δ0ref

4 · 105

4.3

2 · 105 1 · 101 −0.005

0 z[m]

0.005

−0.005

0 z[m]

0.005

Figure 3.15.: Above: Contour plot showing distribution of turbulent kinetic energy. Below: Cross-sections at x0 = −0.02m (lhs) and x0 = 0.005m (rhs) showing the magnitude of velocity gradient |∇~v | which is calculated according to equation 3.8. Solid lines mark the sonic line where M a = 1 and dashdotted lines mark the spots where the horizontal component of velocity is zero.

53

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 3. Approach for Set-up 34.8

y δ0ref

17.4

0

−0.04

−0.02 x0 [m]

0

Figure 3.16.: Streamlines for RR together with pressure iso-lines based on time averaged data. ~b = 0.5 ∗ (~v (n, m) + ~v (m, n) f or m[1, 3] and n[1, 3] q |∇~v | = ~b(1, 1)2 + ~b(1, 2)2 + ~b(1, 3)2 + ~b(2, 1)2 + ~b(2, 2)2 + ~b(2, 3)2 + ~b(3, 1)2 + ~b(3, 2)2 + ~b(3, 3)2

(3.8)

Figure 3.16 gives an impression for the flow through the shock system of a RR. In that case, a convergent-divergent nozzle contour emerges.

54


3.3. Final LES Set-up Following table shows important parameters of the final simulation set-up. Parameter

Dimensions

Values

Definition

FLUID U∞ T∞ ρ∞ p∞ Twall

[ ms ] [K] kg [m 3] [P a] [K]

601.0 100 0.558138 16021.7967 280

Free-stream velocity Free-stream temperature Free-stream density Free-stream pressure 2 Twall = T∞ · (1 + · γ−1 · M∞ ) 2

FLUID-DOMAIN pc

[Pa]

16021.7967

cavity pressure

L h b w xT E

[m] [m] [m] [m] [m]

0.0996279541 0.0003 0.01 0.03941 0.056421

length thickness width wedge length position of wedge trailing edge

STRUCTURE E ν ρs

[Pa] [-] kg [m 3]

1.16 · 1011 0.34 4500

Young’s-Modulus Poisson number material density

Izz A D

[m4 ] [m2 ] [P a · m4 ]

3.573 · 10−14 3.5 · 10−6 0.0041447

area moment of inertia: I = area: A = h · b flexural stiffness: D = EI

L b h L

9.9628 0.003513 0.0353

aspect ratio dimensionless panel thickness

µs

[-] [-] [-]

λp

[-]

48099.556

dynamic pressure: λp =

ρf L ρs h

Table 3.2.: Summary of set-up parameters.

55

h3 b 12

2 L2 ρ∞ U∞ D

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results

4. Interpretation of the LES results 4.1. Panel Figure 4.1 shows the deflection of the panel center for the final LES (solid curve) in comparison with the inviscid simulation (dotted curve).

δy δ0ref

[−]

0 −0.4 −0.8 −1.2 −1.6 0

10

20

30 t·U∞ L

40

50

60

[−]

Figure 4.1.: Panel deflection over simulation time of panel center. Exact measurement position is on the panel surface at x0 = 1.3975 · 10−4 m and z = 9.109 · 10−5 m. The dash-dotted vertical lines mark rotation start at 0.4 ms and rotation stop at 5.4 ms. The solid vertical line indicates the start time for evaluating the displacement for frequency analysis of the final LES. The dotted vertical line represents the evaluation start time of the inviscid simulation. The dashed vertical line marks the time step, by which the behavior of the recirculation bubble is considered as 00 steady state00 at 6.7 ms. The horizontal lines show the dimensionless mean deflection of −1.45 for the final LES (solid) and −1.42 for the inviscid simulation (dotted). The dash-dotted vertical lines mark the rotation start (τstart = 0.4 ms) and rotation stop (τend = 5.4 ms) and the solid vertical line indicates the time step, which is chosen as an exemplary time step for investigations on the instantaneous flow field, as the behavior of the recirculation bubble at this point is already considered as 00 steady state00 .

56

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results The horizontal lines marks the average deflection of the panel center for the final LES at δstatic = −1.6726 mm (solid line) and of the inviscid simulation δstatic = −1.630415 mm (dotted line). It can be already seen, that the panel for the final LES gets excited in contrast to the result of the inviscid simulation , as the amplitude increases. At this point it has to be mentioned again that the figures for the analysis of the inviscid simulation diverge from the figures of chapter 3.2.1, because here only 5 oscillation periods are chosen for the evaluation to be still comparable with the final LES, which has a much shorter simulation time. This means, that the physically correct figures for describing the behaviour of the panel of the inviscid simulation are presented in chapter 3.2.1, while the figures presented here show the correlation between the results for the structure obtained by the final LES and the inviscid simulation. For proving the 00 steady state00 of the panel oscillation after 5.7 ms, there is also done a frequency analysis for the inviscid simulation, evaluating 5 cycles after 30 ms simulation time. The resulting frequency peaks are exactly the same as the ones obtained by considering the 5 oscillation cycles right after rotation stop. That means, there is no transient behaviour of the panel expected after 5.7 ms simulation time. Table 4.1 summarizes the results between the two simulations and additionally the results of the investigation on the oscillations of the recirculation area with and without coupling. The frequencies are classified in frequencies between 1000 Hz and 2000 Hz (fa ), 2000 Hz and 3000 Hz (fb ), 4000 Hz and 5000 Hz (fc ) and 7000 Hz and 8000 Hz (fd ). For the evaluations of only 5 cycles, some frequencies between 2000 Hz and 5000 Hz do not occur in the results of the analysis. As the obtained results already allow a good comparison, no further investigation on the reasons for the missing frequencies are done in this thesis. As before in chapter 3.2.1, the analysis of the panel deflection and oscillation is done by the already mentioned Matlab-tool. Firstly, considering the difference between the exact results of the inviscid simulation and the incorrect result obtained by considering a shorter time-span of the deflection of the inviscid simulation, frequency fa raises by approximately 60% to 1812Hz. This discrepancy would be a big problem, unless the result of the frequency analysis of the inviscid simulation and the final LES would not be almost identical. There is just a difference of 2 Hz, which means about 0.1% deviation between inviscid and final simulation. As a test, there is also done a manual measurement of the frequency for the panel oscillation of the final LES. Averaged over 5 cycles, a mean frequency of 1270 Hz is obtained, which means a 12% deviation from the result of the inviscid

57


Typ

[unit]

Final LES

inviscid simulation

evaluated periods evaluation start-time evaluation end-time

[−] [ms] [ms]

5 5.7 9.6

5 5.9 10.2

δstatic fa fb fc fd errorfa errorfc errorfd

[mm] [Hz] [Hz] [Hz] [Hz] [%] [%] [%]

−1.6726 1814 4535 7256 +60.25 −1.11 +1.71

−1.6304 1812 7256 +60.07 +1.71

28 5.9 30 −1.637 1132 2604 4586 7134 -

LFP of uncoupled simulation [Hz] 1292 LFP of coupled simulation [Hz] 2419 (errors calculated in reference to results of the inviscid simulation) Table 4.1.: Summary of structure results.

simulation (1132 Hz). As the peaks of the oscillation signal of the final LES are hard to identify, this deviation is considered as realistic. Therefore we can assume, the oscillation behaviour of the final LES corresponds with that one of the inviscid simulation. This means a huge benefit for future simulations, as the inviscid simulations are quite cheap, allow detailed parameter studies and still predict the panel behaviour in the viscous simulation very well. The deflection of both, inviscid and final simulation are slightly different. The absolute deviation of the averaged deflection of both simulations is 0.0422 mm, which means, the static deflection of the panel center raises for the final LES by about 3.67% of the boundary-layer thickness at the reference point (δ0ref ). This is mainly caused by the bigger average pressure load caused by the recirculation area, which could be also the reason for the slightly different oscillation frequency. It is also interesting that the frequency of 4535 Hz cannot be found in the analysis of the inviscid simulation, even though it occurs in the one of the final simulation. A reason could be the the missing damping due to the recirculation area. All in all, the frequencies gained by the PSD of the panel oscillation of the final LES are not physically correct, but as they correlate perfectly with the results of the analysis of the panel deflection of an inviscid simulation with a identical set-up, considering

58

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results the same number of oscillation periods, the frequencies obtained by a PSD of a longer time period are assumed being also the physically correct ones for the LES.

δy −δstatic δ0,ref

[−]

0.2 0.1 0 −0.1 −0.2 5

6

7 8 −3 t [s · 10 ]

9

10

Figure 4.2.: For statistical evaluations considered oscillation around the mean deflection over simulation time of panel center (solid line), 1/3-chord point (dashed line) and 1/4-chord point (dotted line). Exact measurement position of all considered points is on the panel surface at z = 9.109 · 10−5 m and at x0 = −0.024867 m (25.2%), x0 = −0.01675 m (33.333%) and x0 = −1.3975 · 10−4 m (50.003%). For completion, Figure 4.2 shows the time section considered for the frequency analysis of specific points on the structure surface. For better comparability, the difference between instantaneous deflection and mean deflection for each measurement point is shown. Especially at the beginning of the oscillation, there is a slight delay between the different points, as the rotation is still not fully finished. Also the amplitudes do not significantly increase until a 00 steady state00 is reached. This can be seen in Figure 4.3, where the local minima of the deflection of the panel center over time are marked.

59


0.2

δy−δstatic δ0,ref

[−]

0.16 0.12 0.08 0.04 0 5

6

7 8 −3 time [s · 10 ]

9

10

Figure 4.3.: Evolution of the amplitude of the panel center. Vertical line marks 6.7 ms of simulation time

1e − 07 2

a ] P SD · f [ PHz

1e − 08 1e − 09 1e − 10 1e − 11 1e − 12 1e − 13 1000


Figure 4.4.: PSD of the panel oscillation at the panel center (solid line), 1/3-chord point (dashed line) and 1/4-chord point (dash-dotted line) for the final LES. The dotted curve represents the PSD of the center point gained by the inviscid simulation. The solid vertical lines highlight the peaks for the results of the final LES, whereas the dotted lines indicate the peaks for the panel center of the inviscid simulation considering a longer oscillation period. Exact measurement position of all considered points is on the panel surface at z = 9.109 · 10−5 m and at x0 = −0.024867 m (25.2%), x0 = −0.01675 m (33.333%) and x0 = −1.3975 · 10−4 m (50.003%).

As shown in Figure 4.4 the peaks for the PSD of all three positions on the panel surface have the same frequency. According to Pasquariello[1], the product of PSD and frequency gives an impression of the oscillation energy. Therefore it can be seen, that the panel oscillation of the structure is more energetic than the result gained by

60

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results the inviscid simulation, which is also an indicator for an excitement of the recirculation bubble for frequencies around 4500 Hz.

61


62


4.2. Flow Field

time = 2.1 ms

time = 5.7 ms

time = 3.1 ms

time = 6.1 ms

time = 3.7 ms

time = 6.2 ms

time = 4.1 ms

time = 6.3 ms

time = 4.4 ms

time = 6.6 ms

time = 5.4 ms rotation f inished

time = 6.7 ms

110

150

190

230

270

Figure 4.5.: Temperature field of coupled simulation for various time steps, considering the section −0.05 < x0 < 0.037. The cell values are averaged in spanwise direction for each time step. Dashed line marks the undeformed structure

63

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results In Figure 4.5, the development of the recirculation area during the wedge rotation is shown qualitatively. It is hard to see, but through the deflection of the panel, there is a slight expansion wave right at the beginning of the panel. Therefore the fluid is accelerated due to this expansion and the pressure after the expansion wave decreases. The pressure in the recirculation bubble still has the same level as it has also for the uncoupled case. The lower pressure level allows the recirculation bubble to extend and to proceed upstream. At the final position of the wedge, flow separation takes place 0.7 mm after the leading edge of the panel, where the local panel-deflection angle is bigger than −0.6◦ (the minus sign indicates convex deflection while plus stands for concave deflection). In Figure 4.6, the point of maximum deflection of the panel can be located at x0 = 0.0043571 mm, which is very close to the theoretical impingement point of the incident shock-wave. Moreover, the maximum deflection angles occur close to the edges of the panel at x0 = −0.0396 m and x0 = 0.04125 m. 4

αdef [◦ ]

2 0 −2 −0.04

−0.02

0

0.02

0.04

x0 [m] dy Figure 4.6.: Local panel deflection angle according to αdef = atan( dx ) · 360 at 6.7 ms 2·π after rotation start. Vertical dashed lines mark points with zero-deflection at x0 = −0.0499635 m, x0 = 0.0043571 m and x0 = 0.128884 m

For comparing the evolution of the recirculation bubble with the uncoupled simulation, Figure 4.7 shows the instantaneous temperature fields of coupled as well as uncoupled simulations at specific time steps after rotation start. In these plots, the extension of the fluid domain due to the panel deflection is neglected, as the focus of this plot is on the global impact of the deflection on the shock structure. As the maximum deflection is about 4% of the domain height, this simplification should be justified.

64


β = 28.8◦

β = 30.1◦

β = 37.5◦

β = 39.8◦

0 uncoupled

y [−] δref

coupled

17.4

17.4

−0.04

−0.02

0

0.02 β = 43.4454◦

x0 [m]

110

150

190

230

270

T [K] Figure 4.7.: Temperature field of both, coupled and uncoupled simulations for different states during the rotation. The angle β represents the angle of the incident shock. The cell values are averaged in spanwise direction for each time step. The coupled plot is restricted to positive values in wall-normal direction.

65

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results In Figure 4.7, it can be also seen that the main differences between both simulations are located at the beginning of the recirculation area. Due to the deflection of the panel, the separation shock moves further upstream than the shock of the uncoupled simulation, as the divergent deflected panel enhances flow separation. As the angle of the separation shock decreases, the general pressure level after the separation shock for the coupled simulation is slightly lower than for the uncoupled. Figure 4.8 compares the wall pressure curves of coupled (blue) and uncoupled (red) simulations at 00 steady state00 , indicating the beginning (Figure 4.10) and size of the separation bubble. The pressure load of the panel (considering on the other side of the wall a constant pressure load pc = 16024.142 P a. 100000 80000 p [P a] 60000 40000 20000 −0.05

0

0.05

x0 [m] Figure 4.8.: Pressure evolution along wall for coupled (blue) and uncoupled (red) simulation, using spanwise averaged flow fields at 6.7 ms simulation time. Dashed vertical lines indicate the leading and trailing edge of the panel.

For determining the separation and reattachment points, Figure 4.9 shows the isocurves where the velocity vector has no streamwise component. The separation of the flow for the coupled simulation takes place at x0 = −0.04872 m and it reattaches at x0 = 0.0043459 m, which almost coincides with the point of maximal deflection of the panel. That means the separation length for the considered time step is for the coupled simulation according to Figure 4.9 Lsep = 0.05306 m while for the uncoupled LES the bubble reaches a length of Lsep = 0.0395607 m. Please mind, that these values represent a spanwise average of an instantaneous flow field. For the time-averaged flow fields, the recirculation area of the coupled LES starts at x0 = −0.04856 m and ends at x0 = 0.00484 m which results in a averaged separation length of Lsep = 0.0534 m. It must be mentioned at this point, that the results are not phase averaged (as the

66

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results 52.2 34.8 y δref

[−]

17.4 0 −17.4 −0.04

−0.02

0

0

x [m] Figure 4.9.: Iso-lines for both, coupled (blue) and uncoupled (red) simulation, with zero streamwise velocity. Considered is a spanwise averaged flow field at 6.7 ms simulation time. Dashed vertical line indicates the leading edge of the panel recirculation bubble is oscillating, for phase-averaging, time-averaging is done for every phase of the oscillation cycle separately. That means for example, only results of time steps are time averaged, where the recirculation bubble has its maximum length). The effects of time-averaging without considering the phases of the oscillating recirculation bubble is not clear and would need further investigations. As this thesis is more focused on the general effects of the coupling and the validation of the design principles, no phase-averaged results are intended as the simulation time would increase. As already mentioned in the chapter before, the uncoupled LES has an average separation length of Lsep = 0.0425 m. Additionally the instantaneous Lsep is now compared with the time-averaged Lsep for getting an impression how much it changes due to the oscillating separation bubble. The difference between the time-averaged separation length and the instantaneous separation length is for the coupled LES 0.6367% and for the uncoupled LES 6.916% (reference is the time-averaged separation length). Of course, this is statistically not significant, but it already indicates a higher agility of the bubble without flexible panel, which can be explained by the pressure gradients caused by the panel deflection. The evaluation of the pressure probes will provide more details on this assumption. In Figure 4.11 and Figure 4.12 Iso-Mach lines indicate the shock architecture of the coupled (blue) and uncoupled (red) simulations for instant time steps right after rotation stop and 1.3 ms ms after rotation stop. At 6.7 ms simulation time (corresponding to 1.3 ms after rotation stop), a 00 steady state00 is reached. It can be seen, that the bubble has for both cases approximately the same height. Even the impinging shock

67


20000

19000

p [P a] 18000 17000 16000 0.03

0.032

0.034 s [m]

0.036

0.038

Figure 4.10.: Pressure elevation along the wall for coupled (blue) and uncoupled (red) simulation. Dashed vertical line indicates the leading edge of the panel. The parameter s represents the run length. dissipates approximately at the same point. For the coupled simulation, the separation shock propagates further upstream. Therefore, the shock angle must be smaller than the one of the uncoupled simulation. It can be also seen that the panel deflection enhances the associated shock propagation, since the separation shock at rotation end is already closer to the final position than in the case without flexible panel. Figure 4.13 shows the distribution of local deflection angle of streamlines according to atan( uv ) (Please mind that in this figure, the color map for the angles is equally spaced for better visualizing differences). The angle of the separation shock is for the coupled simulation 15.8◦ and for the uncoupled simulation 16.75◦ .

68

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 4. Interpretation of the LES results 34.8

y δref

17.4

0 −0.04

−0.02

0 x0 [m]

0.02

0.04

Figure 4.11.: Iso-Mach lines for M a = 1, M a = 1.2 and M a = 2.4 for coupled (blue) and uncoupled (red) simulation at β = 43.4454◦ right after rotation stop (equals 5 ms after rotation start).

34.8

y δref

17.4

0 −0.04

−0.02

0 x [m]

0.02

0.04

0

Figure 4.12.: Iso-mach lines for M a = 1, M a = 1.2 and M a = 2.4 for coupled (blue) and uncoupled (red) simulation at 6.7ms simulation time.

69


P SD · f [·10−6 ·

P a2 ] Hz

4. Interpretation of the LES results

9 8 7 6 5 4 3 2 1 0 1000

10000


1e + 06

1e + 07

uncoupled

Figure 4.14.: PSD of a pressure probes near separation point of the uncoupled LES (red) at x0 = −0.0417 m and coupled LES (blue) at x0 = −0.0482 m. The vertical lines indicate the resonance frequencies of the panel at 1132 Hz, 2601 Hz and 4586 Hz.

17.4

0

coupled

y δref

17.4

−0.04

−0.02

0 x [m]

0.02

0.04

0

−25 −20 −15 −10 0 10 15 16 def lection − angle [◦ ]

17

25

Figure 4.13.: Local deflection angles of streamlines according to atan( uv ). Please mind that the color map for the angles is not equally spaced.

70


60000

p [P a]

50000 40000 30000 20000 10000 0 0

1

2

3

4

−3

t − ttrans [s · 10 ] Figure 4.15.: Pressure signal of a pressure probe of the coupled LES (blue) at x0 = −0.0482 m and uncoupled LES (red) at x0 = −0.0417 m near separation point. The evaluation of the pressure probes close to the separation point of the boundarylayer is shown for the coupled (blue) and uncoupled (red) simulation in Figure 4.14. The PSD is multiplied with the frequency to give an impression of the oscillation energy. Additionally, Figure 4.15 shows the pressure signal over time after 00 steady state00 is reached for the frequency analysis. For the uncoupled simulation, a low-frequency peak can be found at a Strouhal Number based on the separation length of SrLsep = 0.09136 which corresponds to a frequency of 1292 Hz. For the coupled case it increases to SrLsep = 0.2149, corresponding to 2419 Hz. This means, there is a big difference in the behaviour of the recirculation bubble between the simplified case (uncoupled LES) and the final coupled LES, while the panel behaves in the simplified case (inviscid simulation) as well as in the final LES similar. The obtained low-frequency peak is in the range of the resonance frequency fc obtained by the inviscid simulation, which corresponds also to simulations considering viscous fluid. It can not be clarified, if this shift is caused by the static deflection of the panel or the panel oscillation. That means, that even with a larger separation area, which usually goes along with higher inertia (thus lower frequency), the frequency is still increasing compared to the case without flexible panel. In Figure 4.16, the shear stresses are shown for both, coupled and uncoupled simulation. From a global view, the shear stresses increase when considering a deflected panel. Especially the fore part of the separation bubble changes dramatically. The zone with high levels of shear stresses over the separation shock becomes for the coupled

71


0 0

− : −2E − 4

9E − 4

2E − 3

uncoupled coupled

∞

17.4 y δref

[−]

0 17.4 −0.05

0 x [m]

0.05

0

Figure 4.16.: Plot showing time-averaged Reynolds stresses for coupled and uncoupled LES. simulation lower. It is hard to explain the significant difference in the plot without more detailed investigation and phase-averaged results.

72

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 5. Conclusion

5. Conclusion 5.1. Discussion In the framework of this thesis, several coupled simulations with inviscid fluid have been performed to determine a panel design, which has the same resonance frequency as the expected low-frequency peak of the recirculation bubble. The idea is, to increase in this way the probability of an excited panel oscillation. After a preliminary uncoupled LES with a fixed wedge, an uncoupled LES with a rotating wedge was conducted for comparison with the final coupled LES with rotating wedge. As shown in Figure 4.6, the recirculation area adapts its length to the deformation of the panel, as the flow separation takes place close to the leading edge, where the divergent deformation causes an additional negative pressure gradient and the reattachment point is very close to the point of maximal deflection of the panel, after which the structure has a convergent contour. The maximum height of the recirculation area (indicated by the iso-lines for u = 0 in Figure 4.9) as well as the position where the incident shock meets the iso-line for M a = 1, do not seem to be influenced by the panel deflection. Significant is a decrease of the separation shock angle for the larger separation length of the flow of the coupled LES compared to the uncoupled LES. Therefore, also the interaction point between incident shock and separation shock propagates upstream. The reattachment of the boundary-layer takes place at the theoretical impingement point of the shock. As the theoretical impingement point coalesces with the point of maximum panel deflection of the final LES, there can’t be investigated the influence of the convergent curvature on the reattachment point, because the pressure gradient of the shock is much stronger than that one, caused by the panel deformation. Furthermore, there is shown a huge impact of the flexible panel on the oscillation frequency of the recirculation bubble, as the low-frequency peak rises from 1292 Hz up to 2419 Hz which is an increase of more than 87%. The recirculation bubble grows due to the bigger separation length as well as the additional volume evoked by the

73

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 5. Conclusion deflected panel. Another reason for the increase of the frequency can be the deflection of the panel and therefore a divergent-convergent curvature, in which the recirculation bubble is oscillating. Assuming that even if the momentum of inertia of the separation bubble is increased due to the larger recirculation area, the deflection of the panel is limiting the 0 breathing 0 motion of the recirculation bubble and therefore leads to an increased oscillation frequency. The panel size is designed by performing simulations with inviscid fluid and a coarse mesh to find a panel that is oscillating with the same frequency as the expected lowfrequency peak in the LES. In this case, the low-frequency was estimated at about 1300 Hz. In this case, the design approach works very well, as the deviation between the design frequency and the frequency in the final LES is only about 100 Hz. Following table summarizes the computational costs of the different simulations. Simulation

Fluid Cells

Structure Elements

Cores

coupled-inviscid-simulations fixed wedge (coarse) fixed wedge (fine) rotating wedge final LES [coupled]

9900 5120000 114967600 114967600 114967600

100 29700

8 200 2000 2000 3264

CPU-hours 3.30 · 101 1.09 · 105 9.04 · 105 2.95 · 106 3.94 · 106 7.8 · 106

SUM Table 5.1.: Summary of simulations.

5.2. Outlook The gained results indicate a strong coupling between the separation length and the panel length. Therefore it would be interesting to investigate the correlation between these two parameters by varying the panel length to find a mathematical approach for designing optimized panels for manipulating the size and even dynamic the behavior (for example oscillation frequency of the recirculation bubble) in order to enable passive flow-control. Additionally, it would be interesting to see the influence of the impingement point of the shock on the reattachment point. As here the impingement point almost coincides

74

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel 5. Conclusion with the point of maximum panel deflection, it is not possible to investigate whether the separation length before and after the impingement point can be manipulated independently by varying the position of the impingement point. As there can be seen a big difference between coupled and uncoupled simulations considering the oscillation frequency of the recirculation bubble, it would be interesting to evaluate the impact of the volume of the separation area and the static deflection of the panel on the low-frequency peak. This could be done by varying the wedge angle and therefore the shock strength. In this case, the design principle of the panel size to obtain a certain oscillation frequency of the panel delivers good results, but as there is no statistical proof, a general application of this principle cannot be ensured without conducting more tests. For finding more precise design principles to estimate the oscillation frequency of the flexible panel, one could conduct uncoupled simulations of the structure domain by putting a pressure load according to Figure 4.8 on the upper surface, resembling the viscous interaction. Finally, the gained experience should help to find a good set-up for observing targeted effects. This should be achieved by conducting costly simulations or tests using simple design tools for defining parameters.

75

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel A. Domain Optimizer (Matlab)

A. Domain Optimizer (Matlab) 1 2

clear all close all

3 4

video ein = 0

5 6 7 8 9 10 11 12 13

% Domaingre a = 0.1 xa = 0.0300465259 Ximp end = xa + 0.5∗a xd = 0.153395422 deltaMA = 0.0 %Abstnd zwischen Machlinie und Sto in Endzustand mindom = 0.01 %minimale Domaingre minwedge = 0.01 %minimale Keil−Lnge

14 15 16 17 18 19 20 21

%Toleranzen offY = −0.000001041 offX = 0.00001 offZ = 0.00001 tol beta end = 0.0001 step beta end = 0.00009 step Yd = 0.0001

22 23 24

%wedge theta end = 24.5∗pi/180 %Keil−Winkel Endposition

25 26 27 28 29

%Strmungsfeld Mach = 3 alpha mach = asin(1/Mach) u 1 = 601

76


30 31 32 33

dens 1 = 0.558138 temp 1 = 100 press 1 = 287.1∗dens 1∗temp 1 gamma = 1.4

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

%calc beta end = theta end links1 =(tan(beta end−theta end)/tan(beta end)) rechts1 = (1−2/(gamma+1)∗(1−1/((sin(beta end)∗Mach)ˆ2))) test1 = links1−rechts1 yd = mindom wedge length = minwedge while sqrt(test1∗test1)> tol beta end beta end = beta end + step beta end links1 =(tan(beta end−theta end)/tan(beta end)) rechts1 = (1−2/(gamma+1)∗(1−1/((sin(beta end)∗Mach)ˆ2))) test1 = links1−rechts1 if beta end >pi/2 break end end

51 52

RB okay = 0

53 54 55 56 57 58 59 60

61 62 63 64

while RB okay xa+a RB okay = 1 else yd=yd+step Yd end %figure if RB okay == 1 h1 = figure subplot (2,1,1) xlimit=0 if (Xvk start2∗pi break end end beta deg=beta rad∗180/pi Ximp = Xvk + Yvk/(tan(beta rad)) Mn1 = sin(beta rad)∗Mach Mn2 = sqrt((1+Mn1ˆ2∗(gamma−1)/2)/(gamma∗Mn1ˆ2−(gamma−1)/2))

80


159 160

M2 = Mn2/sin(beta rad−theta var) press 2 = press 1∗(2∗gamma/(gamma+1)∗(Mn1ˆ2−1)+1)

161 162 163 164 165 166 167 168 169 170 171

%figure h = figure xlimit=0 if (Xvk start0 l8 = line ([ x1(1) xi(2) ], [y1(1) yi(2) ]) else l8 = line ([ x1(1) x1(2) ], [y1(1) y1(2)]) end

193 194 195 196 197 198

%video M(num)=getframe(h) count = count+1 Lange(num)=sqrt((Xhk−Xvk)∗(Xhk−Xvk)+(Yvk−Yhk)∗(Yvk−Yhk))

199 200 201 202 203

num=num+1

204 205

close

206 207

end

208 209 210 211

movie(M,1,2) movie2avi(M, ’shockmodel.avi’, ’compression’, ’none ’,’ fps ’, 3); end

82

Large-Eddy Simulation of a Turbulent SWBLI over a Flexible Panel B. Structure-displacement analysis

B. Structure-displacement analysis 1 2

clear all ; format(’longE’)

3 4 5

% file file =strcat(’yyy.mon’);

6 7 8

9

%read file [ it ,time,x,y,z,d1,d2,d3,d4,d5,d6,d7] =textread( file ,’%f %f %f %f %f %f %f %f %f %f %f %f’,’headerlines’, 10); L int=1;

10 11 12 13 14

%find y−min [ r ,c]=find(y==min(min(y))) min y=y(r,1)/100 min t=time(r,1)

15 16 17 18

%start index start index=130; %when stable oscillation end index=size(time,1);

19 20 21 22

it =it(start index :end index); time=time(start index:end index)/1000; y=y(start index:end index)/100;

23 24 25

sampling time=mean(diff(time)); sampling freq=1/sampling time;

26 27 28

L=length(y); EM= mean(y);

% mean

83


29

press var = y − EM; % variation

30 31 32 33

34 35 36 37 38 39 40 41 42 43

disp( disp( disp( ) disp( disp( disp( disp( disp( disp( disp( disp( disp( disp(

’−−−’) sprintf (’ start / end indices : %14d / %14d’, start index, end index ) ) sprintf (’ start / end time: %14.5d / %14.5d’, time(1), time(length(time)) ) ’−−−’) sprintf (’ number of samples: %d’, L) ) sprintf (’ sampling time diff : %d [delta0/U inf ]’, sampling time ) ) sprintf (’ sampling frequency: %d [U inf/delta0 ]’, sampling freq ) ) ’−−−’) sprintf (’ sampling Strouhal number: %d [−]’, sampling freq/2 ) ) sprintf (’ new sampling freq =: %d’, sampling freq) ) sprintf (’ min =: %d’, min y) ) sprintf (’ min t =: %d’, min t) ) sprintf (’ mean =: %d’, EM) )

44 45 46 47 48 49 50 51

% displacement figure (1) ; % subplot(211); plot(time,y); title ( file ) ; xlabel (’ time [s ]’) ; ylabel (’ displacement [m]’);

52 53 54 55 56 57 58 59 60 61 62 63

% PSD of displacement in y using welch−algorithmus [Pxx,g] = pwelch(press var ,[],[],[], sampling freq); %subplot(313); intvalue = trapz(g,g.∗Pxx); % weighting PSD such that their integral %intvalue = 1; figure (2) ; % over St becomes unity Pxx norm = Pxx/intvalue; %semilogx(g∗L int,Pxx/intvalue); semilogx(g∗L int,g.∗Pxx/intvalue); %loglog(f,Pxx); %hold all;

84


64 65 66 67 68

hold on; %semilogx( f,abs(fPSD) ); title ( file ) ; xlabel (’ f [Hz]’) ; ylabel (’ Weighted and normalized PSD’);

69 70 71

[pks, locs]=findpeaks(y); [gpks, glocs]=findpeaks(g.∗Pxx/intvalue);

72 73 74

index=1; while index 6; if numb

Large-Eddy Simulation of a Turbulent SWBLI over a

Large-Eddy Simulation of a Turbulent SWBLI over a

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